3.4.3.1 Research Publications and Books

Contents

1. Sample copy of publications

2. Sample copy of books published

2015-16

2015-16

Energy and exergy based thermodynamic analysis of reheat andregenerative Braysson cycle

R. Chandramouli a, *, M.S.S. Srinivasa Rao a, K. Ramji b

a Dept. of Mech. Engg, ANITS (A), Sangivalasa, Visakhapatnam, Andhra Pradesh, Indiab Dept. of Mech. Engg, Andhra University College of Engg (A), Visakhapatnam, Andhra Pradesh, India

a r t i c l e i n f o

Article history:Received 29 December 2014Received in revised form1 July 2015Accepted 4 July 2015Available online 29 July 2015

Keywords:EnergyExergyReheatRegenerationBraysson

a b s t r a c t

The conventional Braysson cycle has not found practical use due to the difficulty in achieving isothermalcompression. To make its implementation a reality, the original cycle has been modified by incorporatingregenerator and a cooler before the final compression process. Reheating was included for augmentingthe power output. Expressions for exergy efficiency and exergy destruction for all the components arederived along with the energy and exergy efficiencies of the complete cycle. The effects of maximumtemperature, pressure ratio and number of compression stages on the cycle efficiencies have beenevaluated. It has been found that the exergy destruction in the combustion chamber and reheater puttogether accounts for more than 55% of the total exergy destruction. The cycle efficiency is maximum atan optimum pressure ratio which itself is found to be a function of maximum temperature in the cycle.The energy and exergy efficiency of the cycle equals the efficiency of normal Braysson cycle at a muchlower pressure ratio. The efficiency achieved through the modified cycle with 2 stages of compression isonly 2.2% less than the efficiency through ideal isothermal compression for a pressure ratio of 3 andturbine inlet temperature of 1200 K.

© 2015 Elsevier Ltd. All rights reserved.

1. Introduction

The Braysson cycle is a hybrid cycle composed of Brayton andErricson cycles. It has high temperature heat addition process - aBrayton cycle and a low temperature heat rejection process - anErricson cycle. A detailed parametric thermodynamic analysis ispresented by Frost et al. [1] who actually proposed this cycle.Thereafter, the Braysson cycle was subjected to numerous studiesand reviews by researchers based on both first and second laws.

Zheng et al. [2] carried out a second law analysis of Brayssoncycle. He has shown that the exergy loss in the combustor is thelargest in the Braysson cycle and both specific work and exergyefficiency of the cycle are larger than those of Brayton cycle. Per-formance optimization of endo-reversible Braysson cycle with heatresistant losses in the hot and cold side heat exchanger is per-formed by using finite time thermodynamics by Zheng et al. [3].Zhou et al. [4] investigated the influence of multi-irreversibilitieson the performance of the Braysson heat engine. Yasin et al. [5]

performed the analysis of endo-reversible Braysson cycle basedon ecological criteria that includes finite rate heat transfer irre-versibility. The ecological objective function is defined as the poweroutput minus the loss power, which is equal to the product ofenvironmental temperature and entropy production rate. Thedesign parameters for maximization of the objective function aredetermined.

Shiyan Zhang et al. [6] presented a novel model of the solar-driven thermodynamic cycle system which consists of a solar col-lector and a Braysson heat engine. They optimized the performancecharacteristics of the system on the basis of linear heat-loss modelof solar collector and the irreversible cycle model of a Braysson heatengine. Srinivas et al. [7] performed second law analysis of anirreversible Braysson cycle. Lanmei et al. [8] investigated an irre-versible solar driven Braysson heat engine by taking into accountthe temperature dependent heat capacity of the working fluid,radiation-convection heat losses of solar collector and irrevers-ibilities resulting from heat transfer and non-isentropic compres-sion and expansion processes. Demos et al. [9] proposed theincorporation of regulated water injection during the finalcompression to maintain constant temperature due to evaporation.They reported that the injection process adopted has a minimal

* Corresponding author. Tel.: þ91 8500215172; fax: þ91 08933 226395.E-mail address: rcmouli5@rediffmail.com (R. Chandramouli).

Contents lists available at ScienceDirect

Energy

journal homepage: www.elsevier .com/locate/energy

http://dx.doi.org/10.1016/j.energy.2015.07.0170360-5442/© 2015 Elsevier Ltd. All rights reserved.

Energy 90 (2015) 1848e1858

Parametric and optimization studies of reheat and regenerativeBraysson cycle

R. Chandramouli a, *, M.S.S. Srinivasa Rao a, K. Ramji b

a Dept.of Mech. Engg, ANITS (A), Sangivalasa, Visakhapatnam, Andhra Pradesh, Indiab Dept. of Mech. Engg, Andhra University College of Engg (A), Visakhapatnam, Andhra Pradesh, India

a r t i c l e i n f o

Article history:Received 14 August 2015Received in revised form10 October 2015Accepted 21 October 2015Available online xxx

Keywords:BrayssonReheatRegenerationSFCNDPO

a b s t r a c t

A detailed parametric and optimization studies of reheat and regenerative Braysson cycle has beencarried out. The effect of compressor and turbine inlet temperatures, temperature rise in a stage of multi-stage compression, individual component efficiencies and exit pressure of reheat turbine on the per-formance has been studied. The effect of perfect cooling after regeneration leads to a gain of 7.4% inmaximum exergy efficiency and 20% in maximum power output. A computer programme has beendeveloped to evaluate the optimum pressure ratio for minimum specific fuel consumption andmaximum power output. It is interesting to note that the optimum pressure ratio for maximum poweroutput and minimum specific fuel consumption are different and they vary by a wide margin. It has beenfurther seen that this optimum pressure ratio is a function of turbine inlet temperature. A thermody-namic system will have degeneracy in operational effectiveness with the decrease in component effi-ciencies due to aging. Hence the variations of optimum pressure ratio with component efficiencies arealso studied and reported in this work. To make the system economically viable, it has been recom-mended to design the system for the operating condition of minimum specific fuel consumption ratherthan for maximum power output.

© 2015 Elsevier Ltd. All rights reserved.

1. Introduction

The gas turbine technology has undergone a lot of developmentin recent years to merit its use in a wide spectrum of applicationsranging from aircraft propulsion, marine propulsion and powerplants. This range of applications is due to some explicit advantagesassociated with the gas turbines. Lot of research has been done toenhance the performance of gas turbines and this has fructifiedwith the inclusion of waste heat recovery through regeneration,turbine reheating, compressor intercooling and introducing steaminto the gas turbine combustor. A power generation cycle has thehighest efficiency, when it runs on a reversible cycle withisothermal heat addition and low temperature isothermal heatrejection. The temperature of heat rejection can be decreased byimplementing a combined cycle power plant in which the heatrejected from Brayton cycle is used to drive the steam turbine cycle.Such cycles are being used the world over due to their higher en-ergy and exergy efficiencies.

Frost et al. [1] proposed an alternative to the combined cyclesand termed it as Braysson cycle. The Braysson cycle is inherently anair driven cycle, and therefore the complexities involved ininstalling and running the heat recovery steam generator,condenser and other auxiliaries of a combined cycle plant aretotally eliminated in this cycle. Braysson cycle is a hybrid of the hightemperature heat addition Brayton cycle and the low temperatureheat rejection Ericsson cycle. The Braysson cycle was subjected tofurther studies based on both the first and the second law analysisby many researchers. Zheng et al. [2] carried out an exergy analysisfor an irreversible Braysson cycle and analyzed the influence ofvarious parameters on its performance. It has been shown that boththe power output and the efficiency of the cycle are greater thanthose of Brayton cycle. Zheng et al. [3] also derived the analyticalformula for power output, efficiency, maximum power output andthe corresponding efficiency of an endo-reversible Braysson cyclewith the heat resistance losses in the hot and cold-side heat ex-changers using finite time thermodynamics. He also analyzed theinfluence of the design parameters on the performance of the cycle.Zheng et al. [4] carried out the optimization of the above parame-ters of endo-reversible Braysson cycle. Furthermore the effects of* Corresponding author. Tel.: þ918500215172; fax: þ8933 226395.

E-mail address: rcmouli5@rediffmail.com (R. Chandramouli).

Contents lists available at ScienceDirect

Energy

journal homepage: www.elsevier .com/locate/energy

http://dx.doi.org/10.1016/j.energy.2015.10.0870360-5442/© 2015 Elsevier Ltd. All rights reserved.

Energy 93 (2015) 2146e2156

See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/283435916

Extractive Text Summarization Using Modified Weighing and Sentence

Symmetric Feature Methods

Article in International Journal of Modern Education and Computer Science · October 2015

DOI: 10.5815/ijmecs.2015.10.05

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DOI: 10.5815/ijmecs.2015.10.05

Copyright © 2015 MECS I.J. Modern Education and Computer Science, 2015, 10, 33-39

Extractive Text Summarization Using Modified

Weighing and Sentence Symmetric Feature

Methods

Selvani Deepthi Kavila Assistant Professor, Department of CSE, Anil Neerukonda Institute of Technology and Sciences, Visakhapatnam, India.

E-mail:selvanideepthi14@gmail.com

Dr.Radhika Y

Associate Professor, Department of CSE, Gitam Institute of Technology, Gitam University, Visakhapatnam, India.

E-mail:yradhikacse@gmail.com

Abstract—Text Summarization is a process that converts

the original text into summarized form without changing

the meaning of its contents. It finds its usefulness in

many areas when the time to go through a large content is

limited. This paper presents a comparative evaluation of

statistical methods in extractive text summarization. Top

score method is taken to be the bench mark for

evaluation. Modified weighing method and modified

sentence symmetric feature method are implemented with

additional characteristic features to achieve a better

performance than the benchmark method. Thematic

weight and emphasize weights are added to conventional

weighing method and the process of weight updation in

sentence symmetric method is also modified in this

paper. After evaluating these three methods using the

standard measures, modified weighing method is

identified as the best method with 80% efficiency.

Index Terms—Text summarization, Top Score Method,

Weighing method, Sentence symmetric feature Method.

I. INTRODUCTION

Text summarization falls under the area of text mining

and information retrieval where the main objective is to

retrieve valued information from text. In the process of

summarization the input could be text documents or

multimedia files such as audio, image or video. Text

Summarization is used to save time in text mining and

information retrieval. Automatic summarization is the

process by which computer program creates a shortened

version of text. The goal of automatic summarization is

reducing the size or volume of source text into a short

version that holds the overall meaning and information

content.

There are two approaches in automatic summarization

systems namely extractive and abstractive. The former

approach works by selecting important

sentences/phrases/subset of existing words. The selection

of important sentences forms the key idea in these

methods. Based on a predefined function, each sentence

is evaluated and most important ones are extracted from

the original text in the original form. On the other hand,

abstractive methods construct an internal semantic

representation of the text. In these techniques, the

intention is to generate a summary which is close to what

a human would generate. Unlike in the extractive

approaches, the sentences are reformed or regenerated

based on the semantic relationships in the original text.

This work focuses on automatic summarization of text

documents using extractive methods.

In extractive approaches, one of the most important

phases in text summarization process is identifying

significant words of the text. Significant words play an

important role in specifying the best sentences for

summary. The top score method[10] extracts significant

sentences by giving score to every sentence based on the

significant words. A combination of techniques like

statistical methods and semantic relationship methods are

used to identify significant words.

The rest of paper is organized as follows: Section 2

describes the Literature survey related on Document

Summarization. Section 3 presents the architecture of the

system and improved methodologies. This section also

presents the comparative study of the proposed methods

with various summarization techniques. Section 4

describes the Results and Performance Analysis followed

by conclusion and future work in section 5.

II. RELATED WORK

A. Back ground work related to document

summarization

Text summarization is the process of reducing the text

with a computer program to create a summary that keep

the most important points of the original document. At

first Text summarization was done by Hans Peter Luhn

[1] (Father of Information Retrieval) in 1958. His main

target is to get summarization of technical literature. It is

based on frequency of most significant words and their

relevant positions. In this method sentence scoring was

done and top scored sentences are extracted.

34 Extractive Text Summarization Using Modified Weighing and Sentence Symmetric Feature Methods

Copyright © 2015 MECS I.J. Modern Education and Computer Science, 2015, 10, 33-39

H.P.Edmundson [2] has proposed a new method for

automatic text summarization in 1969. His method is

based on four different weighting methods i.e. Cue, Title,

Location and Key method. With these four methods he

calculated the sentence scores for every sentence and

marked the highest scoring sentence as the most

important one. The main disadvantage of this method is,

irrelevant data and the longer sentences in the document

are displayed. A. Das et al [3] proposed a neural net

model used to pre-process an input string and match with

the user defined string. They extracted featured words

from the given text with the user defined words .If there

is a match then the value of that word increases. This

process repeats until it attains a constant value and total

sentence score is then calculated. Therefore, a sentence

with higher score will be the first one. Another important

characteristic is to integrate a semantic module to refine

the search words like detecting association among search

words, etc.J Jagadeesh et al [4] proposed Sentence

Extraction Based Single Document Summarization. In

their research they discussed about the techniques to

achieve readable and coherent summaries. Arman Kiani

et al [5] proposed Text summarization using Hybrid

Fuzzy systems which is based on summarizing a text on

the fusion of Genetic System. Saeedeh Gholamrezazadeh

et al [6] presented different types of summarization

methods and a common summarized system was

implemented. They also discussed the most important

issues in evaluating a summary and presented common

criterion for evaluating a summarized system.Ladda

Suanmali et al [7] proposed a Fuzzy logic method for

improving text summarization approach. They

improvised summary by using general statistic method.

Rasim Alguliev et al[8] proposed a sentence based

extraction method by using new functions for finding the

sentence clustering approach. This is most probably used

for document summarization. Vishal Gupta et al [9]

presented a survey on different extractive summarization methods. Maryam Kiabod et al [10] proposed a Top

Score algorithm where they calculate the local and global

scores for the words and also identified the significant

words for the given text. Masrah Azrifah Azmi Murad et

al [11] proposed a similarity method with topic similarity

by using fuzzy sets and probabilities. Based on these

scores they extracted the important sentences from the

given document. Rafeeq Al-Hashemi [12] proposed the

text summarization using extracted keywords. In this

work operation is performed in four stages. In the first

stage pre-processing was done, key phrases are identified

in the second stage, sentences were extracted in the third

stage and in fourth stage summary is produced. Shaidah

Jusoh et al [13] proposed various techniques used in text

summarization like Information retrieval etc. and also

proposed about the applications and challenging issues in

text summarization approach.

B. Existing Summarization tools

There are some summarization tools to generate

summaries .Some of them are:

Free summarizer:

It is a tool that generates the summary based on the

number of sentences required in the summary. The

disadvantage of this tool is that the summary is not

efficient.

Auto summarizer:

It is a tool that also generates the summary on the

number of sentences required in the summary. The

disadvantage of this tool is that semantic relation is

missing in the summary.

Online Summarizer:

It is a tool that generates the summary based on the

threshold value. The summary varies according to the

threshold value. The disadvantage of online summarizer

is when document doesn't contain good summary

sentences it summarizes poorly and also when user

provides url or text it can‘t get the right abstract

document.

Open text summarizer:

It is a tool to summarize texts. The program reads texts

and conforms which sentences are important and which

are not. The Open Text Summarizer is both library and a

command line tool. The main disadvantage with this tool

is it doesn‗t indicate the important sentences because of

the repetition. The main sentences are missing in the

summary.

Text compactor:

It is a tool in which there are three steps to be followed

namely uploading the document, dragging the required

percentage and summarizing the document. Whenever

the input text is too long text compactor unable to

summarize it.

III. PROPOSED SYSTEM

Linguistic roles identification is the first module in this

work where linguistic roles are identified to make the

task of researcher easy. This is performed using methods

of keywords extraction and based is on fonts.

Fig. 1 shows the flow of execution. Initially the

documents are uploaded in IEEE format. In the first step

a document is selected based on the rhetorical roles from

the set of documents which are present in the repository

and the text besides the keywords is extracted. The

extracted text is fed to Text Processing stage where the

whole text is divided into number of sentences and

tokens. Later, it goes to the Intermediate stage where it

performs all the pre-processing steps i.e. Stop word

removal, Stemming etc. After that it calculates sentence

scores for respective algorithms based on their formulas.

Based on the sentence score the sentence extraction is

performed, i.e. the highest ranked sentence will be the

first one in the summary. Then final output of system

generated summary is given. The comparison ratio is

found by comparing the system generated summary with

manual summary by using relevance measures. The final

result i.e. comparison table is measured.

Extractive Text Summarization Using Modified Weighing and Sentence Symmetric Feature Methods 35

Copyright © 2015 MECS I.J. Modern Education and Computer Science, 2015, 10, 33-39

Fig.1. System Architecture

The following important rhetorical roles are used in

this paper.

Abstract:

This Keyword is found after the title of the paper and

names of the authors. It contains the text which is of

around 200 words which gives the essence of the whole

document This abstract can be further summarized so that

user can get the essence of the paper by reading only a

few lines.

Introduction:

This keyword is found after the keywords or index

terms. This contains text which is of 3 to 4 paragraphs. It

gives information related to domain, existing system,

proposed system and the sections that will be further

dealt in the paper. Once the ―Introduction‖ keyword is

identified based on rhetorical roles, the text beside

―Introduction‖ is extracted and it undergoes all the phases

till summarization. The output text of this ―Introduction‖

contains domain of the paper and important points are to

be extracted based on scoring factor.

Conclusion:

This keyword is identified by the word ―conclusion‖.

The text besides this undergoes all the stages and finally

a summarized text will be produced which gives

information about the work done in the paper and also

the future work.

A. Implementation details

In this paper three summarization algorithms are

implemented which mainly focuses on research papers of

the given area. The three algorithms are, as follows:

Top-Score Algorithm

Modified Sentence symmetric feature Algorithm

Modified Weighing method Algorithm

Top score algorithm [10] is an existing well defined

method. In this work sentence symmetric algorithm is

used in a modified way to be compared with the top score

method. The modifications are done to include more

features like thematic weight and emphasize weight.

Weighing method is used in the conventional manner but

the way in which weights are given is changed and also a

graphical matrix representation is used.

B. Modified Sentence Symmetric Feature Method

In Sentence Symmetric feature algorithm the following

attributes are used to calculate the sentence score.

Cue

Key

Title

Location

To calculate the sentence score the formula S= aC+bK+cT+dL is used.

Where C – Cue weight, K – Key weight, T – Title

weight, L – Location weight and a,b,c,d are set of

positive integers in the range [0,1] .

The main disadvantage of using this method is

irrelevant data is also being displayed. To overcome this

disadvantage, a modified version of the above scheme is

used in which instead of calculating the key weight, two

more features are added i.e.,

1. Thematic weight of the sentence.

2. Emphasize weight of the sentence.

So, the Modified Sentence Symmetric Feature consists

of

Fig.2. Data flow diagram of Modified Sentence Symmetric feature

Method

Cue Weight for sentences:

The Cue Weight for sentences is calculated by adding

the cue weight of its constituent words, it is a quantitative

description. This depends up on the hypothesis that has

significant implications for language acquisition, and is

applicable for the specification of a particular sentence

36 Extractive Text Summarization Using Modified Weighing and Sentence Symmetric Feature Methods

Copyright © 2015 MECS I.J. Modern Education and Computer Science, 2015, 10, 33-39

by the its existence or nonexistence of particular cue

words in the cue dictionary.

Total number of cue words present in a sentence s is

denoted by Cwj (Sj) and total number of cue words in the

document is denoted by Cwi.

Thematic Weight for sentences:

Thematic words are defined as most frequent words.

The functions of the thematic words frequencies are

Sentence scores.

Where indicates Total number of thematic words

present in a sentence s is denoted by Thej(Si) and total

number of thematic words present in the document is

denoted by Thei.

Title Weight for sentences:

Here the sentence weight is calculated by the addition

of all the words in the content which are given in the title

and sub title of a text.

Total number of title words present in that sentence s

is denoted by Tij(Si) and total number of title words in

the document is denoted by Tii.

Location Weight for sentences:

The importance of sentence is indicated by its location,

sentences tend to occur at the beginning or in the end of

documents or paragraphs based on the hypothesis. A

greatest correlation is achieved between the human-made

exception and automatic exception by adding the three

latter methods and the results are shown.

Location of the sentence s is denoted by Lj(Si) and

total number of sentences present in the document is

denoted by Si.

The proposed algorithm is presented below.

Table 1. Steps for Modified Sentence Symmetric Feature Method

Algorithm Step 1: Sentence segmentation is performed.

Step 2: for each sentence s do

Step2a:Cue Weight for sentences :

for Cwj in Si do

C = Σ Cwj(Si) /ΣCwi

Step2b:Thematic Weight for sentences :

for Thej in Si do

Th = Σ Thej(Si) /ΣThei

Step 2c:Title Weight for sentences :

for Tij in Si do

T = Σ Tij(Si) /ΣTii

Step 2d:Location Weight for sentences :

for Lj in Si do

L = Σ Lj(Si) /Σ Si

Step2e:Emphasized words Weight for sentences :

for Emj in Si do

E= Σ Emj(Si)

Step 3.End

Step 4.For each sentence do

Sentence Score :

Sf = C + Th + T + L + E

Step 5.End

Step 6.Return sentence score.

C. Modified Weighing Method

a) Pre-processing:

The first step in text summarization involves preparing

text document to be analyzed by the text summarization

algorithm. First of all we perform sentence segmentation

to separate text document into sentences. Then sentence

tokenization is applied to separate the input text into

individual words. Some words in text document do not

play any role in selecting relevant sentences of text for

summary, Such as stop words ("a", "an", the"). For this

purpose, part of speech tagging is used to recognize types

of the text words. Finally, nouns of the text document are

separated.

b) Calculating word local score:

Local score of a word is calculated by using term

frequency and sentence count Term frequency is defined

as frequency of the word normalized by total number of

words. Sentence count is the no of sentences containing

the word normalized by total no of sentences.

c) Title Weight for sentences:

Here the sentence weight is calculated by the addition

of all the words in the content which are given in the title

and sub title of a text.

Total number of title words present in that sentence s

is indicated by Tij(Si) and total number of title words in

the document is indicated by Tii.

Fig.3. Data Flow Diagram of Modified Weighing Method

d) Sentence-to-Sentence Cohesion:

Calculate similarity between each sentence s and each

other sentences of the document and then sum those

identical values, acquiring the fibrous value of this

feature for s. This process is iterated for all sentences.

Sentence weight=∑a[i,j]/∑∑a[p, q]

The proposed algorithm is presented below.

Extractive Text Summarization Using Modified Weighing and Sentence Symmetric Feature Methods 37

Copyright © 2015 MECS I.J. Modern Education and Computer Science, 2015, 10, 33-39

Table 2. Steps for Modified Weighing Method

Algorithm Step 1: Sentence segmentation is performed.

Step 2: for each sentence do

Title word score(f1)= Σ Tij(Si) /ΣTii

Global keyword score(f2)=no of global keywords present

in a sentence

Local keyword score(f3)= no of local key words present

in a sentence

Sentence weight(f4)= Σ a[i,j] /ΣΣ a[p,q]

End.

Step 3:for each sentence do

Sentence score= (f2*s)+(f3*s)+(f4*s)

+ f1

Total no of words in sentence i

Where s=1 for title words

S=0.9 for global keywords

S=0.8 for local keywords.

End

Step 4: Return sentence score.

IV. RESULTS AND PERFORMANCE EVALUATION

Initially selected document is uploaded and the

linguistic roles in it are identified. Later the sentence

scores for the given document are calculated. Next,

extract the sentences of the document based on their

sentence scores.

Once the summarized text for the three algorithms is

achieved then the precision and recall values are

calculated to find the best method.

The performance of the proposed system is evaluated

based on available manual summaries as the dataset using

the evaluation measures. For experimentation, the

summary is generated for different compression rate and

is evaluated on the extractive summary provided in the

dataset using the evaluation measures.

By comparing the average of precision, recall and F-

measure scores of the three algorithms, the best method

among the methods is found to be Modified weighing

method.

The table 3 presents the values collected while

measuring the performance of all the systems.

A. Performance comparison

To test the summarization process, different research

documents have been used as input. The purpose was to

test the context understanding by the summarizers

developed in this work. The table gives the results of

three approaches with their average precision, recall and

f-measure. Therefore it is observed that Modified

Weighing method is the best method among the other

two methods.

Fig.4. Performance Comparison

Table 3. Measuring the Performance for all Three Methods

SN

O DOC NO

MODIFIED WEIGHING METHOD TOP SCORE METHOD MODIFIED SENTENCE

SYMMETRIC METHOD

PRECISI

ON

RECAL

L

F-

MEASUR

E

PRECISIO

N

RECAL

L

F-

MEASUR

E

PRECISIO

N

RECAL

L

F-

MEASUR

E

1 AS001 0.3444 0.2303 0.2759 0.3333 0.1636 0.3078 0.3358 0.1636 0.342

2 AS002 0.5259 0.2939 0.2937 0.4259 0.2039 0.3492 0.4629 0.2196 0.2809

3 AS003 0.4222 0.3755 0.3973 0.3888 0.3175 0.2142 0.5135 0.2755 0.3275

4 AS004 0.5512 0.3017 0.3878 0.5253 0.2615 0.2652 0.4938 0.2812 0.3125

5 AS005 0.4925 0.3125 0.3765 0.4125 0.3218 0.2256 0.5246 0.2615 0.3185

6 AS006 0.4812 0.2725 0.3598 0.4821 0.3025 0.2025 0.5315 0.2912 0.3001

7 AS007 0.5816 0.2985 0.3927 0.3961 0.2827 0.4014 0.4521 0.2127 0.2812

8 AS008 0.4998 0.2935 0.2861 0.5142 0.2569 0.3252 0.4925 0.2412 0.2912

9 AS009 0.4514 0.3885 0.3411 0.5652 0.3599 0.3851 0.3215 0.2231 0.3215

10 AS010 0.4821 0.3012 0.3712 0.5841 0.3321 0.3951 0.3112 0.2489 0.3101

11 AVERAGE 0.4832 0.2968 0.3482 0.4625 0.2806 0.3071 0.4439 0.2418 0.3057

0

0.1

0.2

0.3

0.4

0.5

0.6

Precision

Recall

F-Measure

Summaries Precision Recall f-measure

Modified

Weighing

method

0.4832 0.2968 0.3678

Top score 0.4625 0.2806 0.3488

Modified

Sentence

symmetric

0.4439 0.2418 0.3129

38 Extractive Text Summarization Using Modified Weighing and Sentence Symmetric Feature Methods

Copyright © 2015 MECS I.J. Modern Education and Computer Science, 2015, 10, 33-39

Fig.5. P and R for Modified Weighing Method

Fig.6. P and R for Top score Method

Fig.7. P and R for Modified Sentence Symmetric Method

By comparing Fig. 5 and Fig. 7, Modified Weighing

with Modified Sentence Symmetric Method (MSSM) an

observation can be made that recall value remained same

for an increase in precision for MSSM. Whereas for

Modified Weighing Method, the behaviour of recall with

precision is linear as should be for a perfect system.

By comparing Figure 6 and Figure 5, Top score

method with Modified Weighing method, both are

behaving similarly. But for a text summarization system,

a system with better precision is preferred. And if both

the graphs of Modified Weighing Method and Top score

method are observed, for the Top score method the

precision dipped for a higher recall but in the Modified

Weighing method the increase in precision is consistent.

Hence, an observation can be made that Modified

Weighing Method is a better and consistent method. And

as the average Precision (P) and Recall(R) numbers are

suggesting, the Modified Weighing Method is suitable.

V. CONCLUSION AND FUTURE WORK

This paper mainly focused on summarization of

research papers. Three different algorithms for

summarization are implemented and the performance is

observed. Keywords are used for identifying the

rhetorical roles in the document. For the calculation of

sentence scores and their feature scores for summarizing

the text all these three methods are used based on

statistical approaches. The work with text data is difficult

at times due to vast amount of data to be summarized.

While using extractive methodologies sometimes the

sentences that are not important to be included in the

summary also get included. In the proposed work this

limitation was overcome, by using compression ratio to

find out the important sentences.

The scope of the paper is maintained to Extractive

summarization approaches only. In future, the scope of

this work can be extended to abstractive summarization

approaches, so that the system can be more efficiently

used by all the researchers by giving semantic meanings

to the sentences. Also hybrid approaches of extractive

and abstractive methods can also be tried.

ACKNOWLEDGEMENT

The authors would like to thank the anonymous

reviewers for their careful reading of this paper and for

their helpful comments.

REFERENCES

[1] H.P.Luhn ―The Automatic Creation of Literature

Abstracts‖. IBM Journal of Research and Development,

2(92):159 - 165, 1958.

[2] H. P. EDMUNDSON ―New Methods in Automatic

Extracting‖, Journal of the Association for Computing

Machinery, Vol. 16, No. 2, April 19691 pp. 264~285.

[3] A.Das, M.Marko, A.Probst, M.A.Portal, C.Gersheson

―Neural Net Model For Featured Word Extraction‖, 2002.

[4] Jagadeesh J, Prasad Pingali, Vasudeva Varma, ―Sentence

Extraction Based Single Document Summarization‖

Workshop on Document Summarization, 19th and 20th

March, 2005, IIIT Allahabad.

[5] Arman Kiani B, M. R. Akbarzadeh ―Automatic Text

Summarization Using: Hybrid Fuzzy GA-GP‖, IEEE

International Conference on Fuzzy Systems.Juky 16-21,

2006.

[6] Saeedeh Gholamrezazadeh, Mohsen Amini Salehi,

Bahareh Gholamzadeh, A Comprehensive Survey on Text

Summarization Systems, IEEE 2009.

[7] Ladda Suanmali, Naomie Salim, Mohammed Salem

Binwahlan ―Fuzzy Logic Based Method for Improving

Extractive Text Summarization Using Modified Weighing and Sentence Symmetric Feature Methods 39

Copyright © 2015 MECS I.J. Modern Education and Computer Science, 2015, 10, 33-39

Text Summarization‖, (IJCSIS) International Journal of

Computer Science and Information Security, Vol2 No1

2009.

[8] Rasim ALGULIEV, Ramiz ALIGULIYEV ―Evolutionary

Algorithm for Extractive Text Summarization‖ Intelligent

Information Management 2009, Science Research.

[9] Vishal Gupta, Gurpreet Singh Lehal ―A Survey of Text

Summarization Extractive Techniques‖ Journal Of

Emerging Technologies In Web Intelligence, VOL. 2,

NO. 3, August 2010.

[10] Maryam Kiabod, Mohammad Naderi Dehkordi and Sayed

Mehran Sharafi ―A Novel Method of Significant Words

Identification in Text Summarization‖, Journal Of

Emerging Technologies In Web Intelligence, VOL. 4,

NO. 3, August 2012

[11] Masrah Azrifah Azmi Murad, Trevor Martin ―Similarity-

Based Estimation for Document Summarization using

Fuzzy Sets‖, International Journal of Computer Science

and Security, volume 1 issue 4 2006.

[12] Rafeed Al-Hashemi ―Text Summarization Extraction

System(TSES) Using Extracted Keywords‖, International

Arab Journal e-Technology, vol 1 No 4, June 2010.

[13] Shaidah Jusoh, Hejab M. Alfawareh, Techniques

―Applications and Challenging Issue in Text Mining‖,

IJCSI International Journal of Computer Science

Issues,vol 9, issue 6,November 2012.

Authors’ Profiles

K.Selvani Deepthi is currently Pursuing

PhD (CSE) in Gitam Institute of

Technology from Gitam University and

She is working as Assistant Professor in

Anil Neerukonda Institute of Technology

and Sciences at Visakhaptnam. Her area of

Interest is Natural Language Processing,

Text Mining and Data Mining.

Y.Radhika is doctorate in computer

science and engineering. She is working as

Associate Professor in Gitam Institute of

Technology, Gitam University at

Visakhapatnam. Her area of Interest is Data

Mining.

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Synthesis and Bio'evaluation of 2-hnino-4-amino thiazole Capped Silver Nanoparticles

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INTRODUCTION riol oriJ1,'-!een.rs huilding blocks in natlrral prodrlcts. but are;r lso considered to he extriturdi n ari 11. usetit I scatlbl ds, especi al 11'

in cornbinatorial and ruedrcinal chernistr-v III-15|. FLrrther-inore. thiazolirrcs havc interestinS applications in agricultureas acaricidrs" inseeticides ancl plant growth regulators I I6. I71.

In gcr.rcral thc s1,'ntircsis ot 2-irninothi:rzolinc clcrivativcswas uchieved l-r), titc reactior-r oi thioLrrel derivatir,,es r,vithvlri ou s sLi bsri t utrxl &-brorrokettrnc i rr the ]lrcscuce ol sttrtablcLurse [ 18" l9]. Here also l.he sirrrilar rctrclion proce<lurer rvasadopted tor the s\/nthesls ol 4-arnino-2-irninothiazole lromthe reaclion o1'aroylthiorrlea rvih blonto berrzyI cyanirie.

As it w,its I'uou,n frorl fhe litetature that tr.iethl,larninervurks as best hirsr: trr tl'ris kind o1'transtbrrlrrtiorr [20], thesar.ne blLse v,,as Ltsc-d in this rnethod. 'l'he thiourea cleril'ative s

rvcrt: svnlhcsizcd usin.s the prclior-rs standarri proccr:lrtrc. -['hc

prt.rciuct u'as firllv charucterizer-l using all speclrosr:L)plo analysis(lR. rll Nl\4R. rrC'NI\,lt anrl nlirss). [n thc IR spcctrurn, thecharacleri-rtic peak for NI;rvlsobsener-l ar 334t!1222 crnIancl C=N w,;rs observer-l at 1590- 1550 crn r. In rlI NMR. NH,appear:erl as hroarl singlet at 6 3.5 ppnr.

nxpEEtntnxiCAll the reagelits used rvcle of AR gr"aclc. Sillcr nitlltc

was oblaincrl fronr Nationai llelincrv Pvt. Ltd. and I 0. ll!1ilqueolis solntirrn was used as stclok solution. Sodium boro-hydroxidc: was oLrtainecl frurn lVIerck. Intliit. Organic-liee rvaterr.lus Lr:fd [hrurii:lioLri tlrc rrpcri rtrerrt

2-lminothiazolinc delivatii e s are ilnportallt 5-lnctrbcredhctcrocvclic cornpouncls havc provcr.r to bc a struclr-rral icatLrrcprovidinu a btoacl spectrum of biological lotir.,it1,. 'I'hesc lrepla.v"ing kcy r-ole in tlre dcvclopmcnt ol'riru.us lor hvpcrtcnsion| 1. inflanrnration [2], citncer rheriipies [-11, e/r,. I)ue tr> theirinherent Io,"r,' toxicities and goor.l phtrrnucokinef ic profi le , thesecornpounds have been recognir:ecl as privileued slructuralrnotif in medicinal chemistry. Recently^ 2-irnino-thiazolinesrvelc tirLrnd to have antilirngitI activit!, 4] and skin rvhiteningpt'opelties [5]. The pitithrin (Pti-&) wits rsolated bv screen Eri

chcrnical librarics havins 2-irninothiazolinc skclct<.rn is tlrt- lcailconrpounrl of p53 in activalors anc1 have reccir,ed inr.t-cusingaltclrtion (ir.le to thctr possiblc applicatiuns in se vcr;rl trujor nc Lrirr-

clg:gsnerativedisortters sue h us Alzhe inrer's i,lisease. Prrkinscin'stlisease. stroke, cancers tl-rerapv and otlrcr pathologie -i relaterltc.r valious signaling pafhways [6-9].In this presenr rvolk thederivative of 2-inrino thioz,oline i.c.,N-(,1-arnino13,-5-r.tiphen1,l-

3H-thiazol-2-ylidine)henzarnicle is used as a.rappins itseot roconlfol the size ofsilver nanoparticlcs and stabilize thern. Thecapping agent al-so cnhnnccs thc biological activitl of srlvcrnanopalticles

Synthesis and characterization aI N-(4-:rmiito-3,5-diphenyl -3H-thiazol-2-l,"litline)benzirn-ride : J,I rni r.r orh i azo-line clerivatives are i mportirnt class ol' 5-rnernLrered heterocvol ic

contpouncls, because oltheir rvide applicaliorrs Iil)]. 'l'he,v are

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,,fSW"PillillillZ.:i:1izir,.('\\:

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oor;

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p=

?ol5-lLAnumu[a Ralini,u Muratasetti Nookaraju,b Suman Chirra,u Ajay Kurand Narayanan Venkatathri*a

Titanium aminophosphates are prepared using titanium tetraisopropoxide, orthophosphoric acid andaliphatic amines. The synthesized TNPAP, TNOAP and TNDDAP titanjum aminophosphates arecharacterized using various physicochemicat techniques. The catatytic activity of TNPAP. TNOAP andTNDDAP is studied for the degradation of orange G dye. TNDDAP is found to be most effective catatyst.The optimum conditions requrred for efficient orange G degradation are found to be pH : j.O, IHzOzl :1.0x103M, IOGI :l.Oxl0aMandaTNDDAPcatalystdosageofl5Omg.TNDDAPefficienttycatatyzes the degradation of orange G dye with 973% degradalion after 250 min. These catatysts exhrbitgood reusabitity over five successive cycles, They have potentiat to be used as economicat catatysts fordye degradation for industriaI waste waters.

cially available dyes are azo compounds.n They are very impor-tant pollutants; even at low concentrations they can affect watersources by imparting undesirable colour which reduces thelight penetration through the water column.s Therefore, studieson the decolourisation and detoxification of azo dye effluentshave received increasing attentior-r. Orange G (Fig. f ) isa s)mthetic azo dye used in histology in many stainingformulations.

Synthetic dyes are recalcitrant to removal by conventionalwastewater treatment technologies such as adsorption, pho-todegradation,u-'3 coagulation, flocculation, chemical oxida-tion, electrochemical oxidation and biological process. Theavailable physical and chemical treatment methods havelimited use and are operationally exper-rsive. Research effortsare needed to develop powerful techniques for the removal ofazo dyes fiom aqueous medium to avoid their accumulation.Hence there is a need for efficient water treatmeltt

"Department of Chemistry, Nqtionql Institute of Technology, warangal 506 001,

Telangana, India. E-mail: wnkatathrin@lqhoo.com; Tel: +91-9491 j1 9976bDepartment of Chemistry, AniL Neerukonda Institute of l'echnology and Sctences,

Vishakapatnam - 531 1 62, Andlra Pradesh Fig. 1 Structure of orange G dye

RSC Adr,riances

W:'"':'1,uCite this: RSC Adv.,2015, 5, 106509

Recerved 16th September 2015Accepted 31st October 2015

DOI : 10.1039/c5ra19117c

www.rsc.orgladvances

lntroductionOrganic dyes are major poliutants released into water systemsduring their manufacturing and processing at industrial sites.These compounds are highly coloured and cause seriousproblems ir-r the aquatic environment as they affect photosyn-thetic activity by reducing light penetration. Their presence inlow concentrations such as 1 mg L-1 in eftluent is considered tobe undesirable and needs to be removed before tl-re wastewatercan be discharged into tl-re environment.' In addition, theirpresence in drinking water constitutes a potential humanhealth hazard. They are difficult to degrade due to their complexstructure which makes them mutagenic and carcinogenic.Thus, efflcient colour removal from wastewater involvingphysical, chemical and biological methods has attracted theinterest of environmentalists and researchers. The majoriq, ofthe dyes consumed on an industrial scale have azo, anthra-quinone, triphenylmethane, phthalocyanine, formazin or oxa-

zine functional groups.' These dyes are used extensively intextile industries owing to tl.reir brilliant colour, high wet fast-ness, easy application and minirlum requirement for energyduring the process.

Specifically, azo dyes are non-degradable, carcinogenic,teratogenic and toxic.' Approximatbly 50-7oo/o.of the commer-

Titanium aminophosphates: synthesis,characterization and orange G dye degstudies

z----'=.4-,]J

RSC Adv.2015, s, 106s09-1065i8 I 106509

5;;<.d;,*,.1-.9.,ffi 1,;1

Procedia Technology 21 ( 2015 ) 295 – 302

Available online at www.sciencedirect.com

ScienceDirect

2212-0173 © 2015 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).Peer-review under responsibility of Amrita School of Engineering, Amrita Vishwa Vidyapeetham Universitydoi: 10.1016/j.protcy.2015.10.032

SMART GRID Technologies, August 6-8, 2015

Design of Controller for Automatic Voltage Regulator using Teaching Learning Based Optimization

V. Rajinikantha,* , Suresh Chandra Satapathyb

aDepartment of Electronics and Instrumentation Engineering, St. Joseph’s College of Engineering, Chennai 600119, India bDepartment of Computer Science and Engineering, ANITS, Visakhapatnam 531162, India.

Abstract

In this paper, One Degree Of Freedom (1DOF) and Two Degrees Of Freedom (2DOF) Proportional + Integral + Derivative (PID) controller design is proposed and implemented on the Automatic Voltage Regulator (AVR) system using traditional Teaching Learning Based Optimization (TLBO) algorithm. Minimization of a multi-objective function guides the TLBO algorithm’s exploration until the process converges with an optimal solution. A simulation study is carried to examine the performance of TLBO assisted controller design procedure for three, four and five dimensional searches. The performance of the proposed method is validated with most successful heuristic procedures, such as Particle Swarm Optimization (PSO), Bacterial Foraging Optimization (BFO) and Firefly Algorithm (FA). The result show that, 1DOF PID controller and PID controller with filter offers smooth reference tracking response and the 2DOF PID controller with the Feed Forward (FF) and Feed Back (FB) structure presents reduced time domain and error values compared to the alternatives. © 2015 The Authors. Published by Elsevier Ltd. Peer-review under responsibility of Amrita School of Engineering, Amrita Vishwa Vidyapeetham University.

Keywords: AVR; PID controller; degrees of freedom; teaching learning based optimization; multi-objective function.

1. Introduction

In recent years, Heuristic Algorithm (HA) supported optimization is emerged as a powerful tool for discovering optimal solutions for a variety of engineering optimization problems [1-5]. In this work, newly developed Teaching Learning Based Optimization (TLBO) technique is adopted to solve the controller design problem. The TLBO was originally developed and implemented by Rao et al. to find most favorable solution for the constrained mechanical

* Corresponding author. Tel.: +91 9380593801. E-mail address:rajinikanthv@stjosephs.ac.in

© 2015 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).Peer-review under responsibility of Amrita School of Engineering, Amrita Vishwa Vidyapeetham University

296 V. Rajinikanth and Suresh Chandra Satapathy / Procedia Technology 21 ( 2015 ) 295 – 302

design problems [9]. This algorithm is theoretically similar to the teaching-learning scenario existing in the class room [10, 11]. In the proposed work, PID design problem for the Automatic Voltage Regulator (AVR) is addressed. Even though there exists a number of advanced controller structures, PID and enhanced forms of PID controllers are easy to tune and implement [6-8]. Hence, in this paper One Degree Of Freedom (1DOF) PID and Two Degrees Of Freedom (2DOF) PID controllers are designed and implemented on the benchmark AVR system using the traditional TLBO algorithm. The performance of the TLBO is validated using most successful HAs, such as Particle Swarm Optimization (PSO), Bacterial Foraging Optimization (BFO) and Firefly Algorithm (FA).

2. Automatic Voltage Regulator

Benchmark AVR system widely discussed in the literature is considered in this paper [3-5].

2.1 Principle

Detailed theoretical description about the AVR system can be found in [3]. During power generation process, common troubles, such as dissimilarity of load, limit deviation in transmission system, and turbine oscillation may produce oscillatory output in synchronous generator. This category of electro-mechanical fluctuation affects the firmness of power system. Hence, in modern power generating stations, in order to improve the dynamic stability and to assure the power quality, most of the synchronous generators are outfitted with an excitation unit, which is supervised by an AVR and a Power System Stabilizer (PSS) [4, 5]. Fig. 1. Illustrates the block diagram of the AVR system with linearized intermediate units. During closed loop operation, the controller is responsible to maintain stability, robustness and also to support smooth reference tracking performance based on the set value of terminal voltage.

Fig. 1. Block diagram of the AVR system

2.2 Related previous works

Due to its significance, AVR system is widely considered by most of the researchers. Heuristic algorithm based

approaches are already applied on the AVR system in the literature [3-5]. Most of the researchers proposed the PID controller design for the AVR system and the performance of the controller is validated for reference tracking response.Fig.1. indicates that, the delay time present in the higher order AVR system is very small and designing a suitable controller requires the following assumptions: (i) the system is linear, (ii) external disturbance acting on the system is negligible and (iii) the sensor part is free from the measurement noise. In the proposed work, traditional and enhanced forms of PID controller is considered to regulate AVR system and the controller design process in done using heuristic algorithms. 3. PID Controller

Based on the structure and number of initial parameters to be tuned, PID is classified as One Degree Of Freedom

Controller

-

Vref (s)

+ 11.0

10

s

Vout (s)

14.0

1

s

1

1

s

101.0

1

s

Amplifier Exciter Generator

Sensor

297 V. Rajinikanth and Suresh Chandra Satapathy / Procedia Technology 21 ( 2015 ) 295 – 302

(1DOF) controller, Two Degrees Of Freedom (2DOF) controller and Three Degrees Of Freedom (3DOF) controller [6-8]. In the proposed work, the major aim is to support the reference tracking operation and the considered AVR system is an open-loop stable system. Hence, 1DOF and 2DOF PID structures are considered.

3.1 One DOF structure

One Degree Of Freedom (DOF) PID structure is a commonly used controller structure as given in eqn. (1) and the number of control parameters to be tuner is three, such as Kp, Ki and Kd [8].

sKs

KK)s(C d

ip (1)

1DOF PID some time offers larger overshoot (Mp) and larger settling time (ts) due to the proportional and derivative kick. This drawback can be reduced with 2DOF PID structures.

3.2 Two DOF structure

2DOF controllers are enhanced forms of the traditional 1DOF PID controller. A detailed analysis on the existing 2DOF PID structures is available in [6]. Fig. 2 shows the 2DOF PID structures considered in this work such as (a) PID controller with prefilter, (b) PID with Feed-Forward structure and (c) PID with Feed-Back structure and the corresponding mathematical expressions are presented in Eqn. (2) – (6).

(a) PID controller with prefilter (b) PID with Feed-Forward structure (c) PID with Feed-Back structure

Fig. 2. 2DOF PID structures

1s

1)s(F

f (2)

sK

s

KK)s(C d

ip1 (3)

sK K)s(C dp2 (4)

sK)1(

s

K)1(K)s(C d

ip3 (5)

sK K)s(C dp4 (6)

Eqn. (3) and eqn. (4) shows that, inner loop controller is a traditional PID and outer loop has a PD structure with weighting parameters α and β. Similarly, eqn. (5) shows the PID structure with weighting parameters and eqn. (6) shows the PD controller with α and β.

4. Teaching Learning Based Optimization

TLBO is formulated by imitating the teaching-learning system existing in the classroom scenario and its pseudo code is depicted in Fig. 3. Comparable to other heuristic algorithms, the TLBO also employs a population based approach to obtain the universal solution through the search. A comprehensive explanation about the TLBO can be

F(s) C(s) AVR

R(s) Y(

-

C1(s) AVR R(s) Y

-

C2(s)

-

C1(s) AVR R(s) Y

-

C2(s)

-

298 V. Rajinikanth and Suresh Chandra Satapathy / Procedia Technology 21 ( 2015 ) 295 – 302

found in the recent literature [12,13]. In the proposed work, traditional TLBO is considered to tune the PID controllers for a benchmark AVR system. The TLBO has two essential stages, such as teacher stage and learner stage as shown below:

START;

Initialize algorithm parameters, such as number of learners (N), parameters to be optimized (D), Maximum number of iteration (Miter) and objective function (Jmin) ; Randomly initialize ‘N’ learners for xi (i = 1, 2, … n); Evaluate the performance and select the best solution f(xbest);

WHILE iter = 1:Miter; %TEACHER STAGE % Use f(xbest) as teacher; Sort based on f(x), select other teachers based on : f(x)s = f(xbest) – rand for f(x)s = 2,3, . . . , T;

FOR i = 1:n

Calculate ]12)1,0(rand1[roundT iF ;

)];x.T(x)[1,0(randxx meaniFteacher

iinew

%Calculate objective function for f(xinew)%

If f(xinew) < f(xi), then xi = xi

new; End If % End of TEACHER STAGE%

%STUDENT STAGE %

Arbitrarily Select the learner xi, such that ij ;

If f(xi) < f(xj), then xinew=xi+rand(0,1)(xi-x j);

Else xinew=xi+rand(0,1)(x j - xi);

End If If xi

new is better than xi, then xi= xinew;

End If % End of STUDENT STAGE% End FOR

Set k = k+1; End WHILE Record the controller valus, Jmin, and performance measures; STOP;

Fig.3. Pseudo code for TLBO algorithm

4.1. Other Heuristic Algorithms in this Study

In this paper, heuristic algorithms such as Particle Swarm Optimization (PSO), Bacterial Foraging Optimization (BFO) and Firefly Algorithm (FA) are considered to validate the performance of TLBO.

4.1.1 Particle Swarm Optimization

PSO is a well known heuristic technique, developed by modeling the group activities in flock of birds [8]. Due to its high computational capability, it is widely considered by the researches to solve constrained and unconstrained optimization problem. In this work, PSO with the following mathematical expression is considered:

)SG(RC)SP(RCV.W)1t(V ti

ti22

ti

ti11

ti

ti

(7)

)1()1( tiVtiXtiX

(8)

where tW is inertia weight ( chosen as 0.8), R1 and R2 are random values [0,1], C1 and C2 is allotted as 2.1 and 1.8 correspondingly.

299 V. Rajinikanth and Suresh Chandra Satapathy / Procedia Technology 21 ( 2015 ) 295 – 302

4.1.2 Bacterial Foraging Optimization

BFO is developed by mimicking the foraging scheme of E.coli bacteria. In this paper, the enhanced BFO discussed in [8] is considered.The algorithm values are assigned as:

Number of E.Coli bacteria = N

Nc=2

N; Ns=Nre

3

N; Ned

4

N;Nr=

2

N ; Ped=r

ed

NN

N; datt = Watt =

NsN

; and hrep= Wrep =

(9)

4.1.3 Firefly Algorithm FA is originally discussed by Yang [14]. This technique employs a mathematical representation of the firefly, searching for a mate in the search universe and details of FA can be found in [15 - 17]. The association of an attracted firefly towards a mate can be expressed as:

½) - (rand )XX(eβXX 1ti

tj

d γ0

ti

1ti

2ij (10)

where tiX is early location; 1t

iX is updated location; )XX(eβ ti

tj

d γ0

2ij

is attraction among fireflies; β0 is

preliminary attractiveness; γ is absorption coefficient; α1 is randomization operator and rand is random number [0,1]. In this paper, the following values are chosen for FA parameters: α1= 0.15; β0 = 0.1and γ = 1.

5. Result and Discussions

In this paper, simulation study is performed and implemented using Matlab R2010a software. The following objective function is considered to guide the heuristic search:

ITSE.WITAE.Wt.WM.WJ 43s2p1min (11)

where the weights W1 and W2 are chosen as ‘2’ and the W3 and W4 are chosen as ‘5’ (preference is given to the minimization of ITAE and ITSE), Mp is overshoot, ts is the settling time, ITAE and ITSE are integral time absolute error and integral time squared error respectively. The HA assisted exploration is initiated with a search limit for the 1DOF and 2DOF controller parameters are assigned as follows: For controller parameters: 0 < Kp < 0.5; 0 < Ki < 0.5; and 0 < Kd < 0.5. For filter time constant: 0 < Tf < 0.1; For weighting parameters: 0 < α <1 and 0 < β < 1. In order to perform a fair estimation, all the considered heuristic procedures are assigned with the similar preliminary algorithm parameters as specified below:

Population size (N) is 20; Criterion to terminate the search is Jmin , maximum number of iteration is assigned as100 and simulation time is allocated as 5sec. The controller tuning practice is repeated 10 times for each algorithm with each PID structure and the best Jmin acquired between the trials are selected as the most favorable solution. Firstly, 1DOF PID design procedure is executed with TLBO using a three dimensional search (Kp, Ki, Kd). Later, similar tuning procedure is repeated on the AVR system using other heuristic methods, such as PSO, BFO and FA. For PID with prefilter (FPID) a four dimensional search is considered (τf, Kp, Ki, Kd) and the obtained PID values are presented in Table 1. During this search, the following filter values are attained: τfTLBO = 0.0516; τfPSO = 0.0637; τf BFO = 0.0741 and τf FA = 0.0816. For Feed-forward type 2DOF PID (FFPID) a five dimensional search is proposed (α, β, Kp, Ki, Kd) and the optimal values are shown in Table 1. Similar controller parameters are chosen to analyze the performance using the FBPID controller. Initially, the heuristic algorithm designed 1DOF PID controller is considered to support the reference tracking performance of the AVR system. During the simulation study, it is

NcN

300 V. Rajinikanth and Suresh Chandra Satapathy / Procedia Technology 21 ( 2015 ) 295 – 302

assumed that, the system is free from external disturbances. Fig .4(a) presents the value of the terminal voltage with respect to the simulation time and the corresponding performance measure values are recorded in Table 2. From this table, it is noted that, the TLBO offers smaller Mp and ITAE values compared with alternatives. The FA tuned PID results in better Jmin, ts and ITSE compared with TLBO, PSO and BFO.

Table 1. Optimal controller parameters PID Kp Ki Kd α β

1DOF

TLBO 0.1986 0.1217 0.2683 - - PSO 0.1836 0.1311 0.2088 - - BFO 0.1889 0.1263 0.1862 - - FA 0.1958 0.1261 0.2107 - -

2DOF (FPID)

TLBO 0.2026 0.1257 0.3174 - - PSO 0.2003 0.1247 0.3579 - - BFO 0.1995 0.1295 0.3347 - - FA 0.2102 0.1301 0.3257 - -

2DOF (FF)

TLBO 0.4019 0.3382 0.0161 0.3914 0.0214 PSO 0.3904 0.3188 0.0186 0.4018 0.0311 BFO 0.4038 0.3122 0.0206 0.3882 0.0177 FA 0.3986 0.3117 0.0218 0.4170 0.0184

(a) Reference tracking with 1DOF PID

(b) Reference tracking with FPID

(c) AVR response with 2DOF PID (FB)

(d) AVR response with 2DOF PID (FF)

Fig. 4. Reference tracking response of AVR with various controllers

(a) Reference tracking response

(b) Controller output

Fig. 5. AVR’s response with TLBO tuned controllers

0 1 2 3 4 50

0.2

0.4

0.6

0.8

1

1.2

Time (sec)

Ter

min

al v

olta

ge (

V)

ReferenceTLBO PIDPSO PIDBFO PIDFA PID

0 1 2 3 4 50

0.2

0.4

0.6

0.8

1

1.2

Time (sec)

Ter

min

al v

olta

ge (

V)

ReferenceTLBO FPIDPSO FPIDBFO FPIDFA FPID

0 1 2 3 4 50

0.2

0.4

0.6

0.8

1

1.2

Time (sec)

Ter

min

al v

olta

ge (

V)

ReferenceTLBO FFPIDPSO FFPIDBFO FFPIDFA FFPID

0 1 2 3 4 50

0.2

0.4

0.6

0.8

1

1.2

Time (sec)

Ter

min

al v

olta

ge (

V)

ReferenceTLBO FBPIDPSO FBPIDBFO FBPIDFA FBPID

0 1 2 3 4 50

0.2

0.4

0.6

0.8

1

1.2

Time (sec)

Ter

min

al v

olta

ge (

V)

ReferencePIDFPIDFFPIDFBPID

0 1 2 3 4 5-2

0

2

4

6

8

Time (sec)

Co

ntr

olle

r o

utp

ut

PIDFPIDFFPIDFBPID

301 V. Rajinikanth and Suresh Chandra Satapathy / Procedia Technology 21 ( 2015 ) 295 – 302

From Table 1 and Table 2, the observation is that, the controller values obtained for 1DOF PID with TLBO, BFO, PSO and FA and the corresponding performance measure values are approximately similar. Hence, the performance of the considered algorithms on the 1DOF PID is identical. Fig. 4(b) depicts the reference tracking response of AVR with FPID controller. This controller offers smooth response compared with the 1DOF PID. Table 1 denotes that, the controller value provided by the considered heuristic methods is approximately similar. Table 2 shows that, PSO offers better Mp and ITAE values and FA offers improved Jmin, ts and ITSE. Fig .4(c) and Fig . 4(d) shows the set point tracking performance of AVR for FBPID structure and FFPID structure respectively. As discussed earlier, the controller parameters obtained with FBPID is implemented using FFPID structure. Hence, both the 2DOF PID configuration offers identical performance measures as shown in Table 2. The 2DOF PID designed using TLBO offers negligible Mp with better Jmin, ts, ITAE and ITSE values compared with PSO, BFO and FA. A comparative study is also carried to evaluate the performance of 1DOF and 2DOF PID structures. Fig . 5(a) and Fig. 5(b) shows the AVR terminal voltage and corresponding controller output for the PIDs designed using TLBO. From these figures, it can be observed that, the FPID shows sluggish reference tracking response and fluctuating controller output compared with other controller structures. The FBPID and FFPID structures present enhanced reference tracking with better controller output compared with other PIDs. From this study, it is verified that, even though the number of controller parameters to be tuned is large, the 2DOF PID structure offers better setpoint tracking response and enced controller output compared with traditional PID and FPID controllers.

Table 2. Minimized objective function values

PID Method Jmin Mp ts ITAE ITSE

1DOF

TLBO 6.5787 0.0126 1.6015 0.4469 0.2232 PSO 6.8963 0.0472 1.5007 0.5249 0.2352 BFO 6.6788 0.0424 1.4825 0.4980 0.2278 FA 6.5577 0.0361 1.4825 0.4827 0.2214

2DOF (FPID)

TLBO 7.4911 0.0216 1.7382 0.5160 0.2783 PSO 7.6572 0.0000 1.8806 0.4934 0.2858 BFO 7.6532 0.0216 1.7780 0.5258 0.2850 FA 7.2895 0.0249 1.6411 0.5189 0.2726

2DOF (FF)

TLBO 6.5833 0.0236 1.7243 0.4313 0.1862 PSO 6.8609 0.0000 1.8187 0.4416 0.2031 BFO 6.8617 0.0000 1.7281 0.4787 0.2024 FA 7.0679 0.0000 1.7262 0.5041 0.2190

2DOF (FB)

TLBO 6.6271 0.0241 1.7252 0.4391 0.1866 PSO 6.9119 0.0000 1.8187 0.4530 0.2019 BFO 6.9137 0.0000 1.7281 0.4898 0.2017 FA 7.1189 0.0000 1.7262 0.5152 0.2181

5. Conclusion

In this paper, traditional TLBO based 1DOF and 2DOF PID controller design is proposed for a benchmark AVR system and its performance is validated with PSO, BFO and FA. The simulation study shows that, the controller parameters obtained with the considered heuristic algorithms are approximately similar and all the algorithms shows approximately similar Jmin value, time domain values and error values with the traditional PID and FPID controllers. In addition, the proposed study depicts that, the performance of 2DOF PID is better than PID and FPID structures. The FFPID and FBPID designed with traditional TLBO offers better performance measure values compared with the 2DOF PID controller designed using PSO, BFO and FA.

302 V. Rajinikanth and Suresh Chandra Satapathy / Procedia Technology 21 ( 2015 ) 295 – 302

References [1] Bensenouci A, Besheer AH. Voltage and Power Regulation for a Sample Power System using Heuristics Population Search Based PID Design,

International Review of Automatic Control (IRACO), 2012; 5(6): p.737-748. [2] Besheer AH. Wind Driven Induction Generator Regulation Using Ant system Approach to Takagi Sugeno Fuzzy PID Control, WSEAS

Transaction on Systems and Control, 2011; 6(12):p.427-439. [3] Gaing ZL. A Particle Swarm Optimization Approach for Optimum Design of PID Controller in AVR System, IEEE Transactions on Energy

Conversion, 2004; 19(2):p.384- 391. [4] Bendjeghaba O. Continuous Firefly Algorithm for Optimal Tuning of PID Controller in AVR System, Journal of Electrical Engineering, 2014,

65(1):p. 44-49. [5] Wong, CC, Li SA, Wang HY. 2009. Optimal PID Controller Design for AVR System, Tamkang Journal of Science and Engineering, 2009;

12(3), p.259-270. [6] Araki M, Taguchi H. Two-Degree-of-Freedom PID Controllers, International Journal of Control, Automation, and Systems, 2003;

1(4):p.401-411. [7]Chen CC, Huang HP, Liaw HJ. Set-Point Weighted PID Controller Tuning for Time-Delayed Unstable Processes, Ind. Eng. Chem.

Res., 2008;47 (18): p.6983–6990. [8] Rajinikanth V, Latha K. Setpoint weighted PID controller tuning for unstable system using heuristic algorithm, Archives of Control Sciences,

2012, 22(4): p.481-505. [9] Rao RV, Savsani VJ, Vakharia DP. Teaching–learning-based optimization: A novel method for constrained mechanical design optimization

problems, Computer-Aided Design, 2011; 43(3):p.303–315. [10] Rao RV, Patel V. An improved teaching-learning-based optimization algorithm for solving unconstrained optimization problems, Scientia

Iranica D, 2013; 20(3): p.710–720. [11] Rao RV, Patel V. Comparative performance of an elitist teaching-learning-based optimization algorithm for solving unconstrained

optimization problems, International Journal of Industrial Engineering Computations, 2013; 4(1):p.29-50. [12] Suresh Chandra Satapathy, Anima Naik, Modified Teaching–Learning-Based Optimization algorithm for global numerical optimization—A

comparative study, Swarm and Evolutionary Computation, 2014; 16: p.28-37. [13] Suresh Chandra Satapathy, Anima Naik, Parvathi K. A teaching learning based optimization based on orthogonal design for solving global

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Modelling and Simulation in Engineering, vol. 2014, Article ID 794574, 17 pages. [16] Sri Madhava Raja N, Suresh Manic K, Rajinikanth V. Firefly Algorithm with Various Randomization Parameters: An Analysis, In B.K.

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46: p.1449-1457.

Journal of Advances in Mathematics and Computer Science 25(3): 1-5, 2017; Article no.JAMCS.37595

ISSN: 2456-9968 (Past name: British Journal of Mathematics & Computer Science, Past ISSN: 2231-0851)

_____________________________________

*Corresponding author: E-mail: vijayjaliparthi@gmail.com;

Vibrational Spectra of Carbonyl Sulphide by U(2) Lie Algebraic Method

J. Vijayasekhar1*, M. V. Subba Rao2 and G. Ananda Rao2

1Department of Mathematics, GITAM University, Hyderabad, India.

2Department of Mathematics, ANITS, Visakhapatnam, India.

Authors’ contributions

This work was carried out in collaboration between all authors. Author JV constructed Hamiltonian for the study of vibrational spectra of molecule using Lie algebraic method. Authors MVSR and GAR determined

the parameters using numerical fitting procedure. All authors read and approved the final manuscript.

Article Information

DOI: 10.9734/JAMCS/2017/37595 Editor(s):

(1) Metin Basarir, Professor, Department of Mathematics, Sakarya University, Turkey. Reviewers:

(1) Francisco Bulnes, Mexico. (2) O. J. Oluwadare, Federal University Oye-Ekiti, Nigeria.

Complete Peer review History: http://www.sciencedomain.org/review-history/21801

Received: 21st October 2017 Accepted: 2nd November 2017

Published: 8th November 2017

_______________________________________________________________________________

Abstract

In this paper, we have calculated vibrational frequencies of Carbonyl sulphide (OCS) in fundamental level and at higher overtones by Hamiltonian expression, which is in terms of invariant and Majorana operators, describe stretching vibrations. The Hamiltonian is an algebraic one and so far all the operations in this method, unlike the more well-known differential operators of wave mechanics.

Keywords: Vibrational spectra; carbonyl sulphide; U(2) Lie algebraic method.

AMS classification: 81Q05, 81Q35, 81V45.

1 Introduction U(2) Lie algebraic method has been used to trace the Heisenberg formulation of quantum mechanics [1,2,3,4]. The step up of Lie algebraic method to physical systems was introduced by Iachello and Arima in their initiate work of spectra of atomic nuclei [5-11]. Iachello (1981) presented Lie algebraic method for the study of vibrational spectra of molecules [12]. This method is based on the quantization of the Schrodinger

Original Research Article

Vijayasekhar et al.; JAMCS, 25(3): 1-5, 2017; Article no.JAMCS.37595

2

wave equation with a three dimensional Morse potential function and is described as ro-vibration spectra of diatomic molecules [13]. The essential idea of the Lie algebraic method is that series expansion of the Hamiltonian in terms of a set of operators characterizes the local and normal modes of the system. The Lie algebraic method allows us to analyse experimental ro-vibrational spectra of polyatomic molecules based on the idea of dynamical symmetry. This Lie algebraic method is used to obtain Hamiltonian operator that provides the description of ro-vibrational degrees of freedom of the physical system [14].

2 Structure of Carbonyl Sulphide Carbonyl sulphide (OCS) is a linear molecule consisting of a carbonyl group double bonded to a sulphur atom. This consists of two bonds O-C and C-S. The symmetry point group is Dh.

3 U(2) Lie Algebraic Method The Hamiltonian [15,16] for the molecule (stretching vibrations for n bonds) is of the form

n n

0 i i ij ij ij iji=1 i<j

n

i<j

... . (1)H E AC A C M

The eigenvalues of the Hamiltonian can be evaluated and give a description of n coupled anharmonic vibrations. The couplings in the Hamiltonian are only first order, in the sense that the Majorana operators

ijM annihilate one quantum of vibration in bond j and create one in bond i (or vice versa).

Here i vary from 1 to 2 for two stretching bonds (O-C and C-S) and i i j ij( , , )A A are algebraic parameters,

which are calculated by spectroscopic data. Where iC is an invariant operator of the uncoupled bond with

eigenvalues 2

i i i4( v v )N and the operator ijC for coupled bonds are diagonal with matrix elements

2

i i j j ij i i j j i j i j i j, v ; , v , v ; , v 4 v , ...v v v (2)N N C N N N N

while the Majorana operator ijM has both diagonal and non-diagonal matrix elements

i i j j i i j j i j j i i j

1/2

i i j j i i j j j i i i j j

1/2

i i j j i i j j i j j j i

ij

ij

ij i

1 1

1 1 .

, v ; , v , v ; , v v v – 2v v

, v ; , v , v ; , v v v 1 – v – v 1

, v ; , v , v ; , v v v 1 – v – v

1

N N N N N N

N N N N N N

N N N N

M

M N N

M

... (3)

Where iv (i = 1, 2, ...) are vibrational quantum numbers. The vibron number iN (i =1, 2) for stretching

bonds (O-C and C-S) of molecule will be calculated by the following relation

Vijayasekhar et al.; JAMCS, 25(3): 1-5, 2017; Article no.JAMCS.37595

3

i 1, i 1,2 ... (4)

e

e e

Nx

Here ande e ex are the spectroscopic constants. The initial guess value for the parameter iA (i =1, 2) is

obtained by using the energy equation for the single-oscillator fundamental mode, which is given as,

i iv 1 4 1 ... (5)E A N .

Initial guess for ijA taken as zero. The parameter ij determined from the relation.

ji

ij . ... (6)3

E E

N

To get accurate results a numerical fitting procedure is essential to obtain the parameters i ij,A (i, j =1, 2,

ij) starting from values as given by equations (5) and (6).

4 Results Calculated vibrational frequencies and fitted parameters by Lie algebraic method are as follows:

Table 1. Fitting parameters

Parameters [17] OCS

N1 190

N2 159

A1 -1.13 cm-1

A2 -3.26 cm-1

Aij (O-C bond) -0.25 cm-1

Aij (C-S bond) -1.29 cm-1

Table 2. Vibrational frequencies of carbonyl sulphide

Vibrational mode Vibrational frequencies (cm-1)

Experimental [18] U(2) Lie algebraic method

(1 0 0) 859 859.0317

(0 0 1 ) 2062.2 2061.323

(2 0 0) 1711.1 1712.021

(0 0 2) 4101.4 4100.232

(1 0 1) 2918.1 2918.031

(3 0 0) 2556 2557.934

(0 0 3) 6117.6 6118.469

(2 0 1) 3768.5 3767.008

(1 0 2) 4953.9 4953.156

Vijayasekhar et al.; JAMCS, 25(3): 1-5, 2017; Article no.JAMCS.37595

4

5 Conclusion In this paper, vibrational frequencies of Carbonyl sulphide upto second overtone by U(2) Lie algebraic method calculated and also compared with available experimental data. It has been observed that results from the Lie algebraic method make known near to the exact, consistent with the experimental results.

Competing Interests Authors have declared that no competing interests exist.

References [1] Born M, Heisenberg W, Jordan P. Quantum mechanics II, Z. Phys. 1926;35:557-615.

[2] Born M, Jordan J. A vibrational principle for invariant-tori of fixed frequency. Z. Phys. 1925;34(1):

858-888. [3] Dirac PAM. The fundamental equations of quantum mechanics. Proc. Roy. Soc. A. 1925;109:642-

653. [4] Heisenberg W. Quantum theoretical re-interpretation of kinematic and mechanical relations. Z. Phys.

1925;33:879-893. [5] Arima A, Iachello F. Collective nuclear states representations of a SU(6) group. Phys. Rev. Lett.

1975;35(16):1069-1072. [6] Arima A, Iachello F. Interacting Boson model of collective nuclear states IV. The O(6) limit, Ann.

Phys. (NY). 1979;123(2):468-492. [7] Arima A, Iachello F. Interacting Boson model of collective states I: The vibrational limit, Ann. Phys.

(NY). 1976;99(2):253-317. [8] Arima A, Iachello F. Interacting Boson model of collective states II: The rotational limit, Ann. Phys.

(NY). 1978;111(1):201-238. [9] Chen JQ, Klein A, Ping JL. Point-group symmetrised Boson representation. Algebraic solution for

symmetry-adapted bases of Oh, J. Math. Phys. 1996;37:2400-2421. [10] Iachello F, Arima A. Boson symmetries in vibron nuclei. Phys. Lett. B. 1974;53(4):309-312. [11] Iachello F, Arima AA. The interacting Boson model. Cambridge: Cambridge University Press; 1987. [12] Iachello F. Algebraic methods for molecular rotation-vibration spectra. Chem. Phys. Lett. 1981;78(3):

581-585. [13] Iachello F, Levine RD. Algebraic approach to rotation-vibration spectra. I-diatomic molecules, J.

Chem. Phys. 1982;77:3046-3055. [14] Iachello F, Levine RD. Algebraic theory of molecules. Oxford University Press, Oxford; 1995. [15] Oss S. Algebraic models in molecular spectroscopy. Adv. Chem. Phys. 1996;93:455-649.

Vijayasekhar et al.; JAMCS, 25(3): 1-5, 2017; Article no.JAMCS.37595

5

[16] Vijayasekhar J, Rao KS, Prasad BVSNH. Vibrational frequencies of PH3 and NF3: Lie algebraic method. Orien. J. Chem. 2016;32:1717-1719.

[17] Sarkar NK, Choudhury AJ, Karumuri SR, Bhattacharjee R. A comparative study of the vibrational

spectra of OCS and HCP using the Lie algebraic method. Eur. Phys. J. D. 2009;53:163–171.

[18] Aubanel EE, Wardlaw DM. Application of adiabatic switching to vibrational energies of three dimensional HCO, H2O, and H2CO. J. Chem. Phys. 1987;88:495-517.

_______________________________________________________________________________________ © 2017 Vijayasekhar et al.; This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Peer-review history: The peer review history for this paper can be accessed here (Please copy paste the total link in your browser address bar) http://sciencedomain.org/review-history/21801

JoMME (2016) 12-24 © STM Journals 2016. All Rights Reserved Page 12

Journal of Materials & Metallurgical Engineering ISSN: 2231-3818(online), ISSN: 2321-4236(print)

Volume 6, Issue 3

www.stmjournals.com

Effect of Pulse Current Micro Plasma Arc Welding

Parameters on Pitting Corrosion Rate of AISI 316Ti

Sheets in 3.5 N NaCl Medium

Kondapalli Siva Prasad* Department of Mechanical Engineering, Anil Neerukonda Institute of Technology and Sciences,

Visakhapatnam, Andhra Pradesh, India

Abstract Austenitic stainless steel sheets are used for fabrication of components, which require high

temperature resistance and corrosion resistance such as metal bellows used in expansion

joints in aircraft, aerospace and petroleum industries. When they are exposed to sea water

after welding they are subjected to corrosion as there are changes in properties of the base

metal after welding. The corrosion rate depends on the chemical composition of the base

metal and the nature of welding process adopted. Corrosion resistance of welded joints can be

improved by controlling the process parameters of the welding process. In the present work

Pulsed Current Micro Plasma Arc Welding (MPAW) is carried out on AISI 316Ti austenitic

stainless steel of 0.3 mm thick. Peak current, Base current, Pulse rate and Pulse width are

chosen as the input parameters and pitting corrosion rate of weldment in 3.5N NaCl solution

is considered as output response. Pitting corrosion rate is computed using Linear Polarization

method from Tafel plots. Response Surface Method (RSM) is adopted by using Box-Behnken

Design and total 27 experiments are performed. Empirical relation between input and output

response is developed using statistical software and its adequacy is checked using Analysis of

Variance (ANOVA) at 95% confidence level. The main effect and interaction effect of input

parameters on output response are also studied.

Keywords: Plasma Arc Welding, Austenitic Stainless Steel, Pitting Corrosion rate

*Author for Correspondence E-mail: kspanits@gmail.com

INTRODUCTION Austenitic Stainless Steel (ASS), being the

widest in use of all the stainless steel groups

finds application in the beverages industry,

petrochemical, petroleum, food processing and

textile industries amongst others. It has good

tensile strength, impact resistance and wear

resistance properties. In addition, it combines

these with excellent corrosion resistant

properties [1]. Welding is one of the most

employed methods of fabricating ASS

components. ASS is largely highly weldable;

the higher the carbon content, the harder the

SS and so the more difficult it is to weld. The

problem commonly encountered in welded

ASS joints is intergranular corrosion, pitting

and crevice corrosion in severe corrosion

environments. Weld metals of ASS may

undergo precipitation of (CrFe)23C6 at the grain

boundaries, thus depleting Cr and making the

SS weldment to be preferentially susceptible

to corrosion at the grain boundaries. There

may also be the precipitation of the brittle

sigma Fe-Cr phase in their microstructure if

they are exposed to high temperatures for a

certain length of time as experienced during

welding. High heat input welding invariably

leads to slow cooling. During this slow

cooling time, the temperature range of 700–

850oC stretches in time and with it the greater

formation of the sigma phase [2].

In Pulsed current MPAW process, the

interfuse of metals was produced by heating

them with an arc using a nonconsumable

electrode. It is widely used welding process

finds applications in welding hard to weld

metals such as aluminium, stainless steel,

magnesium and titanium [3]. The increased

use of automated welding urges the welding

procedures and selection of welding

parameters must be more specific for good

weld quality and precision with minimum cost

[4]. The bead geometry plays an important

Effect of Pulse Current Micro Plasma Arc Welding parameters Kondapalli Siva Prasad

JoMME (2016) 12-24 © STM Journals 2016. All Rights Reserved Page 13

role in determining the microstructure of the

welded specimen and the mechanical

properties of the weld [5]. The proper

selection of the input welding parameters

which influence the properties of welded

specimen ensure a high quality joint. Stainless

steels are corrosive resistance in nature finds

diversified application. Even stainless possess

good resistance, they are yet susceptible to

pitting corrosion. The pitting corrosion is a

localized dissolution of an oxide-covered

metal in specific aggressive environments. It is

most common and cataclysmic causes of

failure of metallic structures. The detection

and monitoring of pitting corrosion is an

important task in determining the weld quality.

The pitting corrosion is a random, sporadic

and stochastic process and their prediction of

the time and location of occurrence remains

extremely difficult and undefined [6].

Stainless steels may also suffer from different

forms of metallurgical changes when exposed

to critical temperatures. In welding, the heat

affected zone often experiences temperatures

which cause sufficient microstructural changes

in the welded plates. The precipitation of

chromium nitrides, carbides and carbonitrides

in the parent metal occur under various

welding and environmental conditions and

also depends on the grades of stainless steel.

During pulsed MPAW process, the formation

of coarse grains and inter granular chromium

rich carbides along the grain boundaries in the

heat affected zone deteriorates the mechanical

properties. In the present paper the effect of

welding parameters namely peak current, base

current, pulse rate and pulse width on pitting

corrosion rate of AISI 316Ti sheets are

studied. Linear polarization method is adopted

in measuring the pitting corrosion rate.

WELDING PROCEDURE Weld specimens of 100 x 150 x 0.3 mm size

are prepared from AISI 316Ti sheets and

joined using square butt joint. The chemical

composition and tensile properties of AISI

316Ti stainless steel sheets as provided by M/s

Metallic Bellows (I) Pvt. Ltd, Chennai, India

are presented in Tables 1 and 2. Argon is used

as a shielding gas and a trailing gas to avoid

contamination from outside atmosphere. The

welding conditions adopted during welding are

presented in Table 3. From the earlier works

carried out on Pulsed Current MPAW it was

understood that the peak current, back current,

pulse rate and pulse width are the dominating

parameters which effect the weld quality

characteristics [7–10]. The values of process

parameters used in this study are the optimal

values obtained from our earlier papers [7–10].

Hence peak current, back current, pulse rate

and pulse width are chosen as parameters and

their levels are presented in Table 4. Details

about experimental setup are shown in Figure

1. Four factors and three levels are considered

and according to Box-Benhken Design matrix,

27 experiments are performed as per the

Design matrix shown in Table 5.

Fig. 1: Micro Plasma Arc Welding Setup.

Table 1: Chemical Composition of AISI 316Ti (weight %).

C Si Mn P S Cr Ni N Ti

0.042 0.56 1.57 0.014 0.004 16.99 11.63 0.02 0.32

Journal of Materials & Metallurgical Engineering

Volume 6, Issue 3

ISSN: 2231-3818(online), ISSN: 2321-4236(print)

JoMME (2016) 12-24 © STM Journals 2016. All Rights Reserved Page 14

Table 2: Mechanical Properties of AISI 316Ti.

Elongation

(%)

Yield

Strength

(MPa)

Ultimate Tensile

Strength

(Mpa)

52.34 368.20 612.48

Table 3: Welding Conditions.

Power

source

Secheron Micro Plasma Arc

Machine (Model: PLASMAFIX

50E)

Polarity DCEN

Mode of

operation Pulse mode

Electrode 2% thoriated tungsten electrode

Electrode

Diameter 1 mm

Plasma gas Argon and Hydrogen

Plasma gas

flow rate 6 Lpm

Shielding

gas Argon

Shielding

gas flow

rate

0.4 Lpm

Purging gas Argon

Purging gas

flow rate 0.4 Lpm

Copper

Nozzle

diameter

1 mm

Nozzle to

plate

distance

1 mm

Welding

speed 260 mm/min

Torch

Position Vertical

Operation

type Automatic

Table 4: Process Parameters and their Limits.

Input

Factor Units

Levels

-1 0 +1

Peak Current Amps 6 7 8

Base Current Amps 3 4 5

Pulse rate Pulses/Second 20 40 60

Pulse width % 30 50 70

MEASUREMENT OF PITTING

CORROSION RATE Welded joints of stainless steel are subjected

to pitting corrosion when exposed to different

environments. The pitting corrosion rate

depends upon the type, concentration of the

exposed environment and exposure time of the

welded joint. The details about sample

preparation and testing procedure for

measurement of pitting corrosion rate are

discussed in the following sections.

Surface Preparation for Plating

The welded test specimen surface is polished

with 220 and 600 mesh size emery papers in

the presence of distilled water continuously.

The polished specimen is first rinsed with

distilled water, cleaned with acetone and again

rinsed with distilled water to remove the stains

and grease. Finally the specimen is dried to

remove the moisture content on the surface of

the sample.

Sample Preparation for Corrosion Studies

Once the sample is cleaned, the entire sample

is covered by insulating film and only a cross

sectional area of 225 mm2 is exposed

(Figure 2). The perplex tube (Figure 3) is

attached to the test specimen as described in

chapter.

Fig. 2: Dimensions of Corrosion Test

Specimen.

Fig. 3: Perplex Tube.

Effect of Pulse Current Micro Plasma Arc Welding parameters Kondapalli Siva Prasad

JoMME (2016) 12-24 © STM Journals 2016. All Rights Reserved Page 15

Procedure for Corrosion Studies

The electrochemical cell (test specimen with

tube) is initially washed with distilled water

followed by rinsing with filtered electrolyte

NaCl. Around 100 ml of filtered electrolyte is

poured into the electrochemical cell. The

entire electrode assembly is now placed in the

cell. The reference electrode (standard calomel

electrode) is adjusted in such a way that the tip

of this electrode is very near to the exposed

area of working electrode (test specimen). The

auxiliary platinum electrode is also placed in

the cell. Now the cell assembly has been

connected to the AUTOLAB/PGSTAT12. The

black colored plug has been connected to the

auxiliary electrode, red colored plug to the

working electrode and blue to the reference

electrode. The sample has been exposed to

electrolytic medium for a span of 2 h.

As the start button of the potentiostat is

switched on, the electrode potential changes

continuously, till the reaction between the

electrode and the medium attains equilibrium.

After some time the potential remains nearly

constant without any change. This steady

potential which is displayed on the monitor is

taken as open circuit potential (Erest). Now the

equipment is ready for obtaining the

polarization data.

Potential is scanned cathodically until the

potential is equal to Erest minus the limit

potential. Measurements of potential (E) and

current (I) are made at different intervals and

the data are displayed on the monitor itself as

E versus log I plot. After reaching the cathodic

limit, the scan direction is then reversed.

Similarly anodic polarization data are

obtained. The scan is again reversed and

finally terminated and the cell is isolated from

the potentiostat when the potential reached

Erest. The data recorded gives the Tafel plot

(current versus potential data). Using the

software available corrosion rate, corrosion

current, polarization resistance and Tafel

slopes are evaluated by Tafel plot methods.

The Experimental setup is shown in Figure 4.

Corrosion Testing Methodology

Passive metals may become susceptible to

pitting corrosion when exposed to solutions

having a critical content of aggressive ions

such as chloride. This type of corrosion is

potential-dependent and its occurrence is

observed only above the pitting potential

(Ecorr), which can be used to differentiate the

resistance to pitting corrosion of different

metal/electrolyte systems. The Ecorr value can

be determined electrochemically using both

potentiostatic and potentiodynamic techniques.

Linear Polarization Method

The linear polarization method utilizes the

Tafel extrapolation technique. The

electrochemical technique of polarization

resistance is used to measure absolute

corrosion rate, usually expressed in milli-

inches per year (mpy), which is further

converted in to mm per year. Polarization

resistance can be measured very rapidly,

usually less than ten minutes. Excellent

correlation can often be made between

corrosion rates obtained by polarization

resistance and conventional weight-change

determinations. Polarization resistance is also

referred to as “linear polarization”.

Fig. 4: Pitting Corrosion Setup.

Journal of Materials & Metallurgical Engineering

Volume 6, Issue 3

ISSN: 2231-3818(online), ISSN: 2321-4236(print)

JoMME (2016) 12-24 © STM Journals 2016. All Rights Reserved Page 16

Polarization resistance measurement is

performed by scanning through a potential

range which is very close to the corrosion

potential, Ecorr the potential range is generally

±25 mV about Ecorr.

The resulting current versus potential is

plotted. The corrosion current, Icorr is related to

the slope of the plot through the following

equation.

cacorr

ca

ββ2.3I

ββ

ΔI

ΔE

…….. (1)

where, ∆E/∆I is the slope of the polarization

resistance plot, where ∆E is expressed in volts

and ∆I in µA. This slope has units of

resistance, hence, polarization Resistance.

βa,βc are anode and cathode Tafel constants

(must be determined from a Tafel plot as

shown in Figure 5).

These constants have the units of volts/decade

of current.

Icorr= corrosion current, µA.

Rearranging Eq. (1)

E

I

)(3.2I

ca

cacorr

….. (2)

The corrosion current can be related to the

corrosion rate through the following equation.

Corrosion rate

(mpy) = 0.131(Icorr)(Eq.Wt)/ρ ….. (3)

where, Eq.Wt is the equivalent weight of the

corroding species, ρ is the density of the

corroding species, g/cm3, Icorr is the

corrosion current density, µA/cm2.

Fig. 5: Tafel Plot.

STATISTICAL ANALYSIS The pitting corrosion rates for all the 27 samples are performed and presented in Table 5.

Effect of Pulse Current Micro Plasma Arc Welding parameters Kondapalli Siva Prasad

JoMME (2016) 12-24 © STM Journals 2016. All Rights Reserved Page 17

Table 5: Design Matrix with Experimental Results.

Experiment

No.

Peak Current

(Amps)

Base current

(Amps)

Pulse Rate

(Pulses/second)

Pulse width

(%)

Pitting Corrosion Rate (mm/year)

Experimental Predicted

1 6 3 40 50 0.1443 0.1388

2 8 3 40 50 0.1533 0.1507

3 6 5 40 50 0.1434 0.1437

4 8 5 40 50 0.1492 0.1525

5 7 4 20 30 0.1402 0.1393

6 7 4 60 30 0.1462 0.1412

7 7 4 20 70 0.1449 0.1469

8 7 4 60 70 0.1422 0.1405

9 6 4 20 50 0.1437 0.1444

10 8 4 60 50 0.1542 0.1525

11 6 4 20 50 0.1437 0.1444

12 8 4 20 50 0.1441 0.1439

13 7 3 60 30 0.1470 0.1473

14 7 5 40 30 0.1444 0.1447

15 7 3 40 50 0.1419 0.1413

16 7 5 40 50 0.1460 0.1447

17 6 4 40 30 0.1237 0.1269

18 8 4 40 30 0.1504 0.1525

19 6 4 40 70 0.1453 0.1457

20 8 4 40 70 0.1414 0.1407

21 7 3 20 50 0.1448 0.1465

22 7 5 20 50 0.1537 0.1496

23 7 3 60 50 0.1372 0.1440

24 7 5 60 50 0.1462 0.1476

25 7 4 40 50 0.1398 0.1357

26 7 4 40 50 0.1295 0.1357

27 7 4 40 50 0.1367 0.1357

Empirical Mathematical Modeling

In RSM design, mathematical models are

developed using polynomial equations. The

type of polynomial equation depends on the

problem.

In most RSM problems, the type of the

relationship between the response (Y) and the

independent variables is unknown [11,12].

Thus the first step in RSM is to find a suitable

approximation for the true functional

relationship between the response and the set

of independent variables.

Usually, a low order polynomial is some

region of the independent variables is

employed to develop a relation between the

response and the independent variables. If the

response is well modeled by a linear function

of the independent variables then the

approximating function in the first order

model is

Y = bo+bi xi + …….. (4)

where, bo, bi are the coefficients of the

polynomial and represents noise or error.

If interaction terms are added to main effects

or first order model, then the model is capable

of representing some curvature in the response

function, such as

Y = bo+bi xi + bijxixj+ …….. (5)

Journal of Materials & Metallurgical Engineering

Volume 6, Issue 3

ISSN: 2231-3818(online), ISSN: 2321-4236(print)

JoMME (2016) 12-24 © STM Journals 2016. All Rights Reserved Page 18

A curve results from Eq. (5) by twisting of the

plane induced by the interaction term bijxixj.

There are going to be situations where the

curvature in the response function is not

adequately modeled by Eq. 5. In such cases, a

logical model to consider is

Y = bo+bi xi +biixi2 + bijxixj+ ….. (6)

where, bii represent pure second order or

quadratic effects. Eq. (6) represents a second

order response surface model. Using

MINITAB Ver.14, statistical software, the

significant coefficients are determined and

final model is developed incorporating these

coefficients to estimate the pitting corrosion

rate. In the empirical model only significant

coefficients are considered.

Pitting Corrosion Rate = 0.321023 –

0.031167X1-0.052886X2-0.002493X3

+0.002278X4+ 0.007313X22-0.000383X1X4

where, X1, X2, X3 and X4 are the coded values

of peak current, base current, pulse rate and

pulse width, respectively.

Checking the Adequacy of the Developed

Model for Pitting Corrosion Rate

The adequacy of the developed models is

tested using the ANOVA. As per this

technique, if the calculated value of the Fratio of

the developed model is less than the standard

Fratio (F-table value 4.60) value at a desired

level of confidence of 95%, then the model is

said to be adequate within the confidence

limit. ANOVA test results are presented in

Table 6 for pitting corrosion rate. From Table

6 it is understood that the developed

mathematical models are found to be adequate

at 95% confidence level. Coefficient of

determination ‘R2’ is used to find how close

the predicted and experimental values lie. The

value of ‘R2’ for the above developed models

is found to be about 0.82, which indicates a

good correlation to exist between the

experimental values and the predicted values.

Figure 6 indicates the scatter plots for pitting

corrosion rate of the weld joint and reveals

that the actual and predicted values are close

to each other within the specified limits.

Table 6: ANOVA Test Results for Pitting

Corrosion Rate.

Source DF Seq SS Adj SS Adj MS F

P

Regression 14 0.000918 0.000918 0.000066 3.49

0.018

Linear 4 0.000225 0.000350 0.000088 4.65

0.017

Square 4 0.000355 0.000306 0.000077 4.07

0.026

Interaction 6 0.000339 0.000339 0.000056 3.00

0.050

Residual

Error 12 0.000226 0.000226 0.000019

Lack-of-Fit 9 0.000169 0.000169 0.000019 1.00

0.565

Pure Error 3 0.000057 0.000057 0.000019

Total 26 0.001144

where, SS is the Sum of Squares, MS is the

Mean Squares, DF is the Degree of Freedom,

F is the Fisher’s ratio, P is the probability

ratio.

PREDICTED

EX

PER

IMEN

TA

L

0.1550.1500.1450.1400.1350.130

0.155

0.150

0.145

0.140

0.135

0.130

0.125

0.120

Scatterplot of Corrosion Rate(mm/Year)

Fig. 6: Scatter Plot of Pitting Corrosion Rate.

Effect of Pulse Current Micro Plasma Arc Welding parameters Kondapalli Siva Prasad

JoMME (2016) 12-24 © STM Journals 2016. All Rights Reserved Page 19

RESULTS AND DISCUSSION Effect of welding parameters on pitting

corrosion rate is indicated by the main effect

plot as shown in Figure 7.

Main Effect Plots

From Figure 7, it is understood that the

variation of each individual parameter on

pitting corrosion rate can be assessed.

Pitting corrosion rate increases with the peak

current from 6 to 8 Amps. This is because as

the current increases heat input increases. At

higher heat input, precipitation of (CrFe)23C6at

the grain boundaries takes place, thus

depleting Cr and making the weldment to be

preferentially susceptible to corrosion at the

grain boundaries.

Pitting corrosion rate decreases with the base

current from 3 to 4 Amps, afterwards it

increases. The variation is because, at low base

current generally low peak current will be

used, however, as the purpose of base current

is to maintain the arc, instead of melting the

work piece, the Pitting corrosion rate tends to

decrease as precipitation of (CrFe)23C6 is low.

But when the base current crosses over 4

Amps, corresponding peak current will

increase leading to more precipitation of

(CrFe)23C6 and hence the pitting corrosion rate

increases upto 5 Amps. Pitting corrosion rate

decreases with the pulse rate from 20 to 40

pulses/sec. This may be because, at low pulse

rate, the current variation between base current

and peak current is less, which leads to low

heating of the base metal. However, when the

pulse rate is above 40 pulses/sec, the current

variation between base current and peak

current is high, which leads to more melting of

base metal and precipitation of (CrFe)23C6at

the grain boundaries.

Pitting corrosion rate increases with the pulse

width from 20 to 50%. This is because as the

pulse width increases, the peak current

duration will be more in pulsed mode, leading

to more melting and high correction rate.

When the pulse width crosses 50%, it shows a

negative trend because of high time gap for

cooling the base metal, which leads to lower

precipitation of (CrFe)23C6 at the grain

boundaries.

Contour Plots of Pitting Corrosion Rate of

3.5N NaCl

The simultaneous effect of two parameters at a

time on the output response is generally

studied using contour plots and surface plots.

Contour plots play a very important role in the

study of the response surface. By generating

contour plots using statistical software

(MINITAB Ver.14) for response surface

analysis, the most influencing parameter can

be identified based on the orientation of

contour lines. If the contour patterning of

circular shaped contours occurs, it suggests the

equal influence of both the factors; while

elliptical contours indicate the interaction of

the factors.

Co

rro

sio

n R

ate

(mm

/Y

ea

r)

876

0.148

0.146

0.144

0.142

0.140

543

604020

0.148

0.146

0.144

0.142

0.140

705030

Peak Current(Amps) Base Current (Amps)

Pulse Rate (pulses/sec) Pulse Width(%)

Main Effects Plot for Corrosion Rate(mm/Year)

Fig. 7: Main Effect Plot of Pitting Corrosion Rate.

Journal of Materials & Metallurgical Engineering

Volume 6, Issue 3

ISSN: 2231-3818(online), ISSN: 2321-4236(print)

JoMME (2016) 12-24 © STM Journals 2016. All Rights Reserved Page 20

Figures 8(a) to (f) represent the contour plots

for pitting corrosion rates. From these plots,

the interaction effect between the input

process parameters and output response can be

observed as:

1. Pitting corrosion rate is more sensitive to

change in peak current than in the base

current [Figure 8(a)], since the contour

lines are more diverted towards peak

current.

2. Pitting corrosion rate is sensitive to peak

current than in the pulse rate [Figure 8(b)],

since the contour lines are more diverted

towards peak current.

3. Pitting corrosion rate is more sensitive to

peak current than pulse width [Figure

8(c)], since the contour lines are more

diverted towards peak current.

4. Pitting corrosion rate is more sensitive to

pulse rate than base current [Figure 8(d)],

since the contour lines are more diverted

towards pulse rate.

5. Pitting corrosion rate is more sensitive to

pulse width than base current [Figure

8(e)], since the contour lines are more

diverted towards pulse width.

6. Pitting corrosion rate is more sensitive to

pulse width than pulse rate [Figure 8(f)],

since the contour lines are more diverted

towards pulse width.

From the above welding parameters

considered, it is understood that peak current

is the most important parameter which affects

the pitting corrosion rate of the welded joints.

Peak Current(Amps)

Ba

se C

urr

en

t (A

mp

s)

0.170

0.165

0.160

0.155

0.150

8.07.57.06.56.0

5.0

4.5

4.0

3.5

3.0

Contour Plot of Corrosion Rate

Peak Current(Amps)

Pul

se R

ate

(pul

ses/

sec)

0.29

0.28

0.27

0.26

0.25

8.07.57.06.56.0

60

50

40

30

20

Contour Plot of Corrosion Rate

Fig. 8(a): Contour Plot for Peak Current vs.

Base Current for Corrosion Rate (3.5N NaCl).

Fig. 8(b): Contour Plot for Peak Current vs.

Pulse Rate for Corrosion Rate (3.5N NaCl).

Peak Current(Amps)

Pu

lse

Wid

th(%

)

0.275

0.270

0.265

0.265

8.07.57.06.56.0

70

60

50

40

30

Contour Plot of Corrosion Rate

Base Current (Amps)

Pu

lse

Ra

te (

pu

lse

s/se

c)

0.180

0.165

0.150

0.135

0.120

5.04.54.03.53.0

60

50

40

30

20

Contour Plot of Corrosion Rate

Fig. 8(c): Contour Plot for Peak Current vs.

Pulse Width for Corrosion Rate (3.5N NaCl).

Fig. 8(d): Contour Plot for Base Current vs.

Pulse Rate for Corrosion (3.5N NaCl).

Effect of Pulse Current Micro Plasma Arc Welding parameters Kondapalli Siva Prasad

JoMME (2016) 12-24 © STM Journals 2016. All Rights Reserved Page 21

Base Current (Amps)

Pul

se W

idth

(%)

0.42

0.40

0.38

0.36

0.34

0.32

5.04.54.03.53.0

70

60

50

40

30

Contour Plot of Corrosion Rate

Pulse Rate (pulses/sec)

Pul

se W

idth

(%)

0.40

0.35

0.30

6050403020

70

60

50

40

30

Contour Plot of Corrosion Rate

Fig. 8(e): Contour Plot for Base Current vs.

Pulse Width for Corrosion Rate (3.5N NaCl).

Fig. 8(f): Contour Plot for Pulse Rate vs. Pulse

Width for Corrosion Rate (3.5N NaCl).

5

0.15

4

0.16

0.17

0.18

Base Current (Amps)67 3

8Peak Current(Amps)

Surface Plot of Corrosion Rate

corrosion rate (mm/year)

Fig. 9(a): Surface Plot for Peak Current vs. Base

Current for Corrosion Rate (3.5N NaCl). Fig. 9(b): Surface Plot for Peak Current vs.

Pulse Rate for Corrosion Rate(3.5N NaCl).

600.264

0.270

0.276

0.282

45 Pulse Width(%)67 308

Peak Current(Amps)

Surface Plot of Corrosion Rate

corrosion rate (mm/year)

Fig. 9 (c): Surface Plot for Peak Current vs.

Pulse Width for Corrosion Rate (3.5N NaCl). Fig. 9(d): Surface Plot for Base Current vs.

Pulse Rate for Corrosion Rate (3.5NNaCl).

600.30

0.35

0.40

0.45

45 Pulse Width(%)34 305

Base Current (Amps)

Surface Plot of Corrosion Rate

corrosion rate (mm/year)

60

0.30

0.35

0.40

0.45

45 Pulse Width(%)2040 30

60

Pulse Rate (pulses/sec)

Surface Plot of Corrosion Rate

corrosion rate (mm/year)

Fig. 9(e): Surface Plot for Base Current vs. Pulse

Width for Corrosion Rate (3.5N NaCl). Fig. 9(f): Surface Plot for Pulse Rate vs.

Pulse Width for Corrosion Rate (3.5NNaCl).

Journal of Materials & Metallurgical Engineering

Volume 6, Issue 3

ISSN: 2231-3818(online), ISSN: 2321-4236(print)

JoMME (2016) 12-24 © STM Journals 2016. All Rights Reserved Page 22

Surface Plots

Surface plots help in locating the maximum

and minimum value of the response. The

maximum value of the response is represented

by the apex of the surface plot, whereas the

minimum value is indicated by nadir of the

surface plot. The minimum pitting corrosion

rate is indicated by the nadir of the response

surface, as shown in Figure 9(a) to (f).

Figure 9(a) the minimum pitting corrosion rate

is exhibited by the nadir of the response

surface. It can be seen from the twisted plane

of surface plot that the model contains

interaction. From the response plot, it is

identified that at a peak current of 6 Amps and

base current of 4 Amps, pitting corrosion rate

is minimum. Figure 9(b) depicts that at a peak

current of 6 Amps and pulse rate of 60

pulses/sec, pitting corrosion rate is minimum.

Figure 9(c) shows the three dimensional

response surface plot, it can be seen from the

twisted plane of surface plot that the model

contains interaction. From the response plot, it

is identified that at the peak current of 6 Amps

and pulse width of 30%, pitting corrosion rate

is minimum. Figure 9(d) indicates that at a

base current of 3 Amps and pulse rate of 60

pulses/sec, pitting corrosion rate is minimum.

Figure 9(e) represents that at a base current is

3 Amps and pulse width of 30%, pitting

corrosion rate is minimum. Figure 9(f)

discusses that when pulse rate is 60 pulses/sec

and pulse width of 30%, the pitting corrosion

rate is minimum.

It is clear from the above observations, that for

a peak current of 6 Amps, base current of 3

Amps, pulse rate of 60 pulses/sec and pulse

width of 30% minimum pitting corrosion rate

is achieved.

Microscopic Analysis of Weld Joint

Figures 10(a) and (b) indicate the weld joint

before corrosion and after pitting corrosion.

The dark round spots indicate the area where

pitting corrosion has taken place.

Fig. 10(a): Weld Joint before Corrosion. Fig. 10(b): Weld Joint after Corrosion.

Fig. 11(a): SEM of Base Metal. Fig. 11(b): SEM of Weld Joint after Corrosion.

Effect of Pulse Current Micro Plasma Arc Welding parameters Kondapalli Siva Prasad

JoMME (2016) 12-24 © STM Journals 2016. All Rights Reserved Page 23

Table 7: Chemical Compositions of Base Metal.

Element O Na Si Cl Ti Cr Mn Fe Ni Mo

Weight % 6.08 0.64 0.81 0.36 0.29 14.49 0.75 64.32 9.61 2.65

Atomic % 18.20 1.32 1.38 0.49 0.29 13.35 0.66 55.15 7.84 1.32

Table 8: Chemical Compositions Weld Joint after Corrosion.

Element O Na Si Cl Ti Cr Mn Fe Ni Mo

Weight % 13.47 2.79 0.70 2.46 0.78 12.86 1.13 54.51 9.38 1.91

Atomic % 33.71 4.87 0.99 2.78 0.65 9.90 0.83 39.08 6.39 0.80

Scanning Electron Microscope (SEM) analysis

is carried out to identify the depleting of Cr %

after the weld joint is subjected to pitting

corrosion in 3.5N NaCl solution. Figures 11(a)

and (b) indicate the SEM images of base metal

and weld joint after corrosion and the chemical

compositions. It is observed that depletion of

1.63% (wt.%) of Cr takes place because of

corrosion.

From SEMEDAX, the chemical composition

of base metal and weld joint after corrosion

are shown in Tables 7 and 8.

CONCLUSIONS The following conclusions are drawn from the

experiments performed and statistical analysis.

1. Developed empirical mathematical model

for predicting pitting corrosion rate of

pulsed current MPAW AISI 316Ti sheets

in 3.5 N NaCl medium.

2. The adequacy of the developed model is

checked using ANOVA and from the

scatter plot is understand that the

experimental and predicted values are

close to each other.

3. From the main effect plots, it is

understood that peak current is the

important parameter which influences the

corrosion rate.

4. From the contour plots, it is clear that peak

current is the most important parameter

which affects the pitting corrosion rate of

the welded joints, followed by base

current, pulse rate and pulse width.

5. From the surface plots, it is understood

that for a peak current of 6 Amps, base

current of 3 Amps, pulse rate of 60

pulses/sec and pulse width of 30%

minimum pitting corrosion rate is

achieved.

6. From SEMEDAX it is observed that there

is depletion of 1.63% (wt%) chromium

after corrosion. This is due to high heat

input generated because of welding

current.

7. The developed empirical mathematical

model is valid for the chosen material,

however, the accuracy can be improved by

considering more number of factors and

their levels.

ACKNOWLEDGMENTS The work presented here was conducted with

funding from the University Grants

Commission, Government of India under

Minor Research Project, F.No.MRP-6020/15

(SERO/UGC).

REFERENCES 1. Dillon CP. Corrosion Control in the

Chemical Process Industry, NACE

International, Houston, Texas, 1994.

2. Pickering FB. Stainless Steel ‘84’. The

Institute of Metals: London, 1985, 2p.

3. Cary HB. Modern Welding Technology,

Prentics Hall, New Jersey, 1989.

4. Samati Z. Automatic Pulsed MIG Welding,

Metal Construction. 1986, 38R- 44.

5. Konkol PJ, Koons GF. Optimization of

Parameters for Two Wire AC- ACSAW,

Am Welding J. 1978; 27: 367s–374s.

6. Fong – Yuan Ma, Corrosive Effects of

Chlorides on Metals, Pitting Corrosion,

Nasr Bensalah (Ed.), Intech open, 2012.

7. Siva Prasad K, Ch. Srinivasa Rao,

Nageswara Rao D. Optimization of pulsed

current parameters to minimize pitting

corrosion in pulsed current micro plasma

arc welded AISI 304L sheets using genetic

algorithm, Int J Lean Thinking. 2013; 4(1):

9–19p.

Journal of Materials & Metallurgical Engineering

Volume 6, Issue 3

ISSN: 2231-3818(online), ISSN: 2321-4236(print)

JoMME (2016) 12-24 © STM Journals 2016. All Rights Reserved Page 24

8. Siva Prasad K, Ch. Srinivasa Rao,

Nageswara Rao D. Effect of Welding

Parameters on Pitting Corrosion Rate in

3.5N NaCl of Pulsed Current Micro

Plasma Arc Welded AISI 304L Sheets, J

Manuf Sci Prod. 2013; 13(1-2): 15–23p.

9. Kondapalli Siva Prasad, Ch. Srinivasa

Rao, Nageswara Rao D. Application of

Grey Relational Analysis for Optimizing

Weld pool geometry parameters of Pulsed

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Kondapalli Siva Prasad. Effect of Pulse

Current Micro Plasma Arc Welding

Parameters on Pitting Corrosion Rate of

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12–24p.

SSRG International Journal of Mechanical Engineering (SSRG-IJME) – Special Issue May - 2017

ISSN: 2348 – 8360 www.internationaljournalssrg.org Page 304

Effect of Hank’s solution on sliding wear

behaviour of Cr3C2-NiCr coated

Ti6Al4V alloy M. Raja Roy #1, N. Ramanaiah*2, B. S. K. Sundara Siva Rao #3

#1Sr.Assistant Professor, Department of Mechanical Engineering, ANITS,Visakhapatnam,Andhra Pradesh,India *2Professor, Department of Mechanical Engineering, AUCOE(A)Visakhapatnam, Andhra Pradesh, India

3Former Professor, Department of Mechanical Engineering,AUCOE(A)Visakhapatnam, Andhra Pradesh, India

Abstract — Ti6Al4V alloys are widely used in medical

applications due to their bio compatibility and high

strength. But, Ti6Al4V alloys are poor in wear. Poor

wear resistance results in formation of wear debris in

implants and causing pain and inflammation. In this

work, Cr3C2-NiCr coatings were applied to improve

the hardness and wear resistance. These are deposited

on the substrate (Ti6Al4V) with 100µm, 200µm,

300µm and 400µm thickness using detonation

spray(DS). Pin on disc wear tests have been carried

out on these materials in simulated body environment

(Hank’s solution) with ASTM G-99 standard

specimens. Improvement was observed in hardness

and wear resistance when compared to substrate.

Wear behaviour of the Cr3C2-NiCr coated Ti6Al4V

alloy was studied using Taguchi design of

experiments. Wear resistance improved and it was

better in simulated body environment.

Keywords — Detonation Spray, Ti6Al4V, Surface

Coatings, Wear, Hank’s solution, Taguchi’s

orthogonal array, ANOVA.

I. INTRODUCTION

Mechanical properties and bio-chemical

compatibility makes Ti6Al4V alloy suitable for

orthopedic implant applications [1]. Implants are

subjected to action of sliding and rubbing contact of

articulated surfaces during their service in body.

Generally wear property can be defined as a source of

damage to solid surface by progressive loss of

material, due to relative motion between that surface

and a contacting surface[2]. The property of poor wear

resistance generates wear debris, when the artificial

implant is in contact with the healthy and natural joint,

the accumulated wear debris causes inflammation,

pain and finally loosening of the joint[3]. Hank’s

solution [18] was used to conduct the wear test under

simulated body environment.

Thermal barrier coatings are often deposited

on metals to improve mechanical and tribological

properties. Detonation spray(DS) is a thermal barrier

coating technology expelling the melting or semi-

melting state powder heated by the combustion of fuel

in presence of oxygen to the surface of work piece at a

high speed, which has been extensively used in many

fields, such as aviation, space flight, petroleum,

metallurgy and other chemical and machinery

industries[4].This method gives an extremely good

adhesive strength, low porosity and coating surfaces

with compressive residual stresses.

The present research is carried out with the

aim of determining the wear behavior of Cr3C2-NiCr

coated [6] Ti6Al4V implant alloy. Detonation spray

technique was used to deposit the coating and

thickness was varied as 100µm, 200µm, 300µm and

400µm respectively to study the effect of coating

thickness on wear resistance. Hardness of both

substrates and coated specimens were found by

conducting hardness test. Wear test was performed for

different loads, speeds, sliding distances and coating

thickness using pin-on-disc apparatus. Hank’s solution

was used for simulating body[19] fluid environment

and a comparison is also made between sliding wear

behaviour of Substrate and Coated material in wet

condition.

The design of experiments (DOE) approach

using Taguchi technique has been successfully used

by researchers in the study of wear behavior[16]. The

DOE process is made up of three main phases: the

planning phase, the conducting phase, and the analysis

phase. A major step in the DOE process is the

determination of the combination of factors and levels

which will provide the desired information. Analysis

of the experimental results uses a signal to noise ratio

to aid in the determination of the best process designs.

The Taguchi technique is a powerful design of

experiment tool for acquiring the data in a controlled

way and to analyze the influence of process variable

over some specific variable which is unknown

function of these process variables and for the design

of high quality systems. Taguchi creates a standard

orthogonal array to accommodate the effect of several

factors on the target value and defines the plan of

SSRG International Journal of Mechanical Engineering (SSRG-IJME) – Special Issue May - 2017

ISSN: 2348 – 8360 www.internationaljournalssrg.org Page 305

experiment. The experimental results are analyzed

using analysis of means and variance to study the

influence of parameters [16-17]. A multiple linear

regression model is developed to predict the wear rate

of Ti6Al4V. The major aim of the present

investigation is to analyse the influence of parameters

like load, sliding speed, sliding distance and coating

thickness on sliding wear of Ti6Al4V coated with

Cr3C2-NiCr using Taguchi technique.

II. EXPERIMENTAL WORK

A. Detonation Spray

Precisely measured quantity of the

combustion mixture consisting of oxygen and

acetylene is fed through a tubular barrel closed at one

end. In order to prevent the possible back firing a

blanket of nitrogen gas is allowed to cover the gas

inlets. Simultaneously, a predetermined quantity of the

coating powder is fed into the combustion chamber.

The gas mixture inside the chamber is ignited by a

simple spark plug. The combustion of the gas mixture

generates high pressure detonation wave, which then

propagate through the gas stream. Depending upon the

ratio of the combustion gases, the temperature of the

hot gas stream can go up to 40000C and the velocity of

the shock wave can reach 3500m/sec. The hot gases

generated in the detonation chamber travel down the

barrel at a high velocity and in the process heat the

particles to a plasticizing stage and also accelerate the

particles to a velocity of 1200m/sec. These particles

then come out of the barrel and impact the component

held by the manipulator to form a coating. The high

kinetic energy of the hot powder particles on impact

with the substrate result in a build up of a very dense

and strong coating [10-11].

B. Material and coating deposition

Ti6Al4V was used as substrate and its chemical

composition is given in Table-I. Cr3C2-NiCr was used

as coating material whose chemical composition is

given in Table-II. In the present work, coatings are

performed by 100µm, 200µm , 300µm , 400µm thick

using detonation spray technique. Prior to coating,

Optimum surface roughness was obtained through

Grid blasting with Al2O3 grits for the best adhesion

between coating and substrate. Figure-1 and Figure-2

shows the detonation spray and Grid blasting

equipment used in the present work. The spraying

process parameters for DS are listed in Table-III.

TABLE I

Chemical composition(Weight %) of Ti6Al4V

TABLE II

Chemical composition(Weight %) of Cr3C2-NiCr

TABLE III

DS parameters for Cr3C2-NiCr deposition Oxygen flow rate(slph) 850

Acetelene Fuel(slph) 2440

Nitrogen flow rate(slph) 12

Spray distance 120mm

Gun speed 10mm/sec

Fig 1: Detonation Spray Process

Fig 2: Grid blasting

C. Hank solution

Hank's solution was prepared using high purity

reagents. The chemical composition of the Hank's

solution [20] is shown in Table-IV.

Table IV

Hank solution chemical composition

Component (g/L) Component (g/L)

Nacl 8 Na2HPO4.

2H2O

0.06

KCL 0.4 KH2PO4 0.06

NaHCO3 0.35 MgSO4.7H2O 0.06

CaCl2 0.14 Glucose 1

MgCl2.6H2O 0.1 pH 6.8

D. Hardness

Hardness of substrate and coated material

were found by IS 1586 test procedure as per BIS

standards using Rockwell hardness tester. An average

Ti Al V Fe Cr Mo

Balance 6.53 3.85 0.08 0.01 0.03

Cr3C2 NiCr

75 25

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of five readings is reported.

E. Wear Test

The sliding wear tests were conducted on a

pin on disc wear testing machine using Hank's

solution according to the ASTM - G99 standards.

Cylindrical specimens of size ϕ3mm and 30mm length

made with substrate and coated material was used as

test material. Chrome steel was used as counter face

material. Figure-3 shows the pin on disc wear testing

machine and Figure-4 Shows the specimens prepared

for wear test.

Fig 3: Pin on Disc Wear Testing Machine

Fig 4: Cr3C2 - NiCr Coated Ti6Al4V Specimens

The wear test were carried out by taking the taguchi

design of experiments by considering the load, speed,

distance and coating thickness as factors and each

factor is taken to four levels as shown in Table-V and

Table-VI. Weight loss of the specimens was measured

by using a balance with an accuracy of ±0.0001g.

Table V

Parameters for wear test

Factors Levels

1 2 3 4

Load in N 10 30 40 50

Speed in m/s 0.6 0.9 1.2 1.5

Distance in Km 0.25 0.5 0.75 1

Coating Thickness

in µm

100 200 300 40

0

Table VI

Taguchi Design of Experiments for wear test

Expt

No.

Load

in N

Speed

in

m/sec

Distance

in Km

Thickness

in µm

1 10 0.6 0.25 100

2 10 0.9 0.5 200

3 10 1.2 0.75 300

4 10 1.5 1 400

5 30 0.6 0.5 300

6 30 0.9 0.25 400

7 30 1.2 1 100

8 30 1.5 0.75 200

9 40 0.6 0.75 400

10 40 0.9 1 300

11 40 1.2 0.25 200

12 40 1.5 0.5 100

13 50 0.6 1 200

14 50 0.9 0.75 100

15 50 1.2 0.5 400

16 50 1.5 0.25 300

III. RESULTS AND DISCUSSION

A. Hardness Rockwell hardness (HRC) values for

substrate and coated material were found by IS1586

test procedure as per BIS standards and average of

five readings is reported in Table-VII. Significant

improvement in hardness is achieved through coating

from 100µm to 400µm thickness.

Table VII

Rockwell hardness (HRC) values for substrate and

coated material

Material Coating

Thickness

(Microns)

Rockwell

Hardness

(HRC)

Ti6Al4V Base metal 25.67

Cr3C2 - NiCr

Coated Ti6Al4V

100 35.33

Cr3C2 - NiCr

Coated Ti6Al4V

200 41.33

Cr3C2 - NiCr

Coated Ti6Al4V

300 49.67

Cr3C2 - NiCr

Coated Ti6Al4V

400 53.33

B. Wear Analysis

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ISSN: 2348 – 8360 www.internationaljournalssrg.org Page 307

Wear test was carried out for the Ti6Al4V

substrate using Pin-on disc wear testing machine and

Percentage of weight loss was obtained as 9.77. Effect

of Cr3C2 - NiCr coating on Ti6Al4V substrate was

studying by varying thickness (100µm, 200µm,

300µm, 400µm) using Taguchi design of experiments.

MINITAB software was used for analyzing Taguchi

design. The experimental results were transformed

into signal-to-noise (S/N) ratios. S/N ratio is defined

as the ratio of the mean of the signal to the standard

deviation of the noise. The S/N ratio indicates the

degree of the predictable performance of a product or

process in the presence of noise factors. The S/N ratio

for wear rate using ‘smaller the better’ characteristic,

which can be calculated as logarithmic transformation

of the loss function, is given as:

𝑆 𝑁 = −10 × 𝑙𝑜𝑔 𝑌2

𝑛

Where y is the observed data (wear and surface

roughness) and n is the number of observations. The

above S/N ratio transformation is suitable for

minimization of percentage of weight loss.

C. Sliding wear behaviour in Simulated body

Environment

Hank solution is used to simulated the body

fluid and sliding wear behavior in wet condition was

recorded by conducting experiments as per Taguchi

orthogonal array. The experimental values were

transformed into S/N ratios for measuring the quality

characteristics using MINITAB software. The S/N

ratio obtained for all the experiments are shown in

Table-VIII and response to Signal to Noise ratios are

presented in Table-IX . Maximum weight loss

obtained was listed in Table-X.

Table VIII Percentage of weight loss and S/N Ratio’s for sliding

wear behavior in simulated body fluid

Exp

t

No.

L

N

S

m/se

c

D

m

Thic

k

μm

% of

Weight

loss

S/N

Ratio

1 10 0.6 0.25 100 0.03604

8 28.8624

1

2 10 0.9 0.5 200 0.09395

7 20.5414

3 10 1.2 0.75 300 0.07427

9 22.5826

3

4 10 1.5 1 400 0.02803

4 31.0463

3

5 30 0.6 0.5 300 0.21157

4 13.4907

4

6 30 0.9 0.25 400 0.01145

4 38.8208

3

7 30 1.2 1 100 0.08755

6 21.1542

4

8 30 1.5 0.75 200 0.05657

3 24.9478

8

9 40 0.6 0.75 400 0.11852

2 18.5240

5

10 40 0.9 1 300 0.05042

1 25.9477

3

11 40 1.2 0.25 200 0.04560

4 26.8199

4

12 40 1.5 0.5 100 0.09806

4 20.1697

8

13 50 0.6 1 200 0.19742

2 14.0920

9

14 50 0.9 0.75 100 0.12825

5 17.8384

9

15 50 1.2 0.5 400 0.07857

2 22.0946

1

16 50 1.5 0.25 300 0.14044

1 17.0501

4

Table IX Response Table for Signal to Noise ratios- Smaller is

better (% Weight loss)

Level

Load,

N

Speed

m/sce

Distance

Km

Thickness

Micron

1 25.76 18.74 27.89 22.01

2 24.6 25.79 19.07 21.6

3 22.87 23.16 20.97 19.77

4 17.77 23.3 23.06 27.62

Delta 7.99 7.04 8.81 7.85

Rank 2 4 1 3

Table X Maximum Percentage of weight loss obtained for

substrate and coated specimens in wet condition S.No Percentage of

weight loss

1 Ti6Al4V

substrate

7.25

2 Cr3C2 - NiCr

Coated Ti6Al4V

0.211574

D. Analysis of variance

ANOVA was used to determine the design

parameters significantly influencing the wear rate

(response). Table-XI shows the results of ANOVA for

wear. This analysis was evaluated for a confidence

level of 95%, that is for significance level of α=0.05.

The last column of Table-11 shows the percentage of

contribution of each parameter on the response,

indicating the degree of influence on the result. It can

be observed from the results that significant parameter

for sliding wear in Hank’s solution is Distance

(26.73%) followed by load (23.10%),

Thickness(21.36%) and Speed(15.95%).

SSRG International Journal of Mechanical Engineering (SSRG-IJME) – Special Issue May - 2017

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Table XI

Analysis of Variance for SN ratios- Wear in wet

condition (Hank solution)

So

ur

ce

D

F

Seq

SS

Adj

SS

Ad

j

M

S

F P Contri

bution

Lo

ad 3

149.

24

149.

24

49.

75

1.8

0

0.3

20 23.10

Sp

ee

d

3 103.

05

103.

05

34.

35

1.2

4

0.4

31 15.95

Di

sta

nc

e

3 172.

67

172.

67

57.

56

2.0

8

0.2

81 26.73

Th

ick

nes

s

3 138.

00

138.

00

46.

00

1.6

7

0.3

43 21.36

Re

sid

ual

Err

or

3 82.8

3

82.8

3

27.

61

To

tal 15

645.

79

E. Energy Dispersive X-Ray(EDAX) Analysis

Energy Dispersive X-Ray(EDX) Analysis a

technique used for elemental analysis or chemical

characterization of metals. SEM- EDAX of Cr3C2 -

NiCr Coated Ti6Al4V before and after wear test in

Hank’s Solution are shown in Figure-5 and Figure-6.

It is observed from the EDAX pattern is that, the

coating elements are presented after the wear test. It is

observed from the weight percentages that wear rate is

less in the Hank’s solution.

Fig 1 : SEM - EDAX of Cr3C2 - NiCr Coated

Ti6Al4V before wear test

Table XII Elemental Analysis before wear test

S.No Element Weight % Atomic %

1 C K 43.49 67.15

2 O K 15.55 18.02

3 Al K 0.53 0.36

4 Ti K 1.34 0.52

5 Cr L 39.10 13.95

Total 100 100

Fig 6 : SEM- EDX of Cr3C2 - NiCr Coated Ti6Al4V

after wear test in Hank’s Solution conducted with

load- 30N, sliding distance -0.6 m/min, sliding

distance 0.5 m and coating thickness 300 μm

Table XIII Elemental Analysis after wear test

S.No Element Weight % Atomic %

1 C K 38.40 64.34

2 O K 13.93 17.52

3 Al K 0.10 0.07

4 Ti K 0.81 0.34

5 Cr L 35.86 14.85

6 Fe L 10.90 2.88

Total 100 100

IV CONCLUSIONS

A. Cr3C2 - NiCr coating on Ti6Al4V substrate was

successfully employed using Detonation Spray

technique.

B. Hardness of the substrate is improved from

25.67HRC to 53.33HRC

C. Percentage weight loss was decreased from 9.77

to 0.2115 for Substrate to Coated Specimen in wet

condition respectively.

D. ANOVA results proved that Contributing factors

for sliding wear in Hank’s solution are Distance,

Load, Coating thickness and Speed.

SSRG International Journal of Mechanical Engineering (SSRG-IJME) – Special Issue May - 2017

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E. SEM- EDAX analysis proved that the presence of

coating materials and substrate elements on the top

surface.

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Kaluza-Klein Holographic Cosmological Model inBrans-Dicke Theory of Gravitation

V.U.M.Rao1∗, G.Suryanarayana2 and B.J.M.Rao3

1Department of Applied Mathematics,Andhra University, Visakhapatnam, India.

∗umrao57@hotmail.com2Department of Mathematics, ANITS(A),

Sangivalasa Visakhapatnam, India.suri.maths1729@gmail.com

3Department of Mathematics,SIR C.R.Reddy College, Eluru, India.

drbjmr9@gmail.com

Abstract

Spatially homogeneous KaluzaKlein cosmological model filled with twominimally interacting fields, matter and holographic dark energy componentsin the frame work of Brans-Dicke (Phys. Rev. 124, 925: 1961) scalar-tensortheory of gravitation. To acquire a determinate solution of the field equationswe have used two plausible conditions: (i) scalar expansion is proportional tothe shear scalar of the model and (ii) relation between scalar field (φ) to theaverage scale factor (a(t)) of the model. Some important physical propertiesof our model are also discussed.AMS Subject Classification: 83D05, 83F05Key Words and Phrases: Kaluza-Klein metric, Brans-Dicke theory, Darkmatter, Holographic dark energy.

1 Introduction

Recent observation of the luminosity of type Ia supernovae indicate (Bachall et al.[1]; Perlmutter et al. [2]) an accelerated expansion of the universe and the surveys ofclusters of galaxies show that the density of matter is very much less than the criticaldensity. This observation leads to a new type of matter which violate the strongenergy condition i.e., ρ+ 3p < 0. The matter (fluid) content responsible for such a

International Journal of Pure and Applied MathematicsVolume 117 No. 13 2017, 383-393ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version)url: http://www.ijpam.euSpecial Issue ijpam.eu

383

condition to be satisfied at a certain stage of evaluation of the universe is referredto as dark energy (Sahni and Starobinsky [3]; Peebles and Ratra [4]; Padmanabhan[5]; Copeland et al. [6]). This mysterious fluid is believed to dominate over thematter content of the Universe by 70% and to have enough negative pressure as todrive present day acceleration. Most of the dark energy models involve one or morescalar fields with various actions and with or without a scalar field potential (Maorand Brustein [7]; Cardenas and Campo[8]; Ferreira and Joyce [9]).

In recent times, considerable interest has been stimulated in explaining the ob-served dark energy by the holographic dark energy model (Enqvist et al. [10]; Zhang[11]; Pavon and Zimdahl [12]). An approach to the problem of dark energy arisesfrom the holographic principle stated in the first paragraph. For an effective fieldtheory in a box size L with UV cutoff Λc, the entropy L3Λ3

c . The non-extensivescaling postulated by Bekenstein suggested that quantum theory breaks down inlarge volume (Zhang [11]). To reconcile this breakdown, Cohen et al. [13] pointedout that in quantum field theory a short distance (UV ) cut-off is related to a longdistance (IR) cut-off due to the limit set by forming a black hole. Taking the wholeuniverse into account the largest IR cut-off L is chosen by saturating the inequalityso that we get the holographic dark energy density as ρΛ = 3c2M2

pL−2 (Zhang [11])

where c is a numerical constant and Mp = 1/√

8πG is the reduced Plank mass. Onthe basis of the holographic principle proposed by Fischler and Susskind[14] severalothers have studied holographic model for dark energy (Gong [15]). Employment ofFriedman equation (Setare [16]) ρΛ = 3M2

pH2 where ρ is the total energy density

and taking L = H−1 one can find ρm = 3(1 − c2)M2pH

2. Thus either ρm or ρΛ

behaves like H2.Recently, many authors have investigated various aspects of holographic dark

energy models. Sarkar [17] explored holographic dark energy models in Bianchispace-times with linearly varying deceleration parameter in general relativity. Kiranet al. [18] have discussed minimally interacting holographic dark energy Bianchitype-V models in a scalar-tensor theory of Saez and Ballester. Santhi et al. [19]-[20] discussed spherically symmetric universe with holographic dark energy andgeneralized Chaplygin gas in Brans-Dicke and Saez-Ballester scalar-tensor theories.Rao and Prasanthi [21] studied Kantowski-Sachs holographic dark energy in Brans-Dicke theory of gravitation. Rao and Suryanarayana [22] studied Kantowski-Sachsholographic cosmological model in Saez-Ballester theory of gravitation. Santhi etal. [23]-[25] explored anisotropic modified holographic Ricci dark energy models indifferent theories of gravitation.

Brans-Dicke [26] theory of gravitation is a natural extension of general relativ-ity which introduces an additional scalar field φ besides the metric tensor gij anddimensionless coupling constant ω. The Brans-Dicke field equations for combinedscalar and tensor field are given by

Gij = −8πφ−1Tij − wφ−2

(φ,iφ,j −

1

2gijφ,µφ

,µ

)− φ−1

(φi;j − gijφ,µ;µ

), (1)

andφ,µ;µ = 8πT (3 + 2w)−1 (2)

where Gij = Rij − 12Rgij is the Einstein tensor, Tij(= Tmij + TΛ

ij ) is stress energy

International Journal of Pure and Applied Mathematics Special Issue

384

tensor of matter and dark energy, w is a dimensionless coupling constant, commaand semicolon denote partial and covariant differentiation respectively.Also, we have energy conservation equation as

T ij;j = 0, (3)

which is a consequence of field equations (1) and (2).Several aspects of Brans-Dicke cosmology have been extensively investigated by

many authors. Reddy and Rao [27], Reddy [28], Rao et al. [29]-[30], Rao and Santhi[31]-[32], Rao and Sireesha [33], Naidu et al. [34], Vidya Sagar et al. [35] and Das& Abdulla [36] are some of the authors who have investigated several aspects ofthis theory. Reddy and Vijayalakshmi [37] have obtained Kaluza-Klein dark energymodel in Brans-Dicke theory of gravitation. Rao and Jayasudha [38] have studiedBianchi type-V dark energy model in this theory.

Motivated by the above discussions and investigations, in this paper, we proposeKaluza-Klein cosmological model for minimally interacting holographic dark energyin Brans-dicke theory of gravitation. The plan of the paper is follows. In section 2,we established the Brans-Dicke field equations with the help of Kaluza-Klein metricin the presence of matter and holographic dark energy. In section 3, we obtained thesolution of the field equations. In section 4, we discuss some important propertiesof our model. Some conclusions are presented in the last section.

2 Metric and Field equations

We consider spatially homogeneous five dimensional Kaluza-Klein metric in the form

ds2 = dt2 − A2(t)(dx2 + dy2 + dz2)−B2(t)dψ2 (4)

where A and B are functions of time only .The energy momentum tensors for holographic dark energy and dark matter

(pressure less i.e. ωm=0) are respectively given by

TΛij = (pΛ + ρΛ)uiuj − pΛgij (5)

Tmij = ρmuiuj (6)

here ρm is the energy density of dark matter, ρΛ and pΛ are the energy density andpressure of holographic dark energy respectively.

In comoving coordinate systems, the Brans-Dicke field equations (1)-(2) for themetric (4) with the help of equations (5) and (6) can be written as

2A

A+A2

A2+ 2

AB

AB+B

B+w

2

φ2

φ2+ 2

Aφ

Aφ+Bφ

Bφ+φ

φ= −8πpΛ

φ(7)

3A2

A2+ 3

AB

AB− w

2

φ2

φ2+ 3

Aφ

Aφ+Bφ

Bφ=

8π

φ(ρΛ + ρm) (8)

3A

A+ 3

A2

A2+w

2

φ2

φ2+ 3

Aφ

Aφ+φ

φ= −8πpΛ

φ(9)

International Journal of Pure and Applied Mathematics Special Issue

385

φ+ φ

(3A

A+B

B

)=

8π

3 + 2w[ρΛ + ρm + pΛ], (10)

and the energy conservation equation (3), leads to

ρΛ + ρm +

(3A

A+B

B

)(ρΛ + pΛ + ρm) = 0, (11)

where overhead dot denotes ordinary differentiation with respect to time.Here we are considering the minimally interacting matter and holographic dark

energy components. Hence both the components conserve separately, so that wehave

ρΛ +

(3A

A+B

B

)(ωΛ + 1)ρΛ = 0 (12)

ρm +

(3A

A+B

B

)ρm = 0, (13)

where ωΛ = pΛ

ρΛis the equation of state (EoS) parameter for holographic dark energy.

3 Solutions of Field equations

The equations (7) to (9) is a system of four independent equations with six unknownsA, B, pΛ, ρΛ, ρm and φ. In order to get a deterministic solution we take thefollowing physical conditions, the shear scalar σ is proportional to scalar expansionθ, which leads to the linear relationship between the metric potentials A and B,

B = An (14)

where n is an arbitrary constant.The relation between scalar field φ and average scale factor given by (Pimental

[39]; Johri and Kalyani [40])φ = φ0a

k (15)

where φ0 and k are constant and arbitrary constant respectively.From equations (7), (8), (14) and (15), we get

A =

(k1 + 1)(k2t+ k3)

1k1+1

(16)

B =

(k1 + 1)(k2t+ k3)

nk1+1

(17)

where k1 = 3(n+2)+k(n+3)3

, k2 and k3 are integrating constants.From equations (15)-(17), we get

φ = φ0

(k1 + 1)(k2t+ k3)

k(n+3)3(k1+1)

(18)

International Journal of Pure and Applied Mathematics Special Issue

386

Now the metric (4) can be written as

ds2 = dt2 −

(k1 + 1)(k2t+ k3)

2k1+1

(dx2 + dy2 + dz2)

−

(k1 + 1)(k2t+ k3)

2nk1+1

dψ2 (19)

From equations (7), (9) and (16)-(18), we get

pΛ =−φ0k

22

144π

(k1 + 1)(k2t+ k3)

k(n+3)3(k1+1)

−29n2 + n+ 4

−k(n+ 5)+ k(n+ 3) [(n+ 3)(k(w + 2) + 9)− 6k1 + 12]

(20)

From the equation (13), we get

ρm = ρ0

(k1 + 1)(k2t+ k3)

−(n+3)

(21)

From equations (8), (16)-(18) and (21), we get

ρΛ =φ0k

22

144π

(k1 + 1)(k2t+ k3)

k(n+3)3(k1+1)

−254(n+ 1)

+k(n+ 3)(6− wk(n+ 3))

− ρ0

(k1 + 1)(k2t+ k3)

−(n+3)

(22)

The dark energy EoS ωΛ is given by

ωΛ =

−φ0k22

144π

(k1 + 1)(k2t+ k3)

k(n+3)3(k1+1)

−29n2 + n+ 4

−k(n+ 5)+ k(n+ 3) [(n+ 3)(k(w + 2) + 9)− 6k1 + 12]

φ0k

22

144π

(k1 + 1)(k2t+ k3)

k(n+3)3(k1+1)

−254(n+ 1)

+k(n+ 3)(6− wk(n+ 3))

−ρ0

(k1 + 1)(k2t+ k3)

−(n+3)−1

(23)

Thus the metric (19) together with equations (18) and (20)-(23) constitutesKaluza-Klein minimally interacting holographic dark energy cosmological model inscalar-tensor theory proposed by Brans and Dicke [26].

International Journal of Pure and Applied Mathematics Special Issue

387

4 Some other properties of the model

The volume and average scale factor of the model (19) are respectively given by

V =√−g =

(k1 + 1)(k2t+ k3)

(n+3)(k1+1)

. (24)

a(t) = V 1/4 =

(k1 + 1)(k2t+ k3)

(n+3)4(k1+1)

. (25)

The expression for the expansion scalar θ is given by

θ = ui,i =k2(n+ 3)

(k1 + 1)(k2t+ k3), (26)

and the shear scalar σ2 is given by

σ2 =1

2σijσ

ij =7

18

[k2(n+ 3)

(k1 + 1)(k2t+ k3)

]2

. (27)

The Hubble’s parameter H is given by

H =k2(n+ 3)

4(k1 + 1)(k2t+ k3). (28)

The mean anisotropy parameter Ah is given by

Ah =1

4

4∑

i=1

(Hi −HH

)2

= 3

(n− 1

n+ 3

)2

, (29)

where Hi (i = 1, 2, 3 and 4) are directional Hubble’s parameters.The deceleration parameter q is given by

q =−aaa2

= k + 2. (30)

The sign of q characterizes inflation of the universe. A positive sign of q i.e. q > 0,correspond to decelerating model whereas negative sign of q (particularly 1 ≤ q < 0)indicates accelerating phase or inflationary model. From equation (30), it is observedthat deceleration parameter is negative (i.e., q < 0) for k < −2, hence the obtainedmodel represents accelerating universe.Look back time

∆t =n+ 3

4(k1 + 1)H−1

0

(1− (1 + z)

−4(k1+1)n+3

)(31)

where H0 is present value of Hubbles parameter and z is red-shift.Jerk parameter

j =

...a

aH3=

(n− 4k − 1)(n− 8k − 5)

(n+ 3)2(32)

International Journal of Pure and Applied Mathematics Special Issue

388

Luminosity distance dL = r1a0(1 + z) where r1 =∫ t0t

1a(t)

dt

dL =(1 + z)(n+ 3)

4k1 − n+ 1)H−1

0

(1− (1 + z)

−4k1−n+1)n+3

)(33)

The distance modulus

D(z) = 5 log(dL) + 25 (34)

where dL is given by equation (33).

5 Discussion and Conclusions

In this paper, we have obtained spatially homogeneous Kaluza-Klein cosmologicalmodel filled with two minimally interacting fields, matter and holographic darkenergy components in the frame work of Brans-Dicke theory [26] of gravitation. Thevolume of the model vanishes at t = t∗, where t∗ = −k3

k2and expansion scalar is tends

to infinity, which shows that the Universe starts evolving with zero volume at t∗ withan infinite rate of expansion. Also, the model has no singularity for n > 0. Ourmodel represents accelerated expansion of the universe which is in good agreementwith the recent cosmological observations. Average anisotropic parameter Ah 6= 0for n 6= 1, so our model is anisotropic. We have also obtained expressions for Lookback time, Luminosity distance and Jerk parameter.

Thus the model presented here is anisotropic, shearing and accelerating. Henceour model represents not only the early stage of evolution but also the present uni-verse.

Acknowledgement: We thank anonymous reviwer for comments that improvedthe presentation of manuscript.

References

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[2] S. Perlmutter, et al., Measurements of Ω and Λ from 42 High-Redshift Super-novae, Astrophys. J., 517 (1999), 565-586.

[3] V. Sahni, A.A. Starobinsky, The case for a positive cosmological Λ-term, Int.J. Mod. Phys. D, 9 (2000), 373-443.

[4] P.J.E. Peebles, B.Ratra, The cosmological constant and dark energy Rev. Mod.Phys., 75 (2003), 559.

[5] T.Padmanabhan, Cosmological constantthe weight of the vacuum, Phys. Rep.,380 (2003), 235-320.

[6] E.J.Copeland, M. Sami, S.Tsujikawa, Dynamics of Dark Energy, Int. J. Mod.Phys.D, 15 (2006), 1753-1935.

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[7] I.Maor, R. Brustein, Distinguishing among scalar field models of dark energy,Phys. Rev. D, 67 (2003), 103508.

[8] V.H. Cardenas, S.D.Campo, Scalar field potentials for cosmology, it Phys. Rev.D, 69 (2004), 083508.

[9] P.G.Ferreira, M. Joyce, Cosmology with a primordial scaling field, Phys. Rev.D, 58 (1998), 023503.

[10] K. Enqvist, S. Hannested, M.S.Sloth, Searching for a holographic connectionbetween dark energy and the low l CMB multipoles, J. Cosmol. Astropart.Phys., 2 (2005), 004.

[11] X.Zhang, Statefinder Diagnostic For Holographic Dark Energy Model, Int. J.Mod. Phys. D, 14 (2005), 1597-1606.

[12] D.Pavon, W. Zimdahl, Holographic Dark Energy and Present Cosmic Accel-eration, hep-th/0511053 (2005).

[13] A.G.Cohen, et al., Effective Field Theory, Black Holes, and the CosmologicalConstant, Phys. Rev. Lett., 82 (1999), 4971 .

[14] W.Fischler, L.Susskind, Holography and Cosmology, hep-th/9806039 (1998).

[15] Y.Gong, Extended holographic dark energy, Phys. Rev. D, 70 (2004), 064029.

[16] M.R.Setare, Holographic Chaplygin gas model, Phys. Lett. B, 648 (2007), 329-332.

[17] Sanjay Sarkar, Holographic dark energy with linearly varying deceleration pa-rameter and escaping big rip singularity of the Bianchi type-V universe, Astro-phys. Space Sci., 352 (2014), 859-866.

[18] M. Kiran, et al., Bianchi type-III minimally interacting holographic dark en-ergy model with linearly varying deceleration parameter in Brans-Dicke theory,Astrophys. Space Sci., 354 (2014), 577.

[19] M.V. Santhi, et al., Spherically symmetric universe with holographic dark en-ergy & generalized Chaplygin gas, Prespacetime Journal, 7 (2016), 537-546.

[20] M.V. Santhi, et al., Holographic dark energy model with generalized Chaplygingas in a scalar-tensor theory of gravitation,Prespacetime Journal, 7 (2016),1939-1949.

[21] V. U. M. Rao, U.Y. Divya Prasanthi, Kantowski-Sachs Holographic Dark En-ergy In Brans-Dicke Theory of Gravitation, The African Review of Physics, 11,0001 (2016).

[22] V.U.M.Rao, G.Suryanarayana, Kantowski-Sachs Holographic CosmologicalModel in Saez-Ballester Theory of Gravitation, Prespacetime, 7 (2016), 783-791.

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[23] M.V. Santhi, et al., Anisotropic magnetized holographic Ricci dark energy cos-mological models, Can. J Phys., 95(4) (2017), 381-392.

[24] M.V. Santhi, et al., Bianchi type-V I0 modified holographic Ricci dark energymodel in a scalar-tensor theory, Can. J Phys., 95(2), (2017), 179-183.

[25] M.V. Santhi, et al., Anisotropic modified holographic Ricci dark energy modelin Saez-Ballester Theory of Gravitation,Prespacetime Journal, 7 (2016), 1379-1385.

[26] C.H.Brans, R.H. Dicke, Mach’s Principle and a Relativistic Theory of Gravita-tion, Phys. Rev., 124 (1961), 925.

[27] D.R.K.Reddy, V.U.M.Rao, Field of a charged particle in Brans-Dicke theory ofgravitation, J. Phys. A:Math. Gen., 14 (1981), 1973 .

[28] D.R.K.Reddy, A String cosmological model in Brans-Dicke theory of gravita-tion, Astrophys. Space Sci., 286 (2003), 365-371.

[29] V.U.M Rao, et al., Exact Bianchi type-V perfect fluid cosmological models inBransDicke theory of gravitation , Astrophys. Space Sci., 315 (2008) 211-214.

[30] V.U.M.Rao, et al., Axially symmetric string cosmological models in Brans-Dicke theory of gravitation, Astrophys. Space Sci. 323 (2009), 401-405.

[31] V.U.M Rao, M.Vijaya Santhi, Bianchi Type-II, V III & IX Perfect FluidCosmological Models in Brans Dicke Theory of Gravitation J. Mod. Phys., 2(2011), 1222 -1228.

[32] V.U.M Rao, M.Vijaya Santhi, Five dimensional spherically symmetric cosmo-logical model in Brans-Dicke theory of gravitation, Astrophys Space Sci., 337(2012), 387-392.

[33] V.U.M Rao, K.V.S Sireesha, Axially Symmetric Space-Time with StrangeQuark Matter Attached to String Cloud in BransDicke Theory of Gravitation,Int J Theor Phys., 52 (2013), 1052-1060.

[34] R.L.Naidu, et al., A five dimensional Kaluza-Klein bulk viscous string cos-mological model in Brans-Dicke scalar-tensor theory of gravitation, Astrophys.Space Sci., 347 (2013), 197-201.

[35] T.Vidya Sagar, et al., Bianchi type-III bulk viscous string cosmological modelin Brans-Dicke theory of gravitation Astrophys. Space Sci., 349 (2014), 479-483.

[36] S.Das, A.M. Abdulla, An Interacting Model of Dark Energy in Brans-DickeTheory, Astrophys. Space Sci., 351 (2014), 651-660.

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2016-17

DOI: 10.4018/JGIM.2018070106

Journal of Global Information ManagementVolume 26 • Issue 3 • July-September 2018

Copyright©2018,IGIGlobal.CopyingordistributinginprintorelectronicformswithoutwrittenpermissionofIGIGlobalisprohibited.

78

Mobile Sink as Checkpoints for Fault Detection Towards Fault Tolerance in Wireless Sensor NetworksPritee Parwekar, Anil Neerukonda Institute of Technology and Sciences, Bheemunipatnam, India

Sireesha Rodda, Gitam Univerity, Visakhapatnam, India

Parmeet Kaur, Jaypee Institute of Information Technology, Noida, India

ABSTRACT

AWSNconsistsofalargenumberoflimitedcomputationandstoragecapabilitywirelesssensornodes, which communicate wirelessly. These sensor nodes typical communicate in short rangeandcollaboratetoaccomplishthenetworkfunction.ToincreasetherangeofsensingandwiththeadventofMEMS,mobilesensorsandsinksisthetechnologytheworldismovingto.Thispaperpresentsanetworkofmobilesensorsandasink.Amobilesinkisselectedascheck-pointtohavetherecoverabilityofthenetwork.AFuzzyRulebasedsystem(FRS)isusedtoconstructandselectefficientstaticsensornodeshavingadequateresourcesasCheckPointStorageNodes(CPSNs).TheobjectiveofFRSistoincreasetheprobabilityofrecoveryofcheck-pointeddatasubsequenttoafailure,therebyallowingadistributedapplicationtocompleteitsexecutionsuccessfully.SimulationsshowFRS’sbetterrecoveryprobabilitiesincomparisontoarandomcheck-pointingarrangement.

KeyWoRDSCheckpoint, Fault Detection, Fault Tolerance, Fuzzy Ruleset, Mobile Sink, Wireless Sensor Networks

INTRoDUCTIoN

Wireless sensornetworks (WSN) is consideredasoneof themost important andgame-changingtechnologiesforthecurrentmillenniumasperYimingZhouet.al.(2007).Inthepasttwodecades,therehasbeenanincreasinginterestshownbybothacademiaandindustryacrosstheworldintappingthistechnologytowardsnovelsolutions.AWSNusuallyconsistsofalargenumberoflow-cost,low-power,andmultifunctionalwirelesssensornodes,whichcommunicatewirelesslybuthavelimitedcomputationcapabilities.Thesesensornodestypicallycommunicateinshortrangeandcollaboratetoaccomplishthenetworkfunction,forexample,environmentmonitoring,militarysurveillanceandindustrialprocesscontrol.ThefundamentalphilosophybehindWSNisthat,whilethecapabilityofeachsensornodeislimited,thecumulativepoweroftheentirenetworkissufficientfortherequiredmission.

Basedonthemethodofconnection,theWSNswouldbeeitherconventionalnetworkinwhichthesensorsdeployedin theregionof interestareknownandthenetworkisestablishedwith thedeploymentofthesensors.TheothertypeistheadhocWSNinwhichthenodesconnectwitheach

Journal of Global Information ManagementVolume 26 • Issue 3 • July-September 2018

79

otherinanadhocmannerandthepositionorotherparametersofthesensornodesarenotknowninitiallyat thetimeofthenetworkformation.Suchnetworksaretypicallyestablishedindisastermanagementsiteswherethedynamicsoftheregionofinterestisratherveryactive.

TheWSNstendtooperateinadverseenvironmentalconditionsandarelikelytohavenetworkbreaksordataloss.Faultdetectionandsubsequentlyfaulttolerancehasthereforebecomeanactiveresearchareatoensurethefidelityofthenetwork.

ThefutureWSNapplicationsareexpectedtoincorporateastandardizedmixofhardwareandsoftwaresolutions.Butaswestandtoday,thenetworkdesignersarestilljugglingbetweenthetradeoffsthattheyhavetoadoptsoastominimizethedeploymentcosts,hardwareandsoftwareoverheads,improvesystemreliability,ensuresecurityandmaximizeperformance.Wirelessembeddeddesignersthereforearerequiredtoassessthesetradeoffsandchoosetherighttransducerandbatterytechnology,frequencyofwirelessoperation,outputpowerandnetworkingprotocolsetcsoastoachievebestresultsasproposedinR.Szewczyketal.(2005).Thesensorsareacompact,small,battery-powereddevice,andthereforehavelimitedenergyresource.Therefore,energyconsumptionisacriticalissueinsensornetworks.Weareinterestedinsensornetworksinwhichalargenumberofsensorsaredeployedtoachieveagivengoal.Alldataobtainedbymembersensorsmustbetransmittedtoasinkordatacollector.Thelongerthecommunicationdistance,themoreenergywillbeconsumedduringtransmissionasexplainedbytheauthorW.R.Heinzelmanetal.(2000).Theseclassicissuesarebeingaddressedbynetworkalgorithmsandtheiroptimizationmethods.Whileconventionalmathematicalmethodshavebeenused,latelyinteresthasbuiltintheresearchcommunitytowardsnatureinspiredalgorithmslikePSOanduseoffuzzylogicbasedoptimization.

WithadventofMicroElectroMechanicalSystems(MEMS)asproposedbyI.F.Akyildizetal.(2002),remotecorrespondenceandlowpoweroutlineshavehelpedrapiddevelopmentinthefieldofWirelessSensorNetworks.Thesensorhubsinwirelessnetworkscompriseofpredominantlyfourunitsthataredetecting,correspondence,handlingandcontrolsupply.Asensorsensesthephysicalstateoftheenvironment.Thispaperemploysmobilesinksistocollectthesensedvaluesfromthestaticsensorsdeployedintheregionofinterest(ROI).Thestaticsensorshavelimitedstorageandprocessingcapabilities.Amobilesinkisusedtocollectthedatafromthesensorsaggregates,pre-processthedataandfurthercarriesittobasestationforoffloadingthedata.TheBaseStation(BS)thenfurtheranalysesthedataforthedecision-makingprocess.OncethedataistransferredtotheBS(BaseStation)themobilesinksarefreeforstoringfreshdatafromthesensorsintheROI.Thepossibilityoflossofdataintheprocesshasbeenarrestedinformofafaulttolerancemechanism.Insubsequentparagraphs,thismethodologyhasbeenexplained.Insubsequentparagraphs,thispaperbringsoutamethodologyoffaultdetectionforthedata.AconceptofCheckPointSensorNodes(CPSN)hasbeenintroduced,whichaidinthisendeavor.Thoughthepaperhaslimitedtoapplyingthisconcepttostaticnodes,thesamecanbeextrapolatedtodynamicnodesbyminortweaking.

LITeRATURe SURVey

Routing protocols have been developed to optimize, deployment of sensors, localisation of pre-deployedsensors,clusteringofnodes,identificationofclusterheads,identificationofsinks,Mobilesinksandtheiroptimisedpathtravelandfinallyfaulttoleranceandsecuritywithanultimateaimtoestablishasecure,high–fidelity,longlifewirelesssensornetworks.Researchuptoyear2006wasmoreorlessconcentratedonclusteringandprotocolsforstaticnodes.WithadvancesinMEMSandrobotics,theconceptofdynamicormobilesinkswasintroduced.

PriteeParwekaretal.(2016)bringsoutaprotocolusinghighpacketdeliveryandreliabilityofthenetworkusingfuzzylogicbasedonLinkQualityIndicator(LQI),ReceivedSignalStrengthIndicator(RSSI),andnumberofhopstothebasestation.NazirandHasbullah(2010)proposeaclusterbasedroutingprotocol,MobileSinkBasedRoutingProtocol(MSRP)toovercomethehotspotproblem.Toachievethis,MSRPclustersthenetworkandnominatestheclusterheadbasedontheresidualenergy

Journal of Global Information ManagementVolume 26 • Issue 3 • July-September 2018

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information.ThedynamicmobilesinkarchitectureinMSRPavoidsenergyconcentrationonasmallportionofsensorsandprolongsthenetworklifetime.Thisprotocolalsosetsupthetrajectoryforthesinktomovealong.Basedontheresidualenergyofclusterheads,sinkwillalwaysbeclosertotheclusterheadwithhigherresidualenergyandthoseclusterheadswillrelaydatafromotherclusterheads.

Theproblemofdatacollection in sparse sensornetworks is encountered inmanyscenariossuchasmonitoringphysicalenvironments,animalmigrationsinremote-areas,weatherconditionsinnationalparks,habitatmonitoringonremoteislands,citytrafficmonitoringetc.,asgivenbyauthorP.Juangetal.(2002).TheobjectiveistocollectdatafromsensorsanddeliverittoanaccesspointintheinfrastructureasproposedbyauthorJain,Set.al(2006).Anothercorollaryissuefacedbymobilesinknodesistogatherdatafromstaticsensornodeswhilesinknodeismoving.Asthelocationofthesinkischanging,sensornodesareenabledtosendthedatapackagestothesinkwhensinkisnearby.Therefore,traditionaldatagatheringandroutingschemesarenotsuitableinthiscase.AuthorsJainetal.(2006)presentananalyticalmodeltounderstandthekeyperformancemetricssuchasdatatransfer,latencytothedestination,andpower.

AuthorTian,Ketal.(2010)havebroughtoutthatthemajordisadvantageofbothDataMULEsandRandomWalkisthedeliverydelay.Toimprovetheperformanceofdatagathering,Tianetal.(2010)haveintroducedtheAVRPandTRAILdatagatheringprotocolswhichaccordingtotheauthors,bringsanefficientdatagatheringperformanceandreducesthedeliverydelayinthenetworkswithheavyandlightloadrespectively.TheAVRPprotocolisderivedfromthepreviousanchor-basedVoronoi-ScopingroutingprotocolasgivenbyauthorDubois-Ferriereetal.(2004).Voronoidividesthenetworkintomultipleclusters.Itisefficientinthewaythatthesensorsonlyneedtosendtheirdatatotheclosestanchornodesandtheanchornodescommunicatewiththesinknode.Inordertoadapttoamobilesink,theauthorsimprovedVoronoi-Scopingthatassociatethemobilesinkwiththeanchornodes.AVRPbuildsupadeliverystructureandrefreshesthestructureperiodicallybasedonthemovementofthesink.Therefore,allthesensorsonlyneedtostoretheclosestanchornode’sroutinginformationandAVRPremovestheneedforthedynamicroutingpathinformation,whichisa largedataoverheadintransmission.AVRPstabilizesthedatatransmissionandit issuitablefornetworkswithheavyload.TheTRAILprotocolrecordsthetrailofsinkmovementandthedatagatheringrouteisalongthistrail.Theprotocolstartswitharandomwalkandoncethesinkdetectsadatatransmissionrequestfromsensorsitmovestowardsthemandthetrailsarerecordedandupdatedforthenextdatagatheringroute.Kaur,PandParwekar(2014)hasproposedaFRSforselectionofcheckpointsinmobilegridtoshowasignificantimprovementinfailurerecoverabilityascomparedtoarandomcheckpointingarrangement.

Fromtheliteraturestudy,itisevidentthatenoughoptionsarenowavailabletoefficientlyuseamobilesinktoaggregatedatainaWSN.Thishowever,doesnotensurethatthedatagatheredbythemobilesinkisfinallydeliveredcorrectlytotheBaseStation,whereitfinallymatters.

CoNCePT FoR THe PAPeR

MobileSinks(MS)areroboticdeviceswhichrunonbattery.TheyhaveconsiderablyhigherenergyresourcecomparedtotheothersensorcomponentsinaWSN.ThemobilesinksareexpectedtopatroltheROIandgathersenseddatafromthestaticsensorsdeployedintheROI.Once,dataisgathered,asmallamountofpre-processingofthedataisusuallyundertakenandthenfinallyoncethemobilesinkreachwithinthetransmission/receptionrangeoftheBaseStation,theyoffloadthepre-processeddataandfreethememoryforfurthergatheringofdata.Typically,thisprocessinvolvesmovementofthemobilesinkfromthebasestationinapre-determinedgeographicalregionandconsumesalotofenergy.Further,themobilesinkisalsopronetolossofmobilityduetoanymishapthatcouldoccurwhilsttraversingintheregion.Failureofthemobilesink,thuswouldcompromisetheroleoftheWSN.Certaindatathatiscapturedbythemobilesinkfromthestaticsensorscouldjustbelostwiththelossofthemobilesink.Thebasestationwouldnevercometoknowaboutthisloss,andwillthereforenotbeabletotakeany

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correctiveactioneitherphysicallyorvirtually.Thispaperinvestigatesthesepossibilitiesinthemobilesinkscenario,proposeaconcepttologthetableofcontentsofthemobilesinkataparticulartime,andthenfinallymatchtheTableofContents(TOC)withtheBaseStation,toensurethatallthedatacapturedbythemobilesinkshasfinallyreachedtheBaseStation(seeFigure1).

MeTHoD

Thesensornetworkconsistsofstaticnodeswhicharedeployedatthebeginning.Thedeploymentpattern of static nodes could follow one of the contemporary algorithms so as to maximize thedatacapturedfromtheROI.Thesinksinthenetworkhavebeenpredefinedasmobile.AdifferentalgorithmcanbeusedtodefinethepaththattheMSscouldfollow,soastomaximizetheirenergyandresolutionofdatacollected.Thesinkscapturethetime-stampeddatafromthesensorsanddeliverthesametotheBS.Inthepresentsetup,theMSsarealsoexpectedtogeneratearunningTableofContents(TOC)ofthedatathattheyarecurrentlyinpossessionforonwarddeliverytotheBS.TheTOCisnowrequiredtobesharedwithanotherentityinthenetworkinproximitytothepresentpositionoftheBS.OncetheMSdeliversthedatatotheBS,theTOCofthatdataresidingelsewhereinthenetworkisnolongernecessary.TheTOCismadetoresideonstaticsensor(s)inthenetworkwhohavebeenselectedfuzzilybasedontheirbalanceenergyresourcesandothernetworkmetricslikeRSSIetc.SuchnodesinthenetworkaredefinedasCheckPointSensorNodes(CPSN).EveryMSthatpassesinthevicinityoftheCPSN,exchangestheTOCswiththeCPSNs.Thus,theCPSNsaswellastheMSshaveanincrementallyupdatedinformationonTOCsofmostoftheMSs.ThefollowinginformationexchangedbetweenthemobilesinksandtheCPSNs:

• TOCheldwiththeCPSN;• TOCheldwiththeMS;• TOCofdatareceivedbytheBSfromallthesinks.

TheCPSNsnowfreetheTOCsfromitsmemoryforwhichtheBShasreceiveddata.ThisprocessenablesthenetworkasawholetoidentifythedatathathasfinallyreachedtheBS,andifthereare

Figure 1. Wireless sensor network depicting static sensors in ROI with mobile sinks and remotely located base station

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anygapsinthesame,initiatecorrectiveaction.TheGapsdetectionistheFaultDetectionandthecorrectiveactionforthisdatagapisFaultTolerance.ThispaperhaslimiteditselftothemethodologyforFaultdetection.TheFaulttolerance(presentlyoutofscopeofcurrentpaper)couldbeachievedeitherphysically,thatisbydeployingalternatesinktogatherthemissingdatafromthesensorsofaparticularregionintheROI,orbyvirtualcorrectionstothedatathroughinterpolationalgorithms.Boththesemethodsarebeingconsideredasafuturework.ThemethodwillensurethattheBaseStationwillhaveacompleteknowledgeofperiodicityofthevaluessensedbythenodesfromtheenvironment.TheBaseStationwillalsobeawareofthedatalost(ifany)andwillbeabletotakeaninformeddecisionastointerpolatethemissingdataortodeployanotherlivesinktocapturethelostdata.

FUZZy RULe BASe CoNSTRUCTIoNS

ThemainaimistoidentifysomeofthestaticnodesandselectthemasCPSNnodes.ThispaperproposesuseofFuzzyRuleBaseSystem(FRS)fortheeffectiveCPSNselection.AFRSisconstructedtoderiveweightsfornodesinthesystem.Thereceivedsignalstrength(RSSI),energy,storageavailableandlinkqualityindicator(LQI)isusedasparametertocalculatetheweightofnode.

AnodeisqualifiedasCPSNifithassufficientstorage,substantialamountofenergy,betterreceivedsignalstrengthandgoodlinkqualityindicator.Subsequently,sensorswhichareassignedhighweightsaccordingtotheFRS,areclassifiedasCheckPointSensorNodes-CPSNandthesecanacceptthecheckpointsmobilesinks,MSs.

Thefuzzyrulesystemismodeledbasedonthevaluesoftheparameters.Theparameterstogetherdecidethefitnessofthefuzzyset.Theparameterofstablestoragetakesvaluesintermsofmemorycapacitywhilethebatterypowercantakecrispvaluesintherange(0%,100%).ThemembershipfunctionofstablestoragevaluesareusedfrompaperT.Parketal.(2002),CaoandSinghal(2001).ACPSNisofsize1MBto10MBandamessageisofthesizeofafewbytes.AlargememoryCPSNwouldensurethattheCPSNwillnotgetoverwhelmedwithdataevenincasethedatafromitisnotdownloadedtotheBaseStationbythemobilesinksinashortdurationorlimitediterations.IdeallysuchamemorycapabilitywillensurethattheCPSNwillhavethecapabilitytocapturedataalmosttilltheendofthenetworklife.Triangularmembershipfunctionsarethemostefficientespeciallywhenthevaluesrequiredaretobejustlow,mediumandhighandintermediaterangeisnotofmuchsignificance.Therangeofeachparameterisdividedbyusingthreetriangular-shapedmembershipfunctions:Low,MediumandHighrespectivelyasshowninFigure2,Figure3,Figure4andFigure5.Ahighvalueforbothparametersmakesthefitnessofthefuzzysethigh.TheweightsarecalculatedbasedontheFuzzyrulesasdepictedinTable1andTable2.

Figure 2. Membership function of RSSI

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Table1indicatesfuzzyrulesetforRSSIparameterandTable2indicatesfuzzyrulesetforLQIparameter.LQIisametricofthecurrentqualityofthereceivedsignal,RSSIisasignalstrengthindication.Itdoesnotcareaboutthe‘quality’or‘correctness’ofthesignal.LQIdoesnotcareabouttheactualsignalstrength,butthesignalqualityoftenislinkedtosignalstrength.Thisisbecauseastrongsignalislikelytobelessaffectedbynoiseandthuswillbeseenas‘cleaner’ormore‘correct’

Figure 3. Membership function of LQI

Figure 4. Membership function for stable storage

Figure 5. Membership function for battery power

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bythereceiver.LQIisbestusedasarelativemeasurementofthelinkquality(ahighvalueindicatesabetterlinkthanwhatalowvaluedoes)LQIparameteriscalculatedwiththehelpofRSSIasshownintheAlgorithm1asproposedbyauthorsParwekarandReddy(2013).

SincetheentiresimulationsarerequiredtobedoneinTOSSIMenvironment,arandomnoiseisrequiredtobecreatedwhichwouldotherwisebepresentinnaturalsituations.ApathlossmodelusingGaussiandistributionhasbeenfoundtobethebesttosimulatethis,hence,weusesuchapathlossmodel.

ThepathlossPL(d0).Thepropagationpath-lossfactornindicatestherateofpathlossincreasewiththedistance.Itcanbefoundfromthemeasurementsthatatavalued,thepathlossPL(d)ataparticularlocationisrandomanddistributedlognormally(normalindB)aboutthemeandistance-dependentvalue.Thatis:

PL(d)=PL(d0)+10ηlog10(d/d0)+Xσ (1)

whereXσisazero-meanGaussiandistributedrandomvariable(indB)withstandarddeviationσ(alsoindB).Hereweused0=8andn=3.3.ThevaluesofthisconstantshavebeenadoptedfromsimilarsimulationworkinTinyossimulator(Tossim)inBerkleyuniversity.

We have implemented this approach as a TinyOS module and evaluated through TOSSIMsimulations.InordertoacquireRSSIandLQIvaluesinTOSSIM,theTOSSIMcodeismodifiedandLQIiscalculatedusingRSSIvalue.RSSIistheestimateofthesignalpower.Itiscalculatedover8symbolperiodsandisstoredintheRSSIVALregister.TocalculateLQIfromRSSI,weusedlinearpolynomialmodelwithcoefficients(95%confidencebounds).Thepolynomialis:

Table 1. Fuzzy rule based table using RSSI

Rule RSSI Energy Storage Available Weight

1 Low Low Less Low

2 Low Medium Less Low

3 Low High Less Medium

4 Medium Low Less Low

5 Medium Medium Less Medium

6 Medium High Less High

7 High Low Less Medium

8 High Medium Less High

9 High High Less High

10 Low Low More Low

11 Low Medium More Low

12 Low High More Medium

13 Medium Low More Low

14 Medium Medium More Low

15 Medium High More Medium

16 High Low More Low

17 High Medium More Medium

18 High High More Medium

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Poly(x)=p1x4+p2x

3+p3x2+p4x+p5 (2)

wherexistheRSSIvaluereceivedfromsensornode.Acurvefittingtechniqueisemployedanda4thpolynomialcurveisbestfittedwiththedata,RSSItoLQI,inaleastsquaressense.Algorithm1.1willfindtheLQIvaluefromRSSIasexplainedbyauthorsParwekarandReddy(2013).

SIMULATIoN SeTUP

Assumptions• N-noofsensorsdeployedinROI(Nistakenas200forthesimulation);• AllthesensordeployedinROIarestationarybuthavebeendeployedatrandomlocations;

Table 2. Fuzzy rule based table using LQI

Rule LQI Energy Storage Available Weight

1 Low Low Less Low

2 Low Medium Less Low

3 Low High Less Medium

4 Medium Low Less Low

5 Medium Medium Less Medium

6 Medium High Less High

7 High Low Less Medium

8 High Medium Less High

9 High High Less High

10 Low Low More Low

11 Low Medium More Low

12 Low High More Medium

13 Medium Low More Low

14 Medium Medium More Low

15 Medium High More Medium

16 High Low More Low

17 High Medium More Medium

18 High High More Medium

Algorithm 1. To find the LQI value from RSSI

1.rssi_sen=(rssi_sen>255-45)?255:rssi_sen+452. x =rssi_sen3. p1 = -0.6475,p2 = 3.761,p3 = -5.9974. p4 = 3.418,p5 = 105.95. x = x – 222.66. x = x/10.617. lqi = (p1*pow(x,4.0))+(p2*pow(x,3.0))+(p3*pow(x,2.0))+(p4*x)+p58. lqi = ((lqi>0)?lqi:0.0)9. return lqi

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• Allsinksnodesaremobile(MS);• TheMSreportsthedatatotheBS;• EverysensorinROIissensingaphysicalstateattintervaloftime.

AlltheMSsstoretheirTOCtotheCPSN.Thisensuresthatuponfailure,ifthebasestationdoesnotreceivetheTOC,thefaultwillbedetectedbasedontheTOCresidingontheCPSN.Allthesensornodesareassignedthelocalids.

Data Structures Used by the AlgorithmAt a CPSNMS:willappendtheTOC.TheMSid,thesensorsidandthetimestampwillbeappendedtoTOC.

<MSi,id,timestamp,flag>

EveryentryintheTOCwillhaveaflagfieldassociatedwithit,whichwillbesettozeroinitially.ThesameTOCwillbewiththeMSwhen:

Onreceiptof<id,timestamp>,ActionbyCPSNWhilereceivedmessagesreceivedfromMSOnreceiptof<id,timestamp>Entryofthat<id,timestamp>Deletetheentryofthatsensor<id,timestamp>fromtheMS

ReSULTS

TheproposedapproachisimplementedasaTinyOSinTinyossimulator(Tossim)environmentmodule and evaluated through extensive TOSSIM simulations. A simulation experiment isconductedtoevaluatehowRSSIandLQIvaryoverthedistancewithtwoTelosBmotes.Signalqualityisusuallyconnectedwiththesignalstrength.Furtherifthesignalstrengthisgood,itislessdisturbedbynoise.ThereceivedSignalStrengthIndicator(RSSI)andtheLinkQualityIndicator (LQI)are therefore theprimaryparameterswhichgovern thehealthaswell as thefidelityofthenetwork.TheweightcalculationhasbeendonebyconsideringenergyandstorageavailablethesealsocontributetowardsevaluationasshowninTable1andTable2.ThesetupusedforcollectingRSSIandLQIvaluesisbasedontheTinyOSRSSIDemotutorialTinyos-helptinyos2.0RssiDemo.ThetutorialsoftwareisinherentlycapableofcollectingRSSIvalues.FromRSSItheLQIvaluesarecalculated.

Wehavesimulatedasystemwith200sensorswhicharerandomlydeployedinROIand5mobilesinkswhicharetakingrandomwalk.Thesensorsarerandomlydistributedinarectangularoperationalareaofsize700x700.Thecommunicationrangeofeachnodeistakenas100m.TheprocessalsotakesacheckpointwithafixedtimeintervalofCc=200.ThefailurerateofaMHfollowsanexponentialdistributionwitharateλf=300andonthefailure;theMHinstantlyperformstheproperactionfortherecovery.Themobilenodesmoveandtheirmovementfollowsanexponentialdistributionwithanaverageofλhtakenas0.0033initially.

Sincetheobjectiveoftheproposedalgorithmistoachievefaultdetectioninthenetwork,wedefine a performance metric, Recoverability of the network. Ability to maintain the network infullyoperationalstate is themainaimofanyresearch towardsnetworkoptimization.WedefineRecoverabilityasthepercentageoftimestherecoveryproceduresarecompletedsuccessfullyinagiventimeinterval(seeFigure6).

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CoNCLUSIoN

ThispaperdwellsofFaultDetectionandFaultTolerancewithrespecttothedataacquiredbytheBaseStationinaWirelessSensorEnvironmenthavingstaticsensorsandmobilesinks.Thepaperassumesthatthemobilesinksarepronetolossduetotheirgeographicalenvironment,electronic-mechanicaldamagetoitscircuitryorsimplylossofpower.ConceptofCheckPointStorageNodes(CPSNs)hasbeenintroduced,whichareidentifiedfromtheexistingsetofalreadydeployedstaticsensornodes.ThisprocesshasbeenoptimisedusingFuzzyrulesets.TheCPSNsprovidetheTOCsofsensordataheldwithMobileSinkstotheBasesStationforregularcheckssoastoensuredatafidelity.

FUTURe WoRK

ThepaperhasbeenlimitedtoFaultDetection.Thefurtheractiontowardscompletingahighfidelitywirelesssensornetworkwouldbetointroducefaulttoleranceeitherphysicallyorvirtually.Thisconceptisbeingpondereduponasafuturework.Therearenolimitsenvisagedtowardsscalingtheconcepttoalargerwirelessnetworkoranetworkwithallmobilenodes.Asafuturestudy,theapplicationofthisconceptofCPSNtowardsadhocnetworksisalsoenvisaged.

Figure 6. Comparison of FRS and random scheme for checkpoint selection

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ReFeReNCeS

Akyildiz,I.F.,Su,W.,Sankarasubramaniam,Y.,&Cayirci,E.(2002,March15).Wirelesssensornetworks:Asurvey.Computer Networks,38(4),393–42.doi:10.1016/S1389-1286(01)00302-4

Cao,G.,&Singhal,M.(2001).MutableCheckpoints:ANewCheckpointingApproachforMobileComputingSystems.IEEE Transactions on Parallel and Distributed Systems,12(2),157–172.doi:10.1109/71.910871

Dubois-Ferriere,H.,&Estrin,D.(2004).Efficientandpracticalqueryscopinginsensornetworks.InMobile Ad-hoc and Sensor Systems(pp.564-566).

Heinzelman,W.R.,Chandrakasan,A.,&Balakrishnan,H.(2000)Energy-efficientcommunicationprotocolforwirelessmicrosensornetworks. InProceedings of the 33rd Annual Hawaii International Conference on SystemSciences.

Jain,S.,Shah,R.C.,Bunette,W.,Borriello,G.,&Roy,S.(2006).ExploitingMobilityforEnergyEfficientDataCollectioninSensorNetworks.Mobile Networks and Applications,11(3),327–339.doi:10.1007/s11036-006-5186-9

Juang,P.,Oki,H.,Wang,Y.,Martonosi,M.,Peh,L.S.,&Rubenstein,D.(2002).Energyefficientcomputingforwildlifetracking:Designtradeoffsandearlyexperienceswithzebranet.ACM SIGARCH Computer Architecture News,30(5),96-107.

Kaur,P.,&Parwekar,P. (2014).Fuzzyrulebasedcheckpointingarrangementfor fault tolerance inMobileGrids.InProceedings of the 2014 Seventh International Conference on Contemporary Computing (IC3)(pp.289-329).doi:10.1109/IC3.2014.6897188

Nazir,B.,&Hasbullah,H.(2010).Mobilesinkbasedroutingprotocol(MSRP)forprolongingnetworklifetimeinclusteredwirelesssensornetwork.InProceedings of the 2010 International Conference on Computer applications and industrial electronics (ICCAIE)(pp.624-629).IEEE.

Park,T.,Woo,N.,&Yeom,H.Y.(2002).Anefficientoptimisticmessageloggingschemeforrecoverablemobilecomputingsystems.IEEE Transactions on Mobile Computing, 1(4),265-277.

Parwekar,P.,&Reddy,R. (2013).AnEfficientFuzzyLocalizationApproach inWirelessSensorNetwork.InProceedings of the 2013 IEEE International Conference on Fuzzy Systems (FUZZ). doi:10.1109/FUZZ-IEEE.2013.6622548

Parwekar,P.,&Rodda,S. (2016)FaultTolerance inWirelessSensorNetworks:FindingPrimaryPath. InProceedings of the Second International Conference on Computer and Communication Technologies (pp. 593-604). Springer India.doi:10.1007/978-81-322-2517-1_57

Szewczyk,R.,&Culler,D.(2005)Telos:enablingultra-lowpowerwirelessresearch.InProceedings of theFourth International Symposium on Information Processing in Sensor Networks IPSN ’05.

Tian,K.,Zhang,B.,Huang,K.,&Ma, J. (2010).DataGatheringProtocols forWirelessSensorNetworkswith Mobile Sinks. In Proceedings of the IEEE Global Telecommunications Conference. doi:10.1109/GLOCOM.2010.5684197

Tinyos-helptinyos2.0RssiDemo.(n.d.).Retrievedfromhttp://mail.millennium.berkeley.edu/pipermail/tinyos-help/2008-June/034617.html

Zhou,Y.,Yang,X.,Guo,X.,Zhou,M.,&Wang,L.(2007).ADesignofGreenhouseMonitoring&ControlSystemBasedonZigBeeWirelessSensorNetwork.InProceedings of the Wireless Communications, Networking and Mobile Computing WiCom ’07.

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Pritee Parwekar is pursuing her PhD in Computer Science and Engg from GITAM University, Vishakapatnam. Currently she is working with Department of CSE of ANITS, Vishakapatnam. She has more than 15 years of teaching experience. Her research areas are Sensor network, Cloud Computing and IoT. She is a life Member of CSI. She has reviewed many papers for Springer and IEEE and has already published more than 15 papers with reputed publishers like IEEE and Springer.

Sireesha Rodda is currently working as Professor in Computer Science and Engineering Department, GITAM University, Visakhapatnam. Her research interests include Data Mining, Artificial Intelligence, Sensor Networks and Big Data Analytics.

Parmeet Kaur received a PhD (Comp Engg) from NIT Kurukshetra, an M.Tech in Computer Science from Kurukshetra University, and a BE (Hons) in Computer Science and Engineering from P.E.C., Chandigarh. She is currently working in Jaypee Institute of Information Technology, Noida, and has an academic experience of 15 years. Her research interests include distributed computing, distributed databases, fault tolerance in distributed systems. She has authored several papers in reputed journals and conferences.

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ISSN-z319-21 19

RESEARCH ARTICLEB.Syama Sundar et al, The Experiment, 2016., VoL 35(l)' 2162-2170

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Synthesis of S-Methyl 2-mercapto benzimidazole Capped Silver Nanoparticles

via chemical reduction method & Study of their Biological properties

AbstractlThe morphology and anti bacterial activity of capped Silver Nanoparticles of 5-Methyl 2-mercapto benzimidazole

prepared via chemical reduction is studied. Spherical shaped capped Ag-NPs with size of 40nm are obtained by

treatment of aqueous silver ions with ethanolic solution of 5-Methyl 2-mercapto benzimidazole as capping agent

.The synthesized nanoparticles are characterized by SEM, TEM, XRD, FTIR studies and were tested ior

antibacterial activity. It is observed that the obtained silver nanoparticles are uniform in their shape and sizes and

also shou,n good antibacterial properties.

Key wordst Silver nanoparticles, benzimidazole , capping Agent and Antimicrobial.

Introduction:Chemical reduction is the most frequently applied method for the preparation of silver nano particles (Ag-NPs) as

stable, colloidal dispersions in water or organic solvents.l'2 The commonly used reductants are borohydride, citrate,

ascorbate, and elemental hydrogen.r-rr The reduction of silver ions (Ag*) in aqueous solution generally yields

colloidal silver with particle diameters of several nanometers'

previous studies showeJtnat, use of a strong reductant such as Uo-nyariO., resulted in small particles that were

somewhat monodisperse, but the generation of larger particles was difficult to control.16'17 Use of a weaker reductant

such as citrate, resulted in a slower reduction rate, but the size distribution was far from narrow'r'a' 18 Controlled

synthesis ofAg-Nps is based on a two-step reduction process.'' In this technique a strong reducing agent is used to

produce small Ag particles, which are enlarged in a secondary step by further reduction with a weaker-.leducing

agent.3 Different studies reported the enlargement of particles in the secondary step from about 20-45 nfi to IZO-

lig nm.'t'' Moreover, the initial sol was not reproducible and specialized equipment was needed.s The syntheses of

nanoparticles by chemical reduction methods are thereibre often performed in the presence of stabilizers in order to

prevent unwanted agglomeration of the colloids.

Experimental:Synthesis of Silver Nanoparticles via chemical reduction method:

Synthesis of S-Methyl 2-mercaptobenzimidazole :

3,4-Diaminoroulene (1g, 0.0082mo1) and KOH (0.559, 0.01mol) was dissolved in 1Oml ethanol, then the reaction

mixture was kept in ice bath. To the stirred solution of this reaction mixture, CS2was added (0'6 ml, 0.01mol) in

drop wise manner and stirring is continued for 15min. Latter the reaction mixture was refluxed at 70oC for 2h. The

progress of the reaction was monitored by TLC. After consumption of all the reactants, ethanol was evaporated

under reduced pressure to obtain pale brown coloured residue. The obtained residue was treated with 20% glacial

acetic acid to give glistening pale white crystals, and the mixture is placed in a refrigerator for three hours to

complete the crystallization. The product was collected on a Buckner funnel and dried over night at 400 C. The

dried product was re-crystallized using ethanol.

www.experimentiournal.com 2162

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Catalytic Application of Synthesized Capped Silver Nanoparticles for Reduction ofp-Nitroaniline

R. S.,rneoar"', V. JaceNN.qDHer,r',, \4. paopi,r: ancl B. Sveivrrr Suxnar<j

rI)eparlmentol'Chemistry.Anil Neerukondalnstitut"olTechnologyantlscience.s,,sangiviilasa-S31 162. lndiarDepartnrenr of chernistry. Anclhra University, visakhapatnar,--530 003, Inclia

'\trgi Vernrna Unrversitv. Kadapa-5 l 6 003, Inclia

'*(iorresponcling author: E-nrail: ramar:ajusrd@gmaiLcont

lleceivetl'" 9 June 20[6: Accepted: l5 Septenrher 2016; Publi.rlt,:J ,riline.. 29 Ocroher 2016: AJC-l 8098

:r_,:?ji:^ r:jl],r,of

;)nthesizedlang.ed silver.nanoparricres was perforrned iu ,r," ."a*,in ;;;;;;;;,;;;,;

::I;f':';T,:"i'"=:::::::.:'r'nt sodium botohvdiidc. Th.e11te ol'reduction is observecl wirh nanoparrictes having ditl'ererit .i2.. +0. iolnd 4u + 2 n'h' lhe Product,ohtarned in the.presence or.zo nm.size nanoparlicles is in good yii.la ana produced in less tirne when

;::l:T:i.11 :ii::;fli1ffi;yoreover iiis observed rhat rhis catalvsr sriowed **i*,* ;mlil; i;*;;ffi;;;, il;

;;*, ;;i##anoparricreg,."r".,,," "ill;iH;ir$J irr ,,.i...,-..--.,',.;r., 1:1-:i,rf ,1.i

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INTRODTJCTION

It is always tascinating to us t.hat whir the nanoparliclesare so special and interesting even though these extremelysrnall siructures arc more complicated than that of theirmacroscopic counterparls in handling and synthesis. The answerlies in the unirlue ploperties possessecl by nanostrLlctures.Nanoparticies possess a ver,v l-righ surlace to volume ratio. Inthe nranuf'actuling novel rnaterials. nanotechnology finds aneu.'Iv emerging iietd with numerous applications in science.By rnanipulating size and shape of the particles at nanometrescale ( l to 100 nm) they arc usctl as an efficient catalysts invarious indLrstrial applications, in hcalth nnd claily Iife [1.21,such as better drug delivery r.nerhods f-3.:tl, cherrical depositionfol cnvirontnental pollution cleanup [5,6] nredical irnagin-17.8i and military purposes i9, 101.

EXPERII,ENTAL

S"l,nthesis of Tinosprtra crispa extract bin-moleculescapped nanoparticles (np1) [11]: Tinrsspora crl.rpa plants arerich in bio-rnolecules like alkaloids. glycosides. steroicls,sesclr,riterpenoid. aliphatic cornpounci, esseutial oils. rnjxtrrreof tatty acids and polysaccharicles which have capacity toleduce and stabilize bulk silvel into nano silver.. The finallycut lle-\h plant sterns of Tinospora cri.spa were clried in a hotair or,en at -50-,5-5

oC flot'one rveek ancl then grindeci to porvrler.100 g of this driecl powder ivas boiled in 200 rnl. of distilleclwater for I-5-20 rnin and then cooled. When centrifused this

cooled nrixture at -5000 rprn fbr I0 rrin a yellorv colour super_natant obtained was usecl fbr lurther experinlents.

The sih,er nanoparticles (np I ) were synthesized by stiuing40 rrfl of the supernatant r.r,ith 200 mL of I mlvi of siivernitrate (purchased from Hinreclia cl.rernicals) solutron at roomternperature and the bio-reduced product was monitoreclperiodicalll,' by nsing LiV-vi sible spectrophotorneter. The sizeo['the sl,nthesizcd nanoparticles is fbuncl to be 40 nm.

Synthesis of N-(4-Amino-3,5-diphen yl-3H -thiazol-Z-vlidine)benzamide capped silver nannparticles (np2) [ l2l:By stirring slor,vly 2.5 mL of I0-: iVI AgNO. dilutcil in 7-5 mLof triply distilled organic-ti.ee water rvith a stabilizer 5 rnl- ofI O-'? M A/-(4-arnino-3,5-rliphenyl-3H-thiazoi,2-ylidine)benza_rnide (drssolved in hot EIOLI) for l0 ntin 0t room te lnperature .

2.-5 n[ of ]O'': M KI rvas aclded clroprvise until it yields a green1,ellou, AgI colloid, to this 20 nrg of NaBHa was added andstined for another ?0 nrin. Driring the stirring, the green-yellou,colloidal solution colour changed to nut-brown, then to brolvnand linally to black. The size ol synthesize<i nanoparticlesL,und to bc l0 nnr.

Synthesis of S-rnethyl 2-mercapto benzirnitlazole capperlsilver nanoparticles (np3) | l3l: Thc solution of AgNO. (0. I g,0.00059 mol) dissolvecl in z1 rnl- ol cleionizecl rvarer urixedwith the solution of -5-methyl 2-mercapro benzimiclazole (0. l4g, 0.00089 mol; dissolvecl in 4 mL ethzrnol and was srirreclcontinuously for' t h. To it lreshly prepar.ed aqueous NaBHa(0.04 g, 0.001 I mol. I mI-) added drop by drop, fbilowed byvigorous stirring fbr 2 h until colour. of the reaction u,rixture

#.:i

ffis#/d#lB d,'fr,lf.= nt

Complex Intell. Syst.DOI 10.1007/s40747-016-0022-8

ORIGINAL ARTICLE

Social group optimization (SGO): a new population evolutionaryoptimization technique

Suresh Satapathy1 · Anima Naik1

Received: 2 October 2015 / Accepted: 1 August 2016© The Author(s) 2016. This article is published with open access at Springerlink.com

Abstract Social group optimization (SGO), a population-based optimization technique is proposed in this paper. It isinspired from the concept of social behavior of human towardsolving a complex problem. The concept and the mathe-matical formulation of SGO algorithm is explained in thispaper with a flowchart. To judge the effectiveness of SGO,extensive experiments have been conducted on number ofdifferent unconstrained benchmark functions as well as stan-dard numerical benchmark functions taken from the IEEECongress on Evolutionary Computation 2005 competition.Performance comparisons are made between state-of-the-art optimization techniques, like GA, PSO, DE, ABC andits variants, and the recently developed TLBO. The inves-tigational outcomes show that the proposed social groupoptimization outperforms all the investigated optimizationtechniques in computational costs and also provides opti-mal solutions for most of the functions considered in ourwork. The proposed technique is found to be very simpleand straightforward to implement as well. It is believed thatSGOwill supplement the groupof effective and efficient opti-mization techniques in the population-based category andgive researchers wide scope to choose this in their respectiveapplications.

Keywords Optimization · Evolutionary computation ·Benchmark functions · CEC 2005

B Suresh Satapathysureshsatapathy@ieee.org

1 ANITS, Visakhapatnam, India

Introduction

Population-based optimization algorithms motivated fromnature commonly locate near-optimal solution to optimiza-tion problems. Every population-based algorithm has thecommon characteristics of finding out global solution ofthe problem. A population begins with initial solutions andgradually moves toward a better solution area of searchspace based on the information of their fitness. Over the lastfew decades, numbers of successful population-based algo-rithms have been emerged for solving complex optimizationproblems. Some of the well-known population-based opti-mization techniques are comprehensively cited below, andreaders can refer details in the respective papers. Geneticalgorithms (GAs) [1], being the most popular ones, are basedon genetic science and natural selection operators. The dif-ferential evolution (DE) [2] is based on similar concept ofGA but it offers all solutions an equal chance irrespectiveof their fitness to get selected as parents, unlike GA, andhas found to be recently very well known to optimizationresearchers. Bacteria foraging (BF) [3] based on the socialforaging behavior of Escherichia coli, shuffled frog leap-ing (SFL) [4] inspired by natural memetics providing beautyof local search and global information exchange, simulatedannealing (SA) [5] based on steel annealing process, and antcolony optimization (ACO) [6] motivated from the mannersof real ant colony. A technique based on swarm behaviorsuch as fish schooling and bird flocking in nature knownas Particle Swarm Optimization (PSO) [7] has been widelyresearched and applied to various fields of engineering-alliedsubjects. Artificial bee colony (ABC) [8] algorithm basedon the intelligent foraging behavior of honeybee swarm,the gravitational search algorithm (GSA) [9] based on thelaw of gravity and notions of mass interactions, cuckoosearch [10] inspired by the obligate brood parasitism of

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some cuckoo species by laying their eggs in the nests of otherhost birds (of other species) are gaining popularity as wellamong users. Biogeography-based optimization (BBO) [11]based on the idea of themigration strategy of animals or otherspecies for solving optimization problems, the intelligentwater drops (IWDs) [12] algorithm enthused from observ-ing natural water drops that flow in rivers and find almostoptimal paths to their destination, the firefly algorithm (FA)[13] inspired by the flashing behavior of fireflies in nature, thehoney bee mating optimization (HBMO) [14,15] algorithminspired by the process ofmarriage in real honey bees, the batalgorithm (BA) [16] inspired by the echolocation behavior ofbats are few more population-based techniques in this cate-gory. The harmony search (HS) [17] optimization algorithminspired by the improvising process of composing a piece ofmusic, the big bang–big crunch (BB-BC) optimization [18]based on one of the theories of the evolution of the universe[18], black hole (BH) [19] optimization inspired by the blackhole phenomenon have also been extensively tried success-fully for solving various problems in engineering. Recently,teaching–learning-based optimization (TLBO) [20] based onthe effect of the influence of a teacher on the output of learn-ers in a class is being extensively studied by researchers tosolve a variety of optimization problems in engineering appli-cations. Even though all these algorithms are good enoughfor solving optimization problems, however, issues like find-ing optimal solutions, providing fast convergence withoutover fitting (computational efforts), choosing and controllingalgorithm parameters, algorithm stability and robustness,consistency in providing solutions, adaptability to wide vari-ety of applications, etc., have been the subjects of extensiveresearch in optimization community. To address the afore-mentioned issues, researchers have developed many variantsof the above-mentioned optimization algorithms, and evenhybridization of several algorithms has also been attempted.

In an attempt to address few challenges like computa-tional efforts, optimal solutions and consistency in providingoptimal solutions, this paper proposes a new optimizationtechnique named social group optimization (SGO) based onthe human behavior of learning and solving complex prob-lems.

In thiswork,wehavedone extensive study to further inves-tigate the performance of our proposed SGO algorithm onmany simple benchmark functions as well as benchmarkfunctions from CEC 2005 competitions. Many advancedversions of state-of-the art algorithms like PSO, DE andABC etc., and their variants are simulated to compare theperformances with SGO. Also, the performance of SGOis compared with recently developed TLBO algorithms.Convergence characteristics of SGO are presented in plots.Results are reported in Tables with the mean and standarddeviation values for each algorithm on each function overseveral simulation runs. To compare the significance of the

proposed algorithm, we have doneWilcoxon’s rank-sum sta-tistical tests.

The remaining of the paper is organized as follows: in“Social Group Optimization”, we give a comprehensivedescription of SGO algorithm. The next section “Implemen-tation of SGO for optimization” discusses the implementa-tion of SGO for optimization followed by discussion about“Experimental results”. The paper concludes with furtherresearch in “Conclusion”.

Social group optimization (SGO)

There are many behavioral traits such as honesty, dishonesty,caring, compassion, courage, fear, justness, fairness, toler-ance or respectfulness etc., lying dormant in human beings,which need to be harnessed and channelized in the appropri-ate direction to enable him/her to solve complex tasks in life.Few individuals might have required level of all these behav-ioral traits to be capable of solving, effectively and efficiently,complex problems in life. But very often, complex problemscan be solved with the influence of traits from one personto other or from one group to other groups in the society. Ithas been observed that human beings are great imitators orfollowers in solving any task. Group solving capability hasemerged to be more effective than individual capability inexploiting and exploring different traits of each individual inthe group to solve a given problem. Based upon this concept,a new optimization technique is proposed which is named associal group optimization (SGO).

In SGO, each person (a candidate solution) is empow-ered with some sort of knowledge having a level of capacityfor solving a problem. SGO is another population-basedalgorithm similar to other algorithms discussed in the pre-vious section. For SGO, the population is considered as agroup of persons (candidate solutions). Each person acquiresknowledge and, thereby, possesses some level of capacity forsolving a problem. This is corresponding to the ‘fitness’. Thebest person is the best solution. The best person tries to prop-agate knowledge amongst all persons, which will, in turn,improve the knowledge level of the entire members in thegroup.

The procedure of SGO is divided into two parts. Thefirst part consists of the ‘improving phase’; the second partconsists of the ‘acquiring phase’. In ‘improving phase,’ theknowledge level of each person in the group is enhancedwith the influence of the best person in the group. The bestperson in the group is the one having the highest level ofknowledge and capacity to solve the problem. And in the‘acquiring phase,’ each person enhances his/her knowledgewith the mutual interaction with another person in the groupand the best person in the group at that point in time. The

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basicmathematical interpretation of this concept is presentedbelow.

Let X j , j = 1, 2, 3, . . .N be the persons of social group,i.e., social group contains N persons and each person X j isdefined by X j = (x j1, x j2, x j3, . . . , x j D), where D is thenumber of traits assigned to a person which determines thedimensions of a person and f j , j = 1, 2, . . .N are theircorresponding fitness values, respectively.

Improving phase

The best person (gbest) in each social group tries to propagateknowledge among all persons, whichwill, in turn, help othersto improve their knowledge in the group.

Hence, gbestg = min fi , i = 1, 2, . . .Nat generation g for solvingminimization problem.

(1)

In the improving phase, each person gets knowledge(here knowledge refers to change of traits with the influ-ence of the best person’s traits) from the group’s best (gbest)person. The updating of each person can be computed asfollows:

For i = 1 : NFor j=1:D

Xnewi j =c ∗ Xoldi j +r ∗ (gbest( j)−Xoldi j )End for

End forwhere r is a random number, r ∼ U (0, 1)Accept Xnew if it gives a better fitness than Xold.

(2)

where c is known as self-introspection parameter. Its valuecan be set from 0 < c < 1.

Acquiring phase

In the acquiring phase, a person of social group interactswith the best person (gbest) of that group and also inter-acts randomly with other persons of the group for acquiringknowledge. A person acquires new knowledge if the otherperson has more knowledge than him or her. The best knowl-edgeable person (here known as person having ‘gbest’) hasthe greatest influence on others to learn from him/her. Aperson will also acquire something new from other per-sons if they have more knowledge than him or her in thegroup.

The acquiring phase is expressed as given below:gbest = min f (Xi ), i = 1, 2, . . . N (3)

(Xi ’s are updated values at the end of the improving phase)For i = 1 : N

Randomly select one person Xr ,where i = rIf f (Xi ) < f (Xr )

For j = 1 : DXnewi, j = Xoldi, j + r1 ∗ (

Xi, j − Xr, j)

+ r2 ∗ (gbest j − Xi, j )

End forElse

For j = 1 : DXnewi,: = Xoldi,: + r1 ∗ (

Xr,: − Xi,:)

+r2 ∗ (gbest j − Xi j )

End forEnd If

Accept Xnew if it gives a better fitness function value.End for

(4)

where r1 and r2 are two independent random sequences, r1 ∼U (0, 1) and r2 ∼ U (0, 1) . These sequences are used toaffect the stochastic nature of the algorithm as shown abovein Eq. (4).

For further clarity and ease of implementation, the entireprocess is now presented in an easy-to-understand flowchart(Fig. 1)

Implementation of SGO for optimization

The step-wise procedure for the implementation of SGO isgiven in this section.

Step 1: Enumeration of the problem and Initialization ofparametersInitialize the population size (N), number of gen-erations (g), number of design variables (D), andlimits of design variables (UL , LL). Define theoptimization problem as: Minimize f (X). Sub-ject to = (x1, x2, x3, . . . . . . , xD), so that X j =(x j1, x j2, x j3, . . . . . . , x j D),where f (X) is the objec-tive function, and X is a vector for design variablessuch that LL ,i ≤ x,i ≤ UL ,i .

Step 2: Initialize the populationA random population is generated based on thefeatures (number of parameters) and the size of pop-ulation chosen by user. For SGO, the population sizeindicates the number of persons and the features indi-cate the number of traits of a person. This populationis articulated as:

Population =⎡

⎢⎣

x1,1 x1,2 x1,3 · · · x1,D...

. . ....

xN ,1 xN ,2 xN ,3 · · · xN ,D

⎤

⎥⎦

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Complex Intell. Syst.

Fig. 1 Flow Charts of SGO

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Complex Intell. Syst.

Calculate the fitness of the population f (X).Step 3: Improving Phase

Then, determine gbestg using Eq. (1), which is thebest solution for that iteration. As in the improv-ing phase, each person gets knowledge from theirgroup’s best, i.e., gbest

For i = 1 : NFor j = 1 : D

Xnewi j =c ∗ Xoldi j +r ∗ (gbest ( j)−Xoldi j )

End for

End for

The value of c is self-introspection factor. The valueof c can be empirically chosen for a given problem.We have set it to 0.2 in this work after thoroughstudy of our investigated problems and r is a randomnumber, r ∼ U (0, 1).Accept Xnew if it gives better function value.

Step 4: Acquiring phaseAs explained above, in the acquiring phase, a personof social group interacts with the best person, i.e.,gbest of the group and also interacts randomly withother persons of the group for acquiring knowledge.The mathematical expression is defined in “Acquir-ing phase”.

Step 5: Termination criterionStop the simulation if themaximumgeneration num-ber is achieved; otherwise, repeat from Steps 3–4.

Experimental results

In this paper, the performance of SGO is compared withmany classical population-based optimization techniques aswell as their advanced variants using some basic benchmarkfunctions and 25 test functions proposed in theCEC2005 spe-cial session on real-parameter optimization. A description ofsome basic benchmark functions is given in Appendix, andothers are referred from their respective papers, and a detaileddescription of these 25 CEC2005 test functions can be foundin [21]. In “Experiment 1:SGO vs. GA, PSO, DE, ABC,and TLBO” to “Experiment 8. SGO vs FIPS-PSO, CPSO-H,DMSPSO-LS, CLPSO, APSO, SSG-PSO, SSG-PSO-DFP,SSG-PSO-BFGS, SSG-PSO-NM, SSG-PSO-PS”, we havedescribed the experimentation on basic benchmark func-tions; in “Experiment 9: SGO vs. jDE, SaDE, EPSDE,CoDE, MPEDE , CLPSO, CMA-ES,GL-25 and TLBO”,CEC2005 test functions are experimented; and experimentson composite test functions are discussed and experimentedin “Experiment 10: SGOvs. PSO, CPSO, CLPSO, CMA-ES,G3-PCX, DE, and TLBO using Composite functions”. To

have statistically sound conclusions, Wilcoxon’s rank-sumtest at a 0.05 significance level was conducted on the experi-mental results, and the last three rows of each respective tablesummarize the experimental results.

For comparing the speed of the algorithms, the first thingwe require is a fair time measurement. The number of iter-ations or generations cannot be accepted as a time measure,since the algorithms perform different amount of works intheir inner loops, and they have different population sizes.Hence, we choose the number of fitness function evalu-ations(FEs) as a measure of computation time instead ofgenerations or iterations. Since the algorithms are stochas-tic in nature, the results of two successive runs usually donot match. Hence, we have taken different independent runs(with different seeds of the random number generator) ofeach algorithm.

Finally, we would like to point out that all the experimentcodes are implemented in MATLAB 7. The experiments areconducted on a Pentium 4, 1 GB memory desktop in Win-dows XP 2002 environment.

Experiment 1:SGO vs. GA, PSO, DE, ABC, and TLBO

In this section, for fair comparison of the performances ofalgorithms, the results are directly gained form [22] for GA,PSO, DE and ABC algorithms. However, the simulationshave been carried out by us for TLBO and our proposedSGO algorithm. The common parameter such as populationsize is set to 20 for both TLBO and SGO. The maximumnumber for function evaluation is set to 2000 for TLBO and1000 for SGO. The other specific parameters of algorithmsare given below:

TLBO settings For TLBO, there is no such constant to set.

SGO settings For SGO, there is only one constant self-introspection factor for optimum self-effort c. The value of cis empirically set to 0.2 for better results.

The 25 benchmark functions which are considered forsimulations include many different kinds of problems suchas unimodal, multimodal, regular, irregular, separable, non-separable and multidimensional. All problems are dividedinto four categories such as US, MS, UN, MN, and therange, formulation, characteristics and the dimensions ofthese problems are described in Appendix.

Each of the experiments for TLBO and SGO is repeated30 times (we have taken the same number of experimenta-tions which have been done in [22] to make the comparisonfair) with different random seeds, and the best mean valuesproduced by the algorithms have been recorded. Comparisoncriteria are the mean solution and the standard solution fordifferent independent runs. The mean solution describes theaverage ability of the algorithm to find the global solution,and the standard deviation describes the variation in solution

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from the mean solution. To make the comparison clear, thevalues below 10−12 are assumed to be 0. Also, to have sta-tistically sound conclusions, Wilcoxon’s rank-sum test at a0.05 significance level has been conducted on the experimen-tal results, and the last three rows of Table 1 summarize theresults. The results forGA, PSO,DE andABCare taken fromthe paper [22]. The Table 1 presents the comparison of fitnessvalues of GA, PSO, DE and ABC gained from [22] and forTLBO and SGO computed by us. In Table 1, “NA” standsfor experiment is not conducted for that particular function.The best optimal values are shown in bold face.

From Table 1, it is clear that SGO provides better opti-mal results in as many as 12 functions compared to GA, 5functions compared to PSO, 4 functions compared to DE,3 functions compared to ABC and 2 functions compared toTLBO. It can be arrived at a conclusion here that SGO is verycompetitive compared to ABC and especially with TLBO.However, we have noted that SGO is fasterwhen compared toother algorithms. It takes 1/500th function evaluations com-pared to GA, PSO, DE and ABC for all functions except forRosenbrock wherein it takes 1/50th of total function evalua-tions. And it is computing almost in half of the total functionevaluations compared to TLBO except for Rosenbrock forwhich SGO takes 1/5th number of function evaluation asagainst TLBO. From the above findings, we may arrive ata conclusion that our proposed algorithm not only performsbetter compared to many state-of-the art algorithms like GA,PSO, DE but also very competitive with ABC and TLBOin providing optimal solutions. Importantly, SGO takes lesscomputation efforts compared to all other algorithms inves-tigated in this section.

Experiment 2: SGO vs. HS, IBA, ABC, and TLBO

In this experiment, five different benchmark problems fromKaraboga and Akay [23] are considered, and comparisonis carried out between SGO, the harmony search algorithm(HS), improved bee algorithm (IBA), artificial bee colonyoptimization (ABC) and teaching–learning-based optimiza-tion (TLBO) [24]. To compare the results, the mean solutionand the standard solution for different independent runs aretaken. In our simulation, we run TLBO for maximum 2000FEs with 10 as the population size for all functions exceptRosenbrock, whereas HS, IBA and ABC run for 50,000 FEswith 50 as population size. For Rosenbrock function, TLBOtakes 50,000 FEs with population size of 50. For our opti-mization algorithm, i.e., SGO, maximum FEs are set to 1000with 10 as the population size for all functions except Rosen-brock for which it is set to 50,000 FEs with population sizeof 50. The results are gathered for different independent runsin each case, and the mean and standard deviation are calcu-lated for the results obtained in different runs. Description ofthe functions is given in Appendix.

In this simulation, different dimensions (D) of the bench-mark functions are chosen for study. Values starting from assmall as 5 to 1000 are taken. The results for dimensions 5,10, 30, 50, 100 are directly lifted from [24] for all inves-tigated algorithms and put in Table 2. For SGO algorithm,we have computed results for all dimensions and generatedresults for two large-scale dimensions, such as 500 and 1000,to investigate the performance of SGO for large-dimensionproblems. The maximum number of FEs for SGO is set1/50th of maximum FEs taken for HS, IBA and ABC forall functions except Rosenbrock. And, for Rosenbrock, it is1/5th of HS, IBC and ABC. However, it is exactly half thatof TLBO in all functions [24,25]. It may be emphasized herethat the reduced value of maximum number of FEs for SGOis deliberately chosen to investigate the effectiveness andefficiency over other algorithms. Table 2 shows the resultsfor this experiment. In Table, “NA” stands for experiment isnot conducted for that particular function. The best optimalvalues are shown in bold face. To have statistically soundconclusions, Wilcoxon’s rank-sum test at a 0.05 significancelevel has been conducted on the experimental results, andthe last three rows of Table summarize the results. It canbe seen from Table 2 that SGO has outperformed than allthe algorithms for all the functions in almost all dimensions.This experiment shows that SGO is effective in finding theoptimum solution with increase in dimensions. However, theperformance of other algorithms in higher dimensions hasnot been ascertained in this work. Our preliminary litera-ture study reveals that they do not perform well in higherdimensions.

Experiment 3: SGO vs OEA, HPSO-TVAC, CLPSO,APSO, OLPSO-L and OLPSO-G

In this section, comparisons of SGO versus OEA, HPSO-TVAC (Self-organizing hierarchical particle swarm opti-mizer with time-varying acceleration coefficients) [26],APSO (adaptive particle swarm optimization) [27], CLPSO(comprehensive learning particle swarm optimization) [28],OLPSO (orthogonal learning particle swarmoptimization)-L[29] andOLPSO-G [29] on nine benchmarks listed inAppen-dix are carried out. OEA uses 3.0 × 105 FEs, HPSO-TVAC,CLPSO,APSO,OLPSO-L andOLPSO-G use 2.0×105 FEs,whereas SGO runs for 3× 103 FEs for sphere, schwefel 1.2,schwefel 2.22 function, 1.0×102 FEs for step, 4.0×102 FEsfor rastrigin, noncontinuous rastrigin and griwank, 1.0×103

FEs for Ackley and quartic function. The results of OEA,HPSO-TVAC, CLPSO and APSO are gained from [28] and[27] directly, and for OLPSO-L and OLPSO-G, results aregained from [29] directly and put in Table 3. In Table, “NA”stands for experiment is not conducted for that particularfunction. The best optimal values are shown in bold face. Tohave statistically sound conclusions, Wilcoxon’s rank-sum

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Complex Intell. Syst.

Table 1 Performance comparisons of results of 30 runs obtained by GA, PSO, DE, ABC, TLBO, and SGO algorithms in terms of mean and Std

Name of the functions GA PSO DE ABC TLBO SGO

Step

Mean 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00

Std (0.00e+00)≈ (0.00e+00)≈ (0.00e+00)≈ (0.00e+00)≈ (0.00e+00)≈ 0.00e+00

Sphere

Mean 1.11e+03 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00

Std (7.41e+01)− (0.00e+00)≈ (0.00e+00)≈ (0.00e+00)≈ (0.00e+00)≈ 0.00e+00

Sum Squares

Mean 1.48e+02 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00

Std (1.24e+01)− (0.00e+00)≈ (0.00e+00)≈ (0.00e+00)≈ (0.00e+00)≈ 0.00e+00

Quartic

Mean 1.81e−01 1.16e−03 1.36e−03 3.00e−02 1.20e−03 3.77e−04

Std (2.71e−02)− (2.76e−04)− (4.17e−04)− (4.87e−03)− (3.1134e−04)− 1.46e−04

Beale

Mean 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00

Std (0.00e+00)≈ (0.00e+00)≈ (0.00e+00)≈ (0.00e+00)≈ (0.00e+00)≈ 0.00e+00

Easom

Mean −1 −1 −1 −1 −1 −1

Std (0.00e+00)≈ (0.00e+00)≈ (0.00e+00)≈ (0.00e+00)≈ (0.00e+00)≈ 0.00e+00

Matyas

Mean 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00

Std (0.00e+00)≈ (0.00e+00)≈ (0.00e+00)≈ (0.00e+00)≈ 7(0.00e+00)≈ 0.00e+00

Zakharov

Mean 1.36e−02 0.00e+00 0.00e+00 2.48e−04 0.00e+00 0.00e+00

Std (4.53e−03)− (0.00e+00)≈ (0.00e+00)≈ (1.83e−04)− (0.00e+00)≈ 0.00e+00

Powell

Mean 9.70e+00 1.10e−04 2.17e−07 3.13e−03 0.00e+00 0.00e+00

Std (1.55e+00)− (1.60e−04)− (1.36e−07)− (5.03e−04)− (0.00e+00)≈ 0.00e+00

Schwefel 1.2

Mean 7.40e+03 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00

Std (1.14e+03)− (0.00e+00)≈ (0.00e+00)≈ (0.00e+00)≈ (0.00e+00)≈ 0.00e+00

Schwefel 2.21

Mean NA NA NA NA 0.00e+00 0.00e+00

Std (0.00e+00)≈ 0.00e+00

Schwefel 2.22

Mean 1.1.0e+01 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00

Std (1.39e+00)− (0.00e+00)≈ (0.00e+00)≈ (0.00e+00)≈ (0.00e+00)≈ 0.00e+00

Bohachevsky1

Mean 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00

Std (0.00e+00)≈ (0.00e+00)≈ (0.00e+00)≈ (0.00e+00)≈ (0.00e+00)≈ 0.00e+00

Bohachevsky2

Mean 6.83e−02 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00

Std (7.82e−02)− (0.00e+00)≈ (0.00e+00)≈ (0.00e+00)≈ (0.00e+00)≈ 0.00e+00

Bohachevsky3

Mean 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00

Std (0.00e+00)≈ (0.00e+00)≈ (0.00e+00)≈ (0.00e+00)≈ (0.00e+00)≈ 0.00e+00

Booth

Mean 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00

Std (0.00e+00)≈ (0.00e+00)≈ (0.00e+00)≈ (0.00e+00)≈ (0.00e+00)≈ 0.00e+00

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Table 1 continued

Name of the functions GA PSO DE ABC TLBO SGO

Rastrigin

Mean 5.29e+01 4.39e+01 1.17e+01 0.00e+00 0.00e+00 0.00e+00

Std (4.56e+00)− (1.17e+01)− (2.54e+00)− (0.00e+00)≈ (0.00e+00)≈ 0.00e+00

Noncontinous rastrigin

Mean NA NA NA NA 0.00e+00 0.00e+00

Std (0.00e+00)≈ 0.00e+00

Six Hump Camel back

Mean −1.0316 −1.0316 −1.0316 −1.0316 −1.0316 −1.0316

Std (0.00e+00)≈ (0.00e+00)≈ (0.00e+00)≈ (0.00e+00)≈ (0.00e+00)≈ 0.00e+00

Griewank

Mean 1.06e+01 1.74e−02 1.48e−03 0.00e+00 0.00e+00 0.00e+00

Std (1.16e+00)− (2.08e−02)− (2.96e−03)− (0.00e+00)≈ (0.00e+00)≈ 0.00e+00

Ackley

Mean 1.47e+01 1.65e−01 0.00e+00 0.00e+00 0.00e+00 0.00e+00

Std (1.78e−01)− (4.94e−01)− (0.00e+00)≈ (0.00e+00)≈ (0.00e+00)≈ 0.00e+00

Multimod

Mean NA NA NA NA 0.00e+00 0.00e+00

Std (0.00e+00)≈ 0.00e+00

Weierstrass

Mean NA NA NA NA 0.00e+00 0.00e+00

Std (0.00e+00)≈ 0.00e+00

Elliptic

Mean NA NA NA NA 0.00e+00 0.00e+00

Std (0.00e+00)≈ 0.00e+00

Rosenbrocks

Mean 1.96e+05 1.51e+01 1.82e+01 8.88e−02 2.71e+01 2.70e+01

Std (3.85e+04)− (2.411e+01)+ (5.03e+00)+ (7.74e−02)+ (1.14e+00)− 1.76e−01

− 12 05 04 03 02

+ 00 01 01 01 00

≈ 08 14 15 16 23

Wilcoxon’s rank-sum test at a 0.05 significance level is performed between SGO and each of GA, PSO, DE, ABC and TLBO. “−”, “+”, and “≈”denote that the performance of the corresponding algorithm is worse than, better than, and similar to that of SGO, respectively

test at a 0.05 significance level has been conducted on theexperimental results, and the last three rows of Table sum-marize the results. According to Wilcoxon’s rank-sum test,SGO performs superior than OEA in seven test functionsand comparable to one test function out of eight test func-tions, improved thanHPSO-TVAC in eight test functions andequivalent to one test function out of nine test functions. It isalso found to be better than APSO in eight test functions andequivalent to one test function out of nine test functions. Inour work, we observe that compared to OLPSO-L, our pro-posed SGO is better in two OLPSO-G in four test functionsand equivalent to one test function out of five test functions.Hence, it can be claimed that even though the maximumnumber of fitness evaluations for SGO is less than the other

algorithms, still SGO is either better than or equivalent toother algorithms for each benchmark function according tothe Wilcoxon’s rank-sum test.

Experiment 4: SGO vs JADE, jDE, SaDE, CoDE andEPSDE

The experiments in this section constitute the comparison ofthe SGO algorithm versus SaDE [30], jDE [31], JADE [32],CoDE [33] and EPSDE [34] on nine benchmark functionswhich are listed in Appendix. The results of JADE, jDE andSaDE are gained from [32] directly and put in Table 4. ForCoDE and EPSDE, we have generated results using codesgiven in website Q. Zhang’s homepage:http://dces.essex.ac.

123

Complex Intell. Syst.

Table 2 Performance comparisons of results of 30 runs obtained by HS, IBA, ABC, TLBO and SGO algorithms in terms of mean and Std

Name of function D HS IBA ABC TLBO SGOMean Mean Mean Mean Mean

Sphere 5 3.20e−10 3.91e−17 4.30e−17 5.03e−33 1.58e−67

(2.89e−10)− (1.24e−17)− (1.07e−17)− (1.27e−32)− 1.93e−67

10 6.45e−08 4.95e−17 7.36e−17 2.26e−29 2.01e−66

(3.07e−08)− (2.30e−17)− (4.43e−17)− (3.61e−29)− 7.99e−67

30 7.21e+00 2.92e−16 4.69e−16 6.90e−26 9.19e−66

(3.62e+00)− (6.77e−17)− (1.07e−16)− (3.18e−25)− 9.21e−67

50 5.46e+02 5.39e−16 1.19e−15 8.71e−26 1.66e−65

(1.78e+03)− (1.07e−16)− (4.68e−16)− (1.86e−25)− 1.80e−66

100 1.90e+04− 1.45e−15 1.99e−06 9.42e−26 3.65e−65

(1.78e+03)− (1.63e−16)− (2.26e−06)− (3.70e−25)− 1.58e−66

500 NA NA NA NA 1.96e−64

9.88e−66

1000 NA NA NA NA 4.01e−64

1.69e−65

Rosenbrock 5 5.94e+00 4.55e−01 2.33e−01 1.80e−01 4.20e−06

(6.71e+00)− (1.54e+00)− (2.24e−01)− (8.04e−02)− 2.30e−06

10 6.52e+00 1.10e+01 4.62e−01 5.58e+00 4.15e−02

(8.16e+00)− (2.55e+01)− (5.44e−01)− (6.18e−01)− 1.82e−01

30 3.82e+02 7.57e+01 9.98e−01 2.71e+01 2.30e+01

(5.29e+02)− (1.16e+02)− (1.52e+00)+ (1.14e+00)− 6.76e−01

50 2.47e+04 6.30e+02 4.33e+00 4.78e+01 4.42e+01

(1.02e+04)− (1.20e+03)− (5.48e+00)+ (1.01e+00)− 5.01e−01

100 1.45e+07 6.42e+02 1.12e+02 9.81e+01 9.50e+01

(2.16e+06)− (8.20e+02)− (6.92e+1)− (3.61e−01)− 6.01e−01

500 NA NA NA NA 4.92e+02

6.67e−01

1000 NA NA NA NA 9.89+02

2.12e−01

Ackley 5 2.68e−05 6.35e−10 9.64e−17 0.00e+00 −8.88e−16

(1.24e−05)− (9.77e−11)− (5.24e−17)+ (0.00e+00)+ 0.00e+00

10 2.76e−04 6.71e−02 3.51e−16 0.00e+00 −8.88e−16

(7.58e−05)− (3.61e−01)− (6.13e−17)≈ (0.00e+00)+ 0.00e+00

30 9.43e−01 1.75e+00 3.86e−15 7.11e−16 0.00e+00

(5.63e−01)− (9.32e−01)− (3.16e−15)− (1.82e−15)≈ 0.00e+00

50 5.28e+00 8.43e+00 4.38e−08 1.24e−15 −8.88e−16

(4.03e−01)− (7.70e+00)− (4.65e−08)− (1.95e−15)− 0.00e+00

100 1.32e+01 1.89e+01 1.32e−02 2.13e−15 −8.88e−16

(4.90e−01)− (8.50e−01)− (1.30e−02)− (1.19e−15)− 0.00e+00

500 NA NA NA NA −8.88e−16

0.00e+00

1000 NA NA NA NA −8.88e−16

0.00e+00

123

Complex Intell. Syst.

Table 2 continued

Name of function D HS IBA ABC TLBO SGOMean Mean Mean Mean Mean

Griwank 5 2.60e−02 3.14e+00 4.04e−17 0.00e+00 0.00e+00

(1.38e−02)− (1.41e+00)− (1.12e−17)− (0.00e+00)≈ 0.00e+00

10 1.02e+00 1.04e+00 6.96e−17 0.00e+00 0.00e+00

(3.02e−02)− (1.13e+00)− (4.06e−17)− (0.00e+00)≈ 0.00e+00

30 1.09e+00 6.68e+00 5.82e−06 0.00e+00 0.00e+00

(3.92e−02)− (6.43e+00)− (3.13e−05)− (0.00e+00)≈ 0.00e+00

50 5.81e+00 1.34e+02 5.72e−01 0.00e+00 0.00e+00

(9.16e−01)− (2.41e+01)− (9.22e−01)− (0.00e+00)≈ 0.00e+00

100 1.78e+02 7.93e+02 1.31e+01 0.00e+00 0.00e+00

(1.98e+01)− (7.96e+01)− (6.30e+00)− (0.00e+00)≈ 0.00e+00

500 NA NA NA NA 0.00e+00

0.00e+00

1000 NA NA NA NA 0.00e+00

0.00e+00

Rastrigin 5 6.07e−08 4.58e+00 4.34e−17 0.00e+00 0.00e+00

(5.52e−08)− (2.31e+00)− (1.10e−17)− (0.00e+00)≈ 0.00e+00

10 1.05e−05 2.20e+01 5.77e−17 0.00e+00 0.00e+00

(5.23e−06)− (7.46e+00)− (2.98e−17)− (0.00e+00)≈ 0.00e+00

30 7.40e−01 1.28e+02 4.80e−05 0.00e+00 0.00e+00

(7.00e−01)− (2.49e+01)− (2.43e−04)− (0.00e+00)≈ 0.00e+00

50 3.76e+01 2.72e+02 4.72e−01 0.00e+00 0.00e+00

(4.87e+00)− (3.27e+01)− (4.92e−01)− (0.00e+00)≈ 0.00e+00

100 3.15e+02 6.49e+02 1.46e+01 0.00e+00 0.00e+00

(2.33e+01)− (4.52e+01)− (4.18e+00)− (0.00e+00)≈ 0.00e+00

500 NA NA NA NA 0.00e+00

0.00e+00

1000 NA NA NA NA 0.00e+00

0.00e+00

− 25 25 21 12

+ 00 00 3 02

≈ 00 00 01 11

Wilcoxon’s rank-sum test at a 0.05 significance level is performed between SGO and each of HS, IBA, ABC, and TLBO. “−”, “+”, and “≈” denotethat the performance of the corresponding algorithm is worse than, better than, and similar to that of SGO, respectivelyD dimension

uk/staff/qzhang/. For CoDE, EPSDE and SGO, we haveconsidered population size as 20. The maximum number offitness evaluations for each function is different, and FEs arenoted in bracket of each cell in Table 4. Fitness values areshown in Table 4 in means and standard deviations. In Table,“NA” stands for experiment is not conducted for that particu-lar function. The best optimal values are shown in bold face.To have statistically sound conclusions, Wilcoxon’s rank-sum test at a 0.05 significance level has been conducted onthe experimental results, and the last three rows of Tablesummarize the results. According to Wilcoxon’s rank-sumtest, it can be noted that the performance of SGO is alwaysbetter than all other algorithms except EPSDE in reporting

the optimal value, where SGO performs better than EPSDEin five test functions and equivalent to three test functionsout of eight test functions. So, it is interesting to note thateven though the maximum number of fitness evaluations forSGO is less than the other algorithms, still SGO is betterthan or equivalent with all variants of DE algorithm in thisexperiment according to Wilcoxon’s rank-sum test.

Experiment 5: SGO vs. CABC, GABC, RABC andIABC

In this section, we compare SGO with CABC [35], GABC[36], RABC [37] and IABC [38] on eight benchmark func-

123

Complex Intell. Syst.

Table 3 Performance comparisons of SGO, OEA, HPSO-TVAC, CLPSO, APSO, OLPSO-L and OLPSO-G

Function OEA HPSO-TVAC CLPSO APSO OLPSO-L OLPSO-G SGO

Sphere

Mean 2.48e−30 3.38e−41 1.89e−19 1.45e−150 1.11e−38 4.12e−54 1.46e−205

Std (1.13e−29)− (8.50e−41)− (1.49e−19) (5.73e−150)− (1.28e−38)− (6.34e−54)− 0

Schwefel 2.22

Mean 2.07e−13 6.9e−23 1.01e−13 5.15e−84 7.67e−22 9.85e−30 1.85e−103

Std (2.44e−12)− (6.89e−23)− (6.54e−14)− (1.44e−83)− (5.63e−22)− (1.01e−29)− 2.03e−104

Schwefel 1.2

Mean 1.88e−09 2.89e−07 3.97e+02 1.0e−10 NA NA 4.31e−201

Std (3.726e−9)− (2.97e−07)− (1.42e+02)− (2.13e−10)− 0

Step

Mean 0 0 0 0 NA NA 0

Std (0)≈ (0)≈ (0)≈ (0)≈ 0

Rastrigin

Mean 5.43e−17 2.39 e+00− 2.57e−11 5.8e−15 0 1.07e+00 0

Std (1.68e−16)− (3.71e+00)− (6.64e−11)− (1.01e−14)− (0) ≈ (9.90e−01)− 0

Noncontinuous rastrigin

Mean NA 1.83e+00 1.67e−01 4.14e−16 NA NA 0

Std (2.65e+00)− (3.79e−01)− (1.45e−15)− 0

Ackley

Mean 5.35e−14 2.06e−10 2.01e−12 1.11e−14 4.14e−15 7.98e−15 −8.88e−16

Std (2.94e−13)− (9.45e−10)− (9.22e−13)− (3.55e−15)− (0)≈ (2.03e−15)≈ 1.01e−31

Griewank

Mean 1.32e−02 1.07e−02 6.45e−13 1.67e−02 0 4.83e−03 0

Std (1.56e−02)− (1.14e−02)− (2.07e−12)− (2.41e−02)− (0)≈ (8.63e−03)− 0

Quartic

Mean 3.29e−03 5.54e−02 3.92e−03 4.66e−03 NA NA 5.37e−04

Std (1.09e−03)− (2.08e−02)− (1.14e−03)− (1.70e−03)− 3.91e−05

− 7 8 8 8 2 4

+ 00 00 00 00 00 00

≈ 1 1 1 1 3 1

Wilcoxon’s rank-sum test at a 0.05 significance level is performed between SGO and each of OEA, HPSO-TVAC, CLPSO, APSO, OLPSO-L,OLPSO-G. “−”, “+”, and “≈” denote that the performance of the corresponding algorithm is worse than, better than, and similar to that of SGO,respectively

tions. The parameters of the algorithms are identical to [36].The maximum number of fitness evaluations for each func-tion is different, and FEs are noted in bracket of each cellin Table 5. The results have been summarized in Table 5.The fitness value in terms of mean and standard deviation isreported. The best optimal values are shown in bold face. Tohave statistically sound conclusions, Wilcoxon’s rank-sumtest at a 0.05 significance level has been conducted on theexperimental results, and the last three rows of Table sum-marize the results. It can be observed from Table 5 that SGOperforms better in comparison to all algorithms. So, we cansay that even though the maximum number of fitness evalu-ations for SGO is less than the other algorithms, still SGO isbetter than all variants of ABC algorithm in this experiment.

Experiment 6: SGO vs. TLBO

In this experiment, we compare only TLBO and SGOalgorithms. As TLBO is relatively new compared to otheralgorithms investigated in our work, we have devoted aspecial section to compare our approach with TLBO. Inthis experiment, our main objective is to see how SGOperforms against TLBO in terms of optimal solution andcomputational costs. The common parameter such as pop-ulation size is taken as 20 and maximum number fitnessfunction evaluation is taken as 1,000 for both TLBO andSGO.

We used 25 benchmark problems to test the performanceof the TLBO and our proposed SGO algorithms. The ini-

123

Complex Intell. Syst.

Table4

Performance

comparisons

ofSG

O,JADE,jDE,S

aDE,C

oDEandEPS

DE

Functio

nSaDE(FEs)

jDE(FEs)

JADE(FEs)

CoD

E(FEs)

EPS

DE(FEs)

SGO(FEs)

Sphere

Mean

4.5e

−20(1

.5×

105)

2.5e

−28(1

.5×

105)

1.8e

−60(1

.5×

105)

1.12e−

31(1

.5×

105)

1.53e−

85(1

.5×

105)

0(5.0

×10

3)

Std

(1.90e

−14)

−(3.5e−

28)−

(8.4e−

60)−

(3.45e

−31)

−(9.01e

−86)

−0

Schw

efel2.22

Mean

1.9e

−14(2

.0×

105)

1.5e

−23(2

.0×

105)

1.8e

−25(2

.0×

105)

1.22e−

23(2

.0×

105)

3.18e−

54(2

.0×

105)

0(5.0

×10

3)

Std

(1.10e

−14)

−(1.0e−

23)−

(8.8e−

25)−

(3.90e

−23)

−(3.11e

−54)

−0

Schw

efel1.2

Mean

9.0e

−37(5

.0×

105)

5.2e

−14(5

.0×

105)

5.7e

−61(5

.0×

105)

7.86e−

31(5

.0×

105)

4.81e−

76(5

.0×

105)

0(5.0

×10

3)

Std

(5.40e

−36)

−(1.1e−

13)−

(2.7e−

60)−

(1.86e

−32)

−(1.90e

−76)

−0

Step Mean

9.3e

+02(1

.0×

104)

1.0e

+03(1

.0×

104)

2.9e

+00(1

.0×

104)

3.00e+

00(1

.0×

104)

0(1.0

×10

4)

0(1.0

×10

2)

Std

(1.8e+

02)−

(2.2e+

02)−

(1.2e+

00)−

(1.90e

+00)

−(0)≈

0

Rastrigin

Mean

1.2e

−03(1

.0×

105)

1.5e

−04(1

.0×

105)

1.0e

−04(1

.0×

105)

1.21e−

01(1

.0×

105)

0(1.0

×10

5)

0(4.0

×10

2)

Std

(6.5e−

04)−

(2.0e−

04)−

(6.0e−

05)−

(3.89e

−02)

−(0)≈

0

Schw

efel2.21

Mean

7.4e

−11(5

.0×

105)

1.4e

−15(5

.0×

105)

8.2e

−24(5

.0×

105)

2.44e−

27(5

.0×

105)

1.94e−

2(5

.0×

105)

0(5.0

×10

3)

Std

(1.82e

−10)

−(1.0e−

15)−

(4.0e−

23)−

(1.89e

−27)

−(8.90e

−4)−

0

Ackley

Mean

2.7e

−03(5

.0×

104)

3.5e

−04(5

.0×

104)

8.2e

−10(5

.0×

104)

1.18e−

04(5

.0×

104)

5.36e−

13(5

.0×

104)

4.19e−

14(1.0

×10

3)

Std

(5.1e−

04)−

(1.0e−

04)−

(6.9e−

10)−

(4.90e

−04)

−(4.77e

−14)

−2.85e−

14

Griew

ank

Mean

7.8e

−04(5

.0×

104)

1.9e

−05(5

.0×

104)

9.9e

−08(5

.0×

104)

1.74e−

07(5

.0×

104)

(05.0

×10

4)

0(4.0

×10

2)

Std

(1.2e−

03)−

(5.8e−

05)−

(6.0e−

07)−

(2.33e

−07)

−(0)≈

0

Quartic

Mean

4.8e

−03(3

.0×

105)

3.3e

−03(3

.0×

105)

6.4e

−04(3

.0×

105)

NA

NA

5.37e−

04(1.0

×10

3)

Std

(1.2e−

03)−

(8.5e−

04)−

(2.5e−

04)−

3.91e−

05

−9

99

85

+00

0000

0000

≈00

0000

003

Wilc

oxon’s

rank-sum

test

ata0.05

significancelevelis

performed

betweenSG

Oandeach

ofJA

DE,jD

E,SaDE,CoD

EandEPS

DE.“−

”,“+

”,and“≈

”denote

that

theperformance

ofthe

correspondingalgorithm

isworse

than,b

etterthan,and

similarto

thatof

SGO,respectively

123

Complex Intell. Syst.

Table 5 Performance comparisons of SGO, CABC, GABC, RABC and IABC

Function CABC GABC RABC IABC SGO

Sphere

Mean 2.3e−40 (1.5 × 105) 3.6e−63 (1.5 × 105) 9.1e−61 (1.5 × 105) 5.34e−178 (1.5 × 105) 0 (5.0 × 103)

Std (1.7e−40)− (5.7e−63)− (2.1e−60)− (0)− 0

Schwefel 2.22

Mean 3.5e−30 (2.0 × 105) 4.8e−45 (2.0 × 105) 3.2e−74 (2.0 × 105) 8.82e−127 (2.0 × 105) 0 (5.0 × 103)

Std (4.8e−30)− (1.4e−45)− (2.0e−73)− (3.49e−126)− 0

Schwefel 1.2

Mean 8.4e+02 (5.0 × 105) 4.3e+02 (5.0 × 105) 2.9e−24 (5.0 × 105) 1.78e−65 (5.0 × 105) 0 (5.0 × 103)

Std (9.1e+02)− (8.0e+02)− (1.5e−23)− (2.21e−65)− 0

Step

Mean 0 (1.0×104) 0 (1.0×104) 0 (1.0×104) 0 (1.0×104) 0 (1.0 × 102)

Std 0≈ 0≈ 0≈ 0≈ 0

Rastrigin

Mean 1.3e−00 (5.0 × 104) 1.5e−10 (5.0 × 104) 2.3e−02 (5.0 × 104) 0 (5.0 × 104) 0 (4.0 × 102)

Std (2.7e−00)− (2.7e−10)− (5.1e−01)− 0≈ 0

Schwefel 2.21

Mean 6.1e−03 (5.0 × 105) 3.6e−06 (5.0 × 105) 2.8e−02 (5.0 × 105) 4.98e−38 (5.0 × 105) 0 (5.0 × 103)

Std (5.7e−03)− (7.6e−07)− (1.7e−02)− (8.59e−38)− 0

Ackley

Mean 1.0e−05 (5.0 × 104) 1.8e−09 (5.0 × 104) 9.6e−07 (5.0 × 104) 3.87e−14 (5.0 × 104) 2.47e−15 (1.0 × 103)

Std (2.4e−06)− (7.7e−10)− (8.3e−07)− (8.52e−15)− 1.82e−15

Griewank

Mean 1.2e−04 (5.0 × 104) 6.0e−13 (5.0 × 104) 8.7e−08 (5.0 × 104) 0 (5.0 × 104) 0 (4.0 × 102)

Std (4.6e−04)− (7.7e−13)− (2.1e−08)− 0≈ 0

− 07 07 07 05

+ 00 00 00 00

≈ 01 01 01 04

Wilcoxon’s rank-sum test at a 0.05 significance level is performed between SGO and each of CABC, GABC, RABC and IABC. “−”, “+”, and“≈” denote that the performance of the corresponding algorithm is worse than, better than, and similar to that of SGO, respectively

tial range, formulation, characteristics and the dimensions ofthese problems are listed in Appendix. Each simulation runsfor 30 times. The simulation is terminated on attaining max-imum number of evaluations or obtaining global minimumvalueswith different random seeds. Themean value and stan-dard deviations of fitness value produced by the algorithmshave been recorded in Table 6. At the same time, mean valueand standard deviations of number of fitness evaluation pro-duced by the algorithms have also been recorded in Table7. The best optimal values are shown in bold face. To havestatistically sound conclusions, Wilcoxon’s rank-sum test ata 0.05 significance level has been conducted on the experi-mental results, and the last three rows of Table summarizethe results. It is observed that SGO performs better in 23 testfunctions and equivalent in 2 test functions.

From Tables 6 and 7, it is clear that except step and six-hump camel-back function, in all cases, SGO has shownbetter result than TLBO, and the maximum number of fit-

ness evaluations for easom, bohachevsky1, bohachevsky2,bohachevsky3, rastrigin, noncontinuous rastrigin, multimodand for weierstrass function is less than that of TLBOalgorithm and all these functions have reached to optimalsolution. In both step and six-hump camel-back functioncases, both TLBO and SGO algorithms have performedequivalently and given optimal result, however, in both cases,SGO reaches optimum value with lesser number of fitnessevaluations than TLBO. So, we can say that SGO is betterthanTLBOalgorithm in all cases in this experiment. The con-vergence characteristics of both algorithms have been shownin the graphs below (Fig. 2).

Experiment 7: SGO vs. SAABC, GABC, IABC,ABC/Best1, GPSO, DBMPSO, TCPSO and VABC

In this section, we compare SGO with both ABC and PSOvariants of algorithm such as GABC(Gbest-guided artificial

123

Complex Intell. Syst.

Table 6 Performancecomparisons of SGO and TLBO

Function no. TLBO SGO TLBO SGO

Step Bohachevsky1

Mean 0 0 Mean 6.97e−06 0

Std (0)≈ 0 Std (3.78e−06)− 0

Sphere Bohachevsky2

Mean 3.20e−03 7.82e−031 Mean 5.16e−06 0

Std (2.50e−03)− 1.20e−031 Std (3.44e−06)− 0

Sum Squares Bohachevsky3

Mean 1.20e−03 1.12e−31 Mean 3.13e−06 0

Std (1.50e−03)− 1.25e−33 Std (2.78e−06)− 0

Quartic Booth

Mean 1.46 e−02 2.98e−04 Mean 1.20e−09 5.55e−13

Std (8.90e−03)− 5.67e−05 Std (1.17e−09)− 1.93e−16

Beale Rastrigin

Mean 2.87e−08 2.11e−09 Mean 1.86e+02 0

Std (1.24e−09)− 3.88e−11 Std (3.50e+01)− 0

Easom Noncontinous rastrigin

Mean −9.94e−01 −1 Mean 1.61e+02 0

Std 3.40e−03 −0 Std (2.69e+01)− 0

Matyas Six Hump Camel Back

Mean 8.08e−12 4.91e−41 Mean −1.0316 −1.0316

Std (5.22e−12)− 6.83e−42 Std (6.79e−16)− 6.79e−16

Zakharov Griewank

Mean 7.29e−05 1.24e−33 Mean 1.14e−02 0

Std (4.98e−05)− 1.92e−35 Std (9.50e−03)− 0

Powell Ackley

Mean 6.41e−04 3.72e−32 Mean 3.14e−02 −8.88e−16

Std (3.37e−04)− 4.08e−34 Std (3.11e−02)− 1.01e−31

Schwefel 1.2 Multimod

Mean 3.62e+02 1.15e−26 Mean 9.34e−49 0

Std (1.34e+02)− 2.67e−29 Std (1.31e−48)− 0

Schwefel 2.21 Weierstass

Mean 5.31e−02 2.05e−16 Mean 5.18e−01 0

Std (5.60e−03)− 5.90e−19 Std (1.89e−01)− 0

Schwefel 2.22 Elliptic

Mean 2.25e−02 4.56e−16 Mean 6.69e+02 2.01e−26

Std (2.13e−02)− 1.91e−19 Std (4.56e+02)− 3.34e−29

Rosenbrocks

Mean 29.1979 28.6819

Std (1.1425)− 0.4319

− 11 − 12

+ 00 + 00

≈ 01 ≈ 01

Wilcoxon’s rank-sum test at a 0.05 significance level is performed between SGO and TLBO. “−”, “+”, and“≈” denote that the performance of the TLBO algorithm is worse than, better than, and similar to that ofSGO, respectively

bee colony algorithm) [36], IABC [38], ABC/Best1 [39],SAABC(simulated annealing-based artificial bee colony)[40], VABC(velocity-based artificial bee colony algorithm)

[41], CPSO(Chaotic particle swarm optimization) [42],DBMPSO(particle swarm optimization with double-bottomchaotic maps) [43], TCPSO(two-swarm cooperative particle

123

Complex Intell. Syst.

Table 7 Fitness comparisons of SGO and TLBO

Function no. TLBO SGO TLBO SGO

Step Bohachevsky1

Mean 880 160 Mean 1000 520

Std 80 40 Std 1000 20.2715

Sphere Bohachevsky2

Mean 1000 1000 Mean 1000 560

Std 0 0 Std 0 19.1231

Sum Squares Bohachevsky3

Mean 1000 1000 Mean 1000 560

Std 0 0 Std 0 18.2715

Quartic Booth

Mean 1000 1000 Mean 1000 1000

Std 0 0 Std 0 0

Beale Rastrigin

Mean 1000 1000 Mean 1000 520

Std 0 0 Std 0 0

Easom Noncontinous rastrigin

Mean 1000 640 Mean 1000 560

Std 0 0 Std 0 22.3312

Matyas Six Hump Camel Back

Mean 1000 1000 Mean 680 360

Std 0 0 Std 80 23.5634

Zakharov Griewank

Mean 1000 1000 Mean 1000 600

Std 0 0 Std 0 0

Powell Ackley

Mean 1000 1000 Mean 1000 1000

Std 0 0 Std 0 0

Schwefel 1.2 Multimod

Mean 1000 1000 Mean 1000 640

Std 0 0 Std 0 0

Schwefel 2.21 Weierstass

Mean 1000 1000 Mean 1000 840

Std 0 0 Std 0 33.1231

Schwefel 2.22 Elliptic

Mean 1000 1000 Mean 1000 1000

Std 0 0 Std 0 0

Rosenbrocks

Mean 1000 1000

Std 0 0

swarms optimization) [44] on 23 benchmark functions out ofwhich 15 are multidimensional and 8 are fixed-dimensionalbenchmark functions.All functions are described in [41]. Themaximum number of fitness evaluations is taken as 40,000,and population size is 40, and the parameters of the algo-rithms are identical to [41]. The results of SAABC, GABC,

IABC, ABC/Best1, VABC, CPSO, DBMPSO and TCPSOare gained from [41] for comparison with SGO. The com-parison results are shown in Tables 8, 9, 10 in terms of meansand standard deviations (Std) of the solutions in the 30 inde-pendent runs. Tables 8, 9 show the results for 60 and 100dimensions, respectively, on multidimensional functions,and Table 10 reports the results on the fixed-dimensionalfunctions. The best optimal values are shown in boldface.

As seen from the Tables 8, 9 results, SGO found theglobal optimal values for all the functions except F6, F7,F9, F12, F14 and F15 function for both 60 and 100 dimen-sion. On the other hand, for test functions F9, F12, F14 andF15, the objective values obtained by SGO are extremelyclose to global optima. Again, as seen from Table 10results, SGO found the global optimal values for the func-tions F16, F17 and F21 of fixed-dimensional functions. Onthe other hand, for test functions F19 and F20, the objec-tive values obtained by SGO are extremely close to globaloptima.

To have statistically sound conclusions, Wilcoxon’s rank-sum test at a 0.05 significance level has been conducted onthe experimental results, and the last three rows of Tablesummarize the results. In 60-dimensional case, accordingto Wilcoxon’s rank-sum test, SGO performs superior thanSAABC in 14 test functions and comparable to one test func-tion out of a 5 test functions, improved than GABC in all15 test functions. It is also found to be better than IABC,ABC/Best1,CPSO,DBMPSOandTCPSO in all 15 test func-tions. In our work, we observe that compared to VABC, ourproposed SGO is better in 11 test functions and equivalent to2 test functions out of 15 test functions.

In 100-dimensional case, according to Wilcoxon’s rank-sum test, SGO performs superior than all algorithms exceptVABC algorithm in all 15 multidimensional test functions.Compared to VABC, our proposed SGO is better in 12 testfunctions and equivalent to 1 test functions out of 15 testfunctions.

In fixed-dimensional case, according to Wilcoxon’s rank-sum test, SGO performs superior than SAABC in four testfunctions and equivalent with three test functions out of eighttest functions, improved thanGABC in five test functions andequivalent with two test function out of eight test functions.It is also found to be better than IABC, ABC/Best1, VABC,CPSO, DBMPSO, TCPSO in seven, five, three, two, threeand seven test functions, respectively, out of eight test func-tions and similarly equivalent with one, two, four, five, fourand zero test functions, respectively, out of eight test func-tions.

So, it is interesting to note that the performance of SGO isbetter than other algorithms according to Wilcoxon’s rank-sum test.

123

Complex Intell. Syst.

Fig. 2 Convergencecharacteristics of SGO VsTLBO

0 100 200 300 400 500 600 700 800 900 1000-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05Convergence behaviour of Six Hump Camel Back Function

No. of fitness evaluation

log(

FVs)

TLBOSGO

0 100 200 300 400 500 600 700 800 900 1000-35

-30

-25

-20

-15

-10

-5

0

5Convergence behaviour of Bohachevsky2 function

No.of fitness evaluation

log(

FEs)

TLBOSGO

0 100 200 300 400 500 600 700 800 900 1000-100

-90

-80

-70

-60

-50

-40

-30

-20

-10

0Convergence behaviour of Matyas function

No.of fitness evaluation

log(

FEs)

TLBOSGO

0 100 200 300 400 500 600 700 800 900 1000-35

-30

-25

-20

-15

-10

-5

0

5

10Convergence behaviour of Griewank function

No.of fitness evaluation

log(

FEs)

TLBOSGO

0 100 200 300 400 500 600 700 800 900 1000-40

-30

-20

-10

0

10

20Convergence behaviour of Schwefel 2.22 function

No.of fitness evaluation

log(

FVs)

TLBOSGO

0 100 200 300 400 500 600 700 800 900 1000-40

-35

-30

-25

-20

-15

-10

-5

0

5Convergence behaviour of Schwefel 2.21 function

No.of fitness evaluation

log(

FVs)

TLBOSGO

0 100 200 300 400 500 600 700 800 900 1000-70

-60

-50

-40

-30

-20

-10

0

10

20

30Convergence behaviour of Schwefel 1.2 function

No.of fitness evaluation

log(

FVs)

TLBOSGO

0 100 200 300 400 500 600 700 800 900 1000-80

-70

-60

-50

-40

-30

-20

-10

0

10

20Convergence behaviour of Sphere function

No.of fitness evaluation

log(

FVs)

TLBOSGO

Experiment 8: SGO vs. GPSO, LPSO, FIPS, SPSO,CLPSO, OLPSO and SLPSOA

In this section, the comparison of SGO versus GPSO(globalPSO) [45], LPSO(local PSO) [46], FIPS(fully informedparticle swarm) [47],SPSO(standard for particle swarm

optimization) [48], CLPSO [28], OLPSO [29] and CLP-SOA(scatter learning particle swarm optimization Algo-rithm) [49], on 14 benchmark functions described in paper[49]. The maximum number of fitness evaluations is takenas 200,000, and population size is 40, and the parametersof the algorithms are identical to [49]. The comparison

123

Complex Intell. Syst.

Fig. 2 continued

0 100 200 300 400 500 600 700 800 900 1000-40

-35

-30

-25

-20

-15

-10

-5

0

5Convergence behaviour of Bohachevsky3 Function

No. of fitness evaluation

log(

FVs)

TLBOSGO

0 100 200 300 400 500 600 700 800 900 10002

4

6

8

10

12

14

16

18Convergence behaviour of Rosenbrock Function

No. of fitness evaluation

log(

FVs)

TLBOSGO

0 100 200 300 400 500 600 700 800 900 10000

0.5

1

1.5

2

2.5x 104 Convergence behaviour of Step function

No.of fitness evaluation

(FV

s)

TLBOSGO

0 100 200 300 400 500 600 700 800 900 1000-80

-70

-60

-50

-40

-30

-20

-10

0

10Convergence behaviour of SumSquares function

No.of fitness evaluation

log(

FVs)

TLBOSGO

0 100 200 300 400 500 600 700 800 900 1000-80

-70

-60

-50

-40

-30

-20

-10

0

10Convergence behaviour of Zakharov function

No.of fitness evaluation

log(

FVs)

TLBOSGO

0 100 200 300 400 500 600 700 800 900 1000-35

-30

-25

-20

-15

-10

-5

0

5Convergence behaviour of Weierstrass function

No.of fitness evaluation

log(

FVs)

TLBOSGO

0 100 200 300 400 500 600 700 800 900 1000-35

-30

-25

-20

-15

-10

-5

0

5

10Convergence behaviour of Bohachevsky1 function

No.of fitness evaluation

log(

FEs)

TLBOSGO

results are shown in Table 11 in terms of means andstandard deviations (Std) of the solutions in the 25 inde-pendent runs. The results of GPSO, LPSO, FIPS, SPSO,CLPSO, OLPSO and SLPSOA are gained from [49]. Asseen from Table 11 results, SGO found the global optimalsolution for the functions sphere, schwefel 2.22, rastrigin,griewank, rotated rastrigin and rotated griewank. On theother hand, for the test functions noise, Ackley, generalizedpenalized, generalized penalised1, and rotated Ackley, the

objective values obtained by SGO are extremely close toglobal optima. The best optimal values are shown in boldface.

To have statistically sound conclusions, Wilcoxon’s rank-sum test at a 0.05 significance level has been conducted onthe experimental results, and the last three rows of Tablesummarize the results. According to Wilcoxon’s rank-sumtest, SGOperforms superior thanGPSO,LPSO, FIPS, SPSO,CLPSO, OLPSO and SLPSOA in 13, 14, 13, 12, 12, 6, and 7

123

Complex Intell. Syst.

Fig. 2 continued

0 100 200 300 400 500 600 700 800 900 1000-350

-300

-250

-200

-150

-100

-50

0Convergence behaviour of Easom function

No.of fitness evaluation

log(

FVs)

TLBOSGO

0 100 200 300 400 500 600 700 800 900 1000-60

-50

-40

-30

-20

-10

0

10

20

30Convergence behaviour of Elliptic function

No.of fitness evaluation

log(

FVs)

TLBOSGO

0 100 200 300 400 500 600 700 800 900 1000-700

-600

-500

-400

-300

-200

-100

0

100Convergence behaviour of Multimod function

No.of fitness evaluation

log(

FVs)

TLBOSGO

0 100 200 300 400 500 600 700 800 900 1000-80

-70

-60

-50

-40

-30

-20

-10

0

10Convergence behaviour of Powell function

No.of fitness evaluation

log(

FVs)

TLBOSGO

0 100 200 300 400 500 600 700 800 900 1000-18

-16

-14

-12

-10

-8

-6

-4

-2

0

2Convergence behaviour of Beale function

No.of fitness evaluation

log(

FVs)

TLBOSGO

0 100 200 300 400 500 600 700 800 900 1000-8

-6

-4

-2

0

2

4Convergence behaviour of Quartic function

No.of fitness evaluation

log(

FVs)

TLBOSGO

test functions, respectively, out of 14 test functions. The SGOalgorithm is equivalent with OLPSO in four test functionsand with SLPSOA in two test functions. So, it is interestingto tell according to this experiment that SGO is better thanother algorithms according to Wilcoxon’s rank-sum test.

Experiment 8. SGO vs FIPS-PSO, CPSO-H,DMSPSO-LS, CLPSO, APSO, SSG-PSO,SSG-PSO-DFP, SSG-PSO-BFGS, SSG-PSO-NM,SSG-PSO-PS

To comprehensively compare the performance of SGO withthe performance of SSG-PSO (Superior solutions guidedPSO with the individual level based mutation operator)and its several variants with different local search tech-niques, 21 test benchmark functions of different types wereused, including unimodal functions, multimodal functions,miscalled functions and rotated functions. The detailed infor-

mation of test functions is displayed in [50]. Here, thepopular PSO variants are FIPS-PSO (fully informed PSO),CPSO-H (cooperative based PSO), DMSPSO-LS (dynamicmulti-swarm particle swarm optimizer with local search),CLPSO (comprehensive learning PSO),APSO (adaptive par-ticle swarmoptimization), and different variants of SSG-PSOwith different local search techniques are SSG-PSO-DFP(SSG-PSO with Davidon–Fletcher–Powell method), SSG-PSO-BFGS (SSG-PSO with Broyden–Fletcher–Goldfarb–Shanno method), SSG-PSO-NM (SSG-PSO with Nelder–Mead simplex search), SSG-PSO-PS (SSG-PSOwith PatternSearch).

The maximum number of fitness evaluations is taken as300,000, and population size is 40, and the parameters of thealgorithms are identical to [50]. The comparison results areshown in Table 12 in terms of means and standard deviations(Std) of the solutions in the 30 independent runs. The resultsof all PSO variants and different variants of SSG-PSO withdifferent local search techniques are gained from [50].

123

Complex Intell. Syst.

Table8

Meanandstandard

deviationresults

obtained

bySG

Ocomparedwith

SAABC,G

ABC,IABC,A

BC/Best1,G

PSO,D

BMPS

O,T

CPS

OandVABCon

themultid

imension

albenchm

ark

functio

nsafter30

independentrunsfordim

=60

Functio

nSA

ABC

GABC

IABC

ABC/Best1

CPS

ODBMPS

OTCPS

OVABC

SGO

F1

6.74e−

021.52e−

012.58e−

071.86e+

011.58e+

011.19e+

014.01e+

001.21e−

280

(9.67e

−03)

−(1.42e

−01)

−(1.63e

−07)

−(9.13e

+00)

−(4.38e

+00)

−(3.18e

+00)

−(9.18e

−01)

−(1.67e

−28)

−0

F2

3.59e+

045.99e+

033.83e+

011.03e+

051.05e+

051.03e+

055.02e+

048.73e−

230

(3.23e

+04)

−(9.61e

+03)

−(8.56e

+01)

−(6.99e

+04)

−(2.86e

+04)

−(2.33e

+04)

−(1.60e

+04)

−(1.95e

−22)

−0

F3

2.38e+

002.53e+

003.62e−

021.22e+

014.76e+

014.11e+

012.15e+

015.49e−

190

(1.95e

+00)

−(1.61e

+00)

−(7.69e

−02)

−(2.31e

+00)

−(3.48e

+00)

−(6.17e

+00)

−(6.42e

+00)

−(1.23e

−18)

−0

F4

6.26e+

019.22e+

019.19e+

019.16e+

012.38e+

012.59e+

011.82e+

017.02e−

060

(4.04e

+00)

−(1.22e

+00)

−(2.07e

+00)

−(3.84e

+00)

−(1.49e

+00)

−(3.29e

+00)

−(2.18e

+00)

−(9.88e

−05)

−0

F5

1.81e+

051.76e+

051.48e+

052.01e+

051.35e+

047.58e+

035.16e+

031.19e−

370

(1.99e

+04)

−(2.98e

+04)

−(4.78e

+04)

−(2.75e

+04)

−(2.87e

+03)

−(1.84e

+03)

−(1.84e

+03)

−(2.67e

−37)

−0

F6

4.05e+

051.55e+

042.07e+

034.16e+

051.07e+

064.52e+

053.45e+

051.90e−

1354.4397

(3.13e

+05)

−(1.01e

+04)

−(1.79e

+03)

−(3.13e

+05)

−(4.98e

+05)

−(2.08e

+05)

−(3.32e

+05)

−(1.70e

−13)

+1.0082

F7

1.61e+

031.72e+

031.58e+

031.70e+

032.25e+

021.36e+

021.03e+

028.25e−

280.6667

(9.87e

+01)

−(1.50e

+02)

−(1.64e

+02)

−(8.99e

+01)

−(6.09e

+01)

−(2.57e

+01)

−(4.16e

+01)

−(1.40e

−27)

+2.09e−

05

F8

02.64e+

031.20e+

036.82e+

034.80e+

034.85e+

032.10e+

030

0

(0)≈

(6.81e

+02)

−(5.33e

+02)

−(3.21e

+03)

−(6.35e

+02)

−(1.78e

+03)

−(7.01e

+02)

−(0)≈

0

F9

1.56e+

007.89e−

011.22e−

027.27e−

012.10e+

001.48e+

001.06e+

006.97e−

031.68e−

05

(3.27e

−01)

−(7.63e

−02)

−(5.16e

−03)

−(8.09e

−02)

−(3.93e

−01)

−(3.01e

−01)

−(3.58e

−01)

−(7.58e

−03)

−1.62e−

05

F10

1.11e+

046.37e+

025.44e+

001.78e+

055.41e+

063.08e+

061.79e+

069.72e−

010

(1.15e

+04)

−(1.23e

+03)

−(4.43e

+00)

−(3.78e

+05)

−(1.74e

+06)

−(1.56e

+06)

−(1.33e

+06)

−(2.40e

−09)

−0

F11

3.29e+

022.47e+

021.81e+

022.66e+

023.42e+

023.66e+

022.08e+

020

0

(5.09e

+01)

−(6.76e

+01)

−(2.88e

+01)

−(5.10e

+01)

−(2.48e

+01)

−(6.41e

+01)

−(1.82e

+01)

−(0)≈

0

F12

1.14e+

019.50e+

007.33e+

001.02e+

011.12e+

011.21e+

019.26e+

005.84e−

11−8

.89e

−16

(1.74e

+00)

−(6.26e

−01)

−(1.12e

+00)

−(2.14e

+00)

−(7.97e

−01)

−(6.94e

−01)

−(5.14e

−01)

−(8.76e

−11)

−0

F13

5.05e+

01−1

.27e

+01

−3.88e

+01

7.76e+

005.60e+

015.59e+

011.51e+

01−6

00

(8.14e

+01)

−(1.40e

+01)

−(1.27e

+01)

−(1.99e

+01)

−(9.79e

+00)

−(1.76e

+01)

−(1.02e

+01)

−(0)−

0

F14

7.20e+

065.34e+

053.52e+

056.43e+

042.15e+

025.70e+

021.02e+

033.41e+

031.65e−

06

(7.28e

+06)

−(8.71e

+05)

−(6.80e

+05)

−(4.58e

+04)

−(2.96e

+02)

−(2.30e

+02)

−(1.07e

+02)

−(2.07e

−04)

−5.65e−

06

F15

2.52e+

071.95e+

062.52e+

059.33e+

062.24e+

056.50e+

041.86e+

031.87e+

033.14e−

05

(3.00e

+07)

−(2.97e

+06)

−(3.11e

+05)

−(1.86e

+07)

−(2.14e

+05)

−(4.30e

+04)

−(3.11e

+03)

−(5.57e

−05)

−2.11e−

05

−14

1515

1515

1515

11

+00

0000

0000

0000

02

≈1

0000

0000

0000

02

Wilc

oxon

’srank

-sum

testata0.05

sign

ificancelevelisperformed

betw

eenSG

Oandeach,S

AABC,G

ABC,IABC,A

BC/Best1,C

PSO,D

BMPS

O,T

CPS

OandVABC.“

−”,“

+”,and

“≈”denote

thattheperformance

ofthecorrespondingalgorithm

isworse

than,b

etterthan,and

similarto

thatof

SGO,respectively

123

Complex Intell. Syst.

Table9

Meanandstandard

deviationresults

obtained

bySG

Ocomparing

with

SAABC,G

ABC,IABC,A

BC/Best1,G

PSO,D

BMPS

O,T

CPS

OandVABCon

themultid

imension

albenchm

ark

functio

nsafter30

independentrunsfordim

=100

Functio

nSA

ABC

GABC

IABC

ABC/Best1

CPS

ODBMPS

OTCPS

OVABC

SGO

F1

1.64e+

004.45e+

003.52e−

034.54e+

012.74e+

012.13e+

011.22e+

011.05e−

250

(6.61e

−01)

−(1.79e

+00)

−(4.74e

−03)

−(2.37e

+01)

−(3.79e

+00)

−(1.58e

+00)

−(1.45e

+00)

−(2.34e

−25)

−0

F2

9.96e+

051.44e+

051.24e+

011.46e+

065.57e+

053.58e+

002.13e+

053.67e−

310

(7.98e

+05)

−(8.29e

+04)

−(1.02e

+01)

−(5.47e

+05)

−(1.06e

+05)

−(9.14e

+04)

−(4.70e

+04)

−(8.20e

−31)

−0

F3

2.48e+

011.02e+

011.44e−

013.66e+

018.88e+

017.94e+

015.87e+

012.13e−

120

(7.37e

+00)

−(2.29e

+00)

−(7.90e

−02)

−(1.38e

+01)

−(5.51e

+00)

−(5.13e

+00)

−(5.53e

+00)

−(4.77e

−12)

−0

F4

7.99e+

019.61e+

019.48e+

019.57e+

012.70e+

012.60e+

012.41e+

011.59e−

040

(3.51e

+00)

−(6.72e

−01)

−(1.03+

01)−

(2.77e

−01)

−(2.29e

+00)

−(1.87e

+00)

−(2.97e

+00)

−(2.17e

−04)

−0

F5

4.43e+

055.03e+

053.91e+

055.43e+

054.20e+

042.29e+

041.48e+

041.36e−

330

(7.35e

+04)

−(7.35e

+04)

−(1.21e

+05)

−(7.95e

+04)

−(1.32e

+04)

−(5.51e

+03)

−(2.44e

+03)

−(3.04e

−33)

−0

F6

2.00e+

077.59e+

051.25e+

051.19e+

073.31e+

061.38e+

065.09e+

051.60e−

1194.7977

(2.17e

+07)

−(9.43e

+05)

−(1.66e

+05)

−(6.16e

+06)

−(1.06e

+06)

−(3.53e

+05)

−(1.10e

+05)

−(2.62e

−11)

+1.0316

F7

3.01e+

033.05e+

032.84e+

033.10e+

035.01e+

023.36e+

023.06e+

027.75e−

310.6667

(2.33e

+02)

−(2.53e

+02)

−(1.49e

+02)

−(2.53e

+02)

−(7.71e

+01)

−(5.15e

+01)

−(7.24e

+01)

−(1.73e

−30)

+2.49e−

05

F8

3.22e+

021.47e+

045.27e+

0319.936

1.16e+

048.88e+

035.25e+

030

0

(1.95e

+02)

−(3.51e

+03)

−(2.87e

+03)

−(4.98e

+03)

−(1.34e

+03)

−(8.22e

+02)

−(9.66e

+02)

−(0)≈

0

F9

5.30e+

002.18e+

009.51e−

022.68e+

005.91e+

003.85e+

001.86e+

001.02e−

021.49e−

05

(9.95e

−01)

−(2.21e

−01)

−(3.37e

−02)

−(1.89e

−01)

−(1.42e

+00)

−(7.73e

−01)

−(4.94e

−01)

−(9.49e

−03)

−9.94e−

06

F10

2.29e+

073.40e+

062.65e+

049.85e+

064.47e+

073.55e+

071.45e+

079.83e−

010

(1.67e

+07)

−(7.38e

+06)

−(5.93e

+04)

−(9.94e

+06)

−(1.47e

+07)

−(4.51e

+06)

−(4.32e

+06)

−(4.20e

−08)

−0

F11

6.84e+

026.08e+

024.72e+

027.50e+

026.86e+

026.41e+

025.11e+

023.34e−

140

(6.80e

+01)

−(1.07e

+02)

−(4.20e

+01)

−(7.88e

+01)

−(3.22e

+01)

−(2.53e

+01)

−(1.68e

+01)

−(7.47e

−14)

−0

F12

1.35e+

011.27e+

011.13e+

011.55e+

011.26e+

011.21e+

019.28e+

001.50e−

05−8

.89e

−16

(1.63e

+00)

−(1.64e

+00)

−(1.41e

+00)

−(8.21e

−01)

−(8.02e

−01)

−(6.24e

−01)

−(5.94e

−01)

−(3.33e

−05)

−0

F13

2.44e+

026.11e+

014.02e+

011.40e+

021.23e+

021.25e+

027.00e+

01−1

000

(7.50e

+01)

−(6.08e

+01)

−(2.67e

+01)

−(5.38e

+01)

−(2.66e

+01)

−(3.07e

+01)

−(1.91e

+01)

−(0)−

0

F14

3.28e+

072.33e+

071.18e+

072.13e+

074.90e+

041.85e+

029.51e+

025.72e+

032.06e−

05

(3.49e

+07)

−(2.16e

+07)

−(1.08e

+07)

−(1.91e

+07)

−(4.93e

+04)

−(1.19e

+03)

−(2.47e

+02)

−(4.35e

−06)

−3.62e−

05

F15

9.93e+

074.53e+

075.97e+

064.56e+

071.07e+

065.59e+

059.49e+

033.14e+

031.15e−

04

(9.03e

+07)

−(5.05e

+07)

−(5.07e

+06)

−(3.07e

+07)

−(7.55e

+05)

−(2.36e

+05)

−(5.76e

+03)

−(3.19e

−06)

−2.23e−

04

−15

1515

1515

1515

12

+00

0000

0000

0000

02

≈00

0000

0000

0000

01

Wilc

oxon’srank-sum

testata0.05

significancelevelisperformed

betweenSG

Oandeach,S

AABC,G

ABC,IABC,A

BC/Best1,C

PSO,DBMPS

O,TCPS

OandVABC.“

−”,“

+”,and

“≈”denote

thattheperformance

ofthecorrespondingalgorithm

isworse

than,b

etterthan,and

similarto

thatof

SGO,respectively

123

Complex Intell. Syst.

Table10

Meanandstandard

deviationresults

obtained

bySG

Ocomparing

with

SAABC,G

ABC,IABC,A

BC/Best1,G

PSO,D

BMPS

O,T

CPS

OandVABCon

thefix

ed-dim

ension

albenchm

ark

functio

nsafter30

independentruns

Functio

nSA

ABC

GABC

IABC

ABC/Best1

CPS

ODBMPS

OTCPS

OVABC

SGO

F16

9.20e−

063.42e−

054.77e−

027.09e−

080

00

7.17e−

040

(0.55e

−06)

−(4.79e

−05)

−(1.66e

−02)

−(1.58e

−07)

−(0)≈

(1)≈

(2)≈

(8.76e

−04)

−0

F17

1.16e−

053.38e−

056.00e−

025.95e−

070

00

1.68e−

030

(1.07e

−05)

−(2.83e

−05)

−(4.75e

−02)

−(1.08e

−06)

−(0)≈

(1)≈

(2)≈

(1.35e

−03)

−0

F18

2.68e−

012.68e−

012.69e−

012.68e−

012.68e−

012.68e−

012.68e−

012.95e−

012.68e−

01

(6.73e

−05)

≈(8.37e

−14)

≈(1.14e

−03)

≈(1.24e

−16)

≈(0)≈

(0)≈

(0)≈

(2.85e

−02)

−2.26e−

16

F19

1.80e−

028.83e−

021.63e+

018.77e−

02−1

.14e

−13

−6.82e

−14

−1.14e

−130

1.96e+

01−1

.16e

−14

(3.51e

−02)

−(5.78e

−02)

−(1.95e

+00)

−(1.96e

−01)

−(0)−

(4.76e

−14)

≈(0)−

(3.34e

−01)

−6.06e−

14

F20

1.06e+

023.52e+

011.31e+

021.58e+

013.18e+

011.13e+

01−1

.82e

−13

1.20e+

023.16e−

08

(7.08e

+01)

−(2.21e

+01)

−(2.11e

+01)

−(8.86e

+00)

−(3.30e

+01)

−(1.92e

+01)

−(1.19e

−12)

+(1.50e

−02)

−1.73e−

07

F21

0−4

.78e

−01

−4.83e

−01

−4.95e

−01

−4.90e

−01

−4.98e

−01

−5.00e

−01

−5.00e

−01

0

(0)≈

(1.73e

−02)

−(1.90e

−02)

−(4.70e

−03)

−(6.21e

−17)

−(4.35e

−03)

−(0)−

(0)−

0

F22

3.00e+

003.00e+

003.23e+

003.00e+

003.00e+

003.00e+

003.00e+

003.26e+

013.00

(8.18e

−05)

≈(3.13e

−12)

≈(4.50e

−01)

−(5.50e

−15)

≈(1.06e

−15)

≈(4.97e

−16)

≈(1.02e

−15)

≈(2.93e

−02)

−2.82e−

15

F23

−1.97e

−01

−1.78e

−07

1.61e+

002.97e−

054.43e−

011.70e−

011.86e+

002.11e−

011.23

(8.02e

−02)

+(1.43e

−09)

+(4.21e

−01)

−(4.61e

−05)

+(3.31e

−01)

+(7.42e

−02)

+(2.83e

−01)

−(2.94e

−01)

+6.65e−

04

−04

0507

0503

0203

07

+01

0100

0101

0101

01

≈03

0201

0204

0504

00

Wilc

oxon’srank-sum

testata0.05

significancelevelisperformed

betweenSG

Oandeach,S

AABC,G

ABC,IABC,A

BC/Best1,C

PSO,DBMPS

O,TCPS

OandVABC.“

−”,“

+”,and

“≈”denote

thattheperformance

ofthecorrespondingalgorithm

isworse

than,b

etterthan,and

similarto

thatof

SGO,respectively

123

Complex Intell. Syst.

Table11

Experim

entalresultsof

GPS

O,L

PSO,F

IPS,

SPSO

,CLPS

O,O

LPS

O,S

LPS

OAandSG

Oover

25independentrunson

14testfunctio

nsof

30variableswith

200,000FE

s

Functio

nsGPS

OLPS

OFIPS

SPSO

CLPS

OOLPS

OSL

PSOA

SGO

Sphere

2.05e−

323.34e−

142.42e−

132.29e−

961.58e−

121.11e−

387.30e−

380.00e+

00

(3.56e

−32)

−(5.39e

−14)

−(1.73e

−13)

−(9.48e

−96)

−(7.70e

−13)

−(1.28e

−38)

−(8.14e

−38)

−0.00e+

00

Schw

efel2.22

1.49e−

211.70e−

102.76e−

08−

1.74e−

53−

2.51e−

08−

7.67e−

22−

0.00e+

000.00e+

00

(3.60e

−21)

−(1.39e

−10)

−(9.04e

−09)

−(1.58e

−53)

−(5.84e

−09)

−(56.3e

−22)

−(0.00e

+00)

≈0.00e+

00

Rosenbrock

40.70

28.08

25.12

13.50

11.36

1.26

0.46

18.31

(32.19)−

(21.79)−

(0.51)

−(14.63)+

(9.85)

+(1.40)

+(0.52)

+0.92

Noise

9.32e−

032.28e−

024.24e−

034.02e−

035.85e−

031.64e−

023.14e−

031.02e−

06

(2.39e

−03)

−(5.60e

−03)

−(1.28e

−03)

−(1.66e

−03)

−(1.11−

03)−

(3.25e

−03)

−(1.02e

−03)

−1.62e−

05

Schw

efel

2.48e+

033.16e+

039.93e+

023.14e+

033.82e−

043.82e−

0415.4

3.15e+

03

(2.97e

+02)

+(4.06e

+02)

−(5.09e

+02)

+(7.81e

+02)

+(1.28e

−04)

+(0.00e

+00)

+(3.58)

+779.1223

Rastrigin

26.03

35.07

65.10

41.03

9.09e−

050.00e+

002.09e−

050.00e+

00

(7.27)

−(6.89)

−(13.39)−

(11.09)−

(1.25e

−04)

−(0.00e

+00)

≈(3.20e

−05)

−0.00e+

00

Ackley

1.31e−

148.20e−

082.33e−

073.73e−

023.66e−

074.14e−

152.24e−

16−8

.88e

−16

(2.08e

−15)

−(6.73e

−08)

−(7.19e

−08)

−(0.19)

−(7.57e

−08)

−(0.00e

+00)

−(5.02e

−17)

−0.00e+

00

Griew

ank

2.12e−

021.53e−

039.01e−

127.48e−

039.02e−

100.00e+

000.00e+

000.00e+

00

(2.18e

−02)

−(4.32e

−03)

−(1.84e

−11)

−(1.25e

−02)

−(8.57e

−09)

−(0.00e

+00)

≈(0.00e

+00)

≈0.00e+

00

Generalized

penaliz

ed2.23

e−31

8.10e−

161.96e−

157.47e−

026.45e−

141.57e−

323.32e−

351.78e−

32

(7.07e

−31)

−(1.07e

−15)

−(1.11e

−15)

−(3.11)

−(3.70e

−14)

−(2.79e

−48)

≈(7.76e

−35)

+3.01e−

33

Generalized

penaliz

ed1

1.32

e−03

3.26e−

132.70e−

141.76e−

031.25e−

121.35e−

324.39e−

348.62e−

22

(3.64e

−03)

−(3.70e

−13)

−(1.57e

−14)

−(4.11e

−03)

−(9.45e

−13)

−(5.59e

−48)

+(2.20e

−35)

+2.43e−

21

Rotated

schw

efel

4.61e+

034.50e+

034.41e+

034.57e+

034.39e+

033.13e+

032.72e+

034.00e+

03

(6.21e

+02)

−(3.97e

+02)

−(9.94e

+02)

−(6.28e

+02)

−(3.51e

+03)

−(1.24e

+03)

+(5.95e

+02)

+3.24e+

02

Rotated

rastrigin

60.02

53.36

1.50e+

0243.42

87.14

53.35

40.59

0.00e+

00

(15.98)−

(13.99)−

(14.48)−

(17.38)−

(10.76)−

(13.35)−

(12.46)−

0.00e+

00

Rotated

Ackley

1.93

1.55

3.16e−

079.24e−

025.91e−

054.28e−

152.07e−

14−8

.88e

−16

(0.96)

−(0.45)

−(1.00e

−07)

−(0.32)

−(6.46e

−05)

−(7.11e

−16)

≈(6.68e

−15)

−0.00e+

00

Rotated

griewank

1.80e−

021.68e−

031.28e−

083.05e−

037.96e−

054.19e−

086.50e−

090.00e+

00

(2.41e

−02)

−(3.47e

−03)

−(4.29e

−08)

−(5.70e

−03)

−(7.66e

−05)

−(2.06e

−07)

−(2.31e

−08)

−0.00e+

00

−13

1413

1212

0607

+01

0001

0202

0405

≈00

0000

0000

0402

Wilc

oxon’srank-sum

testata0.05

significancelevelisperformed

betweenSG

Oandeach

GPS

O,L

PSO,FIPS,SP

SO,C

LPS

O,O

LPS

OandSL

PSOA.“

−”,“

+”,and

“≈”denotethattheperformance

ofthecorrespondingalgorithm

isworse

than,b

etterthan,and

similarto

thatof

SGO,respectively

123

Complex Intell. Syst.

Table 12 Experimental results of FIPS-PSO, CPSO-H, DMSPSO-LS, CLPSO APSO, SSG-PSO, SSG-PSO-DFP, SSG-PSO-BFGS, SSG-PSO-NM, SSG-PSO-PS, and SGO over 30 independent runs on 21 test functions of 30 variables with 300,000 FEs

Algorithm Sphere Rosenbrock Schwefel 12 Schwefel 2.21 Schwefel 2.22 Rastrigin Noncontinous rastrigin −/+ /≈MeanStd

MeanStd

MeanStd

MeanStd

MeanStd

MeanStd

MeanStd

FIPS-PSO 0.00e+00 2.52e+01 2.08e+02 6.25e−02 1.64e−09 6.39e+01 5.44e+01 06/00/01

(0.00e+00)≈ (9.08−01)− (8.98e+01)− (4.34e−03)− (5.59e−10)− (1.12e+01)− (2.63e+01)−CPSO-H 2.49e−15 2.77e+01 2.79e+03 5.14e−03 3.92e−08 3.32e−02 2.56e−09 07/00/00

(1.55e−15)− (2.86e+01)− (5.98e+03)− (4.49e−03)− (4.25e−09)− (1.82e−01)− (7.43e−09)−CLPSO 0.00e+00 2.34e+01 1.34e+02 6.61e−03 7.52e−20 0.00e+00 0.00e+00 04/00/03

(0.00e+00)≈ (3.79e+00)− (2.44e+02)− (4.21e−03)− (5.69e−19)− (0.00e+00)≈ (0.00e+00)≈APSO 0.00e+00 2.38e+01 9.91e−03 2.24e−01 4.25e−23 4.52e+00 3.21e+00 06/00/01

(0.00e+00)≈ (7.05e+01)− (5.03e−02)− (6.85e−01)− (7.44e−21)− (1.35e+00)− (6.32e+00)−DMSPSO-LS 0.00e+00 8.99e−11 9.59e−10 4.67e−05 1.09e−18 1.32e+01 3.41e+01 05/01/01

(0.00e+00)≈ (3.54e−11)+ (5.33e−10)− (1.75e−05)− (1.04e−18)− (1.93e+00)− (5.02e+00)−SSG-PSO 0.00e+00 2.12e+01 1.24e+01 2.77e−02 2.75e−25 0.00e+00 0.00e+00 04/00/03

(0.00e+00)≈ (1.84e+00)− (1.35e+01)− (1.47e−02)− (1.96e−25)− (0.00e+00)≈ (0.00e+00)≈SSG-PSO-DFP 0.00e+00 1.84e−01 1.30e−03 1.60e−04 9.65e−25 0.00e+00 0.00e+00 03/01/03

(0.00e+00)≈ (4.35e−01)+ (2.12e−03)− (2.92e−06)− (6.08e−25)− (0.00e+00)≈ (0.00e+00)≈SSG-PSO-BFGS 0.00e+00 5.73e−11 2.57e−14 3.77e−06 1.04e−26 0.00e+00 0.00e+00 03/01/03

(0.00e+00)≈ (8.08e−12)+ (4.58e−14)− (1.63e−06)− (7.54e−26)− (0.00e+00)≈ (0.00e+00)≈SSG-PSO-NM 0.00e+00 8.34e+00 2.46e−02 2.12e−02 1.93e−24 0.00e+00 0.00e+00 03/01/03

(0.00e+00)≈ (2.83e+00)+ (4.14e−02)− (1.11e−02)− (1.21e−24)− (0.00e+00)≈ (0.00e+00)≈SSG-PSO-PS 0.00e+00 6.90e+00 4.16e+01 1.23e−15 9.33e−22 0.00e+00 0.00e+00 03/01/03

(0.00e+00)≈ (1.25e+00)+ (2.31e+00)− (3.25e−16)− (1.23e−22)− (0.00e+00)≈ (0.00e+00)≈SGO 0.00e+00 1.52e+01 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00

0.00e+00 1.31e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00

Ackley Griewank ScaledRosenbrock 100

Scaledrastrigin 10

Scaledrastrigin 1000

Rotatedsphere

RotatedRosenbrock

FIPS-PSO 1.39e−08 2.72e−07 7.37e+04 7.37e+04 4.59e+01 7.54e−13 2.89e+01 07/00/00

(2.98e−09)− (1.18e−06)− (3.14e+05)− (9.23e+00)− (2.38e+01)− (3.26e−13)− (4.15e+00)−CPSO-H 2.44e−05 1.20e−01 3.71+06 1.23e+07 5.65e−05 8.10e−08 1.62e+02 07/00/00

(1.35e−05)− (2.18e−01)− (4.38e+06)− (1.86e−07)− (2.44e−04)− (1.02e−07)− (3.78e+02)−CLPSO 7.77e−14 0.00e+00 4.94e+01 0.00e+00 0.00e+00 4.21e−17 2.64e+01 03/01/03

(1.49e−18)− (0.00e+00)≈ (4.32e+01)− (0.00e+00)≈ (0.00e+00)≈ (6.18e−18)− (1.21e+00)+APSO 6.34e−02 1.42e−02 4.06e+06 1.98e+00 1.49e+01 3.24e−20 7.83e+01 07/00/00

(1.43e+00)− (7.24e−02)− (5.42e+06)− (2.44e+01)− (5.24e+01)− (5.45e−19)− (8.24e+01)−DMSPSO-LS 7.81e−15 0.00e+00 2.56e+01 2.17e+01 3.06e+01 2.59e−30 3.98e−03 05/01/01

(2.80e−15)− (0.00e+00)≈ (1.02e+01)− (6.70e+00)− (6.41e+00)− (1.87e−30)− (2.35e−03)+SSG-PSO 7.25e−15 0.00e+00 3.59e+01 0.00e+00 0.00e+00 5.28e−22 2.53e+01 03/01/03

(1.74e−16)− (0.00e+00)≈ (3.22e+01)− (0.00e+00)≈ (0.00e+00)≈ (8.71e−22)− (6.91e−01)+SSG-PSO-DFP 5.68e−15 0.00e+00 3.19e+01 0.00e+00 0.00e+00 1.91e−23 1.06e−05 03/01/03

(1.78e−15)− (0.00e+00)≈ (3.18e+01)− (0.00e+00)≈ (0.00e+00)≈ (3.04e−23)− (7.88e−04)+SSG-PSO-BFGS 4.97e−15 0.00e+00 2.21e+01 0.00e+00 0.00e+00 2.29e−27 3.98e−10 03/01/03

(1.73e−15)− (0.00e+00)≈ (3.07e+01)− (0.00e+00)≈ (0.00e+00)≈ (3.74e−27)− (1.22e−10)+SSG-PSO-NM 6.85e−15 0.00e+00 3.43e+01 0.00e+00 0.00e+00 5.32e−24 2.40e+01 03/01/03

(3.23e−15)− (0.00e+00)≈ (3.03e+01)− (0.00e+00)≈ (0.00e+00)≈ (4.11e−24)− (1.36e+01)+SSG-PSO-PS 1.25e−14 0.00e+00 1.81e+00 0.00e+00 0.00e+00 1.97e−24 2.50e+01 02/02/03

(6.32e−15)− (0.00e+00)≈ (1.05e+00)+ (0.00e+00)≈ (0.00e+00)≈ (8.65e−25)− (1.41e+01)+SGO −8.88e−16 0.00e+00 1.68e+01 0.00e+00 0.00e+00 0.00e+00 2.82e+01

0.00e+00 0.00e+00 2.45e−01 0.00e+00 0.00e+00 0.00e+00 1.26e−02

123

Complex Intell. Syst.

Table 12 continued

Algorithm Sphere Rosenbrock Schwefel 12 Schwefel 2.21 Schwefel 2.22 Rastrigin Noncontinous rastrigin −/+ /≈MeanStd

MeanStd

MeanStd

MeanStd

MeanStd

MeanStd

MeanStd

Rotatedschwefel 2.21

Rotatedrastrigin

RotatedAckley

Rotatedgriewank

Rotatedellipse

Rotatedtablet

Rotateddiff pow

FIPS-PSO 1.36e−04 1.75e+02 2.24e−08 1.14e−03 1.51e+03 8.45e+02 4.54e+10 07/00/00

(4.89e−05)− (8.79e+00)− (5.60e−09)− (3.00e−03)− (7.14e+02)− (2.14e+02)− (9.01e+10)−CPSO-H 5.43e+01 3.77e+02 1.76e+01 1.66e+00 7.63e+03 1.65e+05 1.6e+10 07/00/00

(7.52e+00)− (1.10e+02)− (3.96e+00)− (2.10e−01)− (6.69e+03)− (4.61e+04)− (4.71e+10)−CLPSO 1.51e−02 1.08e+02 2.76e−03 2.66e−03 4.88e+03 3.35e+02 3.74e+07 07/00/00

(4.64e−03)− (1.36e+01)− (3.25e−03)− (2.12e−03)− (1.38e+03)− (1.34e+02)− (3.45e+07)−APSO 8.05e−01 1.02e+02 3.62e−10 1.72e−02 1.25e+03 7.41e+02 2.91e+07 07/00/00

(1.24e−01)− (1.24e+03)− (9.94e−10)− (2.41e−01)− (2.12e+04)− (8.44e+02)− (4.22e+08)−DMSPSO-LS 1.18e−04 3.10e+01 2.48e−14 7.39e−04 1.27e−09 1.37e−07 1.28e−08 07/00/00

(1.09e−04)− (4.54e+00)− (5.84e−15)− (2.33e−03)− (1.13e−09)− (1.47e−07)− (6.28e−09)−SSG-PSO 7.25e−04 4.65e+01 5.86e−14 1.02e−05 7.72e+02 3.02e+02 5.42e+06 07/00/00

(3.02e−04)− (1.12e+01)− (1.55e−13)− (4.86e−05)− (8.86e+02)− (1.10e+02)− (7.17e+06)−SSG-PSO-DFP 5.38e−06 5.08e+01 4.79e−15 1.11e−16 2.54e−07 1.30e−11 1.45e−06 07/00/00

(8.59e−06)− (1.16e+01)− (1.73e−15)− (3.02e−16)− (1.07e−06)− (4.44e−11)− (9.86e−07)−SSG-PSO-BFGS 8.15e−06 4.10e+01 4.59e−15 1.47e−16 2.44e−16 2.23e−12 8.87e−10 07/00/00

(7.15e−06)− (1.43e+01)− (2.64e−15)− (3.03e−16)− (4.38e−16)− (2.40e−12)− (5.79e−10)−SSG-PSO-NM 4.94e−04 5.38e+00 7.12e−14 1.10e−08 6.17e−01 4.91e−11 7.73e+03 07/00/00

(1.96e−04)− (1.05e+01)− (1.67e−13)− (2.70e−08)− (8.93e−01)− (2.10e−10)− (1.44e+04)−SSG-PSO-PS 4.40e−04 4.44e+01 6.16e−14 1.52e−06 5.13e+01 2.14e−02 3.26e+04 07/00/00

(2.31e−04)− (6,43e+01)− (1.95e−13)− (2.84e−06)− (3.24e+01)− (6.54e−02)− (2.16e+04)−SGO 0.00e+00 0.00e+00 −8.88e−16 0.00e+00 0.00e+00 0.00e+00 0.00e+00

0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00

Wilcoxon’s rank-sum test at a 0.05 significance level is performed between SGO and each FIPS-PSO, CPSO-H, CLPSO,APSO, DMSPSO-LS, SSG-PSO, SSG-PSO-DFP, SSG-PSO-BFGS,SSG-PSO-NM, and SSG-PSO-PS. “−”, “+”, and “≈” denote that the performance of the correspondingalgorithm is worse than, better than, and similar to that of SGO, respectively

As seen from Table 12 results, SGO found the globaloptimal solution for all the test functions except Rosen-brock, Ackley, scaled Rosenbrock 100, rotated Rosenbrockand rotated Ackley. On the other hand, for the test functionsAckley and rotated Ackley, the objective values obtained bySGO are extremely close to global optima. The best optimalvalues are shown in bold face. To have statistically soundconclusions, Wilcoxon’s rank-sum test at a 0.05 significancelevel has been conducted on the experimental results, andthe last three rows of Table summarize the results. Accord-ing toWilcoxon’s rank-sum test, SGOperforms superior thanFIPS-PSO, CPSO-H, DMSPSO-LS, CLPSO, APSO, SSG-PSO, SSG-PSO-DFP, SSG-PSO-BFGS, SSG-PSO-NM, andSSG-PSO-PS in 20, 21, 17, 14, 20, 14, 13, 13, 13 and 12 testfunctions, respectively, and equivalent with 1, 0, 2, 6, 1, 6, 6,6, 6 and 6 test functions, respectively, out of 21 test functions.So, it is interesting to tell according to this experiment thatSGO is better than other algorithms according toWilcoxon’srank-sum test.

Experiment 9: SGO vs. jDE, SaDE, EPSDE, CoDE,MPEDE, CLPSO, CMA-ES,GL-25 and TLBO

To study the performance of proposed SGO, 25 test func-tions proposed in the 2005 special session on real parameteroptimization were used. A detailed description of these testfunctions can be found in [21]. The number of decision vari-ables or dimension of the function was set to 30 for alltest functions. For each algorithm and each test function,25 independent runs were conducted with 300,000 functionevaluations (FEs) as the termination.

SGO was compared with six DE variants, i.e., JADE[32], jDE [31], SaDE [30], EPSDE [34], CoDE [33] andMPEDE[51] and four other approaches, i.e., CLPSO [28],CMA-ES [52], GL-25 [53] and TLBO [24]. In our exper-iments, the parameter settings of these methods were thesame as their original papers. The number of FFs in all thesemethods was 300,000, and each method was run 25 timeson each test function. For the proposed SGO algorithm, we

123

Complex Intell. Syst.

Table 13 Experimental results of JADE, jDE, SaDE, EPSDE, CoDE, MPEDE and SGO over 25 independent runs on 25 test functions of 30variables with 300,000 FEs

Function JADE jDE SaDE EPSDE CoDE MPEDE SGOMean errorStd

Mean errorStd

Mean errorStd

Mean errorStd

Mean errorStd

Mean errorStd

Mean errorStd

F1 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00

(0.00e+00)≈ (0.00e+00)≈ (0.00e+00)≈ (0.00e+00)≈ (0.00e+00)≈ (0.00e+00)≈ (0.00e+00)

F2 1.07e−28 1.11e−06 8.26e−06 4.23e−26 1.69e−15 1.01e−26 6.79e−09

(1.00e−28)+ (1.10e−06)− (1.65e−05)− (4.07e−26)+ (3.95e−15)+ (2.05e−26)+ (4.79e−09)

F3 8.42e+03 1.98e+05 4.27e+05 8.74e+05 1.05e+05 1.01e+01 1.65e+05

(7.26e+03)+ (1.10e+05)− (2.08e+05)− (3.28e+06)− (6.25e+04)+ (8.32e+00)+ (9.07e+04)

F4 1.73e−16 4.40e−02 1.77e+02 3.49e+02 5.81e−03 6.61e−16 2.06e+01

(5.43e−16)+ (1.26e−01)+ (2.67e+02)− (2.23e+03)− (1.38e−02)+ (5.68e−16)+ (1.26e+01)

F5 8.59e−08 5.11e+02 3.25e+03 1.40e+03 3.31e+02 7.21e−06 2.08e+03

(5.23e−07)+ (4.40e+02)+ (5.90e+02)− (7.12e+02)+ (3.44e+02)+ 5.12e−06+ (4.81e+02)

F6 1.02e+01 2.35e+01 5.31e+01 6.38e−01 1.60e −01 9.65e+00 2.63e+00

(2.96e+01)− (2.50e+01)− (3.25e+01)− (1.49e+00)+ (7.85e−01)+ (4.65e+00)− (1.74e+00)

F7 8.07e−03 1.18e−02 1.57e−02 1.77e−02 7.46e−03 2.36e−03 7.00e−03

(7.42e−03)− (7.78e−03)− (1.38e−02)− (1.34e−02)− (8.55e−03)− (1.15e−03)+ (5.60e−03)

F8 2.09e+01 2.09e+01 2.09e+01 2.09e+01 2.01e+01 2.09e+01 2.08e+01

(1.68e−01)− (4.86e−02)− (4.95e−02)− (5.81e−02)− (1.41e−01)+ (5.87e−01)− (5.83e−03)

F9 0.00e+00 0.00e+00 2.39e−01 3.98e−02 0.00e+00 0.00e+00 9.27e+01

(0.00e+00)+ (0.00e+00)+ (4.33e−01)+ (1.99e−01)+ (0.00e+00)+ (0.00e+00)+ (2.15e+01)

F10 2.41e+01 5.54e+01 4.72e+01 5.36e+01 4.15e+01 1.52e+01 9.60e+01

(4.61e+00)+ (8.46e+00)+ (1.01e+01)+ (3.03e+01)+ (1.16e+01)+ (2.98e+00)+ (1.18e+01)

F11 2.53e+01 2.79e+01 1.65e+01 3.56e+01 2.71e+01 2.58e+01 1.72e+01

(1.65e+00)− (1.61e+00)− (2.42e+00)− (3.88e+00)− (1.57e+00)− (3.11e+00)− (1.54e+00)

F12 6.15e+03 8.63e+03 3.02e+03 3.58e+04 3.05e+03 1.17e+03 4.50e+02

(4.79e+03)− (8.31e+03)− (2.33e+03)− (7.05e+03)− (3.80e+03)− (8.66e+02)− (2.12e+01)

F13 1.49e+00 1.66e+00 3.94e+00 1.94e+00 1.57e+00 2.92e+00 3.26e+00

(1.09e−01)+ (1.35e−01)+ (2.81e−01)− (1.46e−01)+ (3.27e−01)+ (6.33e−01)+ (4.57e−01)

F14 1.23e+01 1.30e+01 1.26e+01 1.35e+01 1.23e+01 1.23e+01 1.17e+01

(3.11e−01)− (2.00e−01)− (2.83e−01)− (2.09e−01)− (4.81e−01)− (4.22e−01)− (3.29e−01)

F15 3.51e+02 3.77e+02 3.76e+02 2.12e+02 3.88e+02 3.78e+02 2.75e+02

(1.28e+02)− (8.02e+01)− (7.83e+01)− (1.98e+01)+ (6.85e+01)− (6.32e+01)− (6.56e+01)

F16 1.01e+02 7.94e+01 8.57e+01 1.22e+02 7.37e+01 3.77e+01 1.12e+02

(1.24e+02)+ (2.96e+01)+ (6.94e+01)+ (9.19e+01)− (5.13e+01)+ (5.22e+00)+ (6.22e+01)

F17 1.47e+02 1.37e+02 7.83e+01 1.69e+02 6.67e+01 4.36e+01 1.62e+02

(1.33e+02)− (3.80e+01)+ (3.76e+01)+ (1.02e+02)− (2.12e+01)+ (6.35e+00)+ (5.91e+00)

F18 9.04e+02 9.04e+02 8.68e+02 8.20e+02 9.04e+02 9.04e+02 9.00e+02

(1.03e+00)− (1.08e+01)− (6.23e+01)+ (3.35e+00)+ (1.04e+00)− (1.21e+00)− (0.00e+00)

F19 9.04e+02 9.04e+02 8.74e+02 8.21e+02 9.04e+02 9.04e+02 9.00e+02

8.40e+01 1.11e+00 6.22e+01 (3.35e+00)+ (9.42e−01)− (1.24e+00)− (0.00e+00)

F20 9.04e+02 9.04e+02 8.78e+02 8.22e+02 9.04e+02 9.04e+02 9.00e+02

(8.47e−01)− (1.10e+00)− (6.03e+01)+ (4.17e+00)+ (9.01e−01)− (1.18e+00)− (0.00e+00)

F21 5.00e+02 5.00e+02 5.52e+02 8.33e+02 5.00e+02 5.00e+02 4.79e+02

(4.67e−13)− (4.80e−13)− (1.82e+02)− (1.00+02)− (4.88e−13)− (3.54e−14)− (2.03e+01)

F22 8.66e+02 8.75e+02 9.36e+02 5.07e+02 8.63e+02 8.72e+02 4.68e+02

(1.91e+01)− (1.91e+01)− (1.83e+01)− (7.26e+00)− (2.43e+01)− (2.98e+01)− (2.12e+01)

123

Complex Intell. Syst.

Table 13 continued

Function JADE jDE SaDE EPSDE CoDE MPEDE SGOMean errorStd

Mean errorStd

Mean errorStd

Mean errorStd

Mean errorStd

Mean errorStd

Mean errorStd

F23 5.50e+02 5.34e+02 5.34e+02 8.58e+02 5.34e+02 5.34e+02 5.00e+02

(8.05e+01)− (2.77e−04)− (3.57e−03)− (6.82e+01)− (4.12e−04)− (3.87e−04)− (6.98e+01)

F24 2.00e+02 2.00e+02 2.00e+02 2.13e+02 2.00e+02 2.00e+02 2.00e+02

(2.85e−14)≈ (2.85e−14)≈ (6.20e−13)≈ (1.52e+00)− (2.85e−14)≈ (2.21e−14)≈ (0.00e+00)

F25 2.11e+02 2.11e+02 2.14e+02 2.13e+02 2.11e+02 2.09e+02 2.00e+02

(7.92e−01)− (7.32e−01)− (2.00e+00)− (2.55e+00)− (9.02e−01)− (3.32e−01)− (0.00e+00)

− 15 16 16 14 12 13

+ 8 7 7 10 11 10

≈ 2 2 2 1 2 2

Wilcoxon’s rank-sum test at a 0.05 significance level is performed between SGO and each of JADE, jDE, SaDE, EPSDE, CoDE and MPEDE. “−”,“+”, and “≈” denote that the performance of the corresponding algorithm is worse than, better than, and similar to that of SGO, respectively

Table 14 Experimental results of CLPSO, CMA-ES, GL-25, TLBO and SGO over 25 independent runs on 25 test functions of 30 variables with300,000 FEs

Function CLPSO CMA-ES GL-25 TLBO SGOMean error ± Std Mean error ± Std Mean error ± Std Mean error ± Std Mean error ± Std

F1 0.00e+00 ± 0.00e+00≈ 1.58e−25 ± 3.35e−26− 5.60e−27 ± 1.76e−26− 0.00e+00 ± 0.00e+00≈ 0.00e+00 ± 0.00e+00

F2 8.40e+02 ± 1.90e+02− 1.12e−24 ± 2.93e−25+ 4.04e+01 ± 6.28e+01− 0.00e+00 ± 0.00e+00+ 6.79e−09 ± 4.79e−09

F3 1.42e+07 ± 4.19e+06− 5.54e−21 ± 1.69e−21+ 2.19e+06 ± 1.08e+06− 1.82e+05 ± 4.58e+03≈ 1.65e+05 ± 9.07e+04

F4 6.99e+03 ± 1.73e+03− 9.15e+05 ± 2.16e+06− 9.07e+02 ± 4.25e+02− 0.00e+00 ± 0.00e+00+ 2.06e+01 ± 1.26e+01

F5 3.86e+03 ± 4.35e+02− 2.77e−10 ± 5.04e−11+ 2.51e+03 ± 1.96e+02− 4.60e+03 ± 1.98e+03− 2.08e+03 ± 4.81e+02

F6 4.16e+00 ± 3.48e+00− 4.78e−01 ± 1.32e+00+ 2.15e+01 ± 1.17e+00− 3.47e+01 ± 2.42e+01− 2.63e+00 ± 1.74e+00

F7 4.51e−01+8.47e−02− 1.82e−03 ± 4.33e−03+ 2.78e−02 ± 3.62e−02− 1.63e−02 ± 1.89e−02− 7.00e−03 ± 5.60e−03

F8 2.09e+01 ± 4.41e−02− 2.03e+01 ± 5.72e−01≈ 2.09e+01 ± 5.94e−02− 2.08e+01 ± 4.90e−02≈ 2.08e+01 ± 5.83e−03

F9 0.00e+00± 0.00e+00+ 4.45e+02 ± 7.12e+01− 2.45e+01 ± 7.35e+00+ 2.30e+01 ± 1.14e+00+ 9.27e+01 ± 2.15e+01

F10 1.04e+02 ± 1.53e+01− 4.63e+01 ± 1.16e+01+ 1.42e+02 ± 6.45e+01− 1.09e+02 ± 4.02e+01− 9.60e+01 ± 1.18e+01

F11 2.60e+01 ± 1.63e+00− 7.11e+00 ± 2.14e+00+ 3.27e+01 ± 7.79e+00− 1.77e+01 ± 2.71e+00− 1.72e+01 ± 1.54e+00

F12 1.79e+04 ± 5.24e+03− 1.26e+04 ± 1.74e+04− 6.53e+04 ± 4.69e+04− 1.84e+04 ± 2.17e+04− 4.50e+02 ± 2.12e+01

F13 2.06e+00± 2.15e−01+ 3.43e+00 ± 7.60e−01− 6.23e+00 ± 4.88e+00− 3.01e+00 ± 1.02e+00+ 3.26e+00 ± 4.57e−01

F14 1.28e+01 ± 2.48e−01− 1.47e+01 ± 3.31e−01− 1.31e+01 ± 1.84e−01− 1.30e+01 ± 4.27e−01− 1.17e+01 ± 3.29e−01

F15 5.77e+01± 2.76e+01+ 5.55e+02 ± 3.32e+02− 3.04e+02 ± 1.99e+01− 2.80e+02 ± 7.48e+01− 2.75e+02 ± 6.56e+01

F16 1.74e+02 ± 2.82e+01− 2.98e+02 ± 2.08e+02− 1.32e+02 ± 7.60e+01− 2.31e+02 ± 1.17e+02− 1.12e+02 ± 6.22e+01

F17 2.46e+02 ± 4.81e+01− 4.43e+02 ± 3.34e+02− 1.61e+02 ± 6.80e+01− 2.73e+02 ± 1.21e+01− 1.62e+02 ± 5.91e+00

F18 9.13e+02 ± 1.42e+00− 9.04e+02 ± 3.01e−01− 9.07e+02 ± 1.48e+00− 9.08e+02 ± 4.90e−01− 9.00e+00 ± 0.00e+00

F19 9.14e+02 ± 1.45e+00− 9.16e+02 ± 6.03e+01− 9.06e+02 ± 1.24e+00− 9.09e+02 ± 8.00e−01− 9.00e+00 ± 0.00e+00

F20 9.14e+02 ± 3.62e+00− 9.04e+02 ± 2.71e−01− 9.07e+02 ± 1.35e+00− 9.07e+02 ± 2.24e+00− 9.00e+00 ± 0.00e+00

F21 5.00e+02 ± 3.39e−13≈ 5.00e+02 ± 2.68e−12≈ 5.00e+02 ± 4.83e−13≈ 5.01e+02 ± 1.96e+00− 4.79e+02 ± 2.03e+01

F22 9.72e+02 ± 1.20e+01− 8.26e+02 ± 1.46e+01− 9.28e+02 ± 7.04e+01− 8.89e+02 ± 1.76e+01− 4.68e+02 ± 2.12e+01

F23 5.34e+02 ± 2.19e−04− 5.36e+02 ± 5.44e+00− 5.34e+02 ± 4.66e−04− 5.39e+02 ± 8.45e+00− 5.00e+02 ± 6.98e+01

F24 2.00e+02 ± 1.49e−12≈ 2.12e+02 ± 6.00e+01− 2.00e+02 ± 5.52e−11≈ 2.01e+02 ± 4.00e−01− 2.00e+02 ± 0.00e+00

F25 2.00e+02 ± 1.96e+00− 2.07e+02 ± 6.07e+00− 2.17e+02 ± 1.36e−01− 2.00e+02 ± 0.00e+00≈ 2.00e+02 ± 0.00e+00

− 16 16 22 17

+ 6 7 1 4

≈ 3 2 2 4

Wilcoxon’s rank-sum test at a 0.05 significance level is performed between SGO and each of CLPSO, CMA-ES and GL-25. “−”, “+”, and “≈”denote that the performance of the corresponding algorithm is worse than, better than, and similar to that of SGO, respectively

123

Complex Intell. Syst.

have considered population size to 100, and FEs is samewithother methods and is 300,000. The best optimal values areshown in bold face. To have statistically sound conclusions,Wilcoxon’s rank-sum test at a 0.05 significance level has beenconducted on the experimental results, and the last three rowsof Tables 13 and 14 summarize the experimental results.

According to Wilcoxon’s rank-sum test, it is clear thatSGO performs better than JADE in 15 test functions andequivalent to 2 test functions out of 25 test functions. It isbetter than jDE in 16 test functions and equivalent to 2 testfunctions out of 25 test functions. It can be seen that com-pared to SaDE, it performs better in 16 test functions andequivalent to 2 test functions out of 25 test functions. SGOis better than EPSDE in 14 test functions and equivalent to1 test function out of 25 functions. From the table, it can beverified that SGO is better than CoDE in 12 test functions andequivalent to 2 test functions out of 25 test functions, againbetter than MPEDE in 13 test functions and equivalent to 2test functions out of 25 test functions and better than CLPSOin 16 test functions and equivalent to 3 test functions out of25 test functions. Compared to CMA-ES, our proposed tech-nique is better in 16 test functions and equivalent to 2 testfunctions out of 25 test functions. We have verified that SGOis better than GL-25 in 22 test functions and equivalent to 2test functions out of 25 functions, and better than TLBO in 17test functions and equivalent to 4 test functions out of 25 testfunctions. So, it is interesting to note that the performance ofSGO is better than other algorithms according toWilcoxon’srank-sum test.

Experiment 10: SGO vs. PSO, CPSO, CLPSO,CMA-ES, G3-PCX, DE, and TLBO using compositefunctions

In this experiment, we have considered six composite testfunctions and eight novel algorithms, particle swarm opti-mizer (PSO) [7], cooperative PSO (CPSO) [54], comprehen-sive learning PSO (CLPSO) [28], evolution strategy withcovariance matrix adaptation (CMA-ES) [55], G3 modelwith PCX crossover (G3-PCX) [56], differential evolution(DE) [2], teaching–learning-based optimization [24] andpro-posed SGO for testing their performances. The detaileddescriptions of these functions are given in papers [57] and[25], and the algorithms in their respective papers. Parametersettings for the composite functions are as in [25].

Table 15 shows the results obtained using the eight algo-rithms on six composite functions. For each test function,each algorithm is run 20 times, and themaximumfitness eval-uations are 50,000 for all algorithms. For our proposed algo-rithm, we have considered population size as 100. The meanvalues of the results are recorded in Table 15. The best opti-mal values are shown in bold face. To have statistically soundconclusions, Wilcoxon’s rank-sum test at a 0.05 significance

level has been conducted on the experimental results, and thelast three rows of Table summarize the experimental results.

According to Wilcoxon’s rank-sum test, it is clear thatSGO performs better than PSO in six test functions out ofall six test functions, better than CPSO in six test functionsout of all six test functions but better than CLPSO in threetest functions out of six test functions. SGO is better thanCMA-ES in six test functions out of all six functions, betterthan G3-PCX in five test functions out of six test functionsbut better than DE in three test functions and equivalent toone test function out of six test functions; whereas it is betterthan TLBO in three test functions out of six test functions.So, from Table 15, it is clear that out of seven algorithms, inall cases except CLPSO and TLBO, SGO is showing betterresult; however, with CLPSO and TLBO, SGO is showingequivalent result. From the last column of Table 15, it is alsoclear that SGO sometimes reaches to optimal solution.

Conclusion

This paper proposes a new efficient optimization algorithmthat is inspired by the social behavior of humans towardsolving a complex problem. Whenever a problem/task hasbeen solved by a single person, it becomes too difficult tosolve or the problem may remain unsolvable. But whenthe same problem has been solved by a group of persons,the difficulty becomes easy and the unsolvable problemmay become solvable. In a social group, people are influ-enced by the characteristics (i.e., traits) of the successfulperson, and eventually, they also change/modify their traitsaccordingly and become capable to solve/address complexproblems/situations. This concept has motivated us to pro-pose a new optimization algorithm known as social groupoptimization (SGO). The concept and the mathematical for-mulation of SGO algorithm are explained in this paper witha flowchart. To judge the effectiveness of SGO, extensiveexperiments have been conducted on number of differentunconstrained benchmark functions as well as 25 stan-dard numerical benchmark functions taken from the IEEECongress on Evolutionary Computation 2005 competition.Performance comparisons are made with state-of-the-artoptimization techniques like GA, PSO, DE, ABC and itsvariants and the recently developed TLBO. Different vari-ants of the popular evolutionary optimization techniques arealso taken into consideration for comparing them with SGO.The experimental results show that the proposed social groupoptimization outperforms all investigated optimization tech-niques in computational costs and also provides optimalsolutions for most of the considered functions. One of thebest things in this algorithm is that it is easier to understandand to implement in comparison to other algorithms and theirvariants. It remains to see howSGOworks formulti-objectiveoptimization problems in future.

123

Complex Intell. Syst.

Table15

Performance

comparision

ofalgorithmsPS

O,C

PSO,C

LPS

O,C

MA-ES,

G3-PC

X,D

E,T

LBOandSG

Ousingcompositefunctio

ns

Com

positefunctio

nsPS

OCPS

OCLPS

OCMA-ES

G3-PC

XDE

TLBO

SGO

Bestv

alue

inSG

O

CF1 Mean

1.00e+

02−

1.56e+

02−

5.73e−

08−

1.00e+

02−

6.00e+

01−

6.75e−

02−

3.11e−

01−

1.54e−

250

Std

8.17e+

021.34e+

021.92e+

011.89e+

026.99e+

011.11e−

013.04e−

011.51e−

25

CF2 Mean

1.56e+

02−

2.42e+

02−

1.92e+

01+

1.62e+

02−

9.26e+

01−

2.87e+

01+

1.70e+

01+

4.13e+

010

Std

1.32e+

021.49e+

021.47e+

011.51e+

029.91e+

018.62e+

017.22e+

004.38e+

01

CF3 Mean

1.72e+

02−

3.63e+

02−

1.33e+

02+

2.14e+

02−

3.20e+

02−

1.44e+

02≈

1.24e+

02+

1.40e+

020

Std

3.29e+

011.96e+

022.00e+

017.42e+

011.25e+

021.94e+

016.04e+

014.07e+

01

CF4 Mean

3.14e+

02−

5.22e+

02−

3.22e+

02−

6.16e+

02−

4.93e+

02−

3.25e+

02−

2.94e+

02−

2.82e+

022.23e+

02

Std

2.00e+

011.22e+

022.75e+

016.72e+

021.42e+

021.47e+

013.15e+

012.90e+

01

CF5 Mean

8.35e+

01−

2.56e+

02−

5.37e+

00−

3.59e+

02−

2.60e+

01−

1.08e+

01−

5.18e+

00−

2.39e+

000

Std

1.00e+

021.76e+

022.60e+

001.69e+

024.16e+

012.60e+

001.62e+

001.07e+

00

CF6 Mean

8.61e+

02−

8.53e+

02−

5.01e+

02+

9.00e+

02−

7.72e+

02+

4.91e+

02+

2.30e+

02+

7.44e+

024.01e+

02

Std

1.26e+

021.28e+

027.78e−

018.32e−

021.89e+

023.95e+

024.84e+

011.89e+

02

−06

0603

0605

0303

+00

0003

0001

0203

≈00

0000

0000

0100

Wilc

oxon’srank-sum

testata0.05

significancelevelisperformed

betweenSG

Oandeach

ofPS

O,C

PSO,C

LPS

O,C

MA-ES,

G3-PC

X,D

E,T

LBO.“

−”,“

+”,and

“≈”denotethattheperformance

ofthecorrespondingalgorithm

isworse

than,b

etterthan,and

similarto

thatof

SGO,respectively

123

Complex Intell. Syst.

Open Access This article is distributed under the terms of the CreativeCommons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution,and reproduction in any medium, provided you give appropriate creditto the original author(s) and the source, provide a link to the CreativeCommons license, and indicate if changes were made.

Appendix: Benchmark functions

All problems are divided into four categories such as US,MS, UN, MN, and its range, formulation, characteristics andthe dimensions of these problems are listed in the followingTable 16.

Table 16 Benchmark functions used in experiments 1

Sl. no. Function D C Range Formulation Value

1 Step 30 US [−100, 100] f (x) = ∑Di=1(xi + 0.5)2 fmin = 0

2 Sphere 30 US [−100, 100] f (x) = ∑Di=1 x

2i fmin = 0

3 Sum Squares 30 US [−10, 10] f (x) = ∑Di=1 i x

2i fmin = 0

4 Quartic 30 US [−1.28, 1.28] f (x) = ∑Di=1 i x

4i + random(0, 1) fmin = 0

5 Beale 2 UN [−4.5, 4.5] f (x) = (1.5 − x1 + x1x2)2 + (2.25 − x1+ x1x22 )

2 + (2.625 − x1 + x1x32 )2

fmin = 0

6 Easom 2 UN [−100, 100] f (x) = −cos(x1)cos(x2)exp(−(x1 − π)2

− (x2 − π)2)

fmin = −1

7 Matyas 2 UN [−10, 10] f (x) = 0.26 (x21 + x22 ) − 0.48x1x2 fmin = 0

8 Zakharov 10 UN [−5, 10] f (x) = ∑Di=1 x

2i + (

∑Di=1 0.5i xi )

2

+ (∑D

i=1 0.5i xi )4

fmin = 0

9 Powell 24 UN [−4, 5] f (x) = ∑D/4i=1 (x4i−3 + 10x4i−2)

2

+ 5 (x4i−1 − x4i )2 + (x4i−2 − x4i−1)4

+ 10 (x4i−3 − x4i )4

fmin = 0

10 Schwefel 1.2 30 UN [−100, 100] f (x) = ∑Di=1(

∑ij=1 x j )

2 fmin = 0

11 Schwefel 2.21 30 UN [−100, 100] f (x) = maxi

|xi |, 1 ≤ i ≤ D fmin = 0

12 Schwefel 2.22 30 UN [−10, 10] f (x) = ∑Di=1 |xi | + ∏D

i=1 |xi | fmin = 0

13 Bohachevsky1 2 MS [−100, 100] f (x) = x21 + 2x22 − 0.3 cos(3πx1)− 0.4 cos (4πx2) + 0.7

fmin = 0

14 Bohachevsky2 2 MS [−100, 100] f (x) = x21 +2x22 −0.3 cos(3πx1)∗cos(4πx2)+ 0.3

fmin = 0

15 Bohachevsky3 2 MS [−100, 100] f (x) = x21 +2x22 −0.3 cos((3πx1)+ (4πx2))+ 0.3

fmin = 0

16 Booth 2 MS [−10, 10] f (x) = (x1 + 2x2 − 7)2 + (2x1 + x2 − 5)2 fmin = 0

17 Rastrigin 30 MS [−5.12, 5.12] f (x) = ∑Di=1[x2i − 10 cos(2πxi ) + 10] fmin = 0

18 Noncontinuousrastrigin

30 MS [−5.12, 5.12] f (x) = ∑Di=1[y2i − 10 cos(2πyi ) + 10]

where yi =xi |xi | < 0.5round(2xi )

2 |xi | ≥ 0.5.

fmin = 0

19 Six Hump CamelBack

2 MN [−5, 5] f (x) = 4x21 −2.1x41 + 13 x

61 +x1x2−4x22 +4x42 fmin = −1.03163

20 Griewank 30 MN [−600, 600] f (x) = 14000

∑Di=1 x

2i − ∏D

i=1 cos(xi√i) + 1 fmin = 0

21 Ackley 30 MN [−32, 32] f (x) = −20 exp(−0.2√

1D

∑Di=1 x

2i )

− exp( 1n∑D

i=1 cos(2 ∗ pi ∗ xi ))+ 20 + e

fmin = 0

22 Multimod 30 [−10, 10] f (x) = ∑′Di=1 |xi | ∏D

i=1 |xi | fmin = 0

23 Weierstrass 30 [−0.5, 0.5] f (x) = ∑Di=1(

∑kmaxk=0 [acos(2πb2(xi + 0.5))])

− D∑kmax

k=0 [acos(2πb(xi + 0.5))],where a = 0.5, b = 3, kmax = 20

fmin = 0

24 Elliptic 30 [−100, 100]∑D

i=1(106) i−1

D−1 x2i fmin = 0

25 Rosenbrocks 30 UN [−30, 30]∑D−1

i=1 [100(xi+1 − x2i )2 + (xi − 1)2] fmin = 0

D dimension, C characteristic, U unimodal, M multimodal, S separable, N non-separable

123

Complex Intell. Syst.

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ORIENTAL JOURNAL OF CHEMISTRY

www.orientjchem.org

An International Open Free Access, Peer Reviewed Research Journal

ISSN: 0970-020 XCODEN: OJCHEG

2018, Vol. 34, No.(1):Pg. 559-561

This is an Open Access article licensed under a Creative Commons Attribution-NonCommercial-ShareAlike4.0 International License (https://creativecommons.org/licenses/by-nc-sa/4.0/ ), which permits unrestrictedNonCommercial use, distribution and reproduction in any medium, provided the original work is properly cited.

Vibrational Spectra of Copper Tetramesityl Porphyrin UsingVibron Model

J. VIJAYASEKHAR1*, M. V. SUBBA RAO2 and V. SREERAM3

1Department of Mathematics, GITAM University, Hyderabad, India.2Department of Mathematics, ANITS, Visakhapatnam, Andhra Pradesh, India.

3Department of Chemistry, A.G & S.G. Siddhartha Degree College, Vuyyuru, A.P, India.*Corresponding author E-mail: vijayjaliparthi@gmail.com

http://dx.doi.org/10.13005/ojc/340165

(Received: September 14, 2017; Accepted: December 30, 2017)

ABSTRACT

In this paper we calculated the fundamental level vibrational spectra of Metalloporphyrin(bio molecule) copper tetramesityl porphyrin (Cu(TMP)) using Vibron model.

Keywords: Vibrational spectra, Vibron model, Metalloporphyrins, Cu(TMP).

INTRODUCTION

Group theory is a well known tool thatsimplifies the process of obtaining a variety ofinformation about molecules and their symmetries.Molecules are classified according to theirsymmetry properties and from that one can identify,the molecular symmetry point group. The molecularsymmetry point group of metalloporphyrins isD4h, which contains the principal Cn axis, nperpendicular C2 axis, and the horizontal plane ofsymmetry.

I-IV are pyrrole rings; 1-8 are substituentpositions. X positions are (=CH-) bridges

Fig.1. The strucure of metalloporphyrins

(Brief Communication)

560 VIJAYASEKHAR et al., Orient. J. Chem., Vol. 34(1), 559-561 (2018)

In 2008, Karumuri et al., applied Vibronmodel for coupled anharmonic oscillators todescribe the stretching vibrations of medium sizemolecules and calculated vibrational spectra ofnickel octaethyl porphyrins for the stretching modein Cm−Η. the intervening years extended this model tocalculate stretching vibrational frequencies of nickeltetraphenyl porphyrin and copper octaethyl porphyrinfor different vibrational bands1-9.

Vibron model for MetalloporphyrinsThe general calculation procedure of

vibrational spectra of metalloporphyrin by VibronModel discussed here10, 11. The Hamiltonian for thepolyatomic molecules is of the form Here vary from1 to n for n stretching bonds and (Ai,Aij,λij) arealgebraic parameters, which are determined byspectroscopic data. Where is an invariant operator(uncoupled bonds) with eigenvalues −4(Nivi−vi

2)diagonal matrix elements of the invariant operatorCij(coupled bonds) and diagonal and non-diagonalmatrix elements of Majorana operator Mijobtainedfrom the following relations,

Where vi(i=1,2,3,...) are vibrationalquantum numbers.

The vibron number Ni(i=1,2,3,...) forstretching bonds of molecule will be calculated bythe following relation

Where ωe and ωeχe are spectroscopicconstants. The initial guess value for the parameter

iA is obtained by using the energy equation forthe single-oscillator fundamental mode, which isgiven as,

(5)

Initial guess for Aij may be taken as zero.The parameter λij obtained from the relation

(6)

Table. 1: Vibrational spectra of Cu(TMP)

Symmetry Vibrational Vibrational Species mode frequencies

(cm-1)

A1g (Cm-C) 1235.04348B2g (Cm-C) 1247.03267Eu (Cm-C) 1256.90364

A1g (Cb-H) 1470.00431B2g (Cb-H) 1476.05321Eu (Cb-H) 1470.87451

Table. 2: Algebraic parameters

Algebraic Cm - C Cb - Hparameters

A -2.19234 cm-1 -5.90923 cm-1

Aij -0.98232 cm-1 -2.92012 cm-1

λij(a=3) 0.02421 cm-1 0.80834 cm-1

λij(a=6) 0.20091 cm-1 0.03421 cm-1

N (Dimensionless) 140 44

RESULTS

CONCLUSION

In this paper we have calculated thevibrational frequencies of copper tetramesitylporphyrin (Cu(TMP)) for the stretching modes(Cm-C) and (Cb-H).

(2)

(3)

(4)

561VIJAYASEKHAR et al., Orient. J. Chem., Vol. 34(1), 559-561 (2018)

REFERENCES

1. Karumuri, S. R. Indian J. Phys. 2012, 86(12),1147-1153

2. Karumuri, S. R.; Sekhar, J.V.; Sreeram, V.;Rao, V. U.M.; Rao, M.V.B.; J Mol Spectrosc.2011, 269, 119-123

3. Karumuri, S. R.; Srinivas, G.; Sekhar, J.V.;Sreeram, V.; Rao, V. U.M.; Srinivas, Y.; Babu,K.S.; Kumar, V.S.S.; Hanumaiah, A.; ChinPhys B. 2013, 22(9), 090304 (1-8)

4. Karumuri, S. R.; Sarkar, N.K.; Choudhury, J.;Bhattacharjee, R.; Mol. Phys. 2008, 106(14),1733-1737

5. Karumuri, S. R.; Sarkar, N.K.; Choudhury, J.;Bhattacharjee, R.; Chin Phys Lett. 2009,26(9), 093301 (1-4)

6. Karumuri, S. R. Chin Phys Lett. 2010, 27(10),103301 (1-4)

7. Karumuri, S. R. J Mol Spectrosc. 2010, 259, 86-928. Karumuri, S R. Eur. Phys. J. 2015, 69, 2819. Karumuri, S. R.; Sravani, K.G.; Mol. Phys.

2016, 114(5), 643-64910. Iachello, F.; Levine, R. D. Oxford University

Press, Oxford. 199511. Oss, S. Adv. Chem. Phys. 1996, 93, 455- 649

Journal of King Saud University – Computer and Information Sciences xxx (2018) xxx–xxx

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Efficient and Secure Identity-based Strong Key-Insulated SignatureScheme without Pairings

https://doi.org/10.1016/j.jksuci.2018.08.0111319-1578/ 2018 The Authors. Production and hosting by Elsevier B.V. on behalf of King Saud University.This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

⇑ Corresponding author.E-mail addresses: vasucrypto@andhrauniversity.edu.in (P. Vasudeva Reddy),

rameshamarapu.maths@anits.edu.in (A.R. Babu).

Peer review under responsibility of King Saud University.

Production and hosting by Elsevier

Please cite this article in press as: Vasudeva Reddy, P., et al. Efficient and Secure Identity-based Strong Key-Insulated Signature Scheme without PJournal of King Saud University – Computer and Information Sciences (2018), https://doi.org/10.1016/j.jksuci.2018.08.011

P. Vasudeva Reddy a,⇑, A. Ramesh Babu b, N.B. Gayathri a

aDept. of Engineering Mathematics, Andhra University, Visakhapatnam, A.P, IndiabDept. of Engineering Mathematics, Anil Neerukonda Institute of Technology and Sciences, Visakhapatnam, A.P, India

a r t i c l e i n f o

Article history:Received 27 April 2018Revised 30 July 2018Accepted 24 August 2018Available online xxxx

Keywords:Identity-based signature schemeKey insulation mechanismROM security modelECDLP

a b s t r a c t

Public Key Cryptosystem (PKC) completely relies under the assumption that user’s private key is abso-lutely secure. Exposure of private key may lead to disastrous situations in the communication network.To diminish the damage of private key exposure in PKC, key-insulation mechanism was introduced. Inkey-insulated cryptosystems, a user can update his private key with the help of a physically secure devicefrom time to time. Identity-based cryptosystem alleviates the heavy certificate management problems intraditional PKC. Recently, many Identity-based key insulated signature schemes have been proposed inliterature; however, most of the Identity-based schemes are designed based on the expensive bilinearpairing operation over elliptic curves. Due to the heavy computational cost of a pairing, the pairing basedschemes are less efficient in practice. In order to improve the computational and communicational effi-ciency and to resist the problem of private key exposure in Identity-based signature schemes, we presenta pairing-free key insulated signature scheme in identity based setting. We show that this scheme isunforgeable and achieves strong key insulation property with secure key updates, under the hardnessof the Elliptic Curve Discrete Logarithm Problem (ECDLP). The performance analysis shows that ourscheme is more efficient than the existing schemes. 2018 The Authors. Production and hosting by Elsevier B.V. on behalf of King Saud University. This is anopen access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction

Digital signature is a cryptographic primitive which providesdata integrity, authentication and non-repudiation in digital com-munications. In traditional Public Key Cryptography (PKC), proposedbyDiffe andHellman (1976), each user has a public key and a privatekey. A Certificate Authority (CA) is needed to authenticate the publickey of a user. In multi user environment, the authentication, revoca-tion, storage of public keys leads to lot of key management prob-lems. To avoid such difficulties in PKC, Shamir (1984) came upwith an idea of Identity based Public Key Cryptography (ID-PKC),in which the public key is derived by the user itself using his unique

identity information and the private key is generated by a third partyknown as Private Key Generator (PKG).

The security of any cryptographic scheme relies on the privatekey(s) of the system or the users. The loss/steal/exposure of a pri-vate key leads to the failure of the cryptographic scheme. It couldbe very disastrous and any adversary can steal the private key fromuser’s device easily instead of breaking the underlying computa-tional hard problem. In order to minimize the damage of key expo-sure, Dodis et al. (2002a,b) proposed a method known as keyinsulated mechanism. In this mechanism, a user can periodicallyupdate his private key with the help of a physically secure devicecalled Helper. In Key-Insulated Signature (KIS) schemes, the life-time of temporary signing key is divided in to discrete time peri-ods. Users update their temporary signing key with the help ofhelper. The user generates the signing key for the current time per-iod with the previous signing key and his helper key. Thus a userupdates his private key periodically and the public key associatedwith the signing remains same for the entire life period. This mech-anism of updating the secret key will not allow an adversary toderive secret keys from the device for different time periods evenif a secret key is exposed at a given time period. Thus, a KIS schemeis secure if the master key is stored in helper device. However, it is

airings.

2 P. Vasudeva Reddy et al. / Journal of King Saud University – Computer and Information Sciences xxx (2018) xxx–xxx

important to consider that the helper key exposure in Identitybased KIS i.e., an adversary can forge a signature on behalf of theuser whose helper is untrustworthy. Dodis et al. (2002a,b) pre-sented a method for strong key insulation to provide a securityagainst the leakage of signing key or the leakage of master key.Hence the strong KIS is more secure than a KIS scheme.

1.1. Related work

Following the pioneering work due to Dodis et al. KIS scheme(2002a), many variants of KIS schemes were proposed in the liter-ature (Yum and Lee (2003); Gonzalez-Deleito et al. (2004); Le et al.(2004); Lee et al. (2006); Zhou et al. (2006); Weng et al. (2006);Chen et al. (2007); Weng et al. (2007); Ohtake et al. (2008);Weng et al. (2008a,b); Wan et al. (2009); Wang and Zhang(2011); Chen et al. (2012); Wu et al. (2012); Wan et al. (2013);Gopal and Vasudeva Reddy (2015); Zhu et al. (2015); VasudevaReddy and Gopal (2017); Amarapu and Reddy (2018)). Followingthe pioneer work by Dodis (2002a), Zhou et al. (2006) proposedthe first KIS scheme in ID-based cryptosystem. In the same yearWeng et al. (2006) proved that the Zhou et al. (2006) scheme isnot strong KIS and formalized security notions for identity basedKIS schemes. Also, they proposed an ID-based strong KIS schemein standard model. Ohtake et al. (2008) proposed a strong KISscheme under the assumption of discrete logarithm in ROMmodel.Wu et al. (2012) constructed a novel Identity based KIS schemeunder the Diffe-Hellman assumption which supports batch verifi-cation. Later, Gopal and Vasudeva Reddy (2015) proposed a KISscheme in identity based setting with batch verifications usingbilinear pairings over elliptic curves. Zhu et al. (2015) proposed apairing free KIS scheme in random oracle model. Though it is thefirst identity based KIS scheme in pairing free environment, thetotal computational cost is not efficient with respect to previousKIS schemes. In order to achieve computational efficiency in signa-ture generation and verification process, design of pairing free ID-based KIS scheme in elliptic curve cryptography is more desirable.

1.2. Motivation

In any PKC, to provide the high security, the length of the keysize must be sufficiently large. Larger keys in cryptographicschemes cause the less computational efficiency and require morebandwidth. Thus cryptographic schemes with smaller key size aredesirable. To meet this requirement, Neal Koblitz and Victor Millerindependently proposed the Elliptic Curve Cryptography (ECC)using elliptic curves. ECC has many advantages over PKC, espe-cially, ECC provides high security with smaller keys in size. Forinstance, ECC with 512 bit key provides same level of security asin AES (symmetric algorithm) with 256 bit key and in RSA with15,360 bit key. Though ECC provides much security with shortkeys, the computational cost of a bilinear pairing over elliptic curvegroup is a costly operation, and is significantly expensive thanelliptic curve scalar multiplication operation. Due to the expensiveoperations such as bilinear pairing and point hash functions, mostof the cryptographic schemes are having less efficiency whileimplementing them. In view of this, ECC based schemes withoutpairing operations under general hash function would be moreattractive.

1.3. Our contribution

This paper presents a Pairing Free Identity Based Key InsulatedSignature (PF-IDBKIS) scheme over elliptic curves. This schemesolves the key exposure problem in pairing free IBS schemes. Com-paring to the other mechanisms, the leakage of temporary secretkeys will not affect the security of the remaining time periods.

Please cite this article in press as: Vasudeva Reddy, P., et al. Efficient and SecurJournal of King Saud University – Computer and Information Sciences (2018),

Our scheme achieves the strong key insulation property and secureskey updates. This property assures that the adversary is unable toforge a valid signature even though the helper is malicious. Thisscheme is unforgeable in the ROM model under the hardness ofECDLP. Due to the limitation of bilinear pairings, our schemeachieves high efficiency and is more comfortable for hostile andresource constrained applications. We presented the comparativeanalysis of our scheme with existing Identity based KIS schemesand it shows that the proposed PF-IDBKIS scheme is efficient interms of computational and communicational point of view.

1.4. Organization

This paper is organized as follows. In Section 2 we presentedsome preliminaries. The frame work and security model for ourPF-IDBKIS scheme are presented in Section 3. The proposedPF-IDBKIS scheme and its security analysis are presented inSection 4. Efficiency analysis and some potential applications ofour PF-IDBKIS scheme are presented in Section 5. Conclusions ofthe paper are presented in Section 6.

2. Preliminaries

This section presents some preliminaries related to ellipticcurves and some computational problems.

2.1. Elliptic curve group

Due to the computation, communication and security strengths,ECC plays a very important role in modern cryptography. Here wedefine the elliptic curve group.

Let Eqða; bÞ be a set of elliptic curve points over the prime fieldFq; defined by the non-singular elliptic curve equation: y2 mod q ¼ðx3 þ axþ bÞ mod q with a; b 2 Fq and ð4a3 þ 27b2Þ mod q–0: Theset Gq ¼ fðx; yÞ : x; y 2 Fqg and ðx; yÞ 2 Eqða; bÞ [ fOg is additive cyc-lic group, where the point O is known as ‘‘point at infinity”. Theorder of the elliptic curve over Fq is OðEðFqÞÞ satisfies the relation1 2

ffiffiffiq

p OðEðFqÞÞ qþ 1: The scalar multiplication in Gq isdefined as kP ¼ P þ P þ ::::P ðk timesÞ: Here P 2 Gq is the generatorof order n.

Elliptic Curve Discrete Logarithm Problem: Given a tuple P;Q 2 G;it is computationally infeasible to find the value a 2 Z

q such thatQ ¼ aP with non-negligible probability within probabilistic poly-nomial time.

Notations and their meanings which we use throughout thispaper are presented in the following Table 1.

3. Framework and security model of PF-IDBKIS scheme

3.1. Framework

A PF-IDBKIS scheme consists of the following six polynomialtime algorithms.

Setup: PKG takes the security parameter k 2 Zþ; total number oftime periods N as input and runs the algorithm to generate thecommon system parameters params, ID and master secret key(Msk)s and helper secret key (Hsk)v PKG publishes params, tand keeps s,v secretly. Here v is stored in helper device.

Extract: PKG run this algorithm with inputs params, Msk, Hskand generates initial private key dID;0:

Helper Key Update: This algorithm is run by the helper withinputs identity ID, Hsk, time period indices t; t 1; and gener-ates updated helper key UHKID;t;t1:

e Identity-based Strong Key-Insulated Signature Scheme without Pairings.https://doi.org/10.1016/j.jksuci.2018.08.011

Table 1Notations and their meanings.

Notation Meaning

k; s;v Security parameter, master secret key and helper secret key ofthe system generated by PKG.

Params System Parameter.Msk Master Secret Key: s 2 Z

q

Hsk Helper Secret Key: v 2 Zq

PPub;V Master public key (Mpk) and Master helper key of the system.G Additive cyclic group of prime order qH1;H2;H3 Cryptographic one way hash functionsdID;0 ;dID;t Initial private/signing key and temporary signing key

respectivelyUHKID;t;t1 Updated helper keyt; t 1 Time period indicesPKG Private Key GeneratorAdv1 Perfectly key insulated adversary of PF-IDBKIS schemeAdv2 Strong key insulated adversary of PF-IDBKIS schemen ChallengerX Signature on a messageECDLP Elliptic Curve Discrete Logarithm Problem

P. Vasudeva Reddy et al. / Journal of King Saud University – Computer and Information Sciences xxx (2018) xxx–xxx 3

User Key Update: User runs this algorithm by taking initial pri-vate key dID;t1 for the time period t 1; updated helper keyUHKID;t;t1 for the time period indices t; t 1; and generatesusers private key dID;t for the time period t.

Signature Generation: Signer run this algorithm by taking usersprivate key dID;t for the time period t, and messagem 2 f0;1g asinput and generates a signature XID for the time period t, on amessage m 2 f0;1g:

Signature Verification: For a given tuple ðm; t;XIDÞ with ID,params, the verifier runs this algorithm. This algorithm acceptsthe signature if XID is valid. Rejects otherwise.The above formal model of signature scheme with the principleof key insulated mechanism is depicted in Fig. 1.

3.2. Security model of PF-IDBKIS scheme

Similar to Dodis et al. (2002a,b) and Weng et al. (2008a,b) weformalize the security notions for our PF-IDBKIS scheme by the fol-lowing Game 1 and Game 2. Game 1 describes the standard key-insulated security and is played between challenger and adversary

Fig. 1. Principle of key in

Please cite this article in press as: Vasudeva Reddy, P., et al. Efficient and SecurJournal of King Saud University – Computer and Information Sciences (2018),

Adv1: Game 2 describes the strong key insulated security and isplayed between the challenger and adversary Adv2. In Game 1,adversary Adv1 compromises any users temporary signing key atsome time period but not users helper key. In Game 2, we considera case where the helper is malicious; that is adversary Adv2 canaccess any user’s helper key including target identity but not com-promise the user’s temporary signing key at any time period. Here,we define the key insulated security for our PF-IDBKIS scheme byGame 1 and Game 2.

Game 1: (For the Adversary Adv1)

Setup: In this phase, the Setup algorithm is run by the chal-lenger n to produce params, Msk, Hsk and gives params to theAdv1. The Msk will be kept secretly by the challenger n itself.

- Queries Phase: In this phase, nwill answer the adaptive queriesasked by Adv1 as follows.

Initial Key Extraction Oracle: When Adv1 asks this query onID, n executes the Initial key extraction algorithm and gener-ates the initial private key and returns to Adv1:

Temporary Signing Key Extraction Oracle: When Adv1 asksthis query on ID along with a time period t, n executes theTemporary signing key extraction algorithm and generatesthe updated private key for the time period t and returnsto Adv1:

Sign Oracle: When Adv1 asks this query on ðID;m; tÞ; nreturns the valid signature XID on message m by ID asanswer by performing the signing algorithm.

- Forgery Phase: Finally Adv1 outputs ðID;m;XIDÞ such that the

following restrictions hold.(i) X

ID is a valid signature.(ii) Adv1 can never make Initial Key Extraction queries on

ID; Temporary Signing Key Extraction queries onðID; tÞ; and Sign queries on ðID;mÞ:

Definition1. A PF-IDBKIS scheme is perfectly key insulated andexistential unforgeable if no polynomially bounded adversarycan win the Game 1 with non-negligible probability. We definethe advantage of Adv 0

1s as the probability of winning this Game1. Advantage (Adv1) = Pr [Adv1 wins the PF-IDBKIS_Game 1].

sulated mechanism.

e Identity-based Strong Key-Insulated Signature Scheme without Pairings.https://doi.org/10.1016/j.jksuci.2018.08.011

4 P. Vasudeva Reddy et al. / Journal of King Saud University – Computer and Information Sciences xxx (2018) xxx–xxx

Game 2: (For the Adversary Adv2)

- Setup Phase: The challenger n runs the Setup algorithm. Itgives params to the Adv2:

- Queries Phase: In this phase, challenger will answer thefollowing adaptive queries issued by Adv2 as follows.

Initial Key Extraction Oracle: When Adv2 asks this query on ID, nexecutes the Initial key extraction algorithm and generates theinitial private key and returns to Adv2:

Helper Key Extraction Oracle: When Adv2 asks this query on ID, nruns key extraction algorithm on ID and returns this value toAdv2:

Sign Oracle: Adv2 provides a number t, ID, a message m to signquery and asks n to execute the signing algorithm, and n returnsa valid signature XID to Adv2:

- Forgery Phase: Finally Adv2 outputs a forgery ðID;m;XIDÞ

for a time period t and wins the game if(i) X

ID is a valid signature.(ii) Adv2 is not allowed to make Initial Key Extractionqueries on ID; and Sign queries on ðID;mÞ:

Definition2. A PF-IDBKIS scheme is strong key insulated and exis-tential unforgeable if no polynomially bounded adversary can winthe Game 2 with non-negligible probability.

As in Dodis et al. (2002a,b), we present the following definition3 for the security notion of secure key updates in KIS schemes.

Fig. 2. Schematic presentation of the proposed PF-IDBKIS scheme.

Definition3.

A PF-IDBKIS scheme has secure key-updates if the view of anyadversary Adv making a key update exposure at t; t 1 can beperfectly simulated by an adversary Adv 0 making Key ExtractionOracle queries of Game 1 at periods t 1 and t.

4. Proposed PF-IDBKIS scheme without pairings

The proposed PF-IDBKIS Scheme consists of the following sixalgorithms. The flow of functionalities of these algorithms of ourPF-IDBKIS is described in Fig. 2.

Setup: Given a security parameter k 2 Zþ, total number of timeperiods N, PKG runs this algorithm to generate system parameters.

1. PKG selects a group G with the elements are the points onelliptic curve. Here jGj ¼ q and P be a generator of G. PKG alsoselects hash functions H1 : f0; 1g G G ! Z

q ; Hi : f0; 1gG f0; 1g ! Z

q for i ¼ 2;3:2. PKG randomly selects s 2 Z

q as Msk and v 2 Zq as Hsk and

computes Ppub ¼ sP as the Mpk Also computes V ¼ vP.3. PKG publishes the system parameter as Params ¼ fq ; G ; P ;

Ppub ; V ; H1 ; H2 ; H3g and keeps Msk and Hsk secretly.

Extract: Given a user’s identity ID, PKG runs this algorithm togenerate initial private key.

1. PKG chooses rID 2 Zq at random and computes RID ¼ rIDP:

2. PKG computes the initial private key as dID ; 0 ¼ rID þ s h1 ID þv h2 ID mod q; where h1 ID ¼ H1ð ID ; RID ; Ppub Þ;h2 ID ¼ H2 ð ID ;

V ; 0 Þ:

PKG sends the user’s private key DID ¼ ðdID;0 ;RID Þ to the corre-sponding user securely. The user keeps dID ; 0 secretly and makes RID

as public. (Store the Hsk (v) in a physically secure device).

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Helper key update: The helper (physically secure device) gen-erates the updated helper key UHKID;t;t1 for the user with ID andtime period indices t ; t 1 as follows:

1. Compute V ¼ vP:2. Compute

UHKID;t;t1 ¼ v½H2ð ID ; V ; t Þ H2ð ID ; V ; t 1 Þ mod q:

The device returns the updated helper key UHKID;t;t1 to theuser.

User key update: Given the updated helper key UHKID;t;t1; andtemporary signing key of ID for a time period t 1 i:e:dID ; t1 ; theuser constructs a new temporary signing key of ID for the time per-iod t as dID ; t ¼ UHKID;t;t1 þ dID ; t1 :

Note: At the time period ‘t’, dID ; t is always of the formdID ; t ¼ rID þ sh1ID þ vh2ID ; t :

Signature Generation: To generate a signature on a messagem,for time period t, for the identity ID, the user does as follows.

1. Choose uID 2 Zq and compute UID ¼ uIDP ; and h3ID ¼ H3 ð ID ;

UID ; m Þ: If gcd of ½ðuID þ h3ID Þ1; q ¼ 1; proceed. Else choose

another uID 2 Zq .

2. Compute rID ; t ¼ ðuID þ h3ID Þ1 dID;t mod q.3. The signer outputs the signature on a message m, for the time

period t as XID ¼ ðRID ; UID ;rID Þ i.e. signer outputs ðm ; t ;XID Þ as the signature.

Verification: Given a users identity ID, system parametersparams and a signature tuple ðm ; t ; XID Þ; any verifier can checkthe validity of the signature as follows.

1. Compute h1 ID ¼ H1ð ID ; RID ; Ppub Þ; h2 ID ¼ H2 ð ID ; V ; t Þ andh3 ID ¼ H3 ðm ; ID ; UID Þ:

e Identity-based Strong Key-Insulated Signature Scheme without Pairings.https://doi.org/10.1016/j.jksuci.2018.08.011

P. Vasudeva Reddy et al. / Journal of King Saud University – Computer and Information Sciences xxx (2018) xxx–xxx 5

2. Verify whether the following equation holds or not rID ðUID þh3 IDP Þ ¼ RID þ h1 ID Ppub þ h2 ID V Þ :

If the equation is valid, verifier accepts the signature. Else thesignature will be rejected.

4.1. Security of our PF-ID-KISS

Theorem 1: The proposed PF-IDBKIS scheme is perfectlykey insulated and unforgeable against an adversary Adv1 in therandom oracle model with the assumption that the ECDLP ishard.

Proof: Suppose Adv1 be an adversary who can forge a valid sig-nature with non-negligible probability. We will show how to pro-duce another algorithm n which can solve the ECDLP with the helpof the adversary Adv1. We assume that the challenger n is given arandom instance tuple ðP;Q ¼ sPÞ of the ECDLP. The challenger nanswers to the queries asked by Adv1: The task of n is to com-putes 2 Z

q: For the simulation process, n takes ID as target identityof Adv1 on a message m.

Setup Phase: Challenger n sets Ppub ¼ Q ¼ sP and executes theSetup algorithm to produce necessary parameters of the systemand sends params including Ppub to Adv1.

Query Phase: Adv1 performs the following oracles in an adap-tive manner and the algorithm n will answer to these oracles. Toavoid the conflict of simulation, n need to maintain the initiallyempty lists L1; L2; L3; LKExt; LTSKExt :These lists are used to keep trackof answers to the following queries.

Queries on oracle H1 : H1ð IDi ; Ri ; Ppub Þ : When Adv1 makes aH1 query on the tuple ð IDi ; Ri ; Ppub Þ; n will search whether thetuple ð IDi ; Ri ; Ppub ; l1iÞ is in the list L1: n returns l1i if such tupleexists in the list. If no such tuple exists in L1; n chooses l1i randomlyand inserts in the list L1: Finally, n will return l1i as the answer tothe adversary Adv1:

Queries on oracle H2 : H2ð IDi ; V ; ti Þ : When Adv1 asks a H2

query on ð IDi ; V ; ti Þ; n will check whether the tupleð IDi ; V ; ti ; l2iÞ exists in L2 or not. If such a tuple exists in the listL2; n returns l2i: Otherwise, n picks a random l2i 2 Z

q and returnsl2i: n adds ð IDi ; V ; ti ; l2iÞ to L2:

Queries on oracle H3 : H3ðmi ; IDi ; Ui Þ : When Adv1 makes aquery on H3ðmi ; IDi ; Ui Þ; n will check whether the tupleðmi ; IDi ; Ui ; l3iÞ exists in L3 or not. If such tuple exists in list L3then n gives l3i: Otherwise, n picks a random l3i 2 Z

q and returnsl3i: Finally, n adds ðmi ; IDi ; Ui ; l3iÞ to L3:

Initial Key Extraction Oracle (K Ext(IDi)): Given an identity IDi;

n searches for the tuple ð IDi ; di ; Ri; tÞ in the list LKExt: If such tupleexists in list LKExt; then n gives di: Otherwise, if IDi – ID; n choosesai; bi 2 Z

q and sets di ¼ ai;Ri ¼ biPPub þ aiP h2iV and h1i ¼ bi: Itis clear that ðdi;RiÞ generated in this manner is a valid initial key.Now n adds ð IDi ; di ; Ri;0Þ to the list LKExt and returns di: Also nadds the tuple ð IDi ; Ri ; Ppub ;h1iÞ to the list L1; and ð IDi ; V ; ti ; l2iÞto the list L2: If IDi ¼ ID; n aborts.

Temporary Signing Key Extraction Oracle (TSK Ext (IDi)):Given an identity IDi along with a time period t, n searches forthe tuple of the form ð IDi ; di ; Ri; tÞ in the list LTSKExt: If such tupleexists in list LTSKExt ; then n gives di: Otherwise, if IDi – ID; n findsthe tuple ð IDi ; Ri ; Ppub ;h1iÞ from the list L1;chooses ai 2 Z

q andsets di ¼ ai; Ri ¼ h1iPPub þ aiP h2iV : Now n adds ð IDi ; di ; Ri; tÞto the list LTSKExt for consistency. If IDi ¼ ID; n aborts.

Signing Oracle: When Adv1 makes this query on ð IDi ; mi Þ withtime period ti ; n does as follows:

1. If IDi – ID; n recovers di ¼ ai from LKExt list and h1i ; h2i ; h3i

from L1; L2 &L3 lists respectively. Also n chooses ri 2 Zq and

Please cite this article in press as: Vasudeva Reddy, P., et al. Efficient and SecurJournal of King Saud University – Computer and Information Sciences (2018),

computes Ri ¼ ri P; ri ¼ di h13i ¼ aih

13i , Ui ¼ a1

i h3i ½h1iPpub þh2i V þ Ri ai P

The signature XIDi ; t ¼ ðRIDi; UIDi

; rIDiÞ ¼ ðRi ; Ui ; riÞ is a valid

signature on message mi for a time period ti.n outputsðmi ; ti ; XIDi

Þ as a valid tuple.

2. If IDi ¼ ID ; n chooses ri 2 Zq and sets Ri ¼ ri P and recovers

h1i ; h2i ; h3i from L1; L2 &L3 lists and ri ¼ ri h13i and computes

Ui ¼ ½h1iPpub þ h2i V h3ir1i

The signature XIDi ; t ¼ ðRi ; Ui ; riÞ is a valid signature on mes-sage mi, for the time period ti.n outputs ðmi ; ti ; XiÞ is a valid tuple.

Forgery/output: After forging a valid signature Xi ¼ ðR

i ;Ui ;r

i Þon message m

i under the identity IDi ; for a time period ti, by Adv1,

n recovers the corresponding ð IDi ; Ri ; Ppub ; l1iÞ; ð IDi ; V ; ti ; l2iÞ;ðmi ; IDi ; Ui ; l3iÞ from L1, L2 and L3 lists. If IDi – ID; then n failsand halts. Otherwise if IDi ¼ ID; n computes the value of s as fol-lows. From Forking lemma, if we replay n twice with same randomvalues but different choice of functions H1 ; H2 &H3; Adv1 will out-put another two signatures.

Let XðjÞi ¼ ðR

i ;Ui ;r

ðjÞi Þfor j ¼ 1;2;3: These signatures satisfy the

verification equation and we get

rðjÞi ½u

i þ hðjÞ3i P ¼ ri P þ hðjÞ

1i Ppub þ hðjÞ2i V for j ¼ 1;2;3:

) rðjÞi ½u

i þ hðjÞ3i ¼ ri þ hðjÞ

1i sþ hðjÞ2i v for j ¼ 1;2;3

in which rðjÞi and v are known values, ui, ri and s are unknown val-

ues, which can be found using above three linearly independentequations.

Now we analyze the advantage of algorithm n to solve theECDLP using the following events.

E1: ndoes not terminate while responding to any of Adv 01s Initial

Key Extraction queries.E2: n does not terminate while responding to any of Adv 0

1s Tem-porary Signing Key Extraction queries.E3: Adv1 outputs a valid signature forgery.E4: Adv 0

1s forged signature satisfies for IDi ¼ ID:

The responses for Adv 01s Initial key extraction queries (Tempo-

rary signing key extraction queries, Signing queries res.) are validunless event E1(E1; E2; E3 res.) happens. If Adv1 succeeds in forginga valid signature and E4 does not happen then n can solve theECDLP successfully with a probability PðE1 \ E2 \ E3 \ E4Þ ¼ðdqKEþqTSKEþqS Þð1 dÞ; which is maximized at

d ¼ qKEþqTSKEþqSqKEþqTSKEþqSþ1 : n

0s advantage is e0 satisfies e0 1ðqKEþqTSKEþqSþ1Þe e;

where e is base of natural logarithm and e is a success probabilityof Adv1; qKE; qTSKE; qS respectively are the maximum number ofqueries on key extraction, Temporary signing key extraction,Signing oracles respectively by Adv1:

Theorem 2: In the random oracle model, the proposedPF-IDBKIS scheme is strong key insulated and unforgeable againstan adversary Adv2 with the assumption that the ECDLP is hard.

Proof: The proof is almost same as Theorem 1. Queries on oracleH1;H2;H3 and Initial Key Extraction queries are same as in Theo-rem 1. The only difference is the adversary Adv2 cannot makeany Temporary Signing Key Extraction Query but can adaptivelyask the Helper Key Query.

Helper Key Query: When Adv2 queries on a Helper key for IDi ;

for a time period ti, n returns the helper key v 2 Zq to Adv2.

Signing oracle: For any identity IDi ;n recovers the helper keyv 2 Z

q from Helper key query and chooses ui 2 Zq, ai 2 Z

q andcompute U ¼ uiP and set V ¼ vP and di ¼ ai :

e Identity-based Strong Key-Insulated Signature Scheme without Pairings.https://doi.org/10.1016/j.jksuci.2018.08.011

6 P. Vasudeva Reddy et al. / Journal of King Saud University – Computer and Information Sciences xxx (2018) xxx–xxx

Using these values n makes queries on H1 ; H2 ; H3 and outputsh1i ; h2i ; h3i and sets Ri ¼ ai P h1i Ppub h2i V and

ri ¼ ðui þ h3iÞ1ai:

The signature Xi ¼ ðRi;Ui;riÞ is a valid signature.Similar to Theorem 1, n uses this signature to solve the ECDLP

with a success probability e0 1ðqKEþqSþ1Þe e;where e is base of natural

logarithm.Theorem 3: Our PF-IDBKIS scheme has secure key updates.This theorem follows from the fact that for any time period

indices t 1; t and any identity ID, the updated helper keyUHKID;t;t1 can be derived from dID ; t and dID ; t1 :

5. Performance comparisons

To evaluate the performance of our PF-IDBKIS scheme, we con-sider some cryptographic operations and their conversions whichare mentioned in Table 2. We consider the experimental resultsfrom the literature (Barreto and Kim (2002); Cao et al. (2010);Tan et al. (2010); MIRACL Library). We mentioned all the crypto-graphic operations in terms of modular multiplications. We pre-sent the comparison of our scheme with the related schemes, interms of computational cost in signing and verification processand security point of view, in Table 3.

We now analyze and compare our PF-IBKIS scheme with theexisting and ID-based KIS schemes (Weng et al. (2006); Zhouet al. (2006); Ohtake et al. (2008); Wu et al. (2012); Gopal andVasudeva Reddy (2015); Zhu et al. (2015)) in terms of computationcosts.

5.1. Computation costs

To generate a signature in Weng et al. scheme (2006), signerneeds to execute two scalar multiplications, one map to point hashfunction and one point addition i.e. 2TSM þ 1TMH þ 1TPA: Hence therun time to generate the signature is 20.2554 ms. To verify the sig-nature generated by the signer, a verifier in Weng et al. scheme(2006) needs to execute four bilinear pairings and three map topoint hash functions 4TBP þ 3TMH: Hence the run time to verifythe signature is 101.137 ms and the total run time for Weng

Table 2Conversions of various cryptographic operations.

Notations Description

TMM Modular multiplication operation 1TMM 0:2325msTSM Scalar multiplication over elliptic curves: TSM ¼ 29TMM 6:38msTBP Bilinear pairing: TBP ¼ 87TMM 20:01msTPEX Pairing–based exponentiation: TPEX ¼ 43:5TMM 11:20msTINV Modular inversion operation: TINV ¼ 11:6TMM 2:697msTMH Map to point hash function: 1TMH ¼ 29TMM 6:38msTMX Modular exponentiation operation: TMX ¼ 240TMM 55:20msTPA Elliptic curve point addition: TPA ¼ 0:12TMM 0:0279ms

Table 3Comparison of our PF-IDBKIS scheme.

Scheme Signing Cost Verification co

Weng et al. (2006) 2TSM þ 1TMH þ 1TPA 4TBP þ 3TMH

Zhou et al. (2006) 2TME þ 1TMH 4TBP þ 3TMH

Ohtake et al. (2008) 1TMX þ 1TMM 1TINV þ 2TMX

Wu et al. (2012) 3TSM þ 2TMH 3TBP þ 2TSMþ2TPA þ 2TMH

Gopal and Vasudeva Reddy (2015) 1TPEX þ 2TSM þ 1TPA 3TBP þ 2TSM þZhu et al. (2015) 1TMM 4TMX

Our Scheme 1TSM þ 1TINV 4TSM þ 3TPA

Please cite this article in press as: Vasudeva Reddy, P., et al. Efficient and SecurJournal of King Saud University – Computer and Information Sciences (2018),

et al.’s scheme (2006) is 121.39 ms. Similarly, in Zhou et al. scheme(2006), the run time to generate the signature is 118.34 ms and forverification is 101.13 ms. Hence the total run time for Zhou et al.’sscheme (2006) is 219.4775 ms. In Ohtake et al. scheme (2008), therun time to generate the signature is 56.03 ms and for verificationis 114.297 ms. Hence the total run time for Ohtake et al’s scheme(2008) is 170.327 ms. In Wu et al. scheme (2012), the run timeto generate the signature is 33.945 ms and for verification is88.173 ms. Hence the total run time for Wu et al’s scheme(2012) is 122.1183 ms. In Gopal et al. scheme (2015), the run timeto generate the signature is 23.715 ms and for verification is88.1175 ms. Hence the total run time for Gopal et al’s scheme(2015) is 111.89 ms. In Zhu et al. scheme (2015), the run time togenerate the signature is 0.2325 ms and for verification is223.2 ms. Hence the total run time for Zhu et al’s scheme (2015)is 223.4325 ms. In our proposed scheme, to generate a signature,signer needs to execute one scalar multiplication and one modularinversion operation i.e. 1TSM þ 1TINV : Hence the run time to gener-ate the signature is 9:4395ms: To verify the signature generated bythe signer, a verifier in our scheme needs to execute four scalarmultiplications and three point additions 4TSM þ 3TPA:Hence therun time to verify the signature is 27:0537ms:Hence the total runtime for our scheme is 36.4932 ms.

The computation cost of our PF-IDBKIS scheme is 36:5ms; and is69.93% less than Weng et al. scheme (2006), 83.36% less than Zhouet al. scheme (2006), 78.57% less than Ohtake et al. scheme (2008),70.11% less than Wu et al. scheme (2012), 67.4% less than Gopalet al. scheme (2015) and 83.66% less than Zhu et al. Scheme (2015).

5.2. Communication cost

The communication overhead of the proposed scheme is pre-sented by comparing with the schemes (Weng et al. (2006);Zhou et al (2006); Ohtake et al. (2008); Wu et al. (2012); Gopaland Vasudeva Reddy (2015); Zhu et al. (2015)). To achieve a secu-rity level of 80 bits, in bilinear pairing as well as in ECC, we con-sider the following Table 4.

In schemes (Weng et al. (2006); Zhou et al (2006); Wu et al.(2012)), the length of the signature is 3jGj: Hence the communica-tion cost in these schemes is 3 320 ¼ 960 bits: The communica-tion cost of Gopal and Vasudeva Reddy (2015) scheme isjGj þ jG1j ¼ 1024þ 320 ¼ 1344 bits; The communication cost ofOhtake et al. scheme (2008) is jG1j þ 2jZ

qj ¼ 1024þ 2160 ¼ 1344 bits; and for Zhu et al. scheme (2015) is3jG1j þ jZ

qj ¼ 3 1024þ 160 ¼ 3232bits: But the communicationcost of our proposed scheme is 2jGj þ jZ

qj ¼ 2 320þ 160 ¼800bits:Hence our scheme requires less communication cost thanthe existing KIS schemes in ID-based setting.

The following Table 5 presents the communication overhead ofall schemes in terms of sending a message.

From Tables 3 and 5, it is clear that most of the signatureschemes (Weng et al. (2006); Zhou et al (2006); Wu et al.

st Total cost Securekey-Updates

StrongKey-Insulation

522:12TMM 121:39ms Yes Yes944TMM 219:48ms No No732:6TMM 170:33ms Yes Yes525:24TMM 122:12ms Yes Yes

2TMH 481:62TMM 112ms Yes Yes961TMM 223:43ms Yes Yes156:96TMM 36:5ms Yes Yes

e Identity-based Strong Key-Insulated Signature Scheme without Pairings.https://doi.org/10.1016/j.jksuci.2018.08.011

Fig. 3. Graphical presentation.

Table 5Comparison of the proposed PF-IDBKIS scheme with the related scheme.

Scheme Initial Key Length Updated Key Length Signature Length Number of bytes to send a message

Weng et al. (2006) 2jGj 2jGj 3jGj 120 bytesZhou et al. (2006) jGj 2jGj 3jGj 120 bytesOhtake et al. (2008) 2jZ

qj jZqj jG1j þ 2jZ

qj 168 bytes

Wu et al. (2012) jGj jGj 3jGj 120 bytesGopal and Vasudeva Reddy (2015) jGj jGj jGj þ jG1j 168 bytesZhu et al. (2015) jG1j þ jZ

qj 2jG1j þ jZqj 3jG1j þ jZ

qj 404 bytes

Our Scheme jGj þ jZqj jZ

qj 2jGj þ jZqj 100 bytes

Table 4Length of the group in bilinear pairing and ECC.

Type of the System Type of the Curve Pairing Cyclic group Length of cjpj; jpj Group of order Length of the group

Bilinear Pairing E : y2 ¼ x3 þ x mod p be : G1 G1 ! GT G1 with generatorbP cjpj ¼ 512 bits (64 bytes) bq ¼ 160 bits jG1j ¼ 1024 bits

ECC E : y2 ¼ x3 þ axþ b mod pwhere a; b 2 Z

q:

Without Pairing G with generatorP jpj ¼ 160 bits (20 bytes) q ¼ 160 bits jGj ¼ 320 bits

P. Vasudeva Reddy et al. / Journal of King Saud University – Computer and Information Sciences xxx (2018) xxx–xxx 7

(2012); Gopal and Vasudeva Reddy (2015)) were based on bilinearpairings and there are only two schemes without pairings, of whichOhtake et al. (2008) and Zhu et al. (2015) are not efficient due totheir modular exponentiation operation in signature and verifica-tion algorithms. Hence, of all schemes in the literature, our schemeis the efficient and secure strong key insulated signature schemewithout pairings in identity-based frame work. Also the compar-ison of run time in our proposed scheme with the related schemesis presented graphically in Fig. 3. The graph clearly indicates thatour scheme is more efficient than the existing schemes.

6. Potential applications

In view of the desirable merits, namely free from certificateauthority, free from costly computational operations, key insula-tion, the proposed PF-IDBKIS scheme can be applied to a range ofpractical environments which are troubled by the private keyexposure problem.

Secret handshakes: Secret handshake scheme is a fundamentalcryptographic primitive, which enables the members of a certaingroup to authenticate each other in a private way. This systemallows two members of the same group to authenticate each othersecretly and share a key for the further communication.

Secret Handshake can be used in diverse range of applicationssuch as anonymous routing protocol in ad-hoc network,high-bandwidth digital content protection (HDCP), military secretservices, Wireless Authentication, Sensor Networks and RadioFrequency Identification (RFID) enabled passport privacy. Secret

Please cite this article in press as: Vasudeva Reddy, P., et al. Efficient and SecurJournal of King Saud University – Computer and Information Sciences (2018),

Handshake can also be used for online applications such ase-commerce, e-business and social network by providing privacypreserving authentication. In Secret handshake scheme any twomembers of a group authenticate themselves mutually through asecret secure channel. User’s secret key will be sent by a trustedgroup authority through a secure channel. Though the channel issecure, leakage or exposure of this secret key in authenticationprocess may cause disastrous consequences. Hence the proposedPF-IDBKIS scheme without bilinear pairings can be used to con-struct secret handshakes scheme featured without the problemof key exposure.

Wireless Sensor Networks (Vehicular Ad hoc Networks): The con-tinuous progress of wireless communication technology providesintelligent and efficient transportation system through vehicularad hoc networks (VANETS) to mitigate traffic jams and roadfatalities. This intelligent transportation system improves safetyof passengers and traffic flow. Authentication of these vehiclescan be achieved based on the hypothesis that secret keys are keptperfectly secure. The secret key generated by the Trust Authority(TA), will be send to the Vehicle through a secure channel inauthentication process between vehicle to vehicle or betweenvehicles to Road Side Units (RSU). However, key exposure of thissecret key is inevitable on account of the openness of VANETenvironment. To address this problem, we can adopt the proposedPF-IDBKIS scheme.

7. Conclusions

In this paper, we proposed a new PF-IDBKIS scheme, by inte-grating the key-insulated mechanism with pairing free identitybased signature scheme, in order to solve the private key exposureproblem. This scheme did not use any complex bilinear pairingoperations, which enormously improves the computational effi-ciency. In our PF-IDBKIS scheme user updates his signing key peri-odically with the help of a helper. The proposed scheme canachieve unforgeability, strong key insulation; secure key updatesin random oracle model provided the ECDLP is hard. Performanceanalysis shows that the proposed PF-IDBKIS scheme is more effi-cient in terms of computational and communicational point ofview when comparing to the well-known existing KIS schemes inIdentity-based setting. Also, due to high efficiency, ourscheme can be employed for practical applications where thecomputational resources are limited, for example, Wireless com-munication devices, Smart cards, Vehicular Ad-hoc Networks.

e Identity-based Strong Key-Insulated Signature Scheme without Pairings.https://doi.org/10.1016/j.jksuci.2018.08.011

8 P. Vasudeva Reddy et al. / Journal of King Saud University – Computer and Information Sciences xxx (2018) xxx–xxx

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e Identity-based Strong Key-Insulated Signature Scheme without Pairings.https://doi.org/10.1016/j.jksuci.2018.08.011

2017-18

2017-18

Int J Advanced Design and Manufacturing Technology, Vol. 11/ No. 2/ June – 2017 25

© 2017 IAU, Majlesi Branch

Investigation of Stresses in

U-Shaped Metal Bellow Using

EJMA Standards

Kondapalli Siva Prasad Department of Mechanical Engineering,

Anil Neerukonda Institute of Technology & Sciences, India

E-mail: kspanits@gmail.com

*Corresponding author

Gudla Pavani Department of Mechanical Engineering,

Anil Neerukonda Institute of Technology & Sciences, India

E-mail: pavanireddy.gudla@gmail.com

Received: 5 January 2017, Revised: 1 February 2017, Accepted: 5 February 2017

Abstract: Metal Bellows finds wide application in expansion joints, which are

used in aerospace, chemical plants, power system, heat exchangers, automotive

vehicle parts, piping system, petrochemical plant, refineries, etc. During service

they are subjected to various stresses and exposed to different environments,

which leads to failure. Hence there is a need for proper design of metal bellow as

per the application. The main objective of the paper is to evaluate the stresses

generated in the metal bellow and the cycle life working at different working

pressures. In this paper, the stresses are calculated using Expansion Joint

Manufacturing Association (EJMA) standards and compared with the results

obtained using ANYS software for two different materials namely Inconel 625

and Inconel 718 for the pressure values ranging from 20 to 40 bar.

Keywords: EJMA, Expansion joints, Metal bellow

Reference: Prasad, K. S., Pavanai, G., “Investigation of Stresses in U-Shaped Metal Bellow Using EJMA Standards”, Int J of Advanced Design and Manufacturing Technology, Vol. 11/No. 2, 2017, pp. 25-35.

Biographical notes: Kondapalli Siva Prasad obtained PhD from Andhra University, India in 2003. He is currently working as Associate Professor at the Department of Mechanical Engineering, Anil Neerukonda Institute of Technology & Sciences, India. His current research interest includes Manufacturing and product design. Gudla Pavanai is PG student of Department of Mechanical Engineering, Anil Neerukonda Institute of Technology & Sciences, India.

26 Int J Advanced Design and Manufacturing Technology, Vol. 11/ No. 2/ June – 2017

© 2017 IAU, Majlesi Branch

1 INTRODUCTION

An expansion joint is an assembly designed to safely

absorb the heat-induced expansion and contraction of

construction materials, to absorb vibration, to hold parts

together, or to allow movement due to ground

settlement or earthquakes. Bellow is corrugated part of

the expansion joint which is capable of compensating

large amount of axial, lateral and angular movements as

a single unit. It must be strong enough circumferentially

to withstand the pressure and flexible enough

longitudinally to accept the deflections for which it is

designed, and as repetitively as necessary with a

minimum resistance. This strength with flexibility is a

unique design problem that is not often found in other

components in industrial equipment. Based on the

application the material of the bellow is selected. Its

present requirement is for the aerospace applications at

the bleed air outlet of aircraft engine.

In the field of expansion joints very limited literature is

available. Only few technical books and hand books of

piping includes about the expansion joints which are

used in the piping. But these references are limited up to

the working principle of expansion joints. No text or

reference books include, design of expansion joints, as

this is a specialized area. But all authors are mentioning

the reference of standards developed by EJMA. Since

major contribution in the design of bellows expansion

joint is given by Expansion Joints Manufacturers’

Association (EJMA). EJMA has established the codes

and guidelines for the design of bellows expansion

joints. These codes are available based on membership

of EJMA. Jayesh. B. Khunt and Rakesh. Prajapathi [1]

studied different types of expansion joints used in

industry.

S. H. Gawande et al. [2] performed numerical analysis

to find various characteristics of stresses in U-shaped

metal expansion bellows as per the requirement of

vendor and ASME standards. Lu Zhiming et al. [3]

discussed the effects of axial deformation load on U-

shaped bellows. Brijeshkumar et al. [4] analyzed the

failure of bellows expansion joints made of SS 304.

Zhiming Lu et al. [5] analyzed the failure of metal

bellow made of austenitic stainless steel. Kazuyuki

Tsukimori [6] carried out modeling of creep behavior of

bellows. Norton’s law is used to study the creep

property of bellows. K. Brodzinsko et al. [7] studied the

failure mechanism of LHC cryogenic distribution line.

Hasan Shaikh et al. [8] analyzed the failure of an AM

350 steel bellows. Jinbong Kim [9] analyzed the effect

of geometry on fatigue life for automotive bellows. F.

Elshawesh et al. [10] investigated that the expansion

joint failed as a result of initiation of fatigue cracks at

the corrosion pits that propagated through bellow’s

circumference. Bijayani Panda et al. [11] discussed the

metallurgical factors responsible for failure of bellows

due to stress corrosion cracking. Asril Pramutadi et al.

[12] observed the corrosion behavior conducted on the

bellows of the bellow-sealed valve used in a lithium

circulation loop. Abhay K. Jha et al. [13] observed

various metallurgical features in stainless steel bellows.

Y.Z. Zhu et al. [14] proposed the effect of

environmental medium on corrosion fatigue life.

C. Becht IV [15] predicted the fatigue life of bellows by

partitioning the bellow fatigue data based on a geometry

parameter.

In the above literature review most of the work is done

on bellows made of various grade stainless steels which

are subjected to different types of corrosion such as

fatigue corrosion, liquid droplet erosion, and stress

corrosion when used at high temperatures. Fatigue

analysis of bellows is less concentrated. Based on these

studies, there is a necessity that the materials used

should possess great corrosion resistance at elevated

temperatures. The fatigue life of the bellows is of great

importance as they are subjected repeated loads.

Therefore, these two high temperature nickel-chromium

alloys Inconel 625 and Inconel 718 are used in our

work. These materials have good oxidation resistance,

excellent strength and are easily fabricated. As most of

the bellows are used in corrosive environment use of

these nickel chromium alloys will minimize the failure

due to corrosion. So, these material properties are used

to calculate the stresses produced in bellows. Hence this

work focusses on selection of proper bellow material,

design, calculation of stresses both analytically and

numerically and finally comparing both with the

allowable stress limits.

2 DESIGN OF BELLOW USING EJMA STANDARDS

The design of a bellow is complex and it involves an

evaluation of pressure capacity, stresses due deflection

and pressure, fatigue life, spring forces and instability.

The bellow used in this joint will be tested for two high

temperature materials. The design should be based on

the actual bellow metal temperature expected during

operation. The design values are considered based on

conditions available at the bleed air outlet of aircraft

engine. Detailed design calculations of bellow used in

gimbal joint are shown below.

Fig. 1 Geometry of bellow

Int J Advanced Design and Manufacturing Technology, Vol. 11/ No. 2/ June – 2017 27

© 2017 IAU, Majlesi Branch

Design considerations

Normal Working Pressure = 37 bar

Normal Working Temperature = 650ºC

Angular Moment required = 17.5Nm

Maximum Permissible Deflection = ±6º

Dm= Mean diameter of bellows convolution=50.5mm

Db = Inside diameter of bellows convolution = 42mm

Dc= mean diameter of bellows tangent reinforcing

collar=45 mm

n = number of plies (Assume initially) = 4

t = Bellows nominal material thickness of one ply

(Assume initially) = 0.25mm

tp = Bellows material thickness for one ply corrected for

thinning during forming =0.228mm

w = convolution height minus bellows thickness =

7.25mm

Assume Number of convolutions N = 7

Lb = Bellows convolute length = 43mm

Lt = Bellows tangent length = 6.5mm

Lc= Bellows tangent collar length =6.5mm

tc = Bellows tangent reinforcing collar material

thickness =1mm

q= Pitch =6.143 mm

e = Total equivalent axial moment per convolution = eθ,

since only rotational movement is allowed= eθ= 0.378

mm

k = A factor which considers the stiffening effect of the

attachment weld and the end convolution on the

pressure capacity of the bellows tangent and k value is

calculated by using formula

tD

Lk

b

t

5.1 = 337.1

25.0425.1

5.6

But if k>=1, k should be taken as 1

Hence k =1

2.1 The Stresses induced in bellow

The main causes for the stresses in the bellows are

pressure and initial deflection. Pressure and deflection

causes circumferential and meridional stresses in the

bellows. Stresses due to internal pressure remain largely

unaffected by the number of piles except for the

convolution meridional bending stress, which are

reduced when the total bellows thickness increases. The

deflection stresses are reduced due to thinner material

per ply resulting in an increase in fatigue life. The

equations used below are based on norms followed by

Expansion Joint Manufacturer’s Association (EJMA)

and accepted by ASME (American Society of

Mechanical Engineers).

The following are the stresses

1. Bellows Tangent Circumferential Membrane stress

due to pressure (S1)

2. Primary Collar Circumferential Membrane stress

due to pressure (S11)

3. Circumferential Membrane stress is also induced in

the convolutions (S2)

4. Bellows Meridional Membrane stress due to

pressure (S3)

5. Bellows Meridional Bending stress due to

pressure (S4)

6. Bellows Meridional Membrane stress due to

deflection(S5)

7. Bellows Meridional Bending stress due to

deflection (S6)

For Inconel 625 Material

I. Bellow tangent circumferential membrane stress

due to pressure (S1)

2

12

P D n t L E kth hS

n t E L D n t t k E L Dc c c cth h

Eb = Ec = 16700 kgf/mm2

S1= 3.89kgf/mm2

Su = Ultimate tensile strength of Inconel 625 at design

temperature (650ºC) = 760Mpa or76 kgf/mm2

Sab = allowable material stress of Inconel 625 at design

temperature = 76/2.5 =30 kgf/mm2

Cw = Factor accounting for Welding joint efficiency = 0.7

Effective Sab = CwSab = 21 kgf/mm2

From the above calculations, it is clear that S1 < Sab.

Hence design is safe.

II. Primary collar circumferential membrane stress

due to pressure (S1')

This is the circumference membrane stress induced in

the collar directly due to pressure p

S1'

2

2

P D L E kc ct

n t E L D n t t k L Dc c ctb b

S1’ = 4.25 Kgf/mm

2

Effective Sab = 21 kgf/mm2

S1' < Sab

Hence design is safe.

III. Circumferential membrane stress induced in the

convolutions (S2)

28 Int J Advanced Design and Manufacturing Technology, Vol. 11/ No. 2/ June – 2017

© 2017 IAU, Majlesi Branch

qw

p

m

tn

DPS

2571.0

1

22

(1) S2 = 3.4 kgf/mm2

S2< Sab(Thus design is safe)

IV. Meridional membrane stress in the bellow

convolution is induced due to pressure (S3)

It is a primary stress that follows the longitudinal axis of

the bellows at the crest and root of the convolutions

ptn

WPS

23

S3 =1.52 kgf/mm2

V. Meridional bending stress induced in the bellow

convolution due to pressure (S4)

It is a primary stress that follows the longitudinal axis of

the bellows across the convolutions

p

p

Ct

w

n

pS

2

42

The factor Cp =0.625 2

4 /29.31 mmkgS

Cm = Material strength factor at temperatures below the

creep range

From EJMA standards, Cm =3.0 for bellows in the

formed condition (with cold work)

S3 + S4 = 1.52 + 31.29 = 32.81 kgf/mm2

C m* Sab =3*21=63 kgf/mm2

S3 + S4< Cm * Sab

Hence design is safe.

VI. Meridional membrane stress induced in the

bellow convolution due to deflection(S5)

It is a secondary stress since the applied load is limited

by the deflection. It follows the longitudinal axis of the

bellows.

f

pb

Cw

etES

3

2

52

Eb= 20800 kgf/mm2

(at room temperature)

Cf is a shape factor = 1.38 2

5 /351.0 mmkgS

VII. Meridional bending stress in the bellow

induced due to deflection (S6 )

It is a secondary stress and follows the longitudinal axis

of the bellows. To find the value of the maximum

moment, the convolution is modeled as a fixed guided

strip beam with a concentrated load and a length w.

(2)

d

pb

Cw

etES

263

5

(3) Where

Cd is shape factor = 1.78 2

6 /84.29 mmkgS

The Stresses S5 and S6 are used in the evaluation of

bellows fatigue life.

2.2 Column Squirm (Calculation of Psc)

Column Squirm is defined as a gross lateral shift of the

centre section of the bellows. It results in curvature of

curvature of the bellows centre line. This condition is

mostly associated with bellows which have a relatively

large length to diameter ratio and is analogous to the

buckling of a column under compressive load. The

bellows have to be designed for either elastic or inelastic

condition based on length to diameter ratio.

i.e.

For Lb /Db> =C z, the squirm pressure Psc is evaluated as

qN

fCP iu

sc 2

34.0

For Lb/Db<Cz, the squirm pressure Psc is evaluated as

bz

b

b

yc

scDC

L

qD

SAP

73.01

87.0

Lb /Db = 43/42 = 1.02

Cz = Transition point factor

cby

iu

ADS

qf

272.4

This indicates the value of length to Diameter ratio

where the critical instability pressure transitions to a

maximum value at the length of one convolution which

represents purely inelastic behavior.

Where

Sy = Yield Strength at room temperature of bellows

For Inconel 625

Sy= 49 kgf/mm2 (at room temperature for Inconel 625)

f

pbm

iuCw

ntEDf

3

37.1

= 116.75 kg/mm per

convolution

Db = 42 mm

Ac =cross-sectional area of one bellows convolution

ntwq p 2571.0 = 16.88 mm2

Substituting the values, Cz =0.776

Since Lb / Db ≥ Cz

Int J Advanced Design and Manufacturing Technology, Vol. 11/ No. 2/ June – 2017 29

© 2017 IAU, Majlesi Branch

Psc should be evaluated for elastic region

qN

fCP iu

sc 2

34.0

Cθ =column instability pressure reduction factor based

on initial angular rotation 32 529.0348.1822.11 C

= Ratio of initial to final angular rotation

2907.03.0

bm

m

LD

D

b) Cθ= 0.5713

Psc = 0.236 kgf/mm2

2.4 In-plane Squirm (Calculation of Psi )

This is defined as a shift or rotation of the plane of one

or more convolutions such that the plane of these

convolutions is no longer perpendicular to axis of the

bellows. It is characterized by tilting or warping of one

or more convolutions. The stress induced due to this

squirm is evaluated as follows

2

51.0

K

SP

y

si

where

Psi = Limiting Design Pressure based on Inplane

instability (both ends rigidly supported)

b) K2 = Inplane Instability factor

qw

p

m

tn

DK

2571.0

1

22

=9.189

= Inplane instability stress interaction factor

5.0422 42121

Inplane instability stress ratio=

2

4

3 K

K

Inplane instability factor

2

42

p

p

t

w

n

CK =84.53

=38.13

For Inconel 625

Sy= 49 kgf/mm2

Psi = 0.44 kgf/mm2

For all the materials a factor of safety for limiting stress

of 2.25 is used in the relation for Psc, Psi. As Psc, Psi<

Normal Working pressure (37 bar), theoretically it is

required to go for higher thickness, however there

bellows were tested for burst pressure of 125 bar ‘g’ and

found satisfactory.

2.3 Fatigue Life

Fatigue life of a bellow is a function of the sum of the

meridional pressure stresses range and the total

meridional deflection stresses range. The equation for

fatigue life is as follows. a

t

cbS

cN

where a, b and c are material and manufacturing

constants. These constants are derived from the graph of

total stress range St versus number of cycles Nc.

From EJMA standards, the values of a, b and c are as

a= 3.4, b = 54,000, c = 1.86106

Total Stress St = 0.7 (S3 + S4) + (S5 + S6)

S3 = 1.52 kgf/mm2, S4 = 31.29 kgf/mm

2

For Inconel 625

S5 = 0.351 kgf/mm2

, S6 = 29.84 kgf/mm2

St = 53.158 kgf/mm2=77099.161 psi

Nc (at design temperature) = 3.02×106 cycles

3 NUMERICAL INVESTIF\GATION OF STRESSES

At first the bellow surface model is designed with the

help of CATIA V5 software, which is one of the leading

design software, and after that the surface model is

saved in IGES format and the geometry is imported to

ANSYS software. After importing the geometry, the

material properties are given with a thickness of 1 mm.

The bellow part is analyzed with the help of ANSYS

WORKBENCH 15.0. Figure 2 shows the surface model

and figure 3 shows the meshed ANSYS model with the

thickness given. The loading conditions are given such

that one edge of the bellow is fixed and internal surface

is subjected to pressure varying from 20 bar to 40 bar

for the materials Inconel 625 and Inconel 718. Figure 4

shows the loading conditions of bellow.

Fig. 2 The surface model

30 Int J Advanced Design and Manufacturing Technology, Vol. 11/ No. 2/ June – 2017

© 2017 IAU, Majlesi Branch

Fig. 3 The meshed model

Fig. 4 The loading conditions

2 RESULTS & DICUSSIONS

4.1 Theoretically calculated Values

Theoretical values of design stresses, squirm values and

fatigue life values for the two materials at 4 different

pressures are calculated using EJMA standards.

Table 5.1 Stresses developed at different pressures in Inconel

625

At Pressure 20 Bar 30 Bar 37 Bar 40 Bar

S1(kgf/mm2) 2.10 3.152 3.887 4.20

S1’(kgf/mm

2) 2.30 3.452 4.25 4.602

S2(kgf/mm2) 1.84 2.757 3.4 3.676

S3(kgf/mm2) 0.822 1.23 1.52 1.645

S4(kgf/mm2) 16.91 25.36 31.29 33.81

S5(kgf/mm2) 0.351 0.351 0.351 0.351

S6(kgf/mm2) 29.84 29.84 29.84 29.84

Table 5.2 Stresses developed at different pressures in Inconel

718

At Pressure 20 Bar 30 Bar 37 Bar 40 Bar

S1(kgf/mm2) 2.10 3.152 3.89 4.202

S1’(kgf/mm

2) 2.30 3.452 4.25 4.602

S2(kgf/mm2) 1.838 2.757 3.4 3.675

S3(kgf/mm2) 0.822 1.234 1.52 1.645

S4(kgf/mm2) 16.91 25.36 31.29 33.81

S5(kgf/mm2) 0.337 0.337 0.337 0.337

S6(kgf/mm2) 28.692 28.692 28.692 28.692

4.1.1 Design Stresses induced in Bellow

In the above tables 5.1, and 5.2, the stresses developed

in two high temperature metals at four different

pressures are calculated and it is observed that the

stresses S1, S11,S2, S3 ,S4 are increasing within increase

in pressure whereas the deflection stress S5,S6 are

calculated by using room temperature material

properties which does not have any significant change

with the change in pressure and vary according to the

material used. These stress values are checked with the

allowable stress values and the design is found to be

safe.

4.1.2 Column and In-plane Squirm

A factor of safety for limiting stress of 2.25 is used in

the relation for Psc and Psi. As Psc,Psi< 2.25 times of

working pressure, theoretically it is required to go for

higher thickness but these bellows were tested for burst

pressure of 125 bar and are found satisfactory.

Table 5.3 Squirm values for different materials

INSTABILITY Inconel

625

Inconel

718 Column Squirm

Psc(kgf/mm2)

0.236 0.23

Inplane Squirm Psi

(kgf/mm2)

0.44 1.056

4.1.3 Fatigue Life

Table 5.4 Fatigue life values (number of cycles Nc)

Pressure

(Bar)

(bar)

20 30 37 40

Inconel

625

1.2×108 8.94×106 3.022×106 2.078×106

Inconel

718

2.78×108 1.281×107 3.908×106 2.613×106

Int J Advanced Design and Manufacturing Technology, Vol. 11/ No. 2/ June – 2017 31

© 2017 IAU, Majlesi Branch

From theoretical calculations, it is observed that fatigue

life values depend upon the working pressure as the

pressure increases fatigue life values decreases. Inconel

718 has better fatigue life at all the pressures.

4.1 Comparison of Analytical and Numerical

Stresses due to Internal Pressure for Inconel 625 and

Inconel 718

The analytical stresses obtained from EJMA standards

and numerical stresses obtained from FEA are compared

and tabulated below.

4.2.1 For Inconel 625

Table 5.5 Theoretical and Numerical stresses of Inconel 625

at different pressures STRESS SOURCE 20

Bar

30

Bar

37

Bar

40

Bar

S1(kgf/mm2) EJMA 2.10 3.152 3.887 4.20

FEA 9.21 13.84 16.47 18.18

S11(kgf/mm2) EJMA 2.30 3.45 4.25 4.602

FEA 8.41 13.54 15.72 17.14

S2(kgf/mm2) EJMA 1.84 2.76 3.4 3.675

FEA 11.45 17.29 20.942 23.18

S3(kgf/mm2) EJMA 0.822 1.23 1.52 1.644

FEA 11.14 16.43 20.97 22.85

S4(kgf/mm2) EJMA 16.91 25.36 31.29 33.81

FEA 13.94 20.42 25.35 28.38

From the table 5.5 it is observed for Inconel 625 that all

the stresses obtained by both the approaches are within

the allowable limit and are increasing with increase in

the pressure. The difference in the stress profile is

comparatively large and is due to variation of the

approach methods.

0

5

10

15

20

15 35

CIR

CU

MFE

REN

TIA

L M

EMB

RA

NE

STR

ESS(

KG

F/M

M2)

PRESSURE(BAR)

EJMA

FEA

Fig. 5 Circumferential stresses in bellow tangent (S1) for

Inconel 625.S1 <Sab(21 kgf/mm2)

05

101520

15 35

PR

IMA

RY

CO

LLA

R

CIR

CU

MFE

REN

TIA

L ST

RES

S(K

GF/

MM

2 )

PRESSURE(BAR)

EJMA

FEA

Fig. 6 Primary collar circumferential stress (S11) for

Inconel 625.S11< Sab (21 kgf/mm2)

0

10

20

30

15 35

CIR

CU

MFE

REN

TIA

L M

EMB

RA

NE

STR

ESS

IN

…

PRESSURE(BAR)

EJMA

FEA

Fig. 7 Circumferential membrane stress induced in

convolution (S2) for Inconel 625

S2 < Sab (21 kgf/mm2)

0

5

10

15

20

25

0 50

MER

IDIO

NA

L M

EMB

RA

NE

STR

ESS(

KG

F/M

M2)

PRESSURE

EJMA

FEA

Fig. 8 Meridional membrane stress (S3) for Inconel 625

10

20

30

40

15 25 35 45MER

IDO

INA

L B

END

ING

ST

RES

S(K

GF/

MM

2)

PRESSURE(BAR)

EJMA

FEA

Fig. 9 Meridional bending stress (S4) for Inconel 625

(S3+S4) <Cm×Sab(63 kgf/mm2)

32 Int J Advanced Design and Manufacturing Technology, Vol. 11/ No. 2/ June – 2017

© 2017 IAU, Majlesi Branch

The allowable stress value (Sab) for circumferential

stresses S1, S11 ,S2 is 21 kgf/mm

2. Whereas for

meridional stresses (S3+S4) <Cm×Sab i.e.63 kgf/mm2

for

the design to be safe. Figures 5 to 9 show the

comparison graphs of the stresses in the bellow when

subjected to internal pressure.

Fig. 10 Stress distribution at pressure 20 Bar

Fig. 11 Stress distribution at pressure 30 Bar

Fig. 12 Stress distribution at pressure 37 Bar

Fig. 13 Stress distribution at pressure 40 Bar

When compared to the meridional bending stress (S4) all

the other stresses have considerable variation, but as per

design criteria they are within the allowable limit. The

circumferential membrane stress induced in convolution

(S2) for the pressure 40 bar is slightly above the

allowable stress which states that withstanding the

pressure more than 40 bar there is a necessity to go for

higher thickness of the bellow. Figures 10 to 13 show

the stress distribution for Inconel 625 at different

pressures. It is also incurred from the diagrams that the

maximum and minimum values of stresses developed in

bellow when compared to theoretically calculated

stresses show a close match.

4.2.2 For Inconel 718

From the table 5.6 it is observed for Inconel 718 that all

the stresses obtained by both the approaches are within

the allowable limit and are increasing with increase in

the pressure. The difference in the stress profile is

comparatively large and is due to variation of the

approach methods.

Table 5.6 Analytical and Numerical stresses of Inconel 718 at

different pressures

STRESS SOURCE 20 Bar 30 Bar 37 Bar 40 Bar

S1(kgf/mm2) EJMA 2.10 3.152 3.89 4.202

FEA 9.77 15.06 18.63 20.18

S11(kgf/mm2) EJMA 2.301 3.452 4.25 4.602

FEA 8.39 14.76 17.58 19.51

S2(kgf/mm2) EJMA 1.838 2.757 3.4 3.676

FEA 14.15 21.46 26.36 28.50

S3(kgf/mm2) EJMA 0.822 1.234 1.52 1.645

FEA 13.47 19.29 23.51 23.60

S4(kgf/mm2) EJMA 16.907 25.361 31.29 33.814

FEA 15.12 22.58 29.06 31.37

Int J Advanced Design and Manufacturing Technology, Vol. 11/ No. 2/ June – 2017 33

© 2017 IAU, Majlesi Branch

Figures 14 to 18 show the comparison graphs of the

stresses in the bellow when subjected to internal

pressure in Inconel 718. When compared to the

meridional bending stress (S4), all the other stresses

have considerable variation, but as per design criteria,

they are within the allowable limit.

0

5

10

15

20

15 25 35 45

CIR

CU

MFE

REN

TIA

L M

EMB

RA

NE

STR

ESS(

KG

F/M

M2)

PRESSURE(BAR)

EJMA

FEA

Fig. 14 Circumferential stresses in bellow tangent (S1)for

Inconel 718S1 <Sab(32.34 kgf/mm2)

0

5

10

15

20

15 25 35 45

PR

IMA

RY

CO

LLA

R

CIR

CU

MFE

REN

TIA

L ST

RES

S(K

GF/

MM

2)

PRESSURE(BAR)

EJMA

FEA

Fig 15 Primary collar circumferential stress (S1

1) for

Inconel 718S11< Sab (32.34 kgf/mm2)

0

10

20

30

15 35CIR

CU

MFE

REN

TIA

L M

EMB

RA

NE

STR

ESS

IN

CO

NV

OLU

TIO

N(K

GF/

M…

PRESSURE(BAR)

EJMA

FEA

Fig. 16 Circumferential membrane stress

induced in convolution (S2) for Inconel 718 S2

< Sab (32.34 kgf/mm2)

0

5

10

15

20

25

0 50

MER

IDIO

NA

L M

EMB

RA

NE

STR

ESS(

KG

F/M

M2)

PRESSURE(BAR)

EJMA

FEA

Fig. 17 Meridional membrane stress (S3) for

Inconel 718

10

15

20

25

30

35

40

15 25 35 45

MER

IDO

INA

L B

END

ING

ST

RES

S(K

GF/

MM

2)

PRESSURE(BAR)

EJMA

FEA

Fig. 18 Meridional bending stress (S4) for Inconel

718(S3+S4) <Cm×Sab(97.02 kgf/mm2)

Figures 19 to 22 show the stress distribution for Inconel

718 at different pressures varying from 20 bar to 40 bar.

It is also incurred from the diagrams that the maximum

and minimum values of stresses developed in bellow at

these pressures show a close match.

Fig. 19 Stress distribution at Pressure 20 Bar

34 Int J Advanced Design and Manufacturing Technology, Vol. 11/ No. 2/ June – 2017

© 2017 IAU, Majlesi Branch

Fig. 20 Stress distribution at Pressure 30 Bar

Fig. 21 Stress distribution at Pressure 37 Bar

Fig. 22 Stress distribution at Pressure 40Bar

6 CONCLUSIONS

In this paper the design of the metal bellow and

theoretical evaluation of the stresses is done using

EJMA standards and the numerical evaluation is done

by using ANSYS WORKBENCH software. The results

obtained are compared and there is slight variation in

the stress values and is due to different approaches

followed. From the theoretical calculations it can be said

that the maximum stress value is Meridional bending

stress and this value is checked with the allowable stress

value so that the bellow does not fail.

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Fatigue, Volume.28, 2004, pp.28–32.

[15] C. Becht IV, “Fatigue of Bellows, a new Design

Approach”, International Journal of Pressure Vessels

and Piping, Volume.77, 2000, pp.843-850.

[16] EJMA, “Standards of Expansion Joint Manufacturers

Association, Expansion Joint Manufacturers

Association”, New York, NY, USA, 9th edition, 2008. [17] www.haynesintl.com

ISSN: 2180-1053 Vol. 9 No.2 July – December 2017 87

EFFECT OF PULSE CURRENT MICRO PLASMA ARC WELDING

PARAMETERS ON PITTING CORROSION RATE OF AISI 321

SHEETS IN 3.5 N NACL MEDIUM

Kondapalli Siva Prasad1*

, Chalamalasetti Srinivasa Rao2

1*

Associate Professor, Department of Mechanical Engineering, Anil Neerukonda

Institute of Technology & Sciences, Visakhapatnam, India

2 Professor, Department of Mechanical Engineering, Andhra University,

Visakhapatnam, India

ABSTRACT

Austenitic stainless steel sheets are used for fabrication of components, which

require high temperature resistance and corrosion resistance such as metal

bellows used in expansion joints in aircraft, aerospace and petroleum industries.

When they are exposed to sea water after welding they are subjected to corrosion

as there are changes in properties of the base metal after welding. The corrosion

rate depends on the chemical composition of the base metal and the nature of

welding process adopted. Corrosion resistance of welded joints can be improved

by controlling the process parameters of the welding process. In the present work

Pulsed Current Micro Plasma Arc Welding (MPAW) is carried out on AISI 321

austenitic stainless steel of 0.3 mm thick. Peak current, Base current, Pulse rate

and Pulse width are chosen as the input parameters and pitting corrosion rate of

weldment in 3.5N NaCl solution is considered as output response. Pitting

corrosion rate is computed using Linear Polarization method from Tafel plots.

Response Surface Method (RSM) is adopted by using Box-Behnken Design and

total 27 experiments are performed. Empirical relation between input and output

response is developed using statistical software and its adequacy is checked

using Analysis of Variance (ANOVA) at 95% confidence level. The main effect

and interaction effect of input parameters on output response are also studied.

KEYWORDS: Plasma Arc Welding; Austentic Stainless Steel; Pitting Corrosion Rate

1.0 INTRODUCTION

Austenitic Stainless Steel (ASS), being the widest in use of all the stainless steel groups

finds application in the beverages industry, petrochemical, petroleum, food processing

and textile industries amongst others. It has good tensile strength, impact resistance and

wear resistance properties. In addition, it combines these with excellent corrosion

resistant properties (Dillon, C. P., 1994).

* Corresponding author: kspanits@gmail.com

Journal of Mechanical Engineering and Technology

ISSN: 2180-1053 Vol. 9 No.2 July – December 2017 88

Welding is one of the most employed methods of fabricating ASS components. ASS is

largely highly weldable; the higher the carbon content, the harder the SS and so the

more difficult it is to weld. The problem commonly encountered in welded ASS joints is

intergranular corrosion, pitting and crevice corrosion in severe corrosion environments.

Weld metals of ASS may undergo precipitation of (CrFe)23C6 at the grain boundaries,

thus depleting Cr and making the SS weldment to be preferentially susceptible to

corrosion at the grain boundaries. There may also be the precipitation of the brittle

sigma Fe-Cr phase in their microstructure if they are exposed to high temperatures for a

certain length of time as experienced during welding. High heat input welding

invariably leads to slow cooling. During this slow cooling time, the temperature range

of 700 - 850oC stretches in time and with it the greater formation of the sigma phase

(Pickering, F.B., 1985).

In Pulsed current MPAW process, the interfuse of metals was produced by heating them

with an arc using a non consumable electrode. It is widely used welding process finds

applications in welding hard to weld metals such as aluminium, stainless steel,

magnesium and titanium (H. B. Cary, 1989). The increased use of automated welding

urges the welding procedures and selection of welding parameters must be more

specific for good weld quality and precision with minimum cost (Z. Samati, 1986) . The

bead geometry plays an important role in determining the microstructure of the welded

specimen and the mechanical properties of the weld (P.J. Konkol and G. F.

Koons,1978). The proper selection of the input welding parameters which influence the

properties of welded specimen ensure a high quality joint. Stainless steels are corrosive

resistance in nature finds diversified application. Even stainless posses good resistance,

they are yet susceptible to pitting corrosion. The pitting corrosion is a localized

dissolution of an oxide-covered metal in specific aggressive environments. It is most

common and cataclysmic causes of failure of metallic structures. The detection and

monitoring of pitting corrosion is an important task in determining the weld quality. The

pitting corrosion is a random, sporadic and stochastic process and their prediction of the

time and location of occurrence remains extremely difficult and undefined (Fong –

Yuan Ma, 2012, Rao, P. S, 2004, Srinivasa Rao, P., O. P. Gupta, and S. S. N.

Murty,2005).

Stainless steels may also suffer from different forms of metallurgical changes when

exposed to critical temperatures. In welding, the heat affected zone often experiences

temperatures which cause sufficient microstructural changes in the welded plates. The

precipitation of chromium nitrides, carbides and carbonitrides in the parent metal occur

under various welding and environmental conditions and also depends on the grades of

stainless steel. During pulsed MPAW process, the formation of coarse grains and inter

granular chromium rich carbides along the grain boundaries in the heat affected zone

deteriorates the mechanical properties.

In the present paper the effect of welding parameters namely peak current, base current,

pulse rate and pulse width on pitting corrosion rate of AISI 321 sheets are studied.

Linear polarization method is adopted in measuring the pitting corrosion rate.

Effect of Pulse Current Micro Plasma Arc Welding Parameters on Pitting Corrosion Rate of

AISI 321 Sheets In 3.5 N NACL Medium

ISSN: 2180-1053 Vol. 9 No. 2 July – December 2017 89

2.0 WELDING PROCEDURE

Weld specimens of 100 x 150 x 0.3mm size are prepared from AISI 321 sheets and

joined using square butt joint. The chemical composition and tensile properties of AISI

321 stainless steel sheets are presented in Table .1 & 2. Argon is used as a shielding gas

and a trailing gas to avoid contamination from outside atmosphere. The welding

conditions adopted during welding are presented in Table .3. From the earlier works

(K.Siva Prasad, Ch.Srinivasa Rao, D.Nageswara Rao, 2013,2014) carried out on Pulsed

Current MPAW it was understood that the peak current, back current, pulse rate and

pulse width are the dominating parameters which effect the weld quality characteristics.

The values of process parameters used in this study are the optimal values obtained

from our earlier papers (K.Siva Prasad, Ch.Srinivasa Rao, D.Nageswara Rao,

2013,2014). Hence peak current, back current, pulse rate and pulse width are chosen as

parameters and their levels are presented in Table .4. Details about experimental setup

are shown in Figure1 . Four factors and three levels are considered and according to

Box-Benhken Design matrix, 27 experiments are performed as per the Design matrix

shown in Table 5.

Figure 1. Micro Plasma Arc Welding Setup.

Table 1. Chemical composition of AISI 321 (weight %)

C Si Mn P S Cr Ni N

0.05 0.52 1.30 0.028 0.021 17.48 9.510 0.04

Table 2. Mechanical properties of AISI 321

Elongation

(%)

Yield Strength

(MPa)

Ultimate Tensile Strength

(Mpa)

53.20 272.15 656.30

Welding fixture

Power source

Plasma and shielding

gas cylinders

Journal of Mechanical Engineering and Technology

ISSN: 2180-1053 Vol. 9 No.2 July – December 2017 90

Table 3. Welding conditions

Power source Secheron Micro Plasma Arc Machine (Model:

PLASMAFIX 50E)

Polarity DCEN

Mode of operation Pulse mode

Electrode 2% thoriated tungsten electrode

Electrode Diameter 1mm

Plasma gas Argon & Hydrogen

Plasma gas flow rate 6 Lpm

Shielding gas Argon

Shielding gas flow rate 0.4 Lpm

Purging gas Argon

Purging gas flow rate 0.4 Lpm

Copper Nozzle diameter 1mm

Nozzle to plate distance 1mm

Welding speed 260mm/min

Torch Position Vertical

Operation type Automatic

Table 4. Process parameters and their limits

Levels

Input Factor Units -1 0 +1

Peak Current Amperes 6 7 8

Base Current Amperes 3 4 5

Pulse rate Pulses /Second 20 40 60

Pulse width % 30 50 70

3.0 MEASUREMENT OF PITTING CORROSION RATE

Welded joints of stainless steel are subjected to pitting corrosion when exposed to

different environments. The pitting corrosion rate depends upon the type, concentration

of the exposed environment and exposure time of the welded joint. The details about

sample preparation and testing procedure for measurement of pitting corrosion rate are

discussed in the following sections.

3.1 Surface Preparation for Plating

The welded test specimen surface is polished with 220 and 600 mesh size emery papers

in the presence of distilled water continuously. The polished specimen is first rinsed

with distilled water, cleaned with acetone and again rinsed with distilled water to

remove the stains and grease. Finally the specimen is dried to remove the moisture

content on the surface of the sample.

Effect of Pulse Current Micro Plasma Arc Welding Parameters on Pitting Corrosion Rate of

AISI 321 Sheets In 3.5 N NACL Medium

ISSN: 2180-1053 Vol. 9 No. 2 July – December 2017 91

3.2 Sample Preparation for Corrosion studies

Once the sample is cleaned, the entire sample is covered by insulating film and only a

cross sectional area of 225 mm2 is exposed as shown in Figure 2. The perplex tube as

shown in Figure 3 is attached to the test specimen.

Figure 2. Dimensions of corrosion test specimen.

Figure 3. Setup of perplex tube

3.3 Procedure for Corrosion Studies

The electrochemical cell (test specimen with tube) is initially washed with distilled

water followed by rinsing with filtered electrolyte NaCl. Around 100 ml of filtered

electrolyte is poured into the electrochemical cell. The entire electrode assembly is now

placed in the cell. The reference electrode (standard calomel electrode) is adjusted in

such a way that the tip of this electrode is very near to the exposed area of working

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ISSN: 2180-1053 Vol. 9 No.2 July – December 2017 92

electrode (test specimen). The auxiliary platinum electrode is also placed in the cell.

Now the cell assembly has been connected to the AUTOLAB/PGSTAT12. The black

colored plug has been connected to the auxiliary electrode, red colored plug to the

working electrode and blue to the reference electrode. The sample has been exposed to

electrolytic medium for a span of 2 hours.

As the start button of the potentiostat is switched on, the electrode potential changes

continuously, till the reaction between the electrode and the medium attains equilibrium.

After some time the potential remains nearly constant without any change. This steady

potential which is displayed on the monitor is taken as open circuit potential (Erest).

Now the equipment is ready for obtaining the polarization data.

Potential is scanned cathodically until the potential is equal to Erest minus the limit

potential. Measurements of potential (E) and current (I) are made at different intervals

and the data is displayed on the monitor itself as E vs log I plot. After reaching the

cathodic limit, the scan direction is then reversed. Similarly anodic polarization data is

obtained. The scan is again reversed and finally terminated and the cell is isolated from

the potentiostat when the potential reached Erest. The data recorded gives the Tafel plot

(current vs potential data). Using the software available corrosion rate, corrosion

current, polarization resistance and Tafel slopes are evaluated by Tafel plot methods.

The Experimental setup is shown in Figure 4.

Figure 4. Pitting corrosion setup

3.4 Corrosion Testing Methodology

Passive metals may become susceptible to pitting corrosion when exposed to solutions

having a critical content of aggressive ions such as chloride. This type of corrosion is

potential-dependent and its occurrence is observed only above the pitting potential

(Ecorr), which can be used to differentiate the resistance to pitting corrosion of

Effect of Pulse Current Micro Plasma Arc Welding Parameters on Pitting Corrosion Rate of

AISI 321 Sheets In 3.5 N NACL Medium

ISSN: 2180-1053 Vol. 9 No. 2 July – December 2017 93

differentmetal/electrolyte systems. The Ecorr value can be determined electrochemically

using both potentiostatic and potentiodynamic techniques.

3.5 Linear Polarization method

The linear polarization method utilizes the Tafel extrapolation technique. The

electrochemical technique of polarization resistance is used to measure absolute

corrosion rate, usually expressed in milli-inches per year (mpy), which is further

converted in to mm per year. Polarization resistance can be measured very rapidly,

usually less than ten minutes. Excellent correlation can often be made between

corrosion rates obtained by polarization resistance and conventional weight-change

determinations. Polarization resistance is also referred to as “linear polarization”.

Polarization resistance measurement is performed by scanning through a potential range

which is very close to the corrosion potential, Ecorr the potential range is generally ±25

mV about Ecorr. The resulting current vs. potential is plotted. The corrosion current, Icorr

is related to the slope of the plot through the following equation

cacorr

ca

ββ2.3I

ββ

ΔI

ΔE

(1)

where ∆E/∆I = slope of the polarization resistance plot, where ∆E is expressed

in volts and ∆I in µA. This slope has units of resistance, hence, polarization Resistance.

βa,βc are anode and cathode Tafel constants (must be determined from a Tafel plot as

shown in Figure5).

These constants have the units of volts/decade of current. Icorr= corrosion current, µA.

Rearranging equation (1)

E

I

)(3.2I

ca

ca

corr

(2)

The corrosion current can be related to the corrosion rate through the following

equation.

Corrosion rate (mpy) = 0.131(Icorr)(Eq.Wt)/ρ (3)

where

Eq.Wt = equivalent weight of the corroding species

ρ = density of the corroding species, g/cm3

Icorr = corrosion current density, µA/cm2

Journal of Mechanical Engineering and Technology

ISSN: 2180-1053 Vol. 9 No.2 July – December 2017 94

Figure 5. Modal Tafel plot

4.0 STATISTICAL ANALYSIS

The pitting corrosion rates for all the 27 samples are performed and presented in Table

5.

Table 5. Design matrix with experimental results

Experiment

No.

Peak

Current

(Amperes)

Base

current

(Amperes)

Pulse

Rate

(Pulses/

second)

Pulse

width

(%)

Pitting Corrosion Rate

(mm/year)

Experimental Predicted

1 6 3 40 50 0.16938 0.16298

2 8 3 40 50 0.17676 0.17423

3 6 5 40 50 0.16686 0.16735

4 8 5 40 50 0.17266 0.17702

5 7 4 20 30 0.16366 0.16320

6 7 4 60 30 0.16966 0.16628

7 7 4 20 70 0.16836 0.17087

8 7 4 60 70 0.16566 0.16390

9 6 4 20 50 0.16716 0.16820

10 8 4 60 50 0.17966 0.17672

11 6 4 20 50 0.16716 0.16820

12 8 4 20 50 0.16756 0.16670

13 7 3 60 30 0.17802 0.17666

14 7 5 40 30 0.16786 0.16650

15 7 3 40 50 0.16536 0.16644

16 7 5 40 50 0.17046 0.17002

17 6 4 40 30 0.14716 0.15090

Effect of Pulse Current Micro Plasma Arc Welding Parameters on Pitting Corrosion Rate of

AISI 321 Sheets In 3.5 N NACL Medium

ISSN: 2180-1053 Vol. 9 No. 2 July – December 2017 95

18 8 4 40 30 0.17386 0.17667

19 6 4 40 70 0.16876 0.16885

20 8 4 40 70 0.16486 0.16402

21 7 3 20 50 0.16826 0.16934

22 7 5 20 50 0.17966 0.17530

23 7 3 60 50 0.16166 0.16978

24 7 5 60 50 0.16966 0.17098

25 7 4 40 50 0.16326 0.16119

26 7 4 40 50 0.15916 0.16119

27 7 4 40 50 0.16216 0.16119

4.1 Empirical Mathematical Modeling

In RSM design, mathematical models are developed using polynomial equations. The

type of polynomial equation depends on the problem.

In most RSM problems (M Balasubramanian,V Jayabalan,V Balasubramanian, 2007,

2008) the type of the relationship between the response (Y) and the independent

variables is unknown. Thus the first step in RSM is to find a suitable approximation for

the true functional relationship between the response and the set of independent

variables.

Usually, a low order polynomial is some region of the independent variables is

employed to develop a relation between the response and the independent variables. If

the response is well modeled by a linear function of the independent variables then the

approximating function in the first order model is

Y = bo+bi xi + (4)

where bo, biare the coefficients of the polynomial and represents noise or error.

If interaction terms are added to main effects or first order model, then the model is

capable of representing some curvature in the response function, such as

Y = bo+bi xi + bijxixj+ (5)

A curve results from Equation -5 by twisting of the plane induced by the interaction

term bijxixj. There are going to be situations where the curvature in the response function

is not adequately modeled by Equation -5. In such cases, a logical model to consider is

Y = bo+bi xi +biixi2 + bijxixj+ (6)

where bii represent pure second order or quadratic effects. Equation -3 represents a

second order response surface model. Using MINITAB Ver.14, statistical software, the

significant coefficients are determined and final model is developed incorporating these

coefficients to estimate the pitting corrosion rate. In the empirical model only

significant coefficients are considered.

Journal of Mechanical Engineering and Technology

ISSN: 2180-1053 Vol. 9 No.2 July – December 2017 96

Pitting Corrosion Rate = 0.314280 – 0.016337X1-0.059210X2-0.002214X3

+0.001769X4+ 0.007036X22

+ 0.000299X1X3 -0.000382X1X4

where X1, X2, X3 and X4 are the coded values of peak current, base current, pulse rate

and pulse width respectively.

4.2 Checking the adequacy of the developed model for pitting corrosion rate.

The adequacy of the developed models is tested using the ANOVA. As per this

technique, if the calculated value of the Fratio of the developed model is less than the

standard Fratio(F-table value 4.60) value at a desired level of confidence of 95%, then the

model is said to be adequate within the confidence limit. ANOVA test results are

presented in Table 6.1 for pitting corrosion rate. From Table 6 it is understood that the

developed mathematical models are found to be adequate at 95% confidence level.

Coefficient of determination ‘ R2’ is used to find how close the predicted and

experimental values lie. The value of ‘ R2’ for the above developed models is found to

be about 0.84, which indicates a good correlation to exist between the experimental

values and the predicted values. Figure6 indicate the scatter plots for pitting corrosion

rate of the weld joint and reveal that the actual and predicted values are close to each

other within the specified limits.

Table 6 ANOVA test results for pitting corrosion rate

Source DF Seq SS Adj SS Adj MS F P

Regression 14 0.000949 0.000949 0.000068 3.63 0.016

Linear 4 0.000206 0.000294 0.000074 3.94 0.029

Square 4 0.000338 0.000267 0.000067 3.57 0.039

Interaction 6 0.000404 0.000404 0.000067 3.61 0.028

Residual Error 12 0.000224 0.000224 0.000019

Lack-of-Fit 9 0.000215 0.000215 0.000024 7.96 0.057

Pure Error 3 0.000009 0.000009 0.000003

Total 26 0.001173

Effect of Pulse Current Micro Plasma Arc Welding Parameters on Pitting Corrosion Rate of

AISI 321 Sheets In 3.5 N NACL Medium

ISSN: 2180-1053 Vol. 9 No. 2 July – December 2017 97

PREDICTED

EX

PER

IMEN

TA

L

0.1800.1750.1700.1650.1600.1550.150

0.180

0.175

0.170

0.165

0.160

0.155

0.150

Scatterplot of Corrosion Rate

Figure 6. Scatter plot of pitting corrosion rate

5.0 RESULTS AND DISCUSSIONS

Effect of welding parameters on pitting corrosion rate is indicated by the main effect

plot as shown in Figure7.

5.1 Main effect plots

From Figure 7, it is understood that the variation of each individual parameter on pitting

corrosion rate can be assessed. Pitting corrosion rate increase with the peak current from

6 Amps to 8 Amps. This is because as the current increases heat input increases. At

higher heat input, precipitation of (CrFe)23C6 at the grain boundaries takes place, thus

depleting Cr and making the weldment to be preferentially susceptible to corrosion at

the grain boundaries. Pitting corrosion rate decrease with the base current from 3 Amps

to 4 Amps, afterwards it increases. The variation is because, at low base current

generally low peak current will be used, however as the purpose of base current is to

maintain the arc, instead of melting the workpice, the Pitting corrosion rate tends to

decrease as precipitation of (CrFe)23C6 is low. But when the base current crosses over 4

Amps, corresponding peak current will increase leading to more precipitation of

(CrFe)23C6 and hence the pitting corrosion rate increases upto 5 Amps. Pitting corrosion

rate decrease with the pulse rate from 20 pulses/sec to 40 pulses/sec. This may be

because, at low pulse rate, the current variation between base current and peak current is

less, which leads to low heating of the base metal. However, when the pulse rate is

above 40 pulses/sec, the current variation between base current and peak current is high,

which leads to more melting of base metal and precipitation of (CrFe)23C6 at the grain

boundaries. Pitting corrosion rate increase with the pulse width from 20% to 50 %. This

is because as the pulse with increases, the peak current duration will be more in pulsed

mode, leading to more melting and high correction rate. When the pulse width crosses

Journal of Mechanical Engineering and Technology

ISSN: 2180-1053 Vol. 9 No.2 July – December 2017 98

50 % , it shows a negative trend because of high time gap for cooling the base metal,

which leads to lower precipitation of (CrFe)23C6 at the grain boundaries.

Co

rro

sio

n R

ate

(mm

/Y

ea

r)

876

0.172

0.170

0.168

0.166

0.164

543

604020

0.172

0.170

0.168

0.166

0.164705030

Peak Current(Amps) Base Current (Amps)

Pulse Rate (pulses/sec) Pulse Width(%)

Main Effects Plot for Corrosion Rate(mm/Year)

Figure 7. Main effect plot of pitting corrosion rate

5.2 Contour plots of pitting corrosion rate of 3.5N NaCl

The simultaneous effect of two parameters at a time on the output response is generally

studied using contour plots and surface plots. Contour plots play a very important role

in the study of the response surface. By generating contour plots using statistical

software (MINITAB Ver.14) for response surface analysis, the most influencing

parameter can be identified based on the orientation of contour lines. If the contour

patterning of circular shaped contours occurs, it suggests the equal influence of both the

factors; while elliptical contours indicate the interaction of the factors. Figure’s 8a to 8f

represent the contour plots for pitting corrosion rates. From these plots, the interaction

effect between the input process parameters and output response can be observed as:

(i) From Figure 8a, it is understood that Pitting corrosion rate is more sensitive to

change in peak current than in the base current, since the contour lines are more

diverted towards peak current.

(ii) From Figure 8b, it is understood that Pitting corrosion rate is sensitive to peak

current than in the pulse rate , since the contour lines are more diverted towards

peak current.

(iii) From Figure 8c, it is understood that Pitting corrosion rate is more sensitive to

peak current than pulse width , since the contour lines are more diverted towards

peak current.

(iv) From Figure 8d, it is understood that Pitting corrosion rate is more sensitive to

pulse rate than base current, since the contour lines are more diverted towards

pulse rate.

Effect of Pulse Current Micro Plasma Arc Welding Parameters on Pitting Corrosion Rate of

AISI 321 Sheets In 3.5 N NACL Medium

ISSN: 2180-1053 Vol. 9 No. 2 July – December 2017 99

(v) From Figure 8e, it is understood that Pitting corrosion rate is more sensitive to

pulse width than base current , since the contour lines are more diverted towards

pulse width.

(vi) From Figure 8f, it is understood that Pitting corrosion rate is more sensitive to

pulse width than pulse rate , since the contour lines are more diverted towards

pulse width. From the above welding parameters considered, it is understood

that peak current is the most important parameter which affects the pitting

corrosion rate of the welded joints.

Peak Current(Amps)

Ba

se

Cu

rre

nt

(Am

ps)

0.190.18

0.17

8.07.57.06.56.0

5.0

4.5

4.0

3.5

3.0

Contour Plot of Corrosion Rate

Figure 8a. Contour plot for peak current vs base current for corrosion rate (3.5N NaCl).

Peak Current(Amps)

Pu

lse

Ra

te (

pu

lse

s/

se

c)

0.345

0.330

0.315

0.300

8.07.57.06.56.0

60

50

40

30

20

Contour Plot of Corrosion Rate

Figure 8b. Contour plot for peak current vs pulse rate for corrosion rate (3.5N NaCl).

Journal of Mechanical Engineering and Technology

ISSN: 2180-1053 Vol. 9 No.2 July – December 2017 100

Peak Current(Amps)

Pu

lse

Wid

th(%

)0.280

0.275

0.270

0.265

0.260

8.07.57.06.56.0

70

60

50

40

30

Contour Plot of Corrosion Rate

Figure 8c. Contour plot for peak current vs pulse width for corrosion rate(3.5N NaCl).

Base Current (Amps)

Pu

lse

Ra

te (

pu

lse

s/

se

c)

0.14

0.12

0.10

0.08

5.04.54.03.53.0

60

50

40

30

20

Contour Plot of Corrosion Rate

Figure 8d. Contour plot for base current vs pulse rate for corrosion (3.5N NaCl).

Base Current (Amps)

Pu

lse

Wid

th(%

)

0.38

0.360.34

0.32

0.30

0.28

5.04.54.03.53.0

70

60

50

40

30

Contour Plot of Corrosion Rate

Figure 8e. Contour plot for base current vs pulse width for corrosion rate (3.5N NaCl).

Effect of Pulse Current Micro Plasma Arc Welding Parameters on Pitting Corrosion Rate of

AISI 321 Sheets In 3.5 N NACL Medium

ISSN: 2180-1053 Vol. 9 No. 2 July – December 2017 101

Pulse Rate (pulses/sec)

Pu

lse

Wid

th(%

)

0.39 0.36

0.33

0.30

0.27

6050403020

70

60

50

40

30

Contour Plot of Corrosion Rate

Figure 8f. Contour plot for pulse rate vs pulse width for corrosion rate (3.5N NaCl).

5.3 Surface plots

Surface plots help in locating the maximum and minimum value of the response. The

maximum value of the response is represented by the apex of the surface plot, whereas

the minimum value is indicated by nadir of the surface plot. The minimum pitting

corrosion rate is indicated by the nadir of the response surface, as shown in Figure 9a to

Figure 9f. Figure 9a the minimum pitting corrosion rate is exhibited by the nadir of the

response surface. It can be seen from the twisted plane of surface plot that the model

contains interaction. From the response plot, it is identified that at a peak current of 6

Amps and base current of 4 Amps, pitting corrosion rate is minimum. Figure 9b depicts

that at a peak current of 6 Amps and pulse rate of 20 pulses/second, pitting corrosion

rate is minimum. Figure 9c shows the three dimensional response surface plot it can be

seen from the twisted plane of surface plot that the model contains interaction. From the

response plot, it is identified that at the peak current of 8 Amps and pulse width of 60%,

pitting corrosion rate is minimum. Figure 9d indicates that at a base current of 3 Amps

and pulse rate of 60 pulses/second, pitting corrosion rate is minimum. Figure 9e

represents that at a base current is 3 Amps and pulse width of 30 %, pitting corrosion

rate is minimum. Figure 5.41f discusses that when pulse rate is 60 pulses/second and

pulse width of 30 %, the pitting corrosion rate is minimum. It is clear from the above

observations, that for a peak current of 6 Amps, base current of 3 Amps, pulse rate of 60

pulses/second and pulse width of 30 % minimum pitting corrosion rate is achieved.

Journal of Mechanical Engineering and Technology

ISSN: 2180-1053 Vol. 9 No.2 July – December 2017 102

50.165

4

0.175

0.185

0.195

Base Current (Amps)67 3

8Peak Current(Amps)

Surface Plot of Corrosion Rate

corrosion rate (mm/year)

Figure 9a. Surface plot for peak current vs base current for corrosion rate (3.5N NaCl).

600.30

40

0.32

0.34

0.36

6 Pulse Rate (pulses/sec)7 20

8Peak Current(Amps)

Surface Plot of Corrosion Rate

corrosion rate (mm/year)

Figure 9b.Surface plot for peak current vs pulse rate for corrosion rate (3.5N NaCl).

60

0.26

0.27

0.28

45

0.29

Pulse Width(%)67 30

8Peak Current(Amps)

Surface Plot of Corrosion Rate

corrosion rate (mm/year)

Figure 9c. Surface plot for peak current vs pulse width for corrosion rate (3.5N NaCl)

Effect of Pulse Current Micro Plasma Arc Welding Parameters on Pitting Corrosion Rate of

AISI 321 Sheets In 3.5 N NACL Medium

ISSN: 2180-1053 Vol. 9 No. 2 July – December 2017 103

600.06

0.09

40

0.12

0.15

3 Pulse Rate (pulses/sec)4 20

5

Base Current (Amps)

Surface Plot of Corrosion Rate

corrosion rate (mm/year)

Figure 9d. Surface plot for base current vs pulse rate for corrosion rate (3.5N NaCl)

600.25

0.30

0.35

45

0.40

Pulse Width(%)34 30

5

Base Current (Amps)

Surface Plot of Corrosion Rate

corrosion rate (mm/year)

Figure 9e.Surface plot for base current vs pulse width for corrosion rate (3.5N NaCl).

600.25

0.30

0.35

45

0.40

Pulse Width(%)2040 30

60

Pulse Rate (pulses/sec)

Surface Plot of Corrosion Rate

corrosion rate (mm/year)

Figure 9f. Surface plot for pulse rate vs pulse width for corrosion rate (3.5N NaCl).

Journal of Mechanical Engineering and Technology

ISSN: 2180-1053 Vol. 9 No.2 July – December 2017 104

5.4 Microscopic analysis of weld joint

Figure 10a and 10b indicate the weld joint before corrosion and after pitting corrosion.

The dark round spots indicates the area where pitting corrosion has taken place.

Scanning Electron Microscope (SEM) analysis is carried out to identify the depleting of

Cr % after the weld joint is subjected to pitting corrosion in 3.5N NaCl solution. Figure

11a and 11b indicate the SEM images before and after corrosion and the chemical

compositions. It is observed that depletion of 3.15 % (wt.%) of Cr takes place because

of corrosion.

Figure.10a Weld joint before corrosion

Figure.10b Weld joint after corrosion

100X

100X

Corrosion pit

Effect of Pulse Current Micro Plasma Arc Welding Parameters on Pitting Corrosion Rate of

AISI 321 Sheets In 3.5 N NACL Medium

ISSN: 2180-1053 Vol. 9 No. 2 July – December 2017 105

Figure11a. SEM before corrosion Figure11b. SEM after corrosion

From SEMEDAX, the chemical composition obtained of base metal and weld

joint after corrosion are shown in Table7 and Table8

Table 7. Chemical compositions before corrosion

Element O Na Si Cl Ti Cr Mn Fe Ni Mo

Weight % 6.36 1.46 0.76 1.21 1.25 15.23 1.31 60.30 9.28 2.34

Atomic % 19.87 2.94 1.26 1.58 1.21 13.57 1.11 50.02 7.32 1.13

Table 8. Chemical compositions after corrosion

Element O Na Si Cl Ti Cr Mn Fe Ni Al Cu

Weight % 12.54 16.99 0.35 8.03 0.18 12.08 0.72 42.17 5.32 0.21 0.41

Atomic % 27.04 25.49 0.43 8.79 0.13 8.01 0.45 26.04 3.12 0.27 0.22

6.0 CONCLUSIONS

The following conclusions are drawn from the experiments performed and statistical

analysis. An empirical mathematical model for predicting pitting corrosion rate of

pulsed current MPAW AISI 321 sheets in 3.5 N NaCl medium has been developed.

From the main effect plots, it is understood that peak current is the important parameter

which influences the corrosion rate, followed by base current, pulse rate and pulse

width. Corrosion rate increased gradually with peak current from 6 Amps to 8 Amps,

this is because of more heat input leading wider weld fusion area and higher Heat

Affected Zone (HAZ). Corrosion rate decreased from Base current of 3 Amps to 4

Amps and there after it increased upto 5 Amps. This is because of variation of heat

input. At 4 Amps of Base current the peak current and base current combination is

optimal. Corrosion rate decreased from pulse rate of 20 pulses/sec to 40 pulses/sec and

there after it increased upto 60 pulses/sec. Too low pulse rate leads to over melting of

weld joint and similarly too pulse rate leads to lack of fusion. Corrosion rate increased

from Pulse width of 30% to 50% and there after it decreased to 70%. Too low pulse

width leads to overlapping of weld joint and similarly too high pulse width leads to lack

of fusion and gaps between the weld joint. From the contour plots, it is clear that peak

current is the most important parameter which affects the pitting corrosion rate of the

Journal of Mechanical Engineering and Technology

ISSN: 2180-1053 Vol. 9 No.2 July – December 2017 106

welded joints, followed by base current, pulse rate and pulse width. From the surface

plots, it is understood that for a peak current of 6 Amps, base current of 3 Amps, pulse

rate of 60 pulses/second and pulse width of 30 % minimum pitting corrosion rates are

obtained for both AISI 316Ti and AISI 321. The optimal welding conditions obtained

are out of the 27 combinations as per design matrix; however their values are within the

range of the chosen values of welding variables. From SEMEDAX, it is observed that

there is depletion of depletion of 3.15 % (wt%) chromium after corrosion was noticed in

AISI 321. This is due to high heat input generated because of welding current. The

developed empirical mathematical model is valid for the chosen material, however the

accuracy can be improved by considering more number of factors and their levels.

7.0 REFERENCES

Dillon, C. P., (1994), Corrosion Control in the Chemical Process Industry, NACE

International, Houston,Texas.

Fong – Yuan Ma,(2012), Corrosive Effects of Chlorides on Metals, Pitting Corrosion,

Nasr Bensalah (Ed.),Intech open.

H. B. Cary, (1989),Modern Welding Technology, Prentics Hall, New Jersey.

K.Siva Prasad, Ch.Srinivasa Rao, D.Nageswara Rao,(2013), Optimization of pulsed

current parameters to minimize pitting corrosion in pulsed current micro plasma

arc welded AISI 304L sheets using genetic algorithm, International Journal of

Lean Thinking, 4(1),9-19.

K.Siva Prasad, Ch.Srinivasa Rao, D.Nageswara Rao,(2013), Effect of Welding

Parameters on Pitting Corrosion Rate in 3.5N NaCl of Pulsed Current Micro

Plasma Arc Welded AISI 304L Sheets, Journal of Manufacturing Science and

Production, 13(1-2),15-23. Kondapalli Siva Prasad, Ch.Srinivasa Rao, D.Nageswara Rao,(2013), Application of

Grey Relational Analysis for Optimizing Weld pool geometry parameters of

Pulsed Current Micro Plasma Arc Welded AISI 304L stainless steel sheets,

International Journal of Advanced Design and Manufacturing Technology ,

6(1),pp.79-86.

Kondapalli Siva Prasad, Ch.Srinivasa Rao, D.Nageswara Rao, (2014), Multi-objective

Optimization of Weld Bead Geometry Parameters of Pulsed Current Micro

Plasma Arc Welded AISI 304L Stainless Steel Sheets Using Enhanced Non-

dominated Sorting Genetic Algorithm, Journal of Manufacturing science and

production, 14(2), 79-85.

M Balasubramanian, V Jayabalan, V Balasubramanian, (2007), Response Surface

Approach to optimize the pulsed current gas tungsten arc welding parameters if

Ti-6Al-4V titanium alloy, Metals and Materials International, 13(4), 335-344.

Effect of Pulse Current Micro Plasma Arc Welding Parameters on Pitting Corrosion Rate of

AISI 321 Sheets In 3.5 N NACL Medium

ISSN: 2180-1053 Vol. 9 No. 2 July – December 2017 107

M Balasubramanian,V Jayabalan,V Balasubramanian, (2008), A mathematical model to

predict impact toughness of pulsed current gas tungsten arc welded titanium

alloy, International Journal of Advanced Manufacturing Technology, 35, 852-

858.

P.J. Konkol and G. F. Koons,(1978), Optimization of Parameters for Two Wire AC-

ACSAW, American Welding Journal, 27 , 367s – 374s.

Pickering, F.B., (1985), Stainless Steel ‘84’. The Institute of Metals: pp. 2. London.

Rao, P. S,(2004), Development of arc rotation mechanism and experimental studies on

pulsed GMA welding with and without arc rotation, Diss. Ph. D. thesis,IIT

Kharagpur.

Srinivasa Rao, P., O. P. Gupta, and S. S. N. Murty,(2005), Influence of process

parameters on bead geometry in pulsed gas metal arc welding", IIW

International Congress.

Z. Samati, (1986),Automatic Pulsed MIG Welding, Metal Construction, 38R- 44.

DOI: 10.4018/IJSITA.2017100101

International Journal of Strategic Information Technology and ApplicationsVolume 8 • Issue 4 • October-December 2017

Copyright©2017,IGIGlobal.CopyingordistributinginprintorelectronicformswithoutwrittenpermissionofIGIGlobalisprohibited.

Privacy Preserving Data Mining Using Time Series Data AggregationSivaranjani Reddi, ANITS, Bheemunipatnam, India

ABSTRACT

Thisarticleproposesamechanismtoprovideprivacytominedresultsbyassumingthatthedataisdistributedacrossmanynodes.Thefirstobjectiveincludesminingthequeryresultsbythenodeinacluster,communicatingittotheclusterhead,aggregatingthedatacollectedfromalltheclusternodesandthencommunicatingittothegroupcontroller.Thesecondobjectiveistoincorporateprivacyateachleveloftheclustersnode:clusterheadandthegroupcontrollerlevel.Thefinalobjectiveistoprovideadynamicnetworkfeature,wherethenodescanjoinorleavethedistributednetworkwithoutdisturbingthenetworkfunctionality.TheproposedalgorithmwasimplementedandvalidatedinJavaforitsperformanceintermsofcommunicationcostscomputationalcomplexity.

KeywoRDSData Mining, Encryption, Group Controller, Group Theory, Privacy

INTRoDUCTIoN

Manyreallifeapplicationsofdataminingisfacingproblemstowardstheprivacypreservationofthedata(Anderson,2010;Acs&Castelluccia,2011;Dansana,2012;vanDijketal.,2010;Chowdhuri,2014;Sarkaretal.,2017).Itincludes,firstly,certainattributesofthedataorattributesthatmightleakthepersonalrecognizableinformation.Secondly,thedatacanbesplitacrossmultiplenodeseitherhorizontallyorvertically,andmaynotallowthedatatransfertoanotherside.Finally,usageofdatamodelmighthaverestrictiononrules,andfewrulesmayleadtolawviolationinordertoaccessindividualprofiling.Privacypreservingbaseddatamining(PPDM)(Agrawal,1994)hasarisentodiscusstheabove-mentionedissues.MajorityofthePPDMtechniquesarethemodifiedversionsofthestandarddataminingalgorithms,wherethemodificationincludesthecryptographicmechanismswhichguarantee theprivacyfor theapplication. Inmanycases, restraintsPPDMare:preservingdataaccuracyandretainingminingprocessperformancewhilemaintainingtheprivacyrestrictions.CopiousmethodologiesusedbyPPDMcanbesummarizedbasedonfollowingdimensions:

• Data Distribution:Thisdimensionconcentratesondatadistribution.Theapproachesadopteithercentralizeddatadistributionordecentralizeddatadistribution.Generally,thedatadistributioncanbecategorizedashorizontalandverticaldatadistribution.Whilehorizontaldatasplittingisdiscussedindetailintheforthcomingsections,verticaldistributiondistributesallvaluesfordifferentattributesindifferentplaces.

• Data Alteration:Itisusedtochangetheactualdataintootherformbeforereleasingtothepublicinordertoaccomplishthedataprivacy.Datamodificationmechanismsincludeperturbation,blocking,aggregation(Chenetal.,2014;Wonetal.,2014),swappingandsampling.

1

International Journal of Strategic Information Technology and ApplicationsVolume 8 • Issue 4 • October-December 2017

2

• Privacy Preservation:Assuresthedeliveryofdatatotheintendeddataminebyadaptingdataalterationbeforedelivering.Distributionofdataisdoneamongmorethanonenodewithoutrevealingthedataatindividualsite.Inclassificationphase,wheretheresultswillbegiventodesignatenode,whichdoestheclassification,itchecksfortheoccurrenceofcertainruleswithoutdisclosingthem.

Manyauthorshaveproposedtechniquesinordertoprovidetheconfidentialityindatamining(Aggarwal,2010;Oliveira,2004;Rawat,2015;Fouad&Hassan,2016),ElaineShietal.(2011)hasproposedatimeseriesbasedaggregationmechanisminordertoattaindataprivacy,wheregroupparticipantscanoccasionallyuploadencrypteddatatothegroupaggregator(GA),whoisresponsibletodothesummationondataineverytimeperiodically.Theauthorssuggestedamechanismwhichallowsgroupuserstosubmitencodedvaluestodataaggregator,Afterwardsaggregatorwillperformthesummationonparticipants’valuesineveryperiod,withoutpriorknowledge.Weachievestrongprivacyusingthistechnique.

RongxingLuetal.(2012)hasadvisedanefficientprivacypreservingaggregationmethodbasedon homomorphic Paillier cryptosystem technique, uses a super increasing sequence to structuremulti-dimensional data.ThePaillierCryptosystemcan achievehomomorphicproperties,widelydesirableinmanyprivacy-preservingapplications(Sangetal.,2009;Zhong,2007).Concretely,itiscomprisedofthreealgorithms:keygeneration,encryptionanddecryption.RSAmodulusisusedinkeygeneration,toproducebothpublickeypk=(n,g)andprivatekeysk= λ µ,( ) .Encryptionphaseis used to convert the original message m into ciphertext c=E(m)= g r nm n. mod 2 . Decryptionalgorithmusesciphertextinordertorecovertheplaintextm=D(c)=L c nn( ). mod .modλ µ

2

Asthiscryptosystemisprovablysecureagainstchosenplaintextattack,henceprivacy(Kantarcıoglu,2004;Surekhaetal.,2010,2011,2012,2013,2017;Deyetal.,2016,2017;Rajeswari,2017;Tyagi,2017;Jauvart, 2016; Rao, 2016; Reddi, 2017; Dharavath et al., 2016) is achieved. Compared againsttraditional1-Dimentionaldataaggregationmethods,confirmedthatproposedcansignificantlyreducecomputationalcostandimprovecommunicationefficiency,satisfyingthereal-timehigh-frequencydata collection requirements in smart grid communications. The authors also provided securityanalysistodemonstratingitssecuritystrengthandperformanceanalysisshowsefficiencyimprovement.

RastogiandNath(2010)firstappliesdifferentiallyaggregationfordistributedtime-seriesdata,offers good practical utility even without of trusted server. It has addressed two challenges indataminingapplications,(i)userscanpublishtemporallycorrelatedtime-seriesdatasuchaslocationtraces,webhistoryandhealthdata.(ii)Anuntrustedaggregatorallowstorunaggregatequeriesonthe data. As in Rongxing mechanism this algorithm comprises key generation, encryption anddecryption.Thepublickeyiscomputedbytriplet(m,gp,gpλ )wheregp m b ma m= +( ) . mod1 2 forchosen(a,b)∈ Z xZ

m m* * andprivatekeyλ =β xlcm(p,q),whereβ isarandomnumber.Message

encryptioncanbedonebyc=E(m)=g r nm n. mod 2 .Most of the proposed algorithms have used homomorphic encryption considering only

homomorphicoperationsonciphertext,encryptedusingacommonkey(Bonehetal.,2005;Gentry,2009).Sincetheparticipantsareencryptingthedatabyusingaggregator’spublickey,theaggregatorwillnotonlydecrypttheaggregatestatistics,alsoseparatesindividualnodedata.Bycontrast,theirconstructionallowshomomorphicoperationsonciphertextusingusers’secretkeys.Castellucciaetal.(2009)proposedsymmetrickeyhomomorphicencipherschemewhichallowsanaggregatortodeciphermeanandvarianceofenciphernodedata.However,theyalsoassumeatrustedaggregatorisallowedtodecryptindividualsecretvalues.Yangetal.(2005)proposedacryptographicscheme,allowsanaggregator tocomputetotalonencrypteddatafromparticipants.Magkosetal.(2009)pointedthatthisconstructionsupportsonetimestep,andneedsmorerekeyingoperationstoprovidemultipletimesteps.

International Journal of Strategic Information Technology and ApplicationsVolume 8 • Issue 4 • October-December 2017

3

Theaimofthispaperistoprovideprivacypreserving,allowingonlytheGroupcontroller(GC)toreadtheaggregateddatabutnotanyotherindividualuserinthemodel.Thisisachievedbydoingthefollowingoperations:encryption,dataaggregationbyclusterheadanddecryptionattheGC.Themechanismisefficientbecauseofcomputationalandcommunicationefficiencyduetosingleaggregationondatafortransmission.

Section2givesdistributeddataminingandPPDM.Section3providestheMathematicalgroundworks.Section4describesproposedmodel,afterwardsexperimentationandcomparativeanalysis,finallyconclusion.

DISTRIBUTeD DATAMINING AND PPDM

Distributionofdataamongthenodescanbedoneintwowaysnamelycentralizedanddecentralized.Incentralizedmodel,dataisstored,processedandtheresultisretrievedfortheissuedqueryonasinglenode.ThedisadvantageofthismechanismisthatfortheprocessingofBigdata,itconsumesmoretimefortheuserquery.Inordertoavoidthis,theDistributedDataMining(DDM)modelisused,wheredistributingofdataacrossmultiplenodesmechanismisfollowed.Manyalgorithmsweredevelopedtoaddresseffectivelyretrievingminingresultsfromalldistributednodes,whilemajorconcernisefficiencyincomputationtime.But,mostofthedevelopedalgorithmsdidnotconsidertheprivacyandsecurity intoaccount.Thedatadistributioncanbedonebypartitioningthedataeitherverticallyorhorizontally.Thenextsectiondiscussesaboutboththehorizontalandverticaldataportioningindetail.

Vertical Data Partitioning (Cornell, 1990; Navathe, 1989)Thisisalsoknownasheterogeneousdistribution.Itpartitionsthedataverticallyi.e.,fortheinputdata,someoftheattributesarelocatedononenodeandrestofthemarekeptonmorethanonenode.Thealgorithmforverticalpartitionisgivenasbelow(Vaidya,2000):

Step 1:ReadthedatabaseR,q=readthenoofpartitionsStep 2:P=vertical_partition_database(R)Step 3:Fori=1toqdobeginStep 4:L gen l e itemsets p Pi

i= ∈_ arg _ ( ) ,end

Step 5:L LG = 1

Step 6:Forj=2tondobeginStep 7:C combine local l e L LG G j= _ _ arg ( , )

Step 8:Forallcandidatec Ci

G∈ gen_count(c)Step 9: L L L cG G j= ∈∪ ∪ CG |c count imum port. min sup≥ endStep 10:ReturnLG

Horizontal Data Distribution (Agrawal, 2004)Inorder topartitionthedatabaseontomultiplenodeswithoutdisturbingthedatabasestructure,horizontaldatapartitionispreferable.Ahorizontalpartitionalgorithm(Kantarcıoglu,2004)basicallyworksintwophases.Phase-Iitlogicallysplitsthedatabaseintonon-overlappinghorizontalpartitions,whereeachpartitiongenerateslargeitemsetstogeneratepotentiallylargeitemsets.Inphase-II,actualsupportforitemsetsiscomputed.Thepartitiondimensionsarechosensuchaway,eachpartitionisabletoaccommodatethedatasetsinthemainmemory.Thetimeofexecutiondependsonthesizeofcandidateset.Withincreaseinsize,theexecutiontimeincreases.Anotherparameterthatinfluenceisthedimensionalityoftherecords.Withtheincreaseindimensionality,sizeoftheglobalcandidatesetincreasesalongwiththeexecutiontime.Thealgorithmforhorizontalpartition(Agrawal.S(2004))isgivenasbelow:

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4

Step 1:P=partition_database(R)Step 2:q=numberofpartitionsStep 3:Fori=1toqdobeginStep 4:read_in_partition( p P

i∈ )

Step 5:i

W =gen_large_itemsets( pi),end

Step 6:Fori=2;i

W ≠ φ ,j=1..n;i++do

Step 7:Computei

G

i

j

j nC W=

=1..∪ ,end

Step 8:G

i

G

C W= ∪Step 9:Fori=1tondobeginStep 10:read_in_partion( p P

i∈ )

Step 11:Forallcandidatesc CG∈ gen_count(c pi

, )endStep 12:L c C ccountG G= ∈ ≥ | . min sup

Step 13:ReturnLG

PPDM Block DiagramTheschematicofPPDMisshownbelowFigure1.LetDbethedatabasehavingtheraw/originaldata,node/usermaychooseadistortedparameter.ApplicationdistortionalgorithmonthetworeceivedinputsproducesthedistorteddatabaseD* whichissuppliedtothedataminer.Thedataminerusesthereconstructionalgorithmondistorteddataandtherandomparameter, inordertoretrievetheoriginaldatabase.

MATHeMATICAL PReLIMINARIeS

Let betheabstractcyclicgroup,andp2* isconcretegroupwhichexploitsthefollowingproperties

totheusertosupportbothsmallandlargeplaintextinaninstance.Forgivensecurityparameterλ ,decideprimep=2q+1,where p = λ andqtheprime,Euler’stotientfunctionφ( )p2 isgivenby

φ p pp

pq2 2 11

2( ) = −

= (1)

Showsatotalofφ p pq2 2( ) = elementsarepresentinthegroupp2* .Letx

p∈ * beaninteger

whichissmallerpthenaccordingtoFermat’slittletheoremwehavex pp− ≡1 1mod

i.e.x kpp− = +1 1 forsomeintegerkx kpp p p*( ) ( )− = +1 1 (2)

Raisingbothsidesoftheequation-2tothepowerofpandmodulatewith p2 ,leadsto

Property1:=1 11

2+

=

=∑p

ik p pi

i

p

( . ) mod (3)

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5

Property2:y pp

ip pp p i

i

p

= + = +

=

=∑( ) mod1 1 11

2 (4)

PRoPoSeD SySTeM MoDeL

Theproposedmodelfirstshowshowcryptographictechniquesallowaggregatortodeciphertotalityfrommultipleciphertextencipheredusingdifferentuserkeys.Later,adistributeddatarandomizationprocedure is described guarantees differential privacy; even a subset of participants might becompromised.

ArrangementofthehorizontaldatadistributionamongnclustersisshowninFigure2.Therolespresentinthesystemmodelinclude:

• Trusted Authority (TA):Itsresponsibilityincludeskeydatamanagementanddistributiontonodesinthemodel.Ingeneral,oncethekeydispersaliscompleted,TAdoesn’tparticipateinanyothersucceedingdataaggregationprocess.

• Group Controller(GC):Itwilldodatacollection,processingandanalyzationofdatafromallthenodes.

• Cluster Head(CH):ItisservedasarelayandaggregatorresponsibletocollectthedatafromallthenodestowhichitisconnectedandforwardsthedatatotheGC.

• NodeNi,for 1≤ ≤i n :Eachnodeinthemodelisresponsibleforstoring,retrievingand

returningthedatatotheincomingqueryfromtheGC.

Whenevertheuserwantstominethedata,thequerywillbedeliveredtotheGroupController,fromwhichitwillbecommunicatedtotheClusterHeads,andfinallythequerywillreachtheindividualnodefromtheclusterHead.Privacyprovisionisnottherefromthequerysubmissionbytheusertotillthereceptionbynode.Onceaqueryisreceivedbythenode,itexecutesthequery,andsendsbacktheminedresulttotheClusterHeader.DataaggregationswillbedonebytheClusterHeadaftercollectingthequeryresultfromallthenodesinthecluster,beforeencryptionoftheminedresult.GeneratedciphertextcanbesenttotheGroupController,wherehedoesthedecryptionprocess,thenextractstheresult.Finally,GroupControllerwillcombinealltheextractedplaintextresultsandthensubmitthemtotheuserasafinalresult.

Figure 1. Block diagram of PPDM

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6

PRIVACy PReSeRVING DATA MINING ALGoRITHM

Thealgorithmoftheprivacypreservingdataminingisasfollows:

Step 1:AllthememberswillingtoparticipateinthegroupcommunicationwillregisterthemselvesStep 2:Identifythenumberofclusters,clusterheadandGroupController.Step 3:Distributesecretkeysamongalltheclustermembers,GroupControllerandclusterHead

fromthetrustedthirdparty.Step 4:UponreceivingthequeryfromtheGC,CHbroadcastsquerytoClustermembers.Step 5:Clustermemberswillexecutethequeryandreturntheresultasthereplyintheencrypted

mannertotheClusterHead.Step 6:Afterreceivingthedatafromalltheclustermembers,CusterHeaddoesDataaggregation

andthensenttoGCafterencryptionofaggregateddata.Step 7:GCusesownsecretkeyindecryptionprocess,separatestheaggregateddataandextractsthe

individualresultfromGroupmembers.

Theproposedmechanismmainlyconsistsoffourphases:setup,encryptionatnode,aggregationattheclusterheadanddatadecryptionatGC.

Figure 2. Distributed data mining scenario

International Journal of Strategic Information Technology and ApplicationsVolume 8 • Issue 4 • October-December 2017

7

Setup

S S S p pGC g i

i

n

+ + = −=∑1

0 1mod ( ) (5)

where SiisthesecretkeyofnodeN

ithroughthesecurechannels.TAsends S

GCtoGC, S

gto

clusterhead.

Data encryption at NodesDataencryptionistheoperationdonebyeachnodeindividually.HerethequeryreceivedbytheGCisforwardedtotheclusterhead.Hewillbroadcastthequeryontoallthenodes.Afterreceivingthequery,eachnodeN

iwillexecutethequeryattheirendandextractthequeryresultm

i.Then,assumes

arandomdata xiandcomputesC

ibytakingtheparamvalues.Finally,C

i,theencryptedquery

resulttotheclusterheadiscalculatedasfollows:

C p g H t pi

m x Si i i= + ∗( ) * ( ) mod1 2 (6)

Data encryption at Cluster Head

Oncetheciphertextcifori=1…nisreceivedfromallthenodesN N N N

n= −−−1 2

,, ,

thecluster

headwilldothedataaggregationbyusinghissecretkeysg

,byfollowingthesesteps:

C c H ti

i

nsg=

=∏1

. ( ) (7)

SubstituteciinaboveEquation(7)

= ( ) * * ( ) * ( ) mod11

2+

=∏ p g H t H t pm x s

i

ns

i i i g (8)

= +∑ ∑ ∑= = =

+

( ) * * ( ) modp g H t pm x s si

i

n

ii

n

i gi

n

1 1 1 1 2 (9)

Finally,theresultCwillbecommunicatedtoGC.

Decryption at GCOnreceivingaggregatedcipherdatafromthegateway,GCtriestodecryptthereceivedmessageandseparatesboththeactualmessageandrandomnumberaddedtotheoriginalmessageinordertoprovidethedataprivacyintransit.Stepsfollowedinfulfillingtheoperationinclude:

Step 1:GCuseshissecretkeySc

andcalculatesF

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8

F C H t pSc= . ( ) mod 2 (10)

SubstitutingtheCvalueintheaboveequationresultsin

= +∑ ∑ ∑= = =

+

( ) * * ( ) * ( ) modp g H t H t pm x s s

Sii

n

ii

n

i gi

n

c1 1 1 1 2 (11)

∵ s s s p pi

i

n

g c+ + = −

=∑1

0 1mod ( )

= +∑ ∑= =( ) * modp g pm xi

i

n

ii

n

1 1 1 2 (12)

Step 2:Pollard’smethodisusedtoretrievethesumofrandomvaluesaddedbyindividualnodesbyfollowingsteps.

F F p= ,substitutingFvalueinthisequation

= +∑ ∑= =(( ) . ) modp g pm x

pi

i

n

ii

n

1 1 1 2 (13)

= +∑ ∑= =( ) . modp g p

p m p xii

n

ii

n

1 1 1 2 (14)

∵ ( ) modp pp+ =1 1 2

=g pp xii

n

=∑1 2mod (15)

=h pxi

i

n

=∑1 2mod (16)

GCcanrecover xii

n

=∑ 1byusingShietal.’sschemeorpollard’slambdatechniquewithtime

complexityΟ( )n∆

Step 3:Afterretrieving xii

n

=∑ 1GCcomputes

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9

F =∑=

+∑ ∑

∑= +

∑

=

= =

=

=F

g

p g

g

p px

m x

x

m

ii

n

ii

n

ii

n

ii

n

ii

n

1

1 1

1

11

1 2( ) *( ) mod (17)

= + +

∑

==

=∑ ∑∑

=

11

12

1

p m pm

ii

i

ni i

i

n

i

mii

n

* (18)

= +=∑11

2p m pi

i

n

* mod (19)

Finally, mi

i

n

=∑1

isrecoveredby Fp

−1

GRoUP DyNAMISM PRoPeRTy

Theproposedalgorithmcanprovidethedynamisminthenetwork.Thenetworkwillsupportnode(s)joinandleavefromthenetworkwithoutdisturbingthefunctionality.Thissectionwillbriefaboutthedynamicnetworkstrategyinnetworkmanagement.Thisfeaturehelpsinaddingoneormorenewnodestothenetwork.Italsoallowsremovingthefailurenodesfromthenetwork.

Single Node Join

WhenanynodeNjiswillingtojoininnetwork,theTAarbitrarilytakesasecretkeys

jtoit.Then

clusterheadcomputesthefinalsecretkeybyaddingsjtotheexistingnodes N N N

a a az1 2,

, ,−−− of

theclusterwithsecretkeys s s sa a az1 2,

, ,−−− byusingthefollowingformula.

s s s p pj ai

i

z

ai

i

z

+ = −= =∑ ∑1 1

1mod ( ) (20)

Clustersimplyadds sjtotheexistingclusternodestothesecretkeysandthenmodulatewith

p(p-1)inordertofindthenewencryptionkey,bywhichdataaggregationwillbedoneanddatadecryptionwillbeperformedbytheGCwithoutaffectingthenetwork.

Multiple Nodes Joining

WhenmorenumberofnodesNjforj=1toy,joinsinnetwork,theTArandomlyelectsasecret

keysjforj=1toytothem.Thentheclusterheadwillcomputethefinalsecretkeybyaddings

jto

theexistingnodes N N Na a az1 2,

, ,−−− oftheclusterwithsecretkeys s s sa a az1 2,

, ,−−− byusingthefollowingformula.

International Journal of Strategic Information Technology and ApplicationsVolume 8 • Issue 4 • October-December 2017

10

s s s p pj

j

y

aii

z

ai

i

z

= = =∑ ∑ ∑+ = −1 1 1

1

1mod ( ) (21)

Clustersimplyaddsthesecretkeysofallthenewlyjoinednodessjforj=1toy,totheexisting

clusternodessecretkeys,andthenmodulatewithp(p-1)tocomputethenewencryptionkey,whichis tobeusedindataaggregation.DatadecryptionisperformedbytheGCwithoutaffectingthenetwork.

Single Node Leaving

WhenanynodeNaj

iswillingtoleavefromthenetworkwithsecretkey saj

,thenTAchoosesaarbitrarykeytoallnodesN

akfork=1....n,andfork j≠ havingsecretkeys

akintheclusterexcept

theleavingnode.Itthenassignsthisnewsecretkeysak fork=1...nandk j≠ toeachNak

suchthat

s s s p pai

k

n

ak jk

n

= =∑ ∑= + −1 1

1mod ( ) (22)

Multiple Nodes Leaving from Clusterwhenanynodesarewillingtoleavefromthecluster,thenTAchoosesarandomsecretkeytoallthenodeswhicharelikelytopresentinthecluster,andthenassignsthisnewsecretkeysak fork=1..nandk l l≠ ∈ nodeswillingtoleavetheclustertoeachN

aksuchthat

s s s p pak

k

n

ak all l k

n

k

n

= = ≠=∑ ∑∑= + −1 11

1,

mod ( ) (23)

ReSULTS AND DISCUSSIoN

Theproposedalgorithmwasimplementedbyjava(JDK1.8)andrunonthedesktopwith3.1GHzprocessor,8GBRAM,onwindowsplatformbyusingfollowingparameters(Table1).

Computational CostExperimentationwasdoneonaclusterwithnnodesbyvaryingitfrom200to1000withanincrementof200;averageencryptionatthenodetook35mstoencryptamessage.Figure3andTable2showsthetimespendbytheCHforperformingaggregation,anddecryptionbytheGC.Itshowsthatinboththecases,timeincreaseswithrespecttothenodescountinthecluster.

Communication Cost

When p =512,anycipher text size less thanorequal to1024bits.The featurecomparisonofproposedmethodwiththeexistingmethodsisshowninTable3andTable4.

International Journal of Strategic Information Technology and ApplicationsVolume 8 • Issue 4 • October-December 2017

11

CoNCLUSIoN

Privacyindataminingwasproposedbyusingtimeaggregation.Itprovidestheprivacytomineddataatnodelevel,clusterheadlevelandGClevel.Italsosupportsthenetworkdynamismbyallowingnodesjoining/leavingfromtheclusterswithoutdisturbingtheexistingnetworkfunctionality.Ourprotocol involvespairingsoperationwhosecomputation iscomputationally slower thanmodularexponentiation.Thisprotocolcanusewithsmallandlargeclusterwithtwoormoreclustermembers.Further,thecomputationandcommunicationcostoftheproposedmethodwaspresented.Finally,comparativeanalysiswasdonewithsomeoftheexistingmethods.Thisworkcanbeextendedtovariousapplicationsinmedical,statisticalandinIOTresearchareas.

Figure 3. Computational cost (a). Aggregation at cluster head and (b).decryption by varying n

Table 1. Algorithm parameters

Parameter Value Description

λ 1024

Zp2* Grouporderofφ( )p2

N N=200,400,600,800,1000 Numberofusers

∆ 20 Sensitivityoftheplaintextspacedata

M 0-999 Message

ε 1,2,3 Differentialprivacylevel

International Journal of Strategic Information Technology and ApplicationsVolume 8 • Issue 4 • October-December 2017

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Table 2. Computation time at Cluster Head and Group Controller

Number of Nodes Aggregation time at cluster Head (milli seconds)

Decryption time at Group Controller

(milli seconds)

200 30 38.12

400 32.12 36.43

600 34.0 34.32

800 37.32 32.11

1000 40.12 30.01

Table 3. Feature Comparison with existing methods

Mechanism Features

Shietal.,2011 •Focusedonindividualuserprivacy.•Securedagainstdifferentialattack.•Supportsonedimensionaldataaggregation.

Kantarcıoglu,2004 •Focusedonindividualuserprivacy.•Multi-functionaldataaggregationispossible.•Securedagainstdifferentialattack.•Supportsonedimensionaldataaggregation.

Sangetal.,2009 •Supportsmulti-dimensionaldataaggregation.•Reducescomputationtimebyencryptingmulti-dimensionaldataintoonedimensionalciphertext.•Useshomomorphiccryptosystem.

Dansana,2012 •Securedagainstthedifferentialattack•Usesmodularaddition-basedencryption.

Yang,2005 •Eliminatesplaintextspacelimitation.•Supportsonedimensionaldataaggregation.

Proposed •Suitableforsmallandlargedata•Improvesefficiencyincomputationalandcommunicationcost.•Providesindividualnodeprivacy.•Privacyfordataaggregationispossible.

Table 4. Comparative analysis of properties

Method Operations used Algorithm Phases Dynamic network

Shietal.,2011 Addition Setup,noisyencryptionandaggregatedecryption

No

Sangetal.,2009 Powerfunction,modulation,concatenation,

initialization,reportgenerationrelatingtouser,requestandresponse

No

Yang,2005 Addition,multiplication,gcdandmod

Keygeneration,encryptionanddecryption

No

Dansana,2012 Division,summation Noisegeneration,encryptionanddecryption

No

Proposed Pow,mod,summation, Setup,nodealgorithm,encryptedaggregationatCH,decryptionatGC

Yes

International Journal of Strategic Information Technology and ApplicationsVolume 8 • Issue 4 • October-December 2017

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Sivaranjani Reddi received the B.Tech degree from NIT, Warangal in 2002, M.Tech in computer science from Andhra University in 2005 and PhD in computer science from Andhra University in 2015. She is currently professor in Computer Science Department, ANITS and having a teaching experience of 15+ years. Her Research interests include information security, cyber forensics, image processing and opinion mining. She is the life member of ISTE and CSI

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(International Peer Reviewed Joumal)

Effect of Graphene Oxide on Nano Titania Particles inVisible Light Induced Photocatalytic Degradation of Congo red

Soma Sekhar Ryalil and Paul Douglas Sanasi2*

1. Department of Chemistry, Anil Neerukonda Institute of Technology & Sciences (A), Visakhapatnam-531162, and Department of Chemistry, JNTU, Kakinada -533003, INDIA

2. Dept. of Engg. Chemistry, AU College of Engineering (A), Andhra University,Visakhapatnam-s3 0003, AP, INIIA

Email : spauldouglas.eagchem@auvsp.edu.in

Accepted on 21't November 2017, Published online on27'h November 2017

ABSTRACTIn the present work, visible light induced pholocatalytic degr adation of Congo red dye in aqueous mediumilos investigated by entploying modified nano titania (IVT) particles exfoliated with graphene oxide (GO)parlicles. The composites were synthesized by organic solvent /i,ee controlled hydrolysis of titaniumtefi'crchloride.followed by dispersing an aqueous soltrtion of'graphene oxide. These composites have beencharacterized by X-Ray Difi'action (XRD), Fourier Transform Infra-Red Spectoscopy (FT-IR,) FieldEmission Scanning Electron Microscopy (FE-SEM), and UV-Visible Dffise Reflectance Spectroscopy(UV-Yi.s DRS).Ef/iciency of the composites towurds the photocatalytic degradation o/ Congo red dye wasussessed by analyzing the elfect of nano titania particles with increqse in the content oJ'GO and ffict ofpH 4 the dye solution. Photocatalytic degradation of Congo red dye wa,s enhanced by contriving thecr.tmpo.sites into visible light absorption on grafting GO on the surface of nano titania particles. With qnoptintum increase in the GO content, the photctcatalytic activiQ of the composite,s was improved ond asuperior photocatalytic activity was observed y,ith 10% GO-nqno titania compo,site material.Graphical Abstract:

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Kewvords: Photocatalytic degradation, Nano titania, Graphene oxide, Congo red.

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A Facile Microwave Assisted Neat Synthesis of Bis-amides Using Nano Nickel CopperFerrite as Amicable Catalyst and Study of their Fluorescence Studies

Sarada RI*, Jagannadharao Vr, Govindh 82, padma l\3tDepartment

of Chemistry, Anil Neerukonda Institute of Technolog,, and sciences, Sangivalasa, Visakhapatnam531162, India2Departmentrof H A S, Raghu Insti.tute of Technology, Visakhapatnam, Andhra pradesh,India

' D ep ar t m ent of C h emi s try, An dhr a Univ ers i ty, Vi s akh ap atn am, I n d i a

ABSTRACT

rnicrowavc conditiott't. This Microwave ltadiotiort (MWI) reaction uncler solvent-free conditiotts restitet] in a ..green-chentistry,,prctcedtrre,s,

ond oldeh..t,des in presence oJ'silica metlium to achieve Bis-unitles. This approach ,sl,nthesii o/ the titiect'contpotLnds in tlte absence of co_

.tlnthe,si:ed Bi.t-tmtides were stLh.f ected.for.flLtorescenc,e stLttlie,t.

Keyrvords: Solvent-free, Microwar.e irradiation, Amides, NicuFe,oa, Reusability, Fluorescence

INTRODUCTION

out of large nttmber of organic dyes hitherto knolvn, there is ongoing effort throughout the r.vorld to synthesize new clye laser materials rvhichshow improved stability and better efficiency. Apart from their use in Lasic research in physics and chemistry, they have got wide applicatio,s i,applied fields. r'i2.. environmental science, medical research and defensc eto. Synthesis ofnew laser d1,es is therefore very much required as

- theserrityscrveasitnportsubstinrtioninviewofthepotentialfbrtheirproductiononinclustrial scale.

The ainr ofthis paper was to study Bis-amides as push-pull molecules for first-order nonlinear optics and lasing property. Literature surveyrevealed that bridge-donor-acceptormoiety, were fbund to be lasing [1-al. To this purpose, in the present contest three Bis-amide molecules norvcalled as push-pull molecules were synthesized and charaoterized.

Reccntly, the sylthesis of new dyes rvith increased cross sections and large unconverted fluorescence has opened up a myriad of variousapplications viz., pUsh-pr:ll molecules and lasing molecules [5,6]. Two photon optical power limiting, three dirrensional optical data storage andphoto dynalnic therapy, besides three dimensional irnaging using 1wo photon laser scanning .onfo"cal microscopy. This technique though hashigh order potential applications had not been examine<l thoroughly, due to lack ofdyes whi"ch exhibit high intensity unconverted fluorescence.Thesc investigations will have high utility/applications. When the trvo photon peak occurs at or near g-00 nm, a wave length at which mostorganic and biological materials have large optical transparencl,.

Sr-rrve1' literature t'eveals tlrat Ehrlich et al. had investigated the design of organic two photon materials based on bi-clonor containing Stilbenenrolecltle which exhibit the maxima of their two photon absorption at shorter wave lengih. Reinharl et al.. had synthesized a series ofir,merousuolecules r','ith s1'sterratically varied structul'es, rvhich exhibit more effective two photon cross section and charaoterized in solution usi.g a non-linear tt'anstlission technique. The compounds can be categorized into two basic structural lamilies; viz., Donar/bridge/acceptor andAcccptor/donar-donariAcceptor [7- I 8],

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Oxide-Nano Titania Composites and EvaluationPhotocatalytic Degradation of Rhodamine B

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'Deplrtment of Chentislrl,.;\nil Netrul<ortr-la institute ol..l.echnology & Scicnccs- (A). Visakhapa(nant_531 l(rl. Indjarl)epartnrent or'chemistry, Jawlharrar Nehru Teclrn<,rogicar lJnivcrsity, Kakinada-533 00j, Incria'Depzutrnenl o1-Enginee'ing Chenrislry. AU college ottngineering (A), Andhra u.i"..rlty, Visakhaparnam--5,1, 0u,1, rndiai'(-'.r'rcspondi,g aurhor: E-.lail: sontasckhirr.chernistry(rtratrits.edu.in

Receit,ed: l0 Januar.y 20J g; Accepred:3 Malch 2018:

Keyirords: ()raph.ne Oxide, Nano titani:r, lrhotocatalytic activity, Rhodanrine-B.

Publi,yhed onLitte: 30 April 20l g; AJCr- I 888 1

lightassistecrphotocatalvticdegraclatiernorrhocrarr;;;:;.hffi;;fiffii,:r"i,i;1il*,,]I;i,ffi:llfi:ffffi,:1*'Ji"l[:[:i:conc'iitions rbllowerl bv ultrasonic adclitic,n'f rrre as ;."pr'; i-r;;il;;ffi; ur",,,il;"'ffi;;il;:*nrogy antj srru*ur.al,ropertie:'l'rltecornposirernareri:rrs(r.l.s. rir ri.iz g,,r,h-li'.^;.-;:;:';i;;,:;;;;;;;il,:rjJ*n'1.'r1,,r.,.,.;zcdusineX_raydiiliaction (XID), Fouri"r transibrm intiarerl .p..,.or.npr.,F;ii;, a;v-;;,t;l:.ilirr" r*.1,*r;. il;;iil ,:s DRS). high resolurionl.ntt'nti:si'rrt elcctron micr,isc.p-' (lRl liN4, hctd c,rri.slon ,.,,n,,u,g .i;:;;;;';;;,,,r."r,1 t'Lsilrrr,'-rr'i u ri:ibre spcerrose.py. AB rvas obscrved in 60 rnin ancl orher rp,i"r,, .,riiiiil.. ,o"r" esrablishcd.

Visibrc iight induceci prrotocatarvtic desradation of orsanic,oll.tants prcsent in intl*str.ial r,i,astcu,atJr is a well krl.rvnand cstablished technique in the area ol grecn chernistrv ro,uril'1, 11,351.ruater lll. Anatase fornr ol,iita^ia is il;. ;";;thorou-lhry investigated rnateriar in the riterature ancr seerns tobe a promising photocattrlyst 1itr. the detoxification of organicpolluranrs 2-.11. Orving b irs low roxiciry, high photostarilitvtnd hieh photo-cllic:ienc), if acts as a unique pfiotocatatt,st. flou.cvcr. thcre arc Icu snagging issucs uith this ntatcrial cluc torapid elcctron-horc re conrt,inatitr. in It)'s u.hich resurts in

'*,vquanturn etficienc.y-, a rvide banri gap of 3.2 eV linritins its usiisenrostl,y irr the llV r"egion l5l. l)eveloping a lrereroqeneous photo-catalvst kt br.ing the usage of titania particles in the visible regionis a conyentional routc to enhancc the photoeatal; ric tlcgiaci_arion of organrc clyes like anionic clopirig w,ith C. X ,,nl.f S'iu,r,.r'transrtior rnct,r ions ,r lo.mation of na.o-com;-rosites withsuitablc n.rittcrials to yield high sr.rrface to volumc r.atio 16.7J.E,flitrts have been rn;rde to synthe:size titania_gold natro compo_sites. CdS/CdSe-titania h),bricts, et r,. for inhi trition of electron_hole recttrnbination and ertentling the photocrtalytic processin visiblc region l8|. N4o<Jifving titania with a nano_structuredcarbonaccous srrbstance like graphcnc oxi<Je rCO) has bccn

gaining impr)rLrjllcc to conlrive Visible-light rcsponsrve rnate_rial s.

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IMAGE PREPROCESSING OF ABDOMINAL CT SCAN TO IMPROVE VISIBILITY OF ANY LESIONS IN KIDNEYS

1HIMA BINDU G, 2PRASAD REDDY PVGD, 3M. RAMAKRISHNA MURTY

1Research Scholar, Dept. of Computer Science and Systems Engineering, Andhra University

2Dept. of Computer Science and Systems Engineering, Andhra University 3Dept. of CSE, ANITS(A), Visakhapatnam

Email: 1hima.gopalam@gmail.com, 2prasadreddy.vizag@gmail.com,

3ramakrishna.malla@gmail.com

ABSTRACT

Abdominal CT scan images are widely used in detection of kidney lesions. This paper study is conducted for pre-processing abdominal CT scan images so as to segment the kidney for further analysis of lesion detection. Various noise filters and segmentation techniques have been experimented to select the best filter and segmentation techniques for pre-processing the CT image. The experimental study finds a combination of Median filter followed by Wiener filter more effective to remove different noises present in the CT images. Different segmentation techniques have been run on the test data set of CT images and it is observed that Edge based active contour produced better results than Graph Cut and region based active contours.

Keywords: Medical Imaging, Noise Filters, Segmentation, Active Contour, Region-based, Edge-based, Graph Cut.

1. INTRODUCTION Kidneys are vital organs of human anatomy

which helps in filtering and removing waste products from the blood. Statistics on kidney diseases reveal that cases of kidney cancer have been growing every year and is in the top list of cancers being detected. However, the survival rate increases if detected in early stages and proper treatment is given. Clinical observations along with medical imaging plays a key role in the diagnosis of kidney cancer. The imaging techniques help radiologists in identifying the exact location, size and type/grade of the cancer. Various medical imaging modalities like CT - Computed Tomography, MRI - Magnetic Resonance, US - Ultrasound, other nuclear imaging techniques like PET- Positron Emission Tomography and SPECT - Single Photon Emission Computed Tomography are available for medical imaging. CT images are extensively used in medical diagnosis for its affordability, availability, non-invasive, non-intervention, multi-parameter imaging with better resolution.

In this paper, we study about preprocessing the abdominal CT scan image so as to help in

segmenting the kidneys and detecting the lesions present in the kidney for further analysis and in treatment required to be given to the patient. The preprocessing includes noise filtering, contrast enhancement and segmentation. In this paper, we study various noise removal and segmentation algorithms in CT images and make a comparative study between them. The abdominal CT image contains various organs like liver, spleen, kidney, colon, pancreas, duodenum, inferior vena cava etc. Segmentation of kidneys from abdominal CT scan image is a challenging task as most of the organs in the abdominal CT image are soft tissues with no clear borders. Graph cut, Region based and Edge based active contour segmentation methods are studied in this paper.

2. RELATED LITERATURE STUDY

The work in this paper relates to noise removal and kidney segmentation from abdominal CT images

Kidney lesion do not show symptoms and are detected accidentally from CT scans taken for other ailments. From literature, it is noticed that not much

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work in done on Kidney lesion detection in abdominal CT images. Noise is induced into the CT scan images due to the CT artifacts, improper focus, motion etc. It is very important to remove noise before segmentation of the images. Most common noises that are seen in CT images are Salt and Pepper noise, Speckle noise, Gaussian noise [4,9]. Extensive research has been done for removing noise from the images. The important consideration for selection of noise filter is complete noise removal while preserving edges. Kaur et al. [10] presented performance of various noise filters – Gaussian, Wiener, Mean, Median filters. Sonali et al. [4] have used various filters for preprocessing the image and have proposed median filtering for removing film artifacts. Kumar et al. [6] have proposed use of Median and Weiner filter for removing Salt and Pepper noise and Gaussian noise present in the non-medical images. The paper concludes that Weiner filter works well with Gaussian and speckle noises and Median filter works well with salt and pepper noise. PSNR [4] and MSE [8] are studied in various papers for measuring image quality metrics. Wang et al. [2] has proposed using of metric based on structural similarity, SSIM for image quality assessment.

Another important work in this paper is segmentation. Kidney segmentation from CT scan images is always a challenge as there are many soft tissues with similar intensity surrounding the kidney with unclear borders. Active contours and graph cuts have proved to give promising segmentation results for kidney segmentation. The concept of active contour was first introduced by Kass et al. [17] where contour was evolved for object detection using the energy terms. Since then lot of research has been done on active contours for detection of object boundary. Chan et al. [14] have proposed active contours without edges based on curve evolution techniques, the Mumford-Shah functional for image segmentation and the level sets. This technique handles topology changes well compared to the normal snakes. V. Caselles et.al. [15] have proposed geodesic formulation in active contours for effective boundary detection of objects taking advantage of the gradient differences. A new energy term is introduced to additionally pull the deforming curve towards the boundary of the object. Lei et al. [13] have performed a comparative study of various deformable contour (balloon, topology, distance, gradient vector flow, Geodesic active, etc) methods on various modalities of medical image segmentation, where he suggests that the methods are not mutually exclusive and could be used in

combination for better segmentation results depending on the application. Also the authors recommend incorporation of image or object information into the framework which yields better segmentation results. Kolomaznik [18] has proposed a fast segmentation using both region and edge energy terms where the first phase of segmentation is controlled by statistical shape parameters and the next phase of segmentation is controlled by the image data. Another segmentation technique in the recent years is the graph-cut framework, which applies graph theory for fast segmentation of an image into foreground and background. Anders P. et al. [16] have included prior information obtained from terminal node weights and learning algorithms into the graph cut method to produce efficient segmentation results.

For detection of kidney lesions, it is important to first segment the kidney from the abdominal CT image and then identify the lesion. CT images usually come with noise and can impact the segmentation results. Hence it is very important to remove noise and then segment the kidney for further analysis of kidney lesions. Abdominal CT images consists of many other organs with the same intensity as that kidney and normal segmentation methods used for segmentation of other organs do not perform well with kidney segmentation. So segmentation techniques using active contours and graph-cut are implemented and a comparative study is performed.

Various noise removal filters and image segmentation techniques have been applied in our work to select the best suitable techniques for abdominal CT images. In the next sections, we are going to deal with various noise removal algorithms and segmentation algorithms in CT images. 3. METHODOLOGY

The proposed methodology of Kidney Cancer

detection consists of 4 main phases, Image acquisition, preprocessing of the image to segment kidneys from abdominal CT images, feature extraction from the segmented kidneys, lesion identification and classification. Preprocessing is an important step in our methodology and results of the next phases is highly dependent on the preprocessing phase to produce accurate results in feature extraction, lesion detection and classification phases.

Preprocessing phase consists of noise filtering, contrast enhancement and kidney segmentation. The work in this paper is limited to noise removal

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algorithms and kidney segmentation techniques only. The other techniques used for the remaining phases will be discussed in subsequent work.

Figure:1 Proposed Kidney Cancer Detection Workflow

The research work in this paper aims to find one common suitable noise removal filter for various noises present in the CT image and a suitable kidney segmentation technique that works for images with contrast and without contrast.

MATLAB tool is used throughout the work conducted in this paper.

4. NOISE FILTERING Noise can cause improper segmentation results

and hence it is very important to remove the noise before segmentation. Common noises in CT images are Salt and Pepper noise, Speckle noise, Gaussian noise [1,4,5,9]. Noise can be removed from images using various filters like Mean filter, Gaussian filter, Median filter and Weiner filter, etc.

Mean filter is a linear filter and uses the averaging phenomenon over the neighbourhood region, while Median filter is a non-linear filter and each pixel intensity value in the image is replaced by the median of the pixel intensity values in the selected neighbourhood of the filter [9]. Wiener filter is based on statistical methodology and eliminates the additive noise while preserving the edges and also inverts the blur at the same time [1]. Gaussian filter performs convolution both in spatial

and frequency domain and popularly known for blurring and reducing noise by smoothing the data. A new combination filter, Median followed by Wiener filters has been proposed [3, 6] to remove noise. The Median filter helps to remove impulsive noise while the Wiener filters removes both the blur and additive white Gaussian noise in the image while preserving the edges. Performance of the filters can be measured using image quality assessment metrics such as Mean Squared Error (MSE), Peak Signal to Noise Ratio (PSNR) and Structural Similarity Index (SSIM). For a good quality image, MSE value should be lower and PSNR and SSIM should be higher.

5. NOISE SIMULATION AND ANALYSIS Experiment has been conducted to study the

best filter for noise filtering against different noises present in CT images. For the purpose of study, noise has been added to the acquired images and different filters have been applied sequentially. Below flow diagram depicts the steps involved in applying various noise filters and measuring the image metrics.

Figure:2 Simulation process of applying noise filters and measuring image quality metrics

Experiment study of various noise filters on each of the noises (Salt and Pepper noise, Gaussian noise, speckle noise) is conducted on a test data set of abdominal CT images (both with and without contrast). The noised filtered images and computed values of MSE, PSNR and SSIM from the noise filtered images are presented below.

MSE, PSNR and SSIM are measured on the output images and then the average values are computed and presented in the below tables.

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Figure:3 a) Original Image b) Image simulated with Salt and Pepper Noise. Images after applying: c) Mean filter, d) Median filter, e) Wiener 3x3 filter f) Wiener 5x5 filter g) Gaussian Filter h) Median followed by Wiener filter

Figure:4 a) Original Image b) Image simulated with Gaussian Noise. Images after applying: c) Mean filter, d) Median filter, e) Wiener 3x3 filter f) Wiener 5x5 filter g) Gaussian Filter h) Median followed by Wiener filter

Figure:5 a) Original Image b) Image simulated with Speckle Noise. Images after applying: c) Mean filter, d) Median Filter, e) Wiener 3x3 filter f) Wiener 5x5 filter g) Gaussian Filter h) Median followed by Wiener filter

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Table 1: Comparison of Image Metrics after simulating Salt and Pepper noise and applying various filter

algorithms

Performance Parameters with various Filters for Images with Salt and Pepper Noise

Noise Filters MSE PSNR SSIM

Mean Filter 70.15 29.59 0.770

Median Filter 9.95 39.87 0.983

Wiener3x3 Filter 390.71 22.22 0.632

Wiener5x5 Filter 264.64 23.92 0.675

Gaussian Filter 206.58 24.99 0.686 Median Filter + Wiener Filter 20.52 35.90 0.935

Table 2: Comparison of Image Metrics after simulating Gaussian noise and applying various filter algorithms

Performance Parameters with various Filters for Images with Gaussian Noise

Noise Filters MSE PSNR SSIM

Mean Filter 101.60 28.08 0.553

Median Filter 100.71 28.25 0.579

Wiener3x3 Filter 153.84 26.38 0.474

Wiener5x5 Filter 113.05 27.73 0.556

Gaussian Filter 232.53 24.47 0.366

Median Filter + Wiener Filter 55.76 30.82 0.708

Table 3: Comparison of Image Metrics after simulating Speckle noise and applying various filter algorithms

Performance Parameters with various Filters for Images with Speckle Noise

Noise Filters MSE PSNR SSIM

Mean Filter 55.02 30.97 0.867

Median Filter 89.91 28.70 0.794

Wiener3x3 Filter 114.11 27.64 0.784

Wiener5x5 Filter 91.78 28.59 0.806

Gaussian Filter 125.27 27.28 0.748 Median Filter + Wiener Filter 54.92 30.85 0.869

From the experimental findings, it is observed that for Salt and Pepper noise, Median filter works best followed by the combination filter (Median followed by Wiener filter). Both filters have lesser

MSE and higher PSNR and SSIM values compared to other filters. For Gaussian and Speckle noise, combination filter produces better results in noise removal. Hence combintaion filter (Median followed by Wiener filter) is recommended as the best filter to remove noise as it produces good results for all kinds of noises present in the considered CT images.

6. KIDNEY SEGMENTATION Segmentation is an important step in medical

image processing. It helps to partition the region of interest from the CT image for further analysis. As most of the organs including kidney are soft tissues with unclear borders in the abdominal CT image, semi-automated segmentation methods like Active contours and graph cut are analyzed in the study.

Active contours are popularly known as snake method is used in medical segmentation to draw an outline of an object and thus help in segmenting the region of interest. The active contour takes some initial points known as hinge points as input and these help the contour to traverse and keeps moving until convergence is met[17]. The energy of the active contour is represented as

𝐸 𝑟(𝑠) = ∫ 𝐸 + 𝐸

(1)

where r(s) represents the discrete set of points. 𝐸 is determined by the properties of the contour such as elasticity, stiffness while 𝐸 is determined by image properties such as intensity, contrast and brightness.

The contour starts moving to the neighborhood for a better location determined by the lowest energy and iterates until all points in the contour move to a location of minimum energy. Active contours have two fundamental properties based on the force that drive the contour: similarity and discontinuity. Based on these two properties, region based active contour and edge based active contour algorithms are developed [14, 19-21]. Region based active contours use global information such as intensity, texture from the image for drawing and terminating of the contours at the edges. Chan Vese is one of the popular region based models. Chan Vese model is formulated by minimization of the following energy function and for a given Image I in domain is represented as 𝑬𝑪𝑽 = 𝝀𝟏 ∫ |𝑰(𝒙) − 𝑪𝟏|𝟐𝒅𝒙 +

𝒊𝒏𝒔𝒊𝒅𝒆(𝒄) 𝝀𝟐 ∫ |𝑰(𝒙) − 𝑪𝟐|𝟐𝒅𝒚

𝒐𝒖𝒕𝒔𝒊𝒅𝒆(𝒄)

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where x €Ω (2) C1 and C2 are two constants and define the corresponding mean intensities inside and outside of the contour.

In Edge based active contours, contours evolution is based on the intrinsic geometric measures of the selected image [15]. The geodesic formulation used in the curve evolution draws the deforming curve towards the boundary, thus helping the detection of boundaries of objects which have differences in their gradient.

In Graph Cuts segmentation [14], a graph is generated from the image pixels and is separated into two disjoint sections using selection criteria and these two disjoint sets represent the foreground and background. The graph is represented as G =[𝑉, 𝐸, 𝑊], where V denotes the set of nodes and E denotes the set of edges and W denotes the weight associated with each edge. A cut on the graph partitions V into two disjoint subsets X and Y such that X U Y = V and X ∩ Y = Ф (3) The min-cut formulation partitions graph, G into two disjoint sets so that the sum of the W associated with the edges is minimized. 𝐶 (X, Y) = ∑ 𝑊

, (4) The weights associated with each edge can also be viewed as flow of capacity from source to sink and the maximum amount of flow is equal to the capacity of minimum cut, hence the method is called as max-flow / min-cut graph method. 7. KIDNEY SEGMENTATION

SIMULATION AND ANALYSIS

Noise filtered and contrast enhanced images are considered for kidney segmentation. The three

segmentation techniques discussed above are applied and the results from the segmentation techniques are measured with ground truth obtained from expert radiologist. Quality of segmentation is measured using Dice Coefficient, which is a measurement of similarity between the segmented result and the ground-truth and is defined as [12] Dice Coefficient = 2 * TP / (2 * TP + FP + FN) (5) where, TP is true positive, FP is false positive, FN is false negative

Axial view CT scan images with clear visibility of kidneys are considered for segmentation. Both contrast images (taken after contrast agent is injected to patient) and non-contrast images are considered for the study. Contrast images collected from arterial phase provide better visibility of lesions. For images with kidney lesions, seed points both from the normal and lesion area are considered.

The three segmentation techniques are run on the test dataset of abdominal CT images obtained after noise removal and contrast enhancement. Figure (7-9)b shows segmentation results from region based active contour, Figure (7-9)c shows results of edge based active contour and Figure (7-9)d shows results of graph-cut. Dice coefficient is measured from each of the results and given in the table below.

Figure:6: Simulation process of applying noise filters,

segmentation techniques and measuring dice coefficient

Figure:7: CT Image-1 a). Noise filtered and contrast enhanced CT test images, segmentation from b) Region based

Active Contour c) Edge based Active Contour d) Graph Cut

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Figure:8: CT Image-2 a). Noise filtered and contrast enhanced CT test images, segmentation from b) Region based

Active Contour c) Edge based Active Contour d) Graph Cut

Figure:9: CT Image-3 a). Noise filtered and contrast enhanced CT test images, segmentation from b) Region based

Active Contour c) Edge based Active Contour d) Graph Cut

Dice coefficient is measured from each of the results and given in the table below. Table 4: Average Dice Coefficient for test dataset of CT images measured for different Segmentation Methods

Region based Active Contour

Edge based Active Contour

Graph Cut

0.9078 0.9531 0.9459

For kidney segmentation, all the three segmentation techniques are able to segment the kidney from the CT image. From the segmentation results, it is observed that region based active contour method sometimes crossed the kidney boundary covering neighbouring tissues as well. Edge based active contour and Graph Cut methods produced better results than region based from the sample test data. Also it is observed that the selection of initial points and number of iterations influenced the segmentation results.

8. CONCLUSION In this paper, pre-processing of CT images is

done for kidney lesion detection. As CT images have inherent noise, it is important to remove the noise for better segmentation results. Various noise removal filters are applied on CT images with Salt and pepper noise, Gaussian noise and speckle noise. From the

simulation results, it is observed that Median filter works best for images with salt and pepper noise. Combination filter (Median followed by Wiener filter) also yield good results and close to the results of Median filter. For Gaussian and Speckle noise images, combination filter produces better results in noise removal. This is proved with image metrics showing lower MSE and higher PSNR and SSIM. Hence combination filter (Median followed by Wiener filter) is best suited as it produces good results for all kinds of noises present in the considered CT images.

For kidney segmentation, Region based, Edge based active contours and Graph Cut have been run on the test dataset of CT images. It is observed that region based active contour method sometimes crossed the kidney boundary covering neighbouring tissues as well. Edge based active contour and Graph Cut methods produced better results than region based from the sample test data. Also it is observed that the selection of initial points and number of iterations influenced the segmentation results. Taking the average of dice coefficients over a set of test data, it is observed that dice coefficient of Edge based active contour is produced better segmentation results than Graph Cut and region based active contours.

To conclude, for preprocessing of images for kidney lesion detection, combination filter (Median followed by Wiener filter) is an effective noise removal filter and Edge based active contour is an

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effective segmentation technique for kidney CT images.

9. FUTURE WORK As the kidney could be successfully segmented

from the abdominal CT image, further segmentation is required to identify the lesions present in the Kidney. Clustering algorithms can be explored to identify the lesions. Once the lesions are segmented, temporal and spatial resolution techniques can be used to further classify the lesion and detect the presence of cancer lesions in kidney.

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[5] A.McAndrew, A Computational Introduction to Digital Image Processing, SecondEdition. CRC Press, 2015.

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Journal of Scientific & Industrial Research Vol. 77, January 2018, pp. 24-28

Efficient Identity-Based Parallel Key-Insulated Signature Scheme using Pairings over Elliptic Curves

R B Amarapu1* and P V Reddy2 *1Department of Engineering Mathematics, ANITS (A), Sangivalasa, Visakhapatnam, AP, India

2 Department of Engineering Mathematics, AUCE (A), Andhra University, Visakhapatnam, AP, India

Received 22 November 2016; revised 22 August 2017; accepted 03 November 2017

Many cryptographic schemes were designed under the assumption that the private keys involved in the system are perfectly secure. If the private key of a signer is exposed then security of the system is totally lost. Key-insulation mechanism minimizes the damage caused by the exposure of private keys in cryptographic schemes. To deal with the key-exposure problem in ID-based schemes, private keys have to be updated at very short intervals; but it will increase the risk of helper key exposure. In order to improve the security and efficiency of key insulation mechanism in ID-based signatures, in this paper, we proposed an ID-based parallel key insulated signature scheme. This scheme uses the bilinear pairings over elliptic curves and is provably secure in the ROM model with the assumption that the CDH problem is intractable. This scheme is strong key- insulated and allows frequent key updates without increasing the risk of helper key exposure and hence improve the security of the system. Also our IDPKIS Scheme reduces the computational, communicational complexity and hence the scheme can be deployed on inexpensive, lightweight and mobile devices.

Keywords: Signature Schemes, Key exposure Problem, Key-Insulation, Elliptic Curves, CDH Problem

Introduction Motivated by the concept of the IBC, many ID-

based schemes were designed under the assumption that the private keys involved in the system are perfectly secure. In a signature scheme, if the private key of a signer is exposed then security of the system is totally lost. To deal with this key-exposure problem, in 2002, Dodis et al.1 developed the concept of key-insulated mechanism with an idea by dividing the total life span of the master private key into discrete time periods and user updates their temporary key to encrypt some message in a time period. In 2003, Dodis et al.2 proposed key-insulated signature scheme and then many efforts have been devoted to development of new the key-insulated schemes both in PKI and ID-based settings3-8. In 2006, Hanaoka et al.9 and proposed a parallel key- insulated encryption (IDPKIE) scheme with two distinct helpers which are alternately used to update the secret keys. In the same year 2006, Weng et al.10 proposed an IDPKIE and signature schemes (IDPKIS). In 2008, Weng et al.11 proposed IDPKIS scheme using pairings over elliptic curves. In order to reduce the computational and communicational complexity in key insulated schemes, in this paper, we

proposed an ID-based parallel key-insulated signature scheme (IDPKIS). We prove the security of our scheme under random oracle paradigm without using the Forking lemma and hence the security is tightly connected to the CDH problem12. We compare our scheme with Zhou et al. scheme3 and Weng et al. scheme11. Preliminaries

In this section we briefly review bilinear parings and computational Diffie-Hellman problem which will be used for the construction and security analysis of our scheme.

Bilinear Map Let ( ,+) and ( ,∙) are two cyclic groups of

same prime order with as a generator in . A bilinear pairing is a map ê ∶ × → which satisfy the following. Bilinear: ∀ , ∈ and ∀ , ∈ ∗ , ê( , ) =

ê( , ) Non-Degeneracy: ∃ ∈ , ∋ ê ( , ) ≠ 1. Computable: ∀ , ∈ , ê( , ) can be

calculable using an efficient algorithm. Upon making suitable variations in the Weil or Tate

pairing one can obtain such maps on elliptic curves that are defined on a finite field13, 14.

—————— *Author for Correspondence E-mail: rameshamarapu.maths@anits.edu.in

AMARAPU & REDDY: EFFICIENT IDENTITY-BASED PARALLEL KEY-INSULATED SIGNATURE 25

Computational Diffie-Hellman (CDH) Problem For a given tuple ( , , ) ∈ and anonymous

, ∈ ∗ the CDH problem in is to compute ∈.The advantage of a Probabilistic Polynomial Time

(PPT) algorithm in obtaining the solution of the CDH problem in is = Pr [ ( , , ) =

/ , ∈ ∗]. The CDH assumption assures that for any PPT algorithm is negligible. Syntax of IDPKIS scheme

In this section we present the syntax of our IDPKIS scheme. The lifetime of IDPKIS systems is divided into discrete time periods. The public key of a user is his /her identity and is fixed for all the lifetime, while his secret key is updated in every time period. Every user has two helpers which store ℎ , ℎ The two helper keys are alternately used to update user’s secret/private keys i.e. ℎ is used in odd time periods while ℎ is used for even time periods. At time period t, user obtains an update key , from the jth helper (here ≡ ( 2)). Combining , with the private key , for the previous time period; he can derive the private key , for the current time period and , is used to sign a message during the corresponding time period without further access to the helpers. Moreover, due to the fact that ℎ and ℎ are alternately used, the risk of key exposure for ℎ or ℎ will not be increased, even if this user’s key-updates frequency is doubled. Concretely, an IDPKIS scheme consists of the following five polynomial-time algorithms. System Setup: This algorithm run by the PKG by

taking input a security parameters k and (possibly) a total number of time periods N, outputs a public parameter params and a master key msk.

Extract/Initial Private Key Generation: This algorithm run by the PKG by taking input msk, params and a user’s identity ∈ 0 , 1 ∗ outputs an initial private key , and two helper keys ℎ , ℎ .

Key Update: This consists of two deterministic algorithms. One is run by user helpers, the other is run by user.

Helper key update: This algorithm run by the user’s

helpers by taking as input a time period index t, a user’s Identity and the jth helper key with ≡ ( 2), returns an update key , .

User key update: This algorithm run by the user by taking input a time period index t, a user’s identity

, a private key , and an update key , returns a private key , . Signature Generation: The signing algorithm

takes input a time period index t, a message m and a private key , ; outputs = ( , , ) composed of the time period t and a signature σ.

Signature Verification: The verification algorithm, on input a candidate signature (t, σ) on m and the user’s identity outputs 1 if (t, σ) is a valid signature, and 0 otherwise.

Proposed IDPKIS scheme The proposed IDPKIS scheme consists of five

polynomial time algorithms and the detailed functionalities of these algorithms are presented as follows. System Setup: This algorithm run by the PKG

takes a security parameter ∈ and proceed as follows:

Generate two cyclic groups under addition and under multiplication respectively of same prime

order ≥ 2 , with as a generator of Generate a Bilinear map ê ∶ × → Picks cryptographic hash functions , : 0,1∗ → , : 0,1∗ ×

→ ∗ . Picks random integers and computes =, = ℎ , = ℎ , Also computes = ê , . Here is the system overall public

key, and are public keys of the helpers 0 and 1 respectively.

Publishes the system parameters as =⟨ , , , , ê , , , , , , , ⟩ and keeps the system’s master private key⟨ ⟩ the helpers private keys ℎ , ℎ with itself securely. Initial Private key Generation: Upon receiving

user's identity ∈ 0 , 1 ∗, the PKG computes , as

, = ( ) + ℎ ′ ( , −1) + ℎ ( , 0) ... (1)

PKG sends , as the initial private key for the user . Key Update: This algorithm consists of two

deterministic algorithms. Helper key update: Given an identity ∈ 0 , 1 ∗

and a time period index , the (here ≡ ( 2) helper computes , = ℎ [ ( , ) − ( , − 2)] and sends to user .

User key update: Given an identity ∈ 0 , 1 ∗ a time period index , an update key , and the

26 J SCI IND RES VOL 77 JANUARY 2018

private key , for the previous time period, user computes , = , + , as the private key for time period . Finally the user erases two values , and , . Note that the ≡( 2), ′ ≡ ( − 1)( 2) then the following

equality always holds , = ( ) + ℎ ′ ( , − 1) + ℎ ( , ) ... (2)

Signature Generation: In time period , for a given message and the private key , , the user

chooses ∈ ∗ and computes = , ℎ = ( , , , ), = ℎ , + . The signature on in time period generated by is =( , , ).

Signature Verification: Given a signature on a message and identity , any one can verify this signature as follows: Let ≡ ( 2), ′ ≡( − 1)( 2) : Compute ℎ = ( , , , ); return 1 if the following equality (3) holds; otherwise it returns 0.

ê( , ) = ê , ℎ ( ) ê′, ℎ ( , − 1)

ê(ℎ , ℎ ( , )) ... (3) Analysis of the proposed idpkis scheme

Proof of correctness Theorem 1

The proposed IDPKIS scheme satisfies the proof of correctness

Proof: The proof of correctness of the scheme is justified by verifying the validity of the equation (3). ê( , ) = ê( , ℎ , + )

= ê , ℎ ( ) ê′, ℎ ( , − 1)

ê( , ℎ ( , )) Security Analysis Theorem 2

In the random oracle model, the proposed IDPKIS scheme is strong key-insulated and is existential unforgeable against adaptive chosen message and identity attacks on the feasibility assumption of the CDH problem.

Proof: We will show how to construct an algorithm ℬ against the CDH assumption in group . Suppose ℬ is given a CDH instance( , , ) ∈ . ℬ’s goal is to derive with the help of adversary . ℬ plays the role of ’s challenger and works by interacting with

in a unforgeable under adaptively chosen message and ID- attack game as follows:

Setup: Algorithm ℬ sets = and gives the

public parameters to . Note that the master key is implicitly assigned to be on msk = a which is unknown to ℬ.

Queries: Algorithm ℬ responds to a series of queries made by adaptively as follows.

− queries: When Queries the oracle at a point ∈ 0 , 1 ∗ algorithm ℬ maintains a − list of tuples ( , , , ) and responds to as given below. If the queried already appears on the −list

in a tuple ( , , , ) then ℬ responds with previously defined value as ( ) = ∈ .

Otherwise, ℬ picks ∈ 0 , 1 ∗, that yields 0 with probability and 1 with probability 1 − .

If = 0 then ℬ picks a random value ∈ ∗ and computes ( ) = = ( ), else ( ) =

= . ℬ adds the tuple ( , , , )to the −list and

responds to with ( ) = ∈ . − queries: ℬ maintains a −list which is

initially empty. When queries the tuple ( , ), ℬ first checks whether −list contains a tuple for this input. If it does, the previously defined value is returned. Otherwise, ℬ chooses a random integer ∈

∗ and computes the hash value ( , ) = ∈ and then stores the tuple ( , , , ) in the

−list. − queries: : At any time quires the oracle

at ( , , , ). ℬ maintains a list referred as − list of tuples , , , , ′ and responds to as given below. If the quired tuple ( , , , ) already appears in the − list in a tuple , , , , ′ then ℬ responds with ( , , , ) = ′ ∈ ∗ . Otherwise, ℬ picks a random ′ ∈ ∗

and adds the

tuple , , , , ′ in the − list and responds to with ( , , , ) = ′ ∈ ∗ .

Signature queries: When queries a signature on a message for an identity of a user in a time period , ℬ does the following.

AMARAPU & REDDY: EFFICIENT IDENTITY-BASED PARALLEL KEY-INSULATED SIGNATURE 27

Chooses a random integer ∈ ∗

and computes = .

If the tuple , , , , ′ in the − list, ℬ chooses " ∈ ∗ and tries again. Otherwise, ℬ computes = ′ + + +

and stores in the − list and responds to a with = ( , ) as the Queried signature. All responds to sign Queries are valid; indeed, the output ( , ) is a valid signature on under .

Forgery: Finally, output a forgeable signature ( ∗, ∗) on a message ∗ for an identity ∗ of a user in a time period ∗. ℬ finds the tuple ( , ∗, ∗, ∗) from the −list and proceeds only if ∗ = 0; since = 0, it follows that ( ) = ∗( ). If succeeds in this game, then we have the following.

ê( , ∗) =

ê , ′∗ ∗( ) + ′∗ + ∗ + ∗ )

From this, ℬ can derive as ( ∗) ′∗ −1

[ ∗ − ′∗ − ∗ − ∗ ] and solve the

CDH instance successfully. But the CDHP is computationally infeasible by any polynomial-time bounded algorithm. Therefore, based on the intractability of CDHP, our scheme is provable secure and strong key insulated secure in the random oracle model against the adaptive chosen message and identity attacks. Efficiency analysis

We present the efficiency analysis of our scheme by comparing it with the related and existing IDPKIS scheme in terms of computation and communicational point of view. For comparison, we consider point addition in as , scalar multiplication in as

, exponentiation in as , hash functions as H and computation of bilinear pairing (BP). The following Table 1 gives the comparison of our scheme with Weng et al. scheme11. In the pairing-based signature schemes, the pairing operation is the most time-consuming compared with the other operations

like addition and hash functions etc. Despite a number of attempts to reduce the complexity in evaluation of pairing operation, still the operation is very costly13, 14. From the Table 1, it is clear that our scheme requires 4 pairing operations whereas Weng et al. scheme11 requires 5 Pairing operations which means that the computational complexity of our scheme is less than Weng el al. scheme11. Also, the proposed scheme has 2| | as signature length whereas Weng et al.11 scheme has 4| | as signature length and hence the communicational efficiency is improved greatly than Weng et al. scheme11.

Conclusions In this paper, we presented a new IDPKIS scheme

to deal with key exposure problem in ID-based signatures. The proposed scheme is provable secure and strong key insulated. Also this scheme allows frequent key-updates without increasing the risk of helper key-exposure, hence enhances the security of the system. The proposed scheme requires less number of pairing operations and has the signature size is less than the existing schemes. Thus the proposed IDPKIS scheme is efficient in terms of computation and communication point of view. The security of our scheme is proved under random oracle paradigm without using the Forking lemma and hence the security of the scheme is tightly connected to the CDH problem. Further, as a future work, we extend our scheme to construct a PKIS scheme with multiple helper keys to increase simultaneously the security of both helper keys and user’s without significant loss of efficiency.

References 1 Dodis Y, Katz J, Xu S & Yung M, Key-insulated public key

cryptosystems, Advances in Cryptology, EUROCRYPT'03, LNCS, 2332(2002), 65-82.

2 Dodis Y, Katz J & Xu S and Yung M, Strong key-insulated signature schemes, LNCS, 2567(2003), 130-144.

3 Zhou Y, Cao Z, & Chai Z, ID- based key-insulated signature, in Proc ISPEC 2006, LNCS, 3903 (2006), 226-234.

4 Tsu-yang Wu, Yuh-Min Tseng & Ching-Wen Yu, ID-based key-insulated signature scheme with batch verifications and its novel application, Int J Inno, Comp, Inf and Con, 87(2012), 4797-4810.

5 Zhao H, Duan S, Cheng X, & Hhao R, Key-insulated aggregate signature, Front Cmput Sci, 8(5) (2014), 837–846.

6 Gopal P V S S N & V Reddy P, Efficient ID-based key-insulated signature scheme with batch verifications using bilinear pairings over elliptic curves, J Discrete Math Sci and Cryp, T & F, 18(4) (2015), 385-402.

7 Li J, Du H & Zhang Y, Certificate-based key-insulated signature in the standard model, Security in Comp Sys and Net, Com J, 2(2016), 1-12.

Table1 — Comparison with Weng et. al scheme

Scheme Computational cost for Signature

Generation

Computational cost for Signature

verification.

Communication cost (Signature

Size)

Weng.et.al [11]

1 + 2 + I H + 1AD

4H + 5 BP |4 |

Ours 1 + 2 +1 + 1AD

3 + 4 + 4 BP

|2 |

28 J SCI IND RES VOL 77 JANUARY 2018 8 Rao Y S & Dutta R, Attribute-based key-insulated signature

for boolean formula, Int J Comp Math, T & F, 93(6) (2016), 864–888.

9 Hanaoka G, Hanaoka & Imai H, Parallel key-insulated public key encryption, Proc PKC 06, LNCS 3958 (2006), 105-122.

10 Weng J, Liu S, Chen K & MA C, Identity based parallel key-insulated encryption without random oracles security notions and construction, Proc INDOCRYPT 06, LNCS, 4329 (2006), 409-423.

11 Weng J, Liu S & Chen K., Identity based parallel key-insulated signature: Frame work and Construction, J Res and Pract in Info Tech, 40(1) (2008), 55-68.

12 Hofheinz D & Jager T. Tightly secure signatures and PKE, J Designs, Codes and cryptography, 80(1) (2016), 29-61.

13 IEEE Standard for identity-based Cryptographic techniques using pairing. IEEE Std:1363.3-2013 (2013), 1-151.

14 Enge A & Milan A, Implementing cryptographic pairings at Standard Security levels, SPACE-2014, LNCS, 8804 (2014), 28-46.

ORIGINAL PAPER

Dynamical aspects of anisotropic Bianchi type VI0 cosmological modelwith dark energy fluid and massive scalar field

Y Aditya1, K D Raju2,4, P J Ravindranath3 and D R K Reddy4*1Department of Mathematics, GMR Institute of Technology, Rajam 532127, India

2Department of Mathematics, ANITS (A), Visakhapatnam 531162, India

3Department of Mathematics, Rajiv Gandhi Memorial College of Engineering and Technology, Nandyal, India

4Department of Applied Mathematics, Andhra University, Visakhapatnam 530003, India

Received: 30 May 2019 / Accepted: 25 November 2019

Abstract: In this article, we investigate a spatially homogeneous and anisotropic Bianchi type VI0 cosmological model in

the presence of dark energy fluid and an attractive massive scalar field in Einstein’s theory of gravitation. The field

equations are solved using the following conditions (1) scalar expansion is proportional to shear scalar of the space time

and (2) a power law between the scalar field and the average scale factor of the universe. We have evaluated the

cosmological parameters of the model and discussed their dynamical aspects with reference to the recent discovery of

accelerated expansion of the universe.

Keywords: Anisotropic cosmological model; Dark energy; Massive scalar field; Einstein’s theory

PACS Nos.: 04.50Kd; 98.80.-k; 95.35.?d

1. Introduction

It is a well-established fact that even today, general rela-

tivity is a beautiful physical and geometrical theory which

successfully describes gravitational field. This theory, also,

serves as a useful tool for discussion of cosmological

models of the universe. However, Einstein, himself, poin-

ted out that his theory does not account for some desirable

features. For example, this theory does not, completely,

incorporate Mach’s principle. Also, the modern concept of

accelerated expansion of the universe [1, 2] which is sup-

posed to be driven by dark energy (DE) is not explained by

this theory. Two approaches have been proposed to explain

this mysterious cosmic acceleration. One way is to con-

struct DE models, and another approach is to propose

alternative theories of gravitation. Significant alternatives

to Einstein’s theory are Brans–Dicke [3], Saez–Ballester

[4] scalar–tensor theories of gravity and f(R) and f(R,T)

theories of gravity [5, 6]. Here, R is the scalar curvature

and T is the trace of the energy momentum tensor. Several

DE models in modified theories of gravitation are

reviewed, in detail, in references [7–11]. The other alter-

native theories in which scalar fields play an important role

are Einstein’s theory minimally coupled to zero-mass

scalar fields and attractive massive scalar fields.

Scalar fields are very important in the discussion of

cosmology because they describe interesting phenomena

such as Higgs bosons, dark matter and DE. They also

represent quintessence models of the universe. Apart from

Brans–Dicke and Saez–Ballester scalar fields, mass-less

and massive scalar fields also play a significant role in

physical theories. These fields respond to gravitational

fields and are directly connected to scalar curvature. There

are two types of scalar fields, (1) zero-mass scalar fields

and (2) attractive massive scalar fields. Zero-mass scalar

fields describe long-range interactions, and massive scalar

fields represent short-range interactions. It is said that the

scalar fields help to solve the horizon problem in cosmol-

ogy, and the recent scenario of accelerated expansion of the

universe is caused by scalar fields. Sharif and Jawad [12]

have studied cosmological evolution of interacting new

holographic dark energy and reconstructed various scalar*Corresponding author, E-mail: reddy46@gmail.com; aditya.y@

gmrit.edu.in

Indian J Phys

https://doi.org/10.1007/s12648-020-01722-6

2020 IACS

field models. Jawad and Majeed [13] obtained the corre-

spondence of pilgrim DE with scalar field models. Jawad

[14] investigated interacting modified QCD ghost scalar

field models of dark energy. Santhi et al. [15] discussed

Kantowski–Sachs scalar field cosmological models in

f(R,T) gravity.

There are evidences of spatial anisotropy in the early

stages of our universe. In order to study the realistic picture

of the universe at its early stages of evolution, Bianchi

models which are spatially homogeneous and anisotropic

are quite useful. Bianchi models in general relativity and in

alternative theories of gravitation have been, extensively,

investigated by several authors. Bianchi-type DE models

have also been studied by many researchers. Copeland

et al. [16] and Nojiri and Odintsov [17] presented a nice

review of Bianchi-type DE models in modified theories of

gravitation. Akarsu and Kilinc [18] obtained Bianchi type

III models with anisotropic dark energy. Pradhan et al. [19]

studied Bianchi type VI0 DE model with variable decel-

eration parameter. Amirhashchi [20] investigated Bianchi

type V viscous DE models. Mishra et al. [21] discussed

anisotropic magnetized DE models. Recently, many

authors have discussed various Bianchi-type cosmological

models with different matter distributions in modified

theories of gravitation. Aditya and Reddy [22] and Santhi

et al. [23] studied Bianchi type I and III string cosmolog-

ical models in f(R) gravity. Mishra and Sahoo [24] inves-

tigated Bianchi type VIh model, and Aditya et al. [25]

discussed Bianchi type II, VIII and IX perfect fluid models

in f(R,T) theory of gravity. Sofuoglu [26] studied locally

rotationally symmetric Bianchi type IV universe in the

presence of a cosmic fluid with anisotropic pressure within

Einstein’s theory of gravitation.

Here, we focus our attention on DE models in general

relativity in the presence of massive scalar fields. Bianchi

type V dark energy model with scalar meson fields, in

general relativity, is obtained by Reddy [27], while Naidu

[28] discussed Bianchi type II modified holographic Ricci

dark energy models in the presence of attractive massive

scalar field. Aditya and Reddy [29] discussed dynamics of

Bianchi type III cosmological model in the presence of

anisotropic DE and an attractive massive scalar field.

Reddy et al. [30] investigated dark energy cosmological

model in the presence of anisotropic dark energy fluid

coupled with mass-less scalar field in Kantowski–Sachs

space time in general theory of relativity. Recently, Reddy

and Ramesh [31] presented a new DE model in five-di-

mensional Kaluza–Klein anisotropic space time in the

presence of zero-mass scalar fields in general relativity.

Very recently, Reddy et al. [32] studied dynamics of

Bianchi type II anisotropic DE cosmological model in the

presence of scalar meson fields. Naidu et al. [33] discussed

Bianchi type V dark energy cosmological model in general

relativity in the presence of massive scalar field.

The above discussion inspired us to investigate Bianchi

type VI0 anisotropic DE model in the presence of massive

scalar field in general relativity. Bianchi type VI0 models

are interesting because they are isotropic generalization of

FRW models. Also, by the available evidences, these

models are favored for low-density universes. In Sect. 2,

we derive the field equations of Einstein’s theory with the

help of Bianchi type VI0 metric in the presence of aniso-

tropic DE with an attractive massive scalar field. In Sect. 3,

we present the DE model by solving the field equations.

Section 4 is devoted to the construction of cosmological

parameters of our model and to their physical discussion.

The results are summarized in the last section.

2. Derivation of field equations

Einstein’s field equations with anisotropic DE fluid and an

attractive massive scalar field as sources of gravity are

given by

Rij 1

2gijR ¼ ðTðdeÞ

ij þ TðsÞij Þ ð1Þ

Here, the energy momentum tensor Tij in the presence of

anisotropic DE fluid and an attractive massive scalar field

is given by

Tij ¼ TðdeÞij þ T

ðsÞij ð2Þ

where

TðdeÞij ¼ ðqde þ pdeÞuiuj þ pdegij ð3Þ

TðsÞij ¼ u;iu;j

1

2ðu;ku

;k M2u2Þ: ð4Þ

Here, qde is the DE density, pde is the DE pressure, M is

the mass of the scalar field u which satisfies the Klein–

Gordon equation

giju;ij þM2u ¼ 0 ð5Þ

and comma and semicolon denote ordinary and covariant

differentiations, respectively.

Here, we assume anisotropic and spatially homogeneous

Bianchi type VI0 metric to study the geometry of the uni-

verse, and it can be written as

ds2 ¼ dt2 þ X2dx2 þ Y2e2axdy2 þ Z2e2ax dz2 ð6Þ

where X, Y and Z are functions of time t, and a is a nonzero

constant which can be set equal to unity. The energy–

momentum tensor of anisotropic DE fluid given by Eq. (3)

can be parameterized as

Y Aditya et al.

TðdeÞij ¼ diag½1;xde; ðxde þ cÞ; ðxde þ dÞqde ð7Þ

where we have defined the EoS parameter of DE as

xde ¼pde

qdeð8Þ

And, the skewness parameters c and d are the deviations

from xde along y- and z-axes, respectively.

Now, using comoving coordinates and Eqs. (4) and (7),

the field Eqs. (1), for the metric (6), in the explicit form,

are obtained as

_X _Y

XYþ

_Y _Z

YZþ

_X _Z

XZ 1

X2 qde

_u2

2

þM2u2

2¼ 0 ð9Þ

€Y

Yþ

€Z

Zþ

_Y

Y

_Z

Zþ 1

X2þ xdeqde þ

_u2

2

þM2u2

2¼ 0 ð10Þ

€X

Xþ

€Z

Zþ

_X _Z

XZ 1

X2þ ðxde þ cÞqde þ

_u2

2

þM2u2

2¼ 0

ð11Þ€X

Xþ

€Y

Yþ

_X _Y

XY 1

X2þ ðxde þ dÞqde þ

u2

2

þM2u2

2¼ 0

ð12Þ_Y

Y

_Z

Z¼ 0 ð13Þ

Here, the scalar field and the dark energy fluid are

conserving separately since from the Klein–Gordon

Eq. (5), we get

€uþ _u_X

Xþ

_Y

Yþ

_Z

Z

þM2u ¼ 0 ð14Þ

and from the covariant derivative of Eq. (7), we have the

conservation of energy–momentum tensor for DE fluid as

_qde þ_X

Xþ

_Y

Yþ

_Z

Z

ð1þ xdeÞ ¼ 0 ð15Þ

where an overhead dot denotes differentiation with respect

to t.

We define the following important parameters which

will help in solving the field equations:

Spatial volume is

V ¼ a3ðtÞ ¼ XYZ : ð16Þ

The average Hubble parameter is

H ¼ H1 þ H2 þ H3

3ð17Þ

where H1 ¼ _XX; H2 ¼ _Y

Yand H3 ¼ _Z

Zare the Hubble

parameters in the directions of x-, y- and z-axes,

respectively.

The expansion scalar h and shear scalar r2 are defined as

h ¼ ui;i ¼_X

Xþ

_Y

Yþ

_Z

Zð18Þ

r2 ¼ 1

2rijrij

¼ 1

3

_X

X

2

þ_Y

Y

2

þ_Z

Z

2

_X _Y

XY

_Y _Z

YZ

_Z _X

ZX

" !ð19Þ

The mean anisotropy parameter is

Ah ¼1

3

X3i¼1

Hi H

H

2

: ð20Þ

3. Solutions and the DE model

In this section, we solve the field Eqs. (9)–(15) and present

the corresponding DE model. Integration of Eq. (13),

immediately, yields

Y ¼ c1Z ð21Þ

where c1 is a constant of integration which can be taken as

unity without any loss of generality, so that we get

Y ¼ Z ð22Þ

Now using Eq. (22), the field equations reduce to

2_X _Y

XYþ

_Y

Y2

2

1

X2 qde

_u2

2

þM2u2

2¼ 0 ð23Þ

2€Y

Yþ

_Y

Y2

2

þ 1

X2þ xdeqde þ

_u2

2

þM2u2

2¼ 0 ð24Þ

€X

Xþ

€Y

Yþ

_X _Y

XY 1

X2þ ðxde þ cÞqde þ

_u2

2

þM2u2

2¼ 0

ð25Þ€X

Xþ

€Y

Yþ

_X _Y

XY 1

X2þ ðxde þ dÞqde þ

_u2

2

þM2u2

2¼ 0

ð26Þ

€uþ _u_X

Xþ 2

_Y

Y

þM2u ¼ 0 ð27Þ

_qde þ_X

Xþ 2

_Y

Y

ð1þ xdeÞ ¼ 0 ð28Þ

Now, Eqs. (23)–(28) are five independent equations

[Eq. (28) being the conservation equation] in seven

unknowns X; Yð¼ ZÞ; qde;xde; c; d and u. Hence to find a

determinate solution, we need two more conditions. We

use the following two conditions:

1. The shear scalar of the space time is proportional to the

expansion scalar so that we have (Collins et al. [34])

Y ¼ Xn ð29Þ

Dynamical aspects of anisotropic Bianchi type VI0 cosmological model

where n 6¼ 1 is a positive constant which preserves the

anisotropy of the space time. The physical reason

behind this assumption is warranted from the velocity

redshift relation for extragalactic sources observation.

They suggested that the Hubble expansion of the uni-

verse may attain isotropy when rh is constant (Kan-

towski and Sachs [35]). Collins et al. [34] studied the

physical significance of this condition for a perfect

fluid.

2. Also, in the literature, it is common to use a power law

relation between scalar field u and average scale factor

a of the form (Johri and Sudharsan [36]; Johri and

Desikan [37])

ua ½aðtÞl

where l is a power index. Many authors have

investigated various aspects of this form of scalar

field u (Rao et al. [38]; Santhi et al. [39]; Aditya and

Reddy [40]). In view of the physical significance of

above relation, here, we use the following assumption

to reduce the mathematical complexity of the system

ð2nþ 1Þ_X

X¼ _u

u: ð30Þ

Many authors have constructed cosmological models

using this relation (30). Singh et al. [41, 42] and Aditya and

Reddy [29] studied Bianchi-type cosmological models with

massive scalar fields using relation (30).

Now from Eqs. (25) and (26), we obtain

c ¼ d ð31Þ

Equations (27), (29) and (30), together, yield

u ¼ exp u0t M2t2

2þ u1

ð32Þ

Also, Eqs. (30) and (32) give us

X ¼ expM2t2 2u0t 2u1

2ð2nþ 1Þ

ð33Þ

Y ¼ Z ¼ expnðM2t2 2u0t 2u1Þ

2ð2nþ 1Þ

ð34Þ

Here, u0 and u1 are constants of integration.

Bianchi VI0 dark energy model can now be written,

using Eqs. (6), (33) and (34) as

ds2 ¼ dt2 þ expM2t2 2u0t 2u1

2nþ 1

dx2

þ expnðM2t2 2u0t 2u1Þ

2nþ 1

e2xdy2 þ e2x dz2

ð35Þ

with the scalar field given by Eq. (32) and with the fol-

lowing physical and kinematical parameters.

4. Physical discussion of dynamical parameters

of the model

In this section, we evaluate the physical and dynamical

parameters of the model (35) and discuss their physical

significance.

Spatial volume of the model is

V ¼ expM2t2

2 u0t u1

ð36Þ

The average Hubble parameter is

H ¼ M2t u0

3

ð37Þ

The expansion scalar is

h ¼ 3H ¼ M2t u0 ð38Þ

The shear scalar is

r2 ¼ 1

3

M2t u0

2nþ 1Þ

2

ðn 1Þ2 ð39Þ

The average anisotropy parameter is

Ah ¼8ðn 1Þ2

3ð2nþ 1Þ2ð40Þ

Now using Eqs. (32)–(34) in Eq. (23), we obtain DE

density, in the model, as

qde ¼M2t u0

2nþ 1

2

2nðn 1Þ þ M2

2exp u0t

M2t2

2þ u1

1

2M2t u0

2

exp u0t M2t2

2þ u1

exp

2u0t M2t2 þ 2u1

2nþ 1

ð41Þ

Using Eqs. (32)–(34) in Eq. (24), we find the EoS

parameter of DE as

xde ¼ 1

qde

2M2n

2nþ 1þ 3n2ðM2t u0Þ2

ð2nþ 1Þ2þ u0 M2t

2

"

exp u0t M2t2

2þ u1

þM2

2exp 2u0t M2t2 þ 2u1

þ exp2u0t M2t2 þ 2u1

2nþ 1

ð42Þ

From Eqs. (24), (25) and (32)–(34), we determine the

skewness parameters as

c ¼ d ¼ ðn 1Þqde

ðM2t u0Þ2 þM2

2nþ 1

!ð43Þ

where qde is given by Eq. (41).

Y Aditya et al.

Sahni et al. [43] formulated two important parameters

known as statefinder parameters. In modern cosmology,

these parameters are useful to distinguish various DE

models and are defined as

r ¼ a...

aH3; s ¼ r 1

3 q 12

ð44Þ

In our model, these parameters are calculated as

r ¼ 1þ 9M2

ðM2t u0Þ2; s ¼ 2M2

2M2 þ ðM2t u0Þ2: ð45Þ

The deceleration parameter (DP) plays an important role

in the discussion of the nature of the universe. The universe

decelerates for q[ 0, and it has a constant rate of

expansion for q = 0 and accelerated expansion for

1 q\0. When q ¼ 1, the universe exhibits an

exponential expansion, and for q\ 1, it has super-

exponential expansion. The deceleration parameter is

defined as

q ¼ d

dt

1

H

1: ð46Þ

For our model, it is found as

q ¼ 1þ 3M2

ðM2t u0Þ2

!ð47Þ

The above analytical expressions for cosmological

parameters facilitate the physical discussion of our

model. It may be observed that Eq. (35) describes

Bianchi type VI0 DE model in the presence of an

attractive massive scalar field, and Eq. (32) gives the

scalar field in the model. It is observed that our model is

free from initial singularity, i.e., at t = 0. We can see that

the spatial volume of the model shows an exponential

expansion with an increase in cosmic time. The

kinematical parameters H; h; r2 diverge as t approaches

infinity, while they all give us finite values at t = 0. Our

model is spatially homogeneous and anisotropic throughout

the evolution since the anisotropy parameter of our model

is constant. It can also be observed that when n = 1, the

mean anisotropy parameter Ah, the shear scalar r2 and

skewness parameters vanish. Hence, the universe becomes

isotropic and shear free.

Figure 1 shows the behavior of scalar field u in terms of

cosmic time for various values of u0. We observe that u is

a positive and increasing function throughout the evolution

of the universe. For all three values of u0, scalar field

increases, reaches a maximum value at some point of time

and then finally decreases and attains a constant value.

Also, it can be seen that the scalar field increases with

values of u0. Figure 2 describes the behavior of DE density

qde versus cosmic time for various values of u0. It is

observed that qde is always positive throughout the evo-

lution and increases as scalar field increases. From Fig. 3,

we observe that the model starts from the aggressive

phantom region xde\\ 1 and finally approaches to

Fig. 1 Plot of scalar field versus cosmic time t for

u1 ¼ 4 and M ¼ 0:18

Fig. 2 Plot of energy density of DE versus cosmic time t for

u1 ¼ 4; n ¼ 0:9 and M ¼ 0:18

Fig. 3 Plot of EoS parameter versus cosmic time t for

u1 ¼ 4; n ¼ 0:9 and M ¼ 0:18

Dynamical aspects of anisotropic Bianchi type VI0 cosmological model

xde ¼ 1 for various values of u0. It may be observed that

initially, the scalar field affects the EoS parameter and

finally approaches to a constant value. The behavior of

skewness parameter versus cosmic time is shown in Fig. 4

for different values of u0. Initially, the skewness parameter

is negative and finally vanishes. The scalar field affects the

skewness parameter at the initial epoch and negligible at

present epoch. In Fig. 5, we depicted the behavior of

deceleration parameter of our model versus cosmic time t

for various values u0. It is clear from the figure that the DP

remains in the region q\2 1 for all values of u0. Hence,

our model shows super-exponential expansion of the uni-

verse. It is also observed that as the scalar field increases,

the rate of super-exponential expansion slows down. The

nature of statefinders is plotted in Fig. 6, and we observe

that the model approaches KCDM model, i.e., (r,s) = (1,0)

at late times. It may be noted that the behavior of physical

parameters of our model is almost similar to the behavior

of parameters discussed in Ref. [33]. However, they differ

mathematically, and hence the model is quite new.

5. Conclusions

Scalar fields are very important in the discussion of DE

models in cosmology because of the fact that scalar fields

represent quintessence models. Here, we have presented

Bianchi type VI0 DE model which is spatially homoge-

neous and anisotropic. We have considered anisotropic DE

fluid and an attractive massive scalar field as sources to

obtain Einstein field equations. Exact solution of field

equations is obtained using a relation between metric

potentials and a power law between the average scale

factor of the universe and the scalar field. The corre-

sponding DE model is also presented. The dynamical and

physical aspects of the model are discussed by determining

the cosmological parameters of our model. The following

are some conclusions:

Our model represents a Bianchi type VI0 anisotropic DE

model, in general relativity, in the presence of an attractive

massive scalar field. The model is free from initial singu-

larity and exhibits an exponential expansion starting from

finite volume. This, in fact, leads to inflation.

It is observed that all the physical and dynamical

parameters of our model are finite initially and diverge at

late times (i.e., as t ! 1). The average anisotropy

parameter is constant, and hence our model remains ani-

sotropic. The deceleration parameter indicates that the

universe, described by our model, undergoes a super-ex-

ponential expansion since q\ 1. The DE density is

always positive and increases with cosmic time.

In our model, the scalar field is always positive which

should, in fact, be the case in the present scenario. At this

moment, we compare the work done by other authors on

the scalar field models. In the work of Sharif and Jawad

[12], it is shown that the scalar field increases, and further it

is stated that the large value of u at the present epoch

represents accelerated expansion of the universe. Here, we

Fig. 4 Plot of skewness parameter versus cosmic time t for

u1 ¼ 4; n ¼ 0:9 and M ¼ 0:18

Fig. 5 Plot of deceleration parameter versus cosmic time t for

M ¼ 0:18

Fig. 6 Plot of statefinder parameters for u0 ¼ 0:7 and M ¼ 0:18

Y Aditya et al.

find the results are consistent with those of Sharif and

Jawad [12] on scalar field.

The skewness parameter is negative at the initial epoch

and vanishes at late times, i.e., the anisotropy of the DE

vanishes at late times. Here, it is mentioned that the

behavior of some of the physical parameters of this model

is almost similar to the behavior of parameters discussed in

Naidu et al. [33]. However, they are mathematically

different.

The model varies only in phantom region xde\ 1,

and finally approaches to phantom limit xde ¼ 1(KCDMmodel). The EoS parameter trajectories correspond to dif-

ferent values of u0 start from higher phantom values and

go toward lower phantom values. Also, it is observed that

the model, finally, meets x ¼ 1 boundary. It is interest-

ing to mention, here, that the EoS parameter of our model

corresponds to phantom era of the universe which is a

favorable sign to pilgrim DE conjecture. The works of

Chattopadhyay et al. [44]; Jawad and Majeed [13]; Santhi

et al. [45, 46]; and Aditya and Reddy [47], also, support

this type of behavior of DE.

Statefinders analysis says that the model approaches to

KCDM at late times. Hence, these results show that our

anisotropic DE model with massive scalar field is in

agreement with the current observational data. It is noted

that the scalar field u in the model influences (at the initial

epoch or at late times) all the physical parameters of the

universe. To conclude, we observe that our model gives us

a phantom model which leads to KCDM model of the

universe, and the results obtained, here, are quite in

accordance with the modern cosmological observations.

Acknowledgements We thank the reviewers and the Editorial team

for their positive and constructive comments, which have helped to

improve the quality and presentation of the manuscript.

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Kaluza-Klein dark energy model in Lyra manifold in the presence of massive

scalar field

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Astrophys Space Sci (2019) 364:190 https://doi.org/10.1007/s10509-019-3681-2

ORIGINAL ARTICLE

Kaluza-Klein dark energy model in Lyra manifold in the presenceof massive scalar field

Y. Aditya1 · K. Deniel Raju2,3 · V.U.M. Rao3 · D.R.K. Reddy3

Received: 31 August 2019 / Accepted: 30 October 2019© Springer Nature B.V. 2019

Abstract In this investigation we intend to study the dy-namics of an anisotropic dark energy cosmological modelin the presence of a massive scalar field in a modified Rie-mannian manifold proposed by Lyra (Math. Z. 54:52, 1951)in the background of a five dimensional Kaluza-Klein spacetime. We solve the Einstein field equations using some phys-ically significant conditions and present a deterministic darkenergy cosmological model. We use here the time depen-dent displacement vector field of the Lyra manifold. All thedynamical parameters of the model, namely, average Hub-ble parameter, anisotropy parameter, equation of state pa-rameter, dark energy density, deceleration parameter andstatefinders are evaluated for our model and their physicalrelevance to modern cosmology is discussed in detail.

Keywords Kaluza-Klein model · DE model · Lyramanifold · Massive scalar meson field

1 Introduction

The subject that is attracting several researchers, in mod-ern cosmology, is the accelerated expansion of the universe(Riess et al. 1998; Perlmutter et al. 1999). It has been said

B Y. Adityayaditya2@gmail.com

D.R.K. Reddyreddy_einstein@yahoo.com

1 Department of Mathematics, GMR Institute of Technology,Rajam 532127, India

2 Department of Mathematics, ANITS (A), Visakhapatnam531162, India

3 Department of Applied Mathematics, Andhra University,Visakhapatnam 530003, India

that this is caused by an exotic negative pressure which isknown as dark energy (DE). Several DE models have beenproposed to explain this phenomenon which even today re-mains as mystery. The cosmological constant is supposedto account for this DE. But it has some serious problems.Hence two approaches have been suggested to describe thismysterious concept. One method is to construct DE modeland study their significance in relation to this cosmic infla-tion. Another way is to modify Einstein’s theory of gravi-tation and to construct DE models with a special referenceto the observations of modern cosmology which throws abetter light to explain this scenario.

For this purpose, there have been several modifications ofEinstein’s theory of gravitation by modifying the Einstein-Hilbert action of general relativity and introducing ScalarFields (SFs) into Einstein theory. Introduction of SFs leadsto the well known quintessence models which help to ex-plain the accelerated expansion of the universe. Thus themodified theories of gravitation are f (R) and f (R,T ) the-ories (Nojiri and Odintsov 2003; Harko et al. 2011) andscalar-tensor theories proposed by Brans and Dicke (1961)and Saez and Ballester (1986). Anisotropic DE modelsin the above modified theories of gravitation were inves-tigated by numerous researchers (Copeland et al. 2006;Nojiri et al. 2005; Kiran et al. 2014; Reddy et al. 2014;Aditya et al. 2016; Rao et al. 2018; Aditya and Reddy2018a, 2019).

Here we are interested in the interacting scalar mesonfields. Scalar meson fields are of two types—zero mass SFsand massive SFs. Zero mass SFs describe long range inter-actions while massive SFs represent short range interaction.In fact, this physical significance of SFs leads to immensestudy of SFs. Also, SFs are very important since they repre-sent matter fields with spin less quanta and describe gravita-tional fields. In the literature, there are several investigations

190 Page 2 of 8 Y. Aditya et al.

of cosmological models in the presence of mass less andmassive SFs coupled with different physical sources. Herewe are mainly concerned with models in the presence ofmassive scalar fields. Some note worthy models are obtainedby Naidu (2018), Aditya and Reddy (2018a) and Reddy andRamesh (2019) in the presence of massive SFs.

In an attempt to unify gravity and electromagnetic fieldsseveral modifications of Riemannian geometry have beenproposed. Significant among them is Lyra (1951) geometry.In this geometry a gauge function has been introduced intothe structureless manifold so that displacement field arisesnaturally. The energy conservation is not valid in this the-ory. The displacement field, in this theory plays the samerole as the cosmological constant in general relativity. Sev-eral cosmological models in this particular theory have beendiscussed extensively. The following are relevant and signif-icant to our investigation: Singh and Rani (2015) have dis-cussed Bianchi type-III cosmological models into coupledperfect fluid and attractive massive scalar field as physicalsource in Lyra geometry. Very recently, Reddy et al. (2019)investigated Bianchi type-III DE cosmological model in thepresence of massive scalar field in this geometry.

In order to discuss the early stages of evolution of theuniverse, immediately after big bang, higher dimensionalcosmology plays a vital role. Subsequently, the universehas undergone compactification and we have the presentfour dimensional universe. Witten (1984) and Appelquistet al. (1987) are some of the authors who have studiedhigher dimensional cosmology. In particular, in Kaluza-Klein (Kaluza 1921; Klein 1926) five dimensional geometrythe extra dimension is used to couple the gravity and electro-magnetism. Hence, Kaluza-Klein models gain importance.Kaluza-Klein cosmological models have been discussed byseveral authors in modified theories of gravity (Reddy andLakshmi 2014; Sahoo et al. 2016; Santhi et al. 2016a;Naidu et al. 2018a; Reddy and Aditya 2018; Aditya andReddy 2018b).

The above discussion motivates us to investigate Kaluza-Klein cosmological model in the presence of anisotropicDE fluid coupled with an attractive massive scalar field.The plan of this paper is the following: In Sect. 2, theKaluza-Klein model and the field equations in the presenceof anisotropic DE fluid and massive scalar field are derived.Section 3 presents the solution of the field equations and themodel. Section 4 is devoted to compute all the dynamicalparameters and to present physical discussion. In Sect. 5 theresults are summarized with conclusions.

2 Basic field equations

Here we derive the basic field equations with the help of theKaluza-Klein (KK) metric which is defined as

ds2 = dt2 − A2(dx2 + dy2 + dz2) − B2dψ2 (1)

where A, B are functions of cosmic time t and fifth coordi-nate ψ is space-like. Unlike Wesson (1983), here, the spatialcurvature has been taken as zero (Gron 1988).

We consider the field equations in the normal gauge inLyra manifold as

Rij − 1

2gijR + 3

2

(didj − 1

2gij dkd

k

)= −Tij (2)

where di is the displacement vector field of the manifold(function of time t) defines as

di = [β(t),0,0,0

](3)

here we assume gravitational units so that 8πG = c = 1.The other symbols have their usual meaning. Tij is theenergy-momentum tensor given by

Tij = T deij + T s

ij (4)

where T deij is the energy-momentum of DE given by

T deij = (ρΛ + pΛ)uiuj − pΛgij , uiu

i = 1 (5)

which can also be written as

T deij = diag[ρΛ,−pΛ,−pΛ,−pΛ]. (6)

We assume the anisotropic distribution of DE to ensurethe present acceleration of Universe. Hence the energy-momentum tensor T de

ij can be parameterized as

T deij = [1,−wx,−wy,−wz,−wψ ]ρΛ

= [1,−wΛ,−(wΛ + α),−(wΛ + γ ),−(wΛ + δ)

]ρΛ

(7)

where ωx = ωΛ, ωy = ωΛ + α, ωz = ωΛ + γ and ωψ =ωΛ + δ are the directional equations of equation of state(EoS) parameters on x, y, z and ψ axes respectively. Here,α, γ and δ are the deviations from ωΛ on y, z and ψ axes re-spectively. pΛ and ρΛ being the energy density and pressureof DE fluid, wΛ = pΛ

ρΛis the EoS parameter of DE.

Also

T(s)ij = φ,iφ,j − 1

2

(φ,kφ

′k − M2φ2) (8)

where φ is the massive scalar field, M is the mass of thescalar field (SF). This scalar field satisfies the Klein-Gordonequation, which is given by

gijφ;ij + M2φ = 0. (9)

With the use of Eqs. (3)–(9), the Lyra manifold fieldequations (2) for the KK metric (1), explicitly, can be de-rived as (we use co-moving coordinates)

3

(A2

A+ AB

AB

)− ρΛ − φ2

2− M2φ2

2− 3

4β2 = 0 (10)

Kaluza-Klein dark energy model in Lyra manifold in the presence of massive scalar field Page 3 of 8 190

2A

A+ A2

A2+ 2

AB

AB+ B

B+ wΛρΛ + φ2

2

−M2φ2

2+ 3

4β2 = 0 (11)

2A

A+ A2

A2+ 2

AB

AB+ B

B+ (wΛ + α)ρΛ + φ2

2

−M2φ2

2+ 3

4β2 = 0 (12)

2A

A+ A2

A2+ 2

AB

AB+ B

B+ (wΛ + γ )ρΛ + φ2

2

−M2φ2

2+ 3

4β2 = 0 (13)

3

(A

A+ A2

A2

)+ (wΛ + δ)ρΛ + φ2

2− M2φ2

2+ 3

4β2 = 0

(14)

φ + φ

(3A

A+ B

B

)+ M2φ = 0. (15)

Here an overhead dot indicates differentiation with respectto time t .

The following cosmological parameters are useful tosolve our field equations:

Spatial volume (V ), average scale factor (a(t)), meanHubble parameter (H ) and scalar expansion (θ ) are givenby

V = a3(t) = A3B (16)

H = a

a= 1

4

(3A

A+ B

B

)(17)

θ = 4H =(

3A

A+ B

B

). (18)

Shear scalar σ 2, average anisotropy parameter () and de-celeration parameter (DP) q are given by

σ 2 = 1

2σ ijσij = 1

2

( 4∑

i=1

H 2i − 1

3θ2

)(19)

= 1

4

4∑

i=1

(Hi − H

H

)2

(20)

q = −1 + d

dt

(1

H

). (21)

The nature of expansion of the model can be explained us-ing the DP. For positive value of DP, the model deceleratesin the standard way. If DP vanishes then the model expandswith constant rate. For −1 ≤ q < 0, we get accelerated ex-pansion of the universe. The model exhibits an exponentialexpansion for q = −1 and super exponential expansion forq < −1.

3 Kaluza-Klein DE model

Here, we solve the field equations (10)–(15) and presentKaluza-Klein DE model within the framework of Lyra man-ifold in the presence of massive scalar field.

From Eqs. (11) and (12) we have

α = 0. (22)

From Eqs. (12) and (13) we obtain

α = γ (23)

consequently from Eqs. (22) and (23), we obtain

α = γ = 0. (24)

This is because of the fact that the universe is isotropic in x,y and z directions and hence the deviations from EoS of DEvanished.

Using Eq. (24) in Eqs. (10)–(15) reduce to the followingindependent equations

3

(A2

A+ AB

AB

)− ρΛ − φ2

2− M2φ2

2− 3

4β2 = 0 (25)

2A

A+ A2

A2+ 2

AB

AB+ B

B+ wΛρΛ + φ2

2

−M2φ2

2+ 3

4β2 = 0 (26)

3

(A

A+ A2

A2

)+ (wΛ + δ)ρΛ + φ2

2− M2φ2

2+ 3

4β2 = 0

(27)

φ + φ

(3A

A+ B

B

)+ M2φ = 0. (28)

Now Eqs. (25)–(28) are a system of four independent dif-ferential equations in seven unknowns (A,B,φ,ρΛ,wΛ, δ

and β). Hence, we are free to choose three more mathemati-cal or physical conditions to find a deterministic model. Wechoose the following conditions:

(i) We use the fact that expansion scalar θ is proportionalto shear scalar σ 2, so that we have (Collins et al. 1980)a relation between the metric potentials as follows:

A = Bn (29)

where n is a positive constant which retains theanisotropy of the space-time. The motivation behindconsidering this relation is explained by Thorne (1967).Observations from the velocity-red-shift relation forextragalactic sources suggest that Hubble expansion ofthe universe is isotropic at present within ≈ 30 per cent(Kantowski and Sachs 1966; Kristian and Sachs 1966).

190 Page 4 of 8 Y. Aditya et al.

In particular, the studies of red-shift survey place thelimit as

σ

H≤ 0.3, (30)

in the neighborhood of our present day Galaxy. Collinset al. (1980) have shown that the normal congruence tothe homogeneous expansion satisfies the condition σ

H

is constant.(ii) In recent years, it is quite natural to use a power-law

relation between scalar field φ and average scale factora(t) of the form (Johri and Sudharsan 1989; Johri andDesikan 1994)

φ ∝ [a(t)

]m (31)

where m is a power index. Several researchers havestudied different aspects of this form of scalar field φ

(Rao et al. 2015; Santhi et al. 2016b; Aditya and Reddy2018b). In view of the physical importance of aboverelation, here we assume the following assumption toreduce the mathematical complexity of the system

φ

φ= −(3n + 1)

B

B. (32)

This is a consequence of Eq. (31). This relation (32)has been already taken by many authors and have con-structed cosmological models using this relation. Singh(2005), Singh and Rani (2015), Aditya and Reddy(2018a, 2019) and Naidu et al. (2019) have studiedBianchi type cosmological models with massive scalarfields using the above relation (32).

(iii) In addition to the above, we have taken a power law re-lation between β(t), the displacement vector field andaverage scale factor a(t) given by

β(t) = β0[a(t)

]k (33)

where β0 = 0 and k are positive constants.

Now from Eqs. (28)–(32) we get

φ = exp

(φ0t − M2t2

2+ φ1

)(34)

where φ0 and φ1 are constants of integration.Equations (32) and (33) together yield

A = exp

(n(M2t2 − 2φ0t − 2φ1)

2(3n + 1)

),

B = exp

(M2t2 − 2φ0t − 2φ1

2(3n + 1)

).

(35)

Using Eq. (35) in Eq. (1), the Kaluza-Klein model in thepresence of massive scalar field is given by

ds2 = dt2 − exp

(n(M2t2 − 2φ0t − 2φ1)

3n + 1

)

× (dx2 + dy2 + dz2)

− exp

(M2t2 − 2φ0t − 2φ1

3n + 1

)dψ2 (36)

and the massive scalar field in the model is given by Eq. (34).

4 Dynamical parameters of the model

Dynamical or cosmological parameters (16)–(21) have a sig-nificant role in the discussion of the cosmological models ofthe universe. Hence we evaluate them and present here

V = exp

(M2t2 − 2φ0t − 2φ1

2

)(37)

H =(

M2t − φ0

3

)(38)

θ = M2t − φ0 (39)

σ 2 = 1

2

(M2t − φ0

3n + 1

)2

(n − 1)2 (40)

= 9n2 − 12n + 7

4(3n + 1)2(41)

q = −(

1 + 3M2

(M2t − φ0)2

). (42)

Now from Eqs. (16), (34) and (35) we get

β(t) = β0 exp

(k(M2t2 − 2φ0t − 2φ1)

6

). (43)

Now from Eqs. (25)–(27), (33), (35) and (43) we obtainenergy density ρΛ, EoS parameter ωΛ of DE and skewnessparameter δ as

ρΛ = n(4n + 1)

(3n + 1)2

(M2t − φ0

)2

−[(M2t − φ0)

2 + M2

2

]exp

(2φ0t − M2t2 + 2φ1

)

− 3

4β2

0 exp

(k(M2t2 − 2φ0t − 2φ1)

6

). (44)

wΛ = − 1

ρΛ

2n(n + 1)(M2t − φ0)

2

(3n + 1)2+ nM2

3n + 1

+[(M2t − φ0)

2 − M2

2

]exp

(2φ0t − M2t2 + 2φ1

)

Kaluza-Klein dark energy model in Lyra manifold in the presence of massive scalar field Page 5 of 8 190

+ 3

4β2

0 exp

(k(M2t2 − 2φ0t − 2φ1)

6

)(45)

δ = n(1 − 3n)

ρΛ(3n + 1)2

[M2 − (

M2t − φ0)2]

, (46)

where ρΛ is given by Eq. (44).Several DE models have been formulated for analyzing

the DE phenomenon in the accelerated expansion of the uni-verse. Hence there is a need to distinguish these DE modelsso that one can decide which DE model provides good ex-planation for the present status of the universe. Because ofthe fact that various DE models give almost the same presentvalues of the deceleration and Hubble parameters. Hence,these parameters can not differentiate the DE models com-pletely. For this reason, Sahni et al. (2003) have introducedtwo new dimensionless parameters known as statefinders de-fined as follows:

r =...a

aH 3, s = r − 1

3(q − 12 )

.

For our model the above parameters are obtained as

r = 1 + 27M2

(M2t − φ0)2(47)

s = −6M2

(M2t − φ0)2 + M2. (48)

For (r, s) = (1,0), (1,1) we obtain ΛCDM and CDM

limits, respectively. However, s > 0 and r < 1 shows theDE regions such as phantom and quintessence-like, s < 0and r > 1 indicate the Chaplygin gas. Recently, many au-thors have investigated the statefinders analysis with dif-ferent geometries (Jawad 2014; Singh and Kumar 2016;Santhi et al. 2017; Naidu et al. 2018b; Sharma and Pradhan2019).

5 Physical discussion

Here, we have obtained Kaluza-Klein DE universe (Eq. (36))in the presence of attractive massive scalar field in the frame-work of Lyra manifold. It can be seen that the volume ofour model is non-zero at t = 0, i.e., the model is free fromthe initial singularity. It is clear from Eq. (25) that the vol-ume of the model is exponential function, hence the modelexhibits an exponential expansion from a finite volume ast increases. Also, it is observed that the physical parame-ters H,θ,σ 2 are finite at t = 0 and they tend to infinity ast → ∞. We observed that the anisotropy parameter is in-dependent of the time t . Hence the universe is uniform andspatially homogeneous. In Fig. 1, we have plotted the behav-ior of displacement vector β versus redshift z for different

Fig. 1 Plot of β(t) versus redshift z for M = 1.5, k = 0.18, φ1 = 10,n = 0.9 and β0 = 0.01

Fig. 2 Plot of scalar field versus redshift z for M = 1.5, k = 0.18,φ1 = 10, n = 0.9 and β0 = 0.01

values of φ0. It can be seen form Fig. 1 that β is a decreas-ing and positive function. The function β(t) decreases withincrease in φ0.

In order to study the behavior of physical parameters wehave plotted them in terms of cosmological redshift z. Weused the relation between the redshift z and the average scalefactor a(t) as 1 + z = a0

a. We consider the present value of

average scale factor a0 which has been normalized to one.

Scalar field: In Fig. 2, we have plotted the behavior of mas-sive scalar field φ versus redshift z for different values of φ0.We observed that φ is positive and increasing function forall the three values of φ0. We, also, observe that the scalarfield increases as φ0 increases. It can be observed that thescalar field shows increasing behavior and hence we canconclude that the corresponding kinetic energy decreases.The massive scalar field shows rapid increase from very lowvalues and approaches maximum value. This behavior isquite similar to the behavior of exponential potential which

190 Page 6 of 8 Y. Aditya et al.

Fig. 3 Plot of energy density ρΛ versus redshift z for M = 1.5,k = 0.18, φ1 = 10, n = 0.9 and β0 = 0.01

Fig. 4 Plot of EoS parameter versus redshift z for M = 1.5, k = 0.18,φ1 = 10, n = 0.9 and β0 = 0.01

correspond to cosmological scaling solutions obtained byCopeland et al. (2006) and interacting modified ghost SFmodels of DE constructed by Jawad (2015).

Energy density: Fig. 3 depicts the behavior of energy den-sity of DE ρΛ versus redshift z. It can be observed that ρΛ ispositive and decreasing function. Also, ρΛ increases as theSF increases.

EoS parameter: The EoS parameter of fluid relates its pres-sure p and energy density ρ by the relation, w = p

ρ. Differ-

ent values of EoS parameter correspond to various epochs ofthe universe from early decelerating to present acceleratingexpansion phases. It includes stiff fluid, radiation and mat-ter dominated (dust) for w = 1, w = 1

3 and w = 0 (deceler-ating phases) respectively. Also, it represents quintessencefor −1 < w < −1/3, cosmological constant (vacuum) forw = −1 and phantom for w < −1. Figure 4 describes thebehavior of EoS parameter of DE versus redshift for various

Fig. 5 Plot of skewness parameter versus redshift z for M = 1.5,k = 0.18, φ1 = 10, n = 0.9 and β0 = 0.01

values of φ0. It is observed that for all the three values of φ0

the model starts in quintessence region −1 < wΛ < −1/3,crosses the phantom divided line wΛ = −1 at late times andapproaches the aggressive phantom region wΛ −1. Also,as scalar field increases the EoS parameter of our DE modelapproaches the quintessence region. The trajectories of EoSparameter of DE model coincide with the Planks collabora-tion (Ade et al. 2014) and WMAP nine years observationaldata (Hinshaw et al. 2013) which give the ranges for EoSparameter as

−0.92 ≤ wΛ ≤ −1.26 (Planck + WP + Union 2.1),

−0.89 ≤ wΛ ≤ −1.38 (Planck + WP + BAO),

−0.983 ≤ wΛ ≤ −1.162 (WMAP+ eCMB+BAO+H0).

Skewness parameter: The physical significance of skew-ness parameters is that the amount of anisotropy in the DEfluid. Here the surviving skewness parameter δ is depicted inFig. 5 for various values of φ0. We observed that the skew-ness parameter is positive in the initial epoch and attains anegative value at late times. We can conclude that the DEin our model is anisotropic throughout the evolution of theuniverse and hence it helps to study the anisotropies at smallangular scales which play a key role in the formation of largescale structures of the universe.

Deceleration parameter: The nature of expansion of themodel can be explained using the deceleration parameter(DP). For example, the model decelerates in the standardway for positive value of DP and the model expands withconstant rate as DP vanishes. The model exhibits acceler-ated expansion for −1 ≤ q < 0, an exponential expansionfor q = −1 and super exponential expansion for q < −1.Figure 6 describes the behavior of DP versus redshift z forvarious values of φ0. We observe that DP remains less than

Kaluza-Klein dark energy model in Lyra manifold in the presence of massive scalar field Page 7 of 8 190

Fig. 6 Plot of deceleration parameter versus redshift z for M = 1.5

Fig. 7 Plot of r versus s for M = 1.5 and φ0 = 0.5

−1 and hence we obtain a universe with exponential expan-sion. Also, it can be seen that the model approaches superexponential expansion for q < −1. It can be seen that asEoS parameter of our model attains aggressive phantom re-gion (wde −1) hence we get super exponential expansion.Also, as q < −1 the model expands with super exponentialexpansion. The same phenomenon occurred in our model.

Satefinders: In order to verify the viability of various DEmodels statefinder parameters (r, s) are proposed. Theserepresent well-known DE regions which are given as fol-lows: (r, s) = (1,0), (1,1) represent the ΛCDM and CDM

limit, respectively. However, s > 0 and r < 1 shows the DEregions such as phantom and quintessence-like, s < 0 andr > 1 indicate the Chaplygin gas. In the present study, wedevelop r − s plane for φ0 = 0.5 is shown in Fig. 7. It canbe seen that our DE model corresponds to ΛCDM limit((r, s) = (1,0)) at late times which is in accordance withthe recent observational data. Also, it can be observed thatthe r-s plane correspond to Chaplygin gas model.

6 Conclusion

In this work, we have constructed Kaluza-Klein DE modelwith massive scalar field within the framework of Lyra man-ifold. In order to obtain a deterministic solution of the fieldequations we have used various physically valid conditions.We have computed all the cosmological and kinematicalparameters and discussed their physical significance in thelight of the present cosmological scenario and observations.We summarize our results as follows:

Our Kaluza-Klein DE model with massive scalar field isnon-singular and from a finite volume the model exhibits anexponential expansion leading to early inflation. The decel-eration parameter also confirms that our model starts withexponential expansion (inflation) and attains a super expo-nential expansion at late times. The average anisotropy pa-rameter is constant, the model is uniform throughout and ho-mogeneous. Due to the exponential expansion of the model,all the physical quantities of the model are finite initially andapproach to infinity at late times. The massive scalar field ofour model is positive throughout the evolution of the uni-verse and increases rapidly at present epoch. The behaviorof massive scalar field in our DE model is quite similar tothe behavior of exponential potential which correspond tocosmological scaling solutions obtained by Copeland et al.(2006) and interacting modified ghost SF models of DE con-structed by Jawad (2015). Statefinders plane (r-s plane) anal-ysis shows that the model finally approaches to ΛCDM

limit which is in accordance with the recent observationsand also our DE model corresponds to Chaplygin gas model.The energy density ρΛ of our model is always positive anddecreasing function. It can be seen from the analysis of EoSparameter that the model starts in the quintessence region(−1 < wΛ < −1/3), crosses the phantom divided line andfinally approaches to aggressive phantom region. We ob-served that the skewness parameter is positive in the initialepoch and attains a negative value at late times. We can con-clude that the DE in our model is anisotropic throughoutthe evolution of the universe and hence it helps to study theanisotropies at small angular scales which are play a key rolein the formation of large scale structures of the universe. Inour model, it is observed that the massive scalar field influ-ences all the physical parameters of the model at minimumscale. We hope and believe that the higher dimensional mas-sive scalar field model in Lyra manifold will help to have abetter insight into the understanding of DE which is respon-sible for cosmic acceleration.

Acknowledgements The authors are very much grateful to the re-viewer for constructive comments which certainly improved the qual-ity and presentation of the paper.

Compliance with ethical standards The authors declare that theyhave no potential conflict and will abide by the ethical standards of thisjournal.

190 Page 8 of 8 Y. Aditya et al.

Publisher’s Note Springer Nature remains neutral with regard to ju-risdictional claims in published maps and institutional affiliations.

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Advances in the Theory of Nonlinear Analysis and its Applications 3 (2019) No. 3, 121–140.https://doi.org/10.31197/atnaa.588945Available online at www.atnaa.orgResearch Article

PPF Dependent fixed points of generalized Suzukitype contractions via simulation functionsG. V. R. Babua, M. Vinod Kumarb

aDepartment of Mathematics, Andhra University, Visakhapatnam - 530 003, India.bDepartment of Mathematics, Andhra University, Visakhapatnam - 530 003, India.Permanent Address : Department of Mathematics, Anil Neerukonda Institute of Technology and Science, Sangivalasa,Visakhapatnam-531 162, India.

Abstract

In this paper, we introduce generalized Suzuki type ZG,α,µ,η− contraction with respect to ζ by using thenotion of CG−simulation function introduced by Liu, Ansari, Chandok and Radenović[19] and prove theexistence of PPF dependent fixed points in Banach spaces. We draw some corollaries and an example isprovided to illustrate our main result.

Keywords: α−admissible µ−subadmissible C−class function Suzuki type contraction Razumikhinclass PPF dependent fixed point simulation function CG−simulation function.2010 MSC: 47H10, 54H25.

1. Introduction and Preliminaries

Metric fixed point theory is a suggestive area which includes useful methods, directions, and notions fordealing with various problems. In this area, Banach contraction principle is considered as a fundementalresult. In this principle, Banach proved the existence of fixed points in complete metric spaces in a particularmanner. Due to its importance and way of construction of the proof, many authors attracted and proved itsgeneralizations and extensions by introducing a new function like α−admissible mapping, C−class function,etc., for more details we refer [1, 2, 3, 12, 13, 16, 22, 24, 26].

Recently, Khojasteh, Shukla and Radenović[11] introduced the notion of simulation function in order toexpress different contractivity conditions in a simple, unified manner and they obtained some fixed point

Email addresses: gvr_babu@hotmail.com (G. V. R. Babu ), dravinodvivek@gmail.com (M. Vinod Kumar )

Received July 08, 2019, Accepted: September 09, 2019, Online: September 18, 2019.

G. V. R. Babu, M. Vinod Kumar, Adv. Theory Nonlinear Anal. Appl. 3 (2019), 121–140. 122

results. Later, many authors extended and generalized the simulation function by using different types offunctions, for more details we refer [15, 17, 19, 21, 23, 25].

Throughout this paper, we denote the real line by R, R+ = [0,∞), and N is the set of all natural numbers,Z is the set of intergers.

In 2014, Ansari [1] introduced the concept of C− class function and many authors extended and gener-alized various fixed point results by using C−class functions as a main source in complete metric spaces.

Definition 1.1. [1] A mapping G : R+ × R+ → R is called a C−class function if it is continuous and forany s, t ∈ R+, the function G satisfies the following conditions:(i) G(s, t) ≤ s and(ii) G(s, t) = s implies that either s = 0 or t = 0.We denote the family of all C−class functions by ∆.

Example 1.2. [1] The following functions belong to ∆.(i) G(s, t) = s− t for all s, t ∈ R+.(ii) G(s, t) = ks for all s, t ∈ R+ where 0 < k < 1.(iii) G(s, t) = s

(1+t)r for all s, t ∈ R+ where r ∈ R+.

(iv) G(s, t) = sβ(s) for all s, t ∈ R+ where β : R+ → [0, 1) is continuous.(v) G(s, t) = s− φ(s) for all s, t ∈ R+ where φ : R+ → R+ is continuous

and φ(t) = 0 if and only if t = 0.(vi) G(s, t) = sh(s, t) for all s, t ∈ R+ where h : R+ × R+ → R+ is continuous

such that h(s, t) < 1 for all s, t ∈ R+.

In 2015, Khojasteh, Shukla and Radenović[11] introduced the simulation function as follows.

Definition 1.3. [11] A function ζ : R+ × R+ → R is said to be a simulation function if it satisfies thefollowing conditions:(ζ1) ζ(0, 0) = 0;(ζ2) ζ(t, s) < s− t for all t, s > 0;(ζ3) if tn, sn are sequences in (0,∞) such that lim

n→∞tn = lim

n→∞sn > 0,

then lim supn→∞

ζ(tn, sn) < 0.

We denote the set of all simulation functions in the sense of of Definition 1.3 by ZH .

Example 1.4. [11, 15] Let φi : R+ → R+ be a continuous function with φi(t) = 0 if and only if t = 0 fori = 1, 2, 3. Then the following functions ζ : R+ × R+ → R belong to ZH .(i) ζ(t, s) = s

s+1 − t for all t, s ∈ R+.(ii) ζ(t, s) = λs− t for all t, s ∈ R+ and 0 < λ < 1.(iii) ζ(t, s) = φ1(s)− φ2(t) for all t, s ∈ R+, where φ1(t) < t ≤ φ2(t)

for all t > 0.(iv) ζ(t, s) = s− φ3(s)− t for all t, s,∈ R+.

Definition 1.5. [11] Let (X, d) be a metric space, T : X → X be a mapping and ζ ∈ ZH . Then T is calleda ZH−contraction with respect to ζ if

ζ(d(Tx, Ty), d(x, y)) ≥ 0 (1)

for any x, y ∈ X.

Theorem 1.6. [11] Let (X, d) be a complete metric space and T : X → X be a ZH−contraction withrespect to ζ. Then T has a unique fixed point u in X and for every x0 ∈ X the Picard sequence xn wherexn = Txn−1 for any n ∈ N converges to the fixed point of T .

G. V. R. Babu, M. Vinod Kumar, Adv. Theory Nonlinear Anal. Appl. 3 (2019), 121–140. 123

Definition 1.7. [16] Let T be a self mapping on X and let α : X ×X → R+ be a function. We say that Tis an α−admissible mapping if for any x, y ∈ X with α(x, y) ≥ 1 implies α(Tx, Ty) ≥ 1.

In 2016, Karapınar[15] introduced the notion of α−admissible ZH−contraction with respect to the sim-ulation function ζ and proved the existence of its fixed points in complete metric spaces.

Definition 1.8. [15] Let T be a self-mapping defined on a metric space (X, d). If there exist ζ ∈ ZH andα : X ×X → R+ such that

ζ(α(x, y)d(Tx, Ty), d(x, y)) ≥ 0 (2)

for all x, y ∈ X, then we say that T is an α−admissible ZH−contraction with respect to ζ.

Definition 1.9. [22] Let T : X → X be a mapping and α : X ×X → R+ be a function. We say that T isan α−orbital admissible if

x ∈ X, α(x, Tx) ≥ 1 =⇒ α(Tx, T 2x) ≥ 1. (3)

Furthermore, T is called a triangular α−orbital admissible if T is α−orbital admissible and

x, y ∈ X, α(x, y) ≥ 1 and α(y, Ty) ≥ 1 =⇒ α(x, Ty) ≥ 1. (4)

Theorem 1.10. [15] Let (X, d) be a complete metric space, ζ ∈ ZH and let T : X → X be an α−admissibleZH−contraction with respect to ζ. Suppose that(i) T is triangular α−orbital admissible,(ii) there exists x0 ∈ X such that α(x0, Tx0) ≥ 1,(iii) T is continuous.Then there exists u ∈ X such that Tu = u.

In 2017, Kumum, Gopal and Budhia[17] introduced the notion of Suzuki type ZH−contraction by com-bining the Suzuki type contraction and ZH− contraction and proved the existence of its fixed points incomplete metric spaces.

Definition 1.11. [17] Let (X, d) be a metric space, T : X → X be a mapping and ζ ∈ ZH . Then T is calleda Suzuki type ZH−contraction with respect to ζ if

1

2d(x, Tx) < d(x, y) =⇒ ζ(d(Tx, Ty), d(x, y)) ≥ 0 (5)

for any x, y ∈ X with x 6= y.

Definition 1.12. [17] Let T : X → X be a mapping and x0 ∈ X be aribitrary. Then T is said to possessproperty (K) if for a bounded Picard sequencexn = Txn−1, n = 1, 2, 3, ..., there exist subsequences xmk

and xnk such that lim

k→∞d(xmk

, xnk) = C > 0

where mk > nk > k, k ∈ N then1

2d(xmk−1, xmk

) < d(xmk−1, xnk−1) (6)

holds.

Theorem 1.13. [17] Let (X, d) be a complete metric space, ζ ∈ ZH and T : X → X be a Suzuki typeZH−contraction with respect to ζ. Then T has a unique fixed point u in X and for every x0 ∈ X the Picardsequence xn where xn = Txn−1 for n = 1, 2, 3, ..., converges to the fixed point of T , provided T has property(K).

In 2018, Padcharoen, Kumum, Saipara and Cahipunya[21] introduced the notion of generalized Suzukitype ZH−contraction and proved the existence of its fixed points in complete metric spaces.

G. V. R. Babu, M. Vinod Kumar, Adv. Theory Nonlinear Anal. Appl. 3 (2019), 121–140. 124

Definition 1.14. [21] Let (X, d) be a metric space, T : X → X be a mapping and ζ ∈ ZH . Then T is calledgeneralized Suzuki type ZH−contraction with respect to ζ if

1

2d(x, Tx) < d(x, y) =⇒ ζ(d(Tx, Ty),M(x, y)) ≥ 0 (7)

for any x, y ∈ X with x 6= y, whereM(x, y) = maxd(x, y), d(x, Tx), d(y, Ty), d(x,Ty)+d(y,Tx)2 .

Theorem 1.15. [21] Let (X, d) be a complete metric space, ζ ∈ ZH and T is a generalized Suzuki typeZH−contraction with respect to ζ. Then T has a fixed point.

In 2015, Roldán, Karapınar, Roldán, Martinez[25] modified the Definition 1.3 of simulation function asfollows.

Definition 1.16. [25] A function ζ : R+ × R+ → R is said to be a simulation function if it satisfies thefollowing conditions:(ζ4) ζ(0, 0) = 0;(ζ5) ζ(t, s) < s− t for all t, s > 0;(ζ6) if tn, sn are sequences in (0,∞) such that lim

n→∞tn = lim

n→∞sn > 0

and tn < sn then lim supn→∞

ζ(tn, sn) < 0.

Clearly every simulation function in the sense of Definition 1.3 is also a simulation function in the senseof Definition 1.16. Roldán, Karapınar, Roldán, Martinez[25] shown that its converse is not true(Example3.3, [25]).

In 2018, Liu, Ansari, Chandok and Radenović[19] generalized the simulation function introduced byKhojasteh, Shukla and Radenović[11] by using C−class function as follows.

Definition 1.17. [19] A mapping G : R+×R+ → R has the property CG if there exists an CG ≥ 0 such that(i) G(s, t) > CG implies s > t, and(ii) G(t, t) ≤ CG for all s, t ∈ R+.

Example 1.18. [19] The following functions G : R+ ×R+ → R are functions of ∆ that are from Definition1.1 and having the property CG. For all s, t ∈ R+,(i) G(s, t) = s− t, CG = r, r ∈ R+,

(ii) G(s, t) = s− (2+t)t1+t , CG = 0,

(iii) G(s, t) = s1+kt , k ≥ 1, CG = r

1+k , r ≥ 2.

Definition 1.19. [19]A function ζ : R+ ×R+ → R is said to be a CG−simulation function if it satisfies thefollowing conditions:(ζ7) ζ(0, 0) = 0;(ζ8) ζ(t, s) < G(s, t) for all t, s > 0; here function G : R+ × R+ → R+ is an element of ∆ which hasproperty CG;(ζ9) if tn, sn are sequences in (0,∞) such that lim

n→∞tn = lim

n→∞sn > 0

and tn < sn then lim supn→∞

ζ(tn, sn) < CG.

We denote the set of all CG−simulation functions by ZG.

Example 1.20. We define ζ : R+ × R+ → R by ζ(t, s) = λs− t, where λ ∈ (0, 1) and G : R+ × R+ → R byG(s, t) = s− t for any s, t ∈ R+.Clearly ζ(0, 0) = 0 and G ∈ ∆ with CG = 0.Clearly ζ(t, s) = λs− t < s− t = G(s, t) and hence ζ satisfies (ζ8).If tn, sn are sequences in (0,∞) such that lim

n→∞tn = lim

n→∞sn = k > 0

G. V. R. Babu, M. Vinod Kumar, Adv. Theory Nonlinear Anal. Appl. 3 (2019), 121–140. 125

and tn < sn for all n ∈ N, thenlim supn→∞

ζ(tn, sn) = lim supn→∞

(λsn − tn) = λk − k = (λ− 1)k < 0.

Therefore ζ satisfies (ζ9) and hence ζ ∈ ZG.

Karapınar, Kumam and Salimi [16] introduced the notion of triangular α−admissible mappings as follows.

Definition 1.21. [16] Let T be a self mapping on X and let α : X ×X → R+ be a function. Then T is saidto be a triangular α−admissible mapping if for any x, y, z ∈ X,

α(x, y) ≥ 1 =⇒ α(Tx, Ty) ≥ 1 andα(x, z) ≥ 1, α(z, y) ≥ 1 =⇒ α(x, y) ≥ 1.

In 1977, Bernfeld, Lakshmikantham and Reddy[7] introduced the concept of fixed point for mappings thathave different domains and ranges which is called PPF (Past, Present and Future) dependent fixed point.Furthermore, they gave notion of Banach type contraction for non-self mapping and proved the existenceof PPF dependent fixed points in the Razumikhin class for Banach type contraction mappings, for furtherdetails we refer [5, 6, 9, 10, 14, 18].

Let (E, ||.||E) be a Banach space and we denote it simply by E. Let I = [a, b] ⊆ R and E0 = C(I, E),the set of all continuous functions on I equipped with the supremum norm ||.||E0

and we define it by||φ||E0

= supa≤t≤b

||φ(t)||E for φ ∈ E0.

For a fixed c ∈ I, the Razumikhin classRc of functions inE0 is defined byRc =φ ∈ E0/ ||φ||E0

= ||φ(c)||E.

Clearly every constant function from I to E belongs to Rc so that Rc is a non-empty subset of E0 .

Definition 1.22. [7] Let Rc be the Razumikhin class of continuous functions in E0. We say that(i) the class Rc is algebraically closed with respect to the difference if

φ− ψ ∈ Rc whenever φ, ψ ∈ Rc.(ii) the class Rc is topologically closed if it is closed with respect to the

topology on E0 by the norm ||.||E0.

The Razumikhin class of functions Rc has the following properties.

Theorem 1.23. [4] Let Rc be the Razumikhin class of functions in E0. Then(i) for any φ ∈ Rc and α ∈ R, we have αφ ∈ Rc.(ii) the Razumikhin class Rc is topologically closed with respect to the norm

defined on E0.(iii) ∩Rc

c∈[a,b]= φ ∈ E0/φ : I → E is constant .

Definition 1.24. [7] Let T : E0 → E be a mapping. A function φ ∈ E0 is said to be a PPF dependent fixedpoint of T if Tφ = φ(c) for some c ∈ I.

Definition 1.25. [7] Let T : E0 → E be a mapping. Then T is called a Banach type contraction if thereexists k ∈ [0, 1) such that||Tφ− Tψ||E ≤ k ||φ− ψ||E0

for all φ, ψ ∈ E0.

Theorem 1.26. [7] Let T : E0 → E be a Banach type contraction. Let Rc be algebraically closed with respectto the difference and topologically closed. Then T has a unique PPF dependent fixed point in Rc.

Definition 1.27. Let c ∈ I. Let T : E0 → E and α : E ×E → R+ be two functions. Then T is said to be aαc−admissible mapping if for any φ, ψ ∈ E0,

α(φ(c), ψ(c)) ≥ 1 =⇒ α(Tφ, Tψ) ≥ 1. (8)

Definition 1.28. Let c ∈ I. Let T : E0 → E and µ : E × E → R+ be two functions. Then T is said to be aµc−subadmissible mapping if for any φ, ψ ∈ E0,

µ(φ(c), ψ(c)) ≤ 1 =⇒ µ(Tφ, Tψ) ≤ 1. (9)

G. V. R. Babu, M. Vinod Kumar, Adv. Theory Nonlinear Anal. Appl. 3 (2019), 121–140. 126

Ćirić, Alsulami, Salimi and Vetro[8] introduced the concept of triangular αc−admissible mapping withrespect to µc as follows.

Definition 1.29. [8] Let c ∈ I and T : E0 → E. Let α, µ : E × E → R+ be two functions. Then T is saidto be a triangular αc−admissible mapping with respect to µc if for any φ, ψ, ϕ ∈ E0,

(i) α(φ(c), ψ(c)) ≥ µ(φ(c), ψ(c)) =⇒ α(Tφ, Tψ) ≥ µ(Tφ, Tψ)and

(ii) α(φ(c), ψ(c)) ≥ µ(φ(c), ψ(c)), α(ψ(c), ϕ(c)) ≥ µ(ψ(c), ϕ(c))=⇒ α(φ(c), ϕ(c)) ≥ µ(φ(c), ϕ(c)).

(10)

Note that if µ(x, y) = 1 for any x, y ∈ E, then we say that T is a triangularαc−admissible mapping and if α(x, y) = 1 for any x, y ∈ E, then we say thatT is a triangular µc−subadmissible mapping.

Lemma 1.30. [8] Let T be a triangular αc−admissible mapping with respect to µc. We define the sequenceφn by Tφn = φn+1(c) for all n ∈ N∪0, where φ0 ∈ Rc is such that α(φ0(c), Tφ0) ≥ µ(φ0(c), Tφ0). Thenα(φm(c), φn(c)) ≥ µ(φm(c), φn(c)) for all m,n ∈ N with m < n.

If µ(x, y) = 1 for any x, y ∈ E in Lemma 1.30, we get the following lemma.

Lemma 1.31. Let T be a triangular αc−admissible mapping. We define the sequence φn by Tφn = φn+1(c)for all n ∈ N∪0, where φ0 ∈ Rc is such that α(φ0(c), Tφ0) ≥ 1. Then α(φm(c), φn(c)) ≥ 1 for all m,n ∈ Nwith m < n.

If α(x, y) = 1 for any x, y ∈ E in Lemma 1.30, we get the following lemma.

Lemma 1.32. Let T be a triangular µc−subadmissible mapping. We define the sequence φn by Tφn =φn+1(c) for all n ∈ N ∪ 0, where φ0 ∈ Rc is such that µ(φ0(c), Tφ0) ≤ 1. Then µ(φm(c), φn(c)) ≤ 1 for allm,n ∈ N with m < n.

We use the following proposition to prove Lemma 1.34.

Proposition 1.33. If an and bn are two real sequences, bn is bounded, then lim inf(an + bn) ≤lim inf an + lim sup bn.

Lemma 1.34. Let φn be a sequence in E0 such that ||φn − φn+1||E0→ 0 as n → ∞. If φn is not a

Cauchy sequence, then there exists an ε > 0 and two subsequences φmk and φnk

of φn withmk > nk > ksuch that||φnk

− φmk||E0≥ ε, ||φnk

− φmk−1||E0< ε and

i) limk→∞

||φnk− φmk+1||E0

= ε, ii) limk→∞

||φnk+1 − φmk||E0

= ε,

iii) limk→∞

||φnk− φmk

||E0= ε, iv) lim

k→∞||φnk+1 − φmk+1||E0

= ε.

Proof. If φn is not a Cauchy sequence then there exists an ε > 0 and two subsequences φmk and φnk

with mk > nk > k satisfying

||φnk− φmk

||E0≥ ε. (11)

We choose mk, the least positive integer satisfying (11). Then we have

||φnk− φmk

||E0≥ ε and ||φnk

− φmk−1||E0< ε. (12)

We now prove (i).By triangular inequality we haveε ≤ ||φnk

− φmk||E0≤ ||φnk

− φmk+1||E0+ ||φmk+1 − φmk

||E0.

G. V. R. Babu, M. Vinod Kumar, Adv. Theory Nonlinear Anal. Appl. 3 (2019), 121–140. 127

Now by applying Proposition 1.33 with ak = ||φnk− φmk+1||E0

andbk = ||φmk+1 − φmk

||E0we have

ε ≤ lim infk→∞

||φnk− φmk+1||E0

. (13)

(since ||φn − φn+1||E0→ 0 as n→∞)

By triangular inequality we have||φnk

− φmk+1||E0≤ ||φnk

− φmk−1||E0+ ||φmk−1 − φmk

||E0

< ε+ ||φmk−1 − φmk||E0

. (by (12))On applying limit superior as k →∞ we get

lim supk→∞

||φnk− φmk+1||E0

≤ ε. (14)

(since ||φn − φn+1||E0→ 0 as n→∞)

From (13) and (14) we getε ≤ lim inf

k→∞||φnk

− φmk+1||E0≤ lim sup

k→∞||φnk

− φmk+1||E0≤ ε.

Thereforelimk→∞

||φnk− φmk+1||E0

= ε. (15)

Hence (i) holds.We now prove (ii).By triangular inequality we haveε ≤ ||φnk

− φmk||E0≤ ||φnk

− φnk+1||E0+ ||φnk+1 − φmk

||E0.

Now by applying Proposition 1.33 with ak = ||φnk+1 − φmk||E0

andbk = ||φnk

− φnk+1||E0we have

ε ≤ lim infk→∞

||φnk+1 − φmk||E0

. (16)

(since ||φn − φn+1||E0→ 0 as n→∞)

By triangular inequality we have||φnk+1 − φmk

||E0≤ ||φnk+1 − φnk

||E0+ ||φnk

− φmk+1||E0+ ||φmk+1 − φmk

||E0.

On applying limit superior as k →∞ we get

lim supk→∞

||φnk+1 − φmk||E0≤ ε. (17)

(from (15) and ||φn − φn+1||E0→ 0 as n→∞ )

From (16) and (17) we getε ≤ lim inf

k→∞||φnk+1 − φmk

||E0≤ lim sup

k→∞||φnk+1 − φmk

||E0≤ ε.

Thereforelimk→∞

||φnk+1 − φmk||E0

= ε. (18)

This proves (ii).We now prove (iii).From (11) we have ||φnk

− φmk||E0≥ ε.

On applying limit inferior as k →∞ we get

lim infk→∞

||φnk− φmk

||E0≥ ε. (19)

By triangular inequality we have||φnk

− φmk||E0≤ ||φnk

− φnk+1||E0+ ||φnk+1 − φmk

||E0.

On applying limit superior as k →∞ we get

lim supk→∞

||φnk− φmk

||E0≤ ε. (20)

G. V. R. Babu, M. Vinod Kumar, Adv. Theory Nonlinear Anal. Appl. 3 (2019), 121–140. 128

(from (18) and ||φn − φn+1||E0→ 0 as n→∞ )

From (19) and (20) we getε ≤ lim inf

k→∞||φnk

− φmk||E0≤ lim sup

k→∞||φnk

− φmk||E0≤ ε.

Thereforelimk→∞

||φnk− φmk

||E0= ε. (21)

Hence (iii) holds.We now prove (iv).By triangular inequality we haveε ≤ ||φnk

− φmk||E0≤ ||φnk

− φnk+1||E0+ ||φnk+1 − φmk+1||E0

+ ||φmk+1 − φmk||E0

.Now by applying Proposition 1.33 with ak = ||φnk+1 − φmk+1||E0

andbk = ||φnk

− φnk+1||E0+ ||φmk+1 − φmk

||E0we have

ε ≤ lim infk→∞

||φnk+1 − φmk+1||E0. (22)

(since ||φn − φn+1||E0→ 0 as n→∞)

By triangular inequality we have||φnk+1 − φmk+1||E0

≤ ||φnk+1 − φnk||E0

+ ||φnk− φmk

||E0+ ||φmk

− φmk+1||E0.

On applying limit superior as k →∞ we get

lim supk→∞

||φnk+1 − φmk+1||E0≤ ε. (23)

(from (21) and ||φn − φn+1||E0→ 0 as n→∞ )

From (22) and (23) we getε ≤ lim inf

k→∞||φnk+1 − φmk+1||E0

≤ lim supk→∞

||φnk+1 − φmk+1||E0≤ ε.

Thereforelimk→∞

||φnk+1 − φmk+1||E0= ε, (24)

so that (iv) holds.

In Section 2, we introduce different types of Suzuki type ZH−contractions(ZG−contractions) by using simulation functions in ZH(ZG.) Also, we define generalized Suzuki typeZG,α,µ,η−contraction with respect to ζ in Banach spaces. In Section 3, we prove the existence of PPFdependent fixed points of generalized Suzuki type ZG,α,µ,η−contraction with respect to ζ. In Section 4 wedraw some corollaries and an example is provided to illustrate our main result.

2. Suzuki type ZH−contractions

We denoteΨ = η | η : R+ → R+ is continuous, nondecreasing and

η(t) = 0 ⇐⇒ t = 0.

Definition 2.1. Let c ∈ I. Let T : E0 → E be a function and ζ ∈ ZH . Then T is called a ZH−contractionwith respect to ζ if

ζ(||Tφ− Tψ||E , ||φ− ψ||E0) ≥ 0 (25)

for any φ, ψ ∈ E0.

Remark 2.2. It is clear from the definition of simulation function that ζ(t, s) < 0 for all t ≥ s > 0.Therefore, if T is a ZH−contraction with respect to ζ then

||Tφ− Tψ||E < ||φ− ψ||E0 (26)

for any φ, ψ ∈ E0. Therefore every ZH−contraction mapping is contractive and hence it is continuous.

G. V. R. Babu, M. Vinod Kumar, Adv. Theory Nonlinear Anal. Appl. 3 (2019), 121–140. 129

Definition 2.3. Let c ∈ I. Let T : E0 → E be a function and ζ ∈ ZH . Then T is called Suzuki typeZH−contraction with respect to ζ if

1

2||φ(c)− Tφ||E < ||φ− ψ||E0 =⇒ ζ(||Tφ− Tψ||E , ||φ− ψ||E0) ≥ 0 (27)

for any φ, ψ ∈ E0 with φ 6= ψ.

Remark 2.4. It is clear from the definition of simulation function that ζ(t, s) < 0 for all t ≥ s > 0.Therefore, if T is a Suzuki type ZH−contraction with respect to ζ then

1

2||φ(c)− Tφ||E < ||φ− ψ||E0 =⇒ ||Tφ− Tψ||E < ||φ− ψ||E0 (28)

for any φ, ψ ∈ E0 with φ 6= ψ.

Definition 2.5. Let c ∈ I. Let T : E0 → E be a function and ζ ∈ ZH . Then T is called generalized Suzukitype ZH−contraction with respect to ζ if

1

2||φ(c)− Tφ||E < ||φ− ψ||E0 =⇒ ζ(||Tφ− Tψ||E ,M(φ, ψ)) ≥ 0 (29)

for any φ, ψ ∈ E0 with φ 6= ψ, whereM(φ, ψ) = max||φ− ψ||E0 , ||φ(c)− Tφ||E , ||ψ(c)− Tψ||E ,

||φ(c)−Tψ||E+||ψ(c)−Tφ||E2 .

Remark 2.6. It is clear from the definition of simulation function that ζ(t, s) < 0 for all t ≥ s > 0.Therefore, if T is a generalized Suzuki type ZH−contraction with respect to ζ then

1

2||φ(c)− Tφ||E < ||φ− ψ||E0 =⇒ ||Tφ− Tψ||E < M(φ, ψ) (30)

for any φ, ψ ∈ E0 with φ 6= ψ, whereM(φ, ψ) = max||φ− ψ||E0 , ||φ(c)− Tφ||E , ||ψ(c)− Tψ||E ,

||φ(c)−Tψ||E+||ψ(c)−Tφ||E2 .

Definition 2.7. Let c ∈ I. Let T : E0 → E be a function and ζ ∈ ZH . If there exists α : E × E → R+ suchthat

12 ||φ(c)− Tφ||E < ||φ− ψ||E0 =⇒

ζ(α(φ(c), ψ(c))||Tφ− Tψ||E ,M(φ, ψ)) ≥ 0(31)

for any φ, ψ ∈ E0 with φ 6= ψ, whereM(φ, ψ) = max||φ− ψ||E0 , ||φ(c)− Tφ||E , ||ψ(c)− Tψ||E ,

||φ(c)−Tψ||E+||ψ(c)−Tφ||E2 ,

then we say that T is a generalized Suzuki type ZH,α−contraction with respect to ζ.

Definition 2.8. Let c ∈ I. Let T : E0 → E be a function and ζ ∈ ZH . If there exist α : E × E → R+ andη ∈ Ψ such that

12 ||φ(c)− Tφ||E < ||φ− ψ||E0 =⇒

ζ(α(φ(c), ψ(c))η(||Tφ− Tψ||E), η(M(φ, ψ))) ≥ 0(32)

for any φ, ψ ∈ E0 with φ 6= ψ, whereM(φ, ψ) = max||φ− ψ||E0 , ||φ(c)− Tφ||E , ||ψ(c)− Tψ||E ,

||φ(c)−Tψ||E+||ψ(c)−Tφ||E2 ,

then we say that T is a generalized Suzuki type ZH,α,η−contraction with respect to ζ.

Remark 2.9. If η is the identity mapping in Definition 2.8 then T is a generalized Suzuki type ZH,α−contractionwith respect to ζ.

G. V. R. Babu, M. Vinod Kumar, Adv. Theory Nonlinear Anal. Appl. 3 (2019), 121–140. 130

Definition 2.10. Let c ∈ I. Let T : E0 → E be a function and ζ ∈ ZG. If there exist α, µ : E × E → R+

and η ∈ Ψ such that

12µ(φ(c), ψ(c))||φ(c)− Tφ||E < ||φ− ψ||E0 =⇒

ζ(α(φ(c), ψ(c))η(||Tφ− Tψ||E), η(M(φ, ψ))) ≥ CG(33)

for any φ, ψ ∈ E0 with φ 6= ψ, whereM(φ, ψ) = max||φ− ψ||E0 , ||φ(c)− Tφ||E , ||ψ(c)− Tψ||E ,

||φ(c)−Tψ||E+||ψ(c)−Tφ||E2 ,

then we say that T is a generalized Suzuki type ZG,α,µ,η−contraction with respect to ζ.

Remark 2.11. If T is a generalized Suzuki type ZG,α,µ,η−contraction with respect to ζ then

12µ(φ(c), ψ(c))||φ(c)− Tφ||E < ||φ− ψ||E0 =⇒

α(φ(c), ψ(c))η(||Tφ− Tψ||E) < η(M(φ, ψ)))(34)

for any φ, ψ ∈ E0 with φ 6= ψ, whereM(φ, ψ) = max||φ− ψ||E0 , ||φ(c)− Tφ||E , ||ψ(c)− Tψ||E ,

||φ(c)−Tψ||E+||ψ(c)−Tφ||E2 .

For, we assume that M(φ, ψ) > 0. Then η(M(φ, ψ)) > 0.If there exist φ, ψ ∈ E0 such that either α(φ(c), ψ(c)) = 0 or ||Tφ − Tψ||E = 0 then the inequality (34) istrivial.

Suppose that α(φ(c), ψ(c)) 6= 0 and ||Tφ− Tψ||E 6= 0 for any φ, ψ ∈ E0. By (ζ8), we getCG ≤ ζ(α(φ(c), ψ(c))η(||Tφ− Tψ||E), η(M(φ, ψ)))

< G(η(M(φ, ψ)), α(φ(c), ψ(c))η(||Tφ− Tψ||E)).Now from (i) of Definition 1.17 of property CG, we get the inequality (34).

Remark 2.12. (i) If µ(x, y) = 1 for any x, y ∈ E in the inequality (33) then T is called a generalized Suzukitype ZG,α,η−contraction with respect to ζ.(ii) If α(x, y) = 1 and µ(x, y) = 1 for any x, y ∈ E in the inequality (33)

then T is called a generalized Suzuki type ZG,η−contraction with respectto ζ.

(iii) If α(x, y) = 1 = µ(x, y) for any x, y ∈ E and η = identity in theinequality (33) then T is called a generalized Suzuki type ZG−contraction with respect to ζ.

3. Existence of PPF dependent fixed points

Theorem 3.1. Let c ∈ I. Let T : E0 → E be a function satisfying the following conditions:(i) T is a generalized Suzuki type ZG,α,µ,η−contraction with respect to ζ,(ii) T is a triangular αc−admissible mapping and triangular µc−subadmissible mapping,(iii) Rc is algebraically closed with respect to the difference,(iv) if φn is a sequence in E0 such that φn → φ as n→∞,

α(φn(c), φn+1(c)) ≥ 1 and µ(φn(c), φn+1(c)) ≤ 1 for any n ∈ N ∪ 0then α(φn(c), φ(c)) ≥ 1 and µ(φn(c), φ(c)) ≤ 1 for any n ∈ N ∪ 0 and

(v) there exists φ0 ∈ Rc such that α(φ0(c), Tφ0) ≥ 1 and µ(φ0(c), Tφ0) ≤ 1.Then T has a PPF dependent fixed point in Rc.

Moreover, if α(x, y) ≥ 1, µ(x, y) ≤ 1 for any x, y ∈ E and if T is one-one then T has a unique PPFdependent fixed point in Rc.

Proof. From (v), we have φ0 ∈ Rc such that α(φ0(c), Tφ0) ≥ 1 andµ(φ0(c), Tφ0) ≤ 1. Let φn be a sequence in Rc defined by

Tφn = φn+1(c) and ||φn+1 − φn||E0 = ||φn+1(c)− φn(c)||E , (35)

G. V. R. Babu, M. Vinod Kumar, Adv. Theory Nonlinear Anal. Appl. 3 (2019), 121–140. 131

for any n = 0, 1, 2, 3....Since T is traingular αc−admissible and triangular µc−subadmissible mappings, by Lemma 1.31 and Lemma1.32 we have

α(φm(c), φn(c)) ≥ 1 and µ(φm(c), φn(c)) ≤ 1 (36)

for any m,n ∈ N with m < n.If there exists n ∈ N ∪ 0 such that φn = φn+1 then Tφn = φn+1(c) = φn(c) and hence φn ∈ Rc is a PPFdependent fixed point of T .Suppose that φn 6= φn+1 for any n ∈ N ∪ 0.We considerM(φn, φn+1) = max||φn − φn+1||E0 , ||φn(c)− Tφn||E , ||φn+1(c)− Tφn+1||E ,

||φn(c)−Tφn+1||E+||φn+1(c)−Tφn||E2

= max||φn − φn+1||E0 , ||φn+1 − φn+2||E0. (37)

Clearly12µ(φn(c), φn+1(c))||φn(c)− Tφn||E ≤ 1

2 ||φn(c)− φn+1(c)||E= 1

2 ||φn − φn+1||E0

< ||φn − φn+1||E0 .From (33), we have

CG ≤ ζ(α(φn(c), φn+1(c))η(||Tφn − Tφn+1||E), η(M(φn, φn+1))). (38)

Suppose that M(φn, φn+1) = ||φn+1 − φn+2||E0 .Clearly α(φn(c), φn+1(c))η(||Tφn − Tφn+1||E) > 0 and η(M(φn, φn+1)) > 0.From (38), we haveCG ≤ ζ(α(φn(c), φn+1(c))η(||φn+1 − φn+2||E0), η(||φn+1 − φn+2||E0))

< G(η(||φn+1 − φn+2||E0), α(φn(c), φn+1(c))η(||φn+1 − φn+2||E0)).(by (ζ8))Now by the property CG and (36), we getη(||φn+1 − φn+2||E0) > α(φn(c), φn+1(c))η(||φn+1 − φn+2||E0)

≥ η(||φn+1 − φn+2||E0),a contradiction.ThereforeM(φn, φn+1) = ||φn − φn+1||E0 and hence ||φn+1 − φn+2||E0 < ||φn − φn+1||E0 .Therefore the sequence ||φn − φn+1||E0 is a monotonically decreasingsequence in R+ and hence it is convergent.Let lim

n→∞||φn − φn+1||E0 = k (say). Suppose that k > 0.

Clearly η(||φn − φn+1||E0) > 0.From (38), we haveCG ≤ ζ(α(φn(c), φn+1(c))η(||Tφn − Tφn+1||E), η(M(φn, φn+1)))

= ζ(α(φn(c), φn+1(c))η(||Tφn − Tφn+1||E), η(||φn − φn+1||E0))< G(η(||φn − φn+1||E0), α(φn(c), φn+1(c))η(||Tφn − Tφn+1||E)).

Now by the property CG, we get

η(||φn − φn+1||E0) > α(φn(c), φn+1(c))η(||Tφn − Tφn+1||E) (39)

= α(φn(c), φn+1(c))η(||φn+1 − φn+2||E0)≥ η(||φn+1 − φn+2||E0).

On applying limits as n→∞, we getη(k) ≥ lim

n→∞α(φn(c), φn+1(c))η(||φn+1 − φn+2||E0) ≥ η(k).

Thereforelimn→∞

α(φn(c), φn+1(c))η(||φn+1 − φn+2||E0) = η(k) > 0. (40)

G. V. R. Babu, M. Vinod Kumar, Adv. Theory Nonlinear Anal. Appl. 3 (2019), 121–140. 132

Clearly limn→∞

M(φn, φn+1) = k.

Since η is continuous we havelimn→∞

η(M(φn, φn+1)) = η(k) > 0. (41)

On applying limit superior as n→∞ to (38), we getCG ≤ lim sup

n→∞ζ(α(φn(c), φn+1(c))η(||Tφn − Tφn+1||E), η(M(φn, φn+1)))

= lim supn→∞

ζ(α(φn(c), φn+1(c))η(||φn+1 − φn+2||E0), η(||φn − φn+1||E0))

< CG (from (39), (40), (41) and (ζ9)),a contradiction.Therefore k = 0 and hence

limn→∞

||φn − φn+1||E0 = 0. (42)

We now show that the sequence φn is a Cauchy sequence in Rc.Suppose that the sequence φn is not a Cauchy sequence.Then there exists an ε > 0 and two subsequences φmk

and φnk of φn with mk > nk > k such that

||φnk− φmk

||E0 ≥ ε , ||φnk− φmk−1||E0 < ε and from Lemma 1.34 we have

limk→∞

||φnk− φmk

||E0 = ε (43)

andlimk→∞

||φnk− φmk+1||E0

= ε = limk→∞

||φnk+1 − φmk||E0

= limk→∞

||φnk+1 − φmk+1||E0.

Since η is continuous, we getlimk→∞

η(||φnk+1 − φmk+1||E0) = η(ε) > 0. (44)

We considerM(φnk

, φmk) = max||φnk

− φmk||E0 , ||φnk

(c)− Tφnk||E , ||φmk

(c)− Tφmk||E ,

||φnk(c)−Tφmk

||E+||φmk(c)−Tφnk

||E2

= max||φnk− φmk

||E0 , ||φnk− φnk+1||E0 , ||φmk

− φmk+1||E0 ,||φnk

−φmk+1||E0+||φmk

−φnk+1||E02 .

On applying limits as k →∞, we getlimk→∞

M(φnk, φmk

) = ε and hence

limk→∞

η(M(φnk, φmk

)) = η(ε) > 0. (45)

From (44) and (45), there exists k1 ∈ N such that

η(M(φnk, φmk

)) >η(ε)

2> 0 for any k ≥ k1 (46)

andη(||Tφnk

− Tφmk||E) = η(||φnk+1 − φmk+1||E0) > η(ε)

2 > 0 for any k ≥ k1.From (36), we get

α(φnk(c), φmk

(c))η(||Tφnk− Tφmk

||E) ≥ η(||Tφnk− Tφmk

||E) > 0 (47)

for any k ≥ k1.Suppose that there exists k ≥ k1 such that ||φnk

− φnk+1||E0 > ||φnk− φmk

||E0 .On applying limits as k →∞, we get 0 ≥ ε, a contradiction.Therefore

||φnk− φnk+1||E0 ≤ ||φnk

− φmk||E0 (48)

G. V. R. Babu, M. Vinod Kumar, Adv. Theory Nonlinear Anal. Appl. 3 (2019), 121–140. 133

for any k ≥ k1.Now for any k ≥ k1, we have12µ(φnk

(c), φmk(c))||φnk

(c)− Tφnk||E ≤ 1

2 ||φnk− φnk+1||E0 (since nk < mk )

≤ 12 ||φnk

− φmk||E0

< ||φnk− φmk

||E0 .From (33), we get

CG ≤ ζ(α(φnk(c), φmk

(c))η(||Tφnk− Tφmk

||E), η(M(φnk, φmk

))) (49)

< G(η(M(φnk, φmk

)), α(φnk(c), φmk

(c))η(||Tφnk− Tφmk

||E)).(from (46),(47) and (ζ8))

Now by the property CG, we get

η(M(φnk, φmk

)) > α(φnk(c), φmk

(c))η(||Tφnk− Tφmk

||E) (50)

≥ η(||Tφnk− Tφmk

||E) = η(||φnk+1 − φmk+1||E0).On applying limits as k →∞, we get

limk→∞

α(φnk(c), φmk

(c))η(||Tφnk− Tφmk

||E) = η(ε) > 0. (51)

On applying limit superior as k →∞ to (49), by (50) and (ζ9) we getCG ≤ lim sup

k→∞ζ(α(φnk

(c), φmk(c))η(||Tφnk

− Tφmk||E), η(M(φnk

, φmk)))

< CG,a contradiction.Therefore the sequence φn is a Cauchy sequence in Rc.Since E0 is complete, there exists φ∗ ∈ E0 such that φn → φ∗ as n→∞.Since Rc is topologically closed, we have φ∗ ∈ Rc.We now show that Tφ∗ = φ∗(c). Suppose that Tφ∗ 6= φ∗(c).From (36), we have α(φn(c), φn+1(c)) ≥ 1 and µ(φn(c), φn+1(c)) ≤ 1for any n ∈ N ∪ 0. From (iv), we getα(φn(c), φ∗(c)) ≥ 1 and µ(φn(c), φ∗(c)) ≤ 1 for any n ∈ N ∪ 0.First we show that either

12µ(φn(c), φ∗(c))||φn(c)− Tφn||E < ||φn − φ∗||E0

or12µ(φn+1(c), φ

∗(c))||φn+1(c)− Tφn+1||E < ||φn+1 − φ∗||E0 holdsfor any n ∈ N ∪ 0.Suppose that there exists m ∈ N ∪ 0 such that

1

2µ(φm(c), φ∗(c))||φm(c)− Tφm||E ≥ ||φm − φ∗||E0 (52)

and1

2µ(φm+1(c), φ

∗(c))||φm+1(c)− Tφm+1||E ≥ ||φm+1 − φ∗||E0 . (53)

From (52), we have||φm − φ∗||E0 ≤ 1

2µ(φm(c), φ∗(c))||φm(c)− Tφm||E≤ 1

2 ||φm(c)− Tφm||E .Therefore2||φm − φ∗||E0 ≤ ||φm(c)− φ∗(c)||E + ||φ∗(c)− Tφm||E

= ||φm − φ∗||E0 + ||φ∗ − φm+1||E0

and hence||φm − φ∗||E0 ≤ ||φm+1 − φ∗||E0

G. V. R. Babu, M. Vinod Kumar, Adv. Theory Nonlinear Anal. Appl. 3 (2019), 121–140. 134

≤ 12µ(φm+1(c), φ

∗(c))||φm+1(c)− Tφm+1||E (by 53)≤ 1

2 ||φm+1 − φm+2||E0 .Clearly||φm+1 − φm+2||E0 < ||φm − φm+1||E0

≤ ||φm − φ∗||E0 + ||φ∗ − φm+1||E0

≤ 12 ||φm+1 − φm+2||E0 + 1

2 ||φm+1 − φm+2||E0

= ||φm+1 − φm+2||E0 ,a contradiction.Therefore either

12µ(φn(c), φ∗(c))||φn(c)− Tφn||E < ||φn − φ∗||E0

or12µ(φn+1(c), φ

∗(c))||φn+1(c)− Tφn+1||E < ||φn+1 − φ∗||E0

holds for any n ∈ N ∪ 0.Case (i): Suppose that 1

2µ(φn(c), φ∗(c))||φn(c)− Tφn||E < ||φn − φ∗||E0 .From (33), we get

CG ≤ ζ(α(φn(c), φ∗(c))η(||Tφn − Tφ∗||E0), η(M(φn, φ∗))). (54)

We considerM(φn, φ

∗) = max||φn − φ∗||E0 , ||φn(c)− Tφn||E0 , ||φ∗(c)− Tφ∗||E ,||φn(c)−Tφ∗||E+||φ∗(c)−Tφn||E

2 .If M(φn, φ

∗) = 0 then Tφ∗ = φ∗(c), a contradiction.Therefore M(φn, φ

∗) > 0 and hence η(M(φn, φ∗)) > 0.

If η(||Tφn − Tφ∗||E0) = 0 then Tφn = Tφ∗ and hence Tφ∗ = φn+1(c).On applying limits as n→∞, we get Tφ∗ = φ∗(c), a contradiction.Therefore η(||Tφn − Tφ∗||E) > 0 and henceα(φn(c), φ∗(c))η(||Tφn − Tφ∗||E) > 0.On applying limits to M(φn, φ

∗) as n→∞, we getlimn→∞

M(φn, φ∗) = ||φ∗(c)− Tφ∗||E .

Since η is continuous, we have

limn→∞

η(M(φn, φ∗)) = η(||φ∗(c)− Tφ∗||E) > 0. (55)

From (54), we haveCG ≤ ζ(α(φn(c), φ∗(c))η(||Tφn − Tφ∗||E), η(M(φn, φ

∗)))< G(η(M(φn, φ

∗)), α(φn(c), φ∗(c))η(||Tφn − Tφ∗||E)). (by (ζ8))Now by the property CG, we get

η(M(φn, φ∗)) > α(φn(c), φ∗(c))η(||Tφn − Tφ∗||E) (56)

≥ η(||Tφn − Tφ∗||E) = η(||φn+1(c)− Tφ∗||E).On applying limits as n→∞, we get

limn→∞

α(φn(c), φ∗(c))η(||Tφn − Tφ∗||E) = η(||φ∗(c)− Tφ∗||E) > 0. (57)

On applying limit superior as n→∞ to (54), by (56) and (ζ9) we getCG ≤ lim sup

n→∞ζ(α(φn(c), φ∗(c))η(||Tφn − Tφ∗||E), η(M(φn, φ

∗))) < CG,

a contradiction.Case (ii): Suppose that

12µ(φn+1(c), φ

∗(c))||φn+1(c)− Tφn+1||E < ||φn+1 − φ∗||E0 .From (33), we get

CG ≤ ζ(α(φn+1(c), φ∗(c))η(||Tφn+1 − Tφ∗||E0), η(M(φn+1, φ

∗))) (58)

G. V. R. Babu, M. Vinod Kumar, Adv. Theory Nonlinear Anal. Appl. 3 (2019), 121–140. 135

We considerM(φn+1, φ

∗) = max||φn+1 − φ∗||E0 , ||φn+1(c)− Tφn+1||E0 , ||φ∗(c)− Tφ∗||E ,||φn+1(c)−Tφ∗||E+||φ∗(c)−Tφn+1||E

2 .If M(φn+1, φ

∗) = 0 then Tφ∗ = φ∗(c), a contradiction.Therefore M(φn+1, φ

∗) > 0 and hence η(M(φn+1, φ∗)) > 0.

If η(||Tφn+1 − Tφ∗||E) = 0 then Tφn+1 = Tφ∗ and hence φn+2(c) = Tφ∗.On applying limits as n→∞, we get Tφ∗ = φ∗(c), a contradiction.Therefore η(||Tφn+1 − Tφ∗||E) > 0 and henceα(φn+1(c), φ

∗(c))η(||Tφn+1 − Tφ∗||E) > 0On applying limits to M(φn+1, φ

∗) as n→∞, we getlimn→∞

M(φn+1, φ∗) = ||φ∗(c)− Tφ∗||E .

Since η is continuous, we have

limn→∞

η(M(φn+1, φ∗)) = η(||φ∗(c)− Tφ∗||E) > 0. (59)

From (58), we haveCG ≤ ζ(α(φn+1(c), φ

∗(c))η(||Tφn+1 − Tφ∗||E), η(M(φn+1, φ∗)))

< G(η(M(φn+1, φ∗)), α(φn+1(c), φ

∗(c))η(||Tφn+1 − Tφ∗||E)). (by (ζ8))Now by the property CG, we get

η(M(φn+1, φ∗)) > α(φn+1(c), φ

∗(c))η(||Tφn+1 − Tφ∗||E) (60)

≥ η(||Tφn+1 − Tφ∗||E) = η(||φn+2(c)− Tφ∗||E).On applying limits as n→∞, we get

limn→∞

α(φn+1(c), φ∗(c))η(||Tφn+1 − Tφ∗||E) = η(||φ∗(c)− Tφ∗||E) > 0. (61)

On applying limit superior to (58) as n→∞, by (60) and (ζ9) we getCG ≤ lim sup

n→∞ζ(α(φn+1(c), φ

∗(c))η(||Tφn+1 − Tφ∗||E), η(M(φn+1, φ∗)))

< CG,a contradiction.

Therefore from Case(i) and Case (ii), we conclude that Tφ∗ = φ∗(c) and hence φ∗ ∈ Rc is a PPFdependent fixed point of T .

Suppose that T is one-one, α(x, y) ≥ 1 and µ(x, y) ≤ 1 for any x, y ∈ E.We now show that T has a unique PPF dependent fixed point in Rc.Let φ, ψ ∈ Rc be two PPF dependent fixed points of T.Then Tφ = φ(c) and Tψ = ψ(c).Since Rc is algebraically closed with respect to the difference, we have||φ− ψ||E0 = ||φ(c)− ψ(c)||E . Suppose that φ 6= ψ.If ||Tφ− Tψ||E = 0 then Tφ = Tψ.Since T is one-one we have φ = ψ, a contradiction.Therefore ||Tφ− Tψ||E 6= 0 and hence ||Tφ− Tψ||E > 0.Clearly η(||Tφ− Tψ||E) > 0 and hence α(φ(c), ψ(c))η(||Tφ− Tψ||E) > 0.Clearly 0 = 1

2µ(φ(c), ψ(c))||φ(c)− Tφ||E < ||φ− ψ||E0 .From (33), we get

CG ≤ ζ(α(φ(c), ψ(c))η(||Tφ− Tψ||E), η(M(φ, ψ))). (62)

We considerM(φ, ψ) = max||φ− ψ||E0 , ||φ(c)− Tφ||E , ||ψ(c)− Tψ||E ,

||φ(c)−Tψ||E+||ψ(c)−Tφ||E2

= max||φ− ψ||E0 ,||φ(c)−ψ(c)||E+||ψ(c)−φ(c)||E

2

G. V. R. Babu, M. Vinod Kumar, Adv. Theory Nonlinear Anal. Appl. 3 (2019), 121–140. 136

= max||φ− ψ||E0 , ||φ− ψ||E0 = ||φ− ψ||E0 and henceη(M(φ, ψ)) > 0.By (62) and (ζ8), we getCG < G(η(M(φ, ψ)), α(φ(c), ψ(c))η(||Tφ− Tψ||E)).Now by the property CG, we getη(M(φ, ψ)) > α(φ(c), ψ(c))η(||Tφ − Tψ||E) ≥ η(||Tφ − Tψ||E) and which implies that η(||φ − ψ||E0) >η(||Tφ− Tψ||E)

= η(||φ(c)− ψ(c)||E)= η(||φ− ψ||E0),

a contradiction.Therefore φ = ψ and hence T has a unique PPF dependent fixed point in Rc.

4. Corollaries and Examples

Corollary 4.1. Let c ∈ I. Let T : E0 → E be a function satisfying the following conditions:(i) T is a generalized Suzuki type ZG,α,η−contraction with respect to ζ,(ii) Rc is algebraically closed with respect to the difference,(iii) T is a triangular αc−admissible mapping,(iv) there exists φ0 ∈ Rc such that α(φ0(c), Tφ0) ≥ 1 and(v) if φn is a sequence in E0 such that φn → φ as n→∞,

α(φn(c), φn+1(c)) ≥ 1 for any n ∈ N ∪ 0 then α(φn(c), φ(c)) ≥ 1for any n ∈ N ∪ 0.

Then T has a PPF dependent fixed point in Rc. Moreover, if T is one-one and α(x, y) ≥ 1 for any x, y ∈ Ethen T has a unique PPF dependent fixed point in Rc.

Proof. By taking µ(x, y) = 1 for any x, y ∈ E in Theorem 3.1 we obtain the desired result.

Corollary 4.2. Let c ∈ I. Let T : E0 → E be a function satisfying the following conditions:(i) T is a generalized Suzuki type ZG,η−contraction with respect to ζ,(ii) Rc is algebraically closed with respect to the difference.Then T has a PPF dependent fixed point in Rc. Moreover, if T is one-one then T has a unique PPF dependentfixed point in Rc.

Proof. By taking α(x, y) = 1 for any x, y ∈ E in Corollary 4.1 we obtain the desired result.

Corollary 4.3. Let c ∈ I. Let T : E0 → E be a function satisfying the following conditions:(i) T is a generalized Suzuki type ZH,α,η−contraction with respect to ζ,(ii) Rc is algebraically closed with respect to the difference,(iii) T is a triangular αc−admissible mapping,(iv) there exists φ0 ∈ Rc such that α(φ0(c), Tφ0) ≥ 1 and(v) if φn is a sequence in E0 such that φn → φ as n→∞,

α(φn(c), φn+1(c)) ≥ 1 for any n ∈ N ∪ 0 then α(φn(c), φ(c)) ≥ 1for any n ∈ N ∪ 0.

Then T has a PPF dependent fixed point in Rc. Moreover, if T is one-one and α(x, y) ≥ 1 for any x, y ∈ Ethen T has a unique PPF dependent fixed point in Rc.

Proof. By taking µ(x, y) = 1 for any x, y ∈ E,G(s, t) = s− t for any s, t ∈ R+ and CG = 0 in Theorem 3.1we obtain the desired result.

Corollary 4.4. Let c ∈ I. Let T : E0 → E be a function satisfying the following conditions:(i) T is a generalized Suzuki type ZH,α−contraction with respect to ζ,(ii) Rc is algebraically closed with respect to the difference and

G. V. R. Babu, M. Vinod Kumar, Adv. Theory Nonlinear Anal. Appl. 3 (2019), 121–140. 137

(iii) T is a triangular αc−admissible mapping,(iv) there exists φ0 ∈ Rc such that α(φ0(c), Tφ0) ≥ 1 and(v) if φn is a sequence in E0 such that φn → φ as n→∞,

α(φn(c), φn+1(c)) ≥ 1 for any n ∈ N ∪ 0 then α(φn(c), φ(c)) ≥ 1for any n ∈ N ∪ 0.

Then T has a PPF dependent fixed point in Rc. Moreover, if T is one-one and α(x, y) ≥ 1 for any x, y ∈ Ethen T has a unique PPF dependent fixed point in Rc.

Proof. By taking η = Identity mapping in Corollary 4.3 we obtain the desired result.

Corollary 4.5. Let c ∈ I. Let T : E0 → E be a function satisfying the following conditions:(i) T is a generalized Suzuki type ZH−contraction with respect to ζ,(ii) Rc is algebraically closed with respect to the difference.Then T has a PPF dependent fixed point in Rc. Moreover, if T is one-one then T has a unique PPF dependentfixed point in Rc.

Proof. By taking α(x, y) = 1 for any x, y ∈ E in Corollary 4.4 we obtain the desired result.

Example 4.6. Let E = R, c = 1 ∈ I = [12 , 2] ⊆ R, E0 = C(I, E).We define T : E0 → E by

Tφ =

φ(c)16 if 0 ≤ φ(c) ≤ 1φ(c)8 if otherwise,

for any φ ∈ E0.We define η : R+ → R+ by η(x) = 2x for any x ∈ R+. Clearly η ∈ Ψ.We define ζ : R+ × R+ → R by ζ(t, s) = λs− t, where λ ∈ (0, 1), CG = 0 andG : R+ × R+ → R by G(s, t) = s− t for any s, t ∈ R+.Clearly ζ ∈ ZG.(Example 1.20).Let φ, ψ ∈ E0 be such that φ 6= ψ.Assume that

1

2||φ(c)− Tφ||E < ||φ− ψ||E0 . (63)

Case (i): Suppose that Tφ = φ(c)16 and Tψ = ψ(c)

16 .

Clearly ||Tφ− Tψ||E = 116 ||φ(c)− ψ(c)||E which implies that

η(||Tφ− Tψ||E) = η( 116 ||φ(c)− ψ(c)||E) = 1

8 ||φ(c)− ψ(c)||E≤ 1

8 [||φ(c)− Tφ||E ] + 18 [||Tφ− ψ(c)||E ]

< 14 ||φ− ψ||E0 + 1

8 [||Tφ− ψ(c)||E ] (by 63)≤ 1

4 ||φ− ψ||E0 + 18 [||Tφ− Tψ||E + ||Tψ − ψ(c)||E ]

= 14 ||φ− ψ||E0 + 1

8 [ 116 ||φ(c)− ψ(c)||E + ||Tψ − ψ(c)||E ]

≤ 14 ||φ− ψ||E0 + 1

8 [ 116 ||φ− ψ||E0 + ||Tψ − ψ(c)||E ]

≤ 14M(φ, ψ) + 1

8 [ 116M(φ, ψ) +M(φ, ψ)]

= [14 + 1128 + 1

8 ]M(φ, ψ)= 49

128M(φ, ψ) = 49256 η(M(φ, ψ)).

Therefore

η(||Tφ− Tψ||E) <49

256η(M(φ, ψ)). (64)

Case (ii): Suppose that Tφ = φ(c)8 and Tψ = ψ(c)

8 .

Clearly ||Tφ− Tψ||E = 18 ||φ(c)− ψ(c)||E which implies that

η(||Tφ− Tψ||E) = η(18 ||φ(c)− ψ(c)||E) = 14 ||φ(c)− ψ(c)||E

≤ 14 [||φ(c)− Tφ||E ] + 1

4 [||Tφ− ψ(c)||E ]< 1

2 ||φ− ψ||E0 + 14 [||Tφ− ψ(c)||E ] (by 63)

G. V. R. Babu, M. Vinod Kumar, Adv. Theory Nonlinear Anal. Appl. 3 (2019), 121–140. 138

≤ 12 ||φ− ψ||E0 + 1

4 [||Tφ− Tψ||E + ||Tψ − ψ(c)||E ]= 1

2 ||φ− ψ||E0 + 14 [18 ||φ(c)− ψ(c)||E + ||Tψ − ψ(c)||E ]

≤ 12 ||φ− ψ||E0 + 1

4 [18 ||φ− ψ||E0 + ||Tψ − ψ(c)||E ]≤ 1

2M(φ, ψ) + 14 [18M(φ, ψ) +M(φ, ψ)]

= [12 + 132 + 1

4 ]M(φ, ψ)= 25

32M(φ, ψ) = 2564 η(M(φ, ψ)).

Therefore

η(||Tφ− Tψ||E) <25

64η(M(φ, ψ)). (65)

Case (iii): Suppose that Tφ = φ(c)16 and Tψ = ψ(c)

8 .

Clearly ||Tφ− Tψ||E = ||φ(c)16 −ψ(c)8 ||E which implies that

η(||Tφ− Tψ||E) = η(||φ(c)16 −ψ(c)8 ||E) = ||φ(c)8 −

ψ(c)4 ||E

≤ ||φ(c)8 −Tφ8 ||E + ||Tφ8 −

ψ(c)4 ||E

< 14 ||φ− ψ||E0 + ||φ(c)128 −

ψ(c)128 ||E + ||ψ(c)128 −

ψ(c)4 ||E (by 63)

≤ 14 ||φ− ψ||E0 + 1

128 ||φ− ψ||E0 + 31128 ||ψ(c)||E

< 14 ||φ− ψ||E0 + 1

128 ||φ− ψ||E0 + 78 ||ψ(c)||E

= 14 ||φ− ψ||E0 + 1

128 ||φ− ψ||E0 + ||Tψ − ψ(c)||E≤ 1

4M(φ, ψ) + 1128M(φ, ψ) +M(φ, ψ)

= [14 + 1128 + 1]M(φ, ψ)

= 161128M(φ, ψ) = 161

256 η(M(φ, ψ)).Therefore

η(||Tφ− Tψ||E) <161

256η(M(φ, ψ)). (66)

Case (iv): Suppose that Tφ = φ(c)8 and Tψ = ψ(c)

16 .

Clearly ||Tφ− Tψ||E = ||φ(c)8 −ψ(c)16 ||E which implies that

η(||Tφ− Tψ||E) = η(||φ(c)8 −ψ(c)16 ||E) = ||φ(c)4 −

ψ(c)8 ||E

≤ ||φ(c)4 −Tφ4 ||E + ||Tφ4 −

ψ(c)8 ||E

< 12 ||φ− ψ||E0 + ||Tφ4 −

ψ(c)8 ||E (by 63)

≤ 12 ||φ− ψ||E0 + ||φ(c)32 −

ψ(c)32 ||E + ||ψ(c)32 −

ψ(c)8 ||E

< 12 ||φ− ψ||E0 + 1

32 ||φ− ψ||E0 + 332 ||ψ(c)||E

≤ 12 ||φ− ψ||E0 + 1

32 ||φ− ψ||E0 + 1516 ||ψ(c)||E

= 12 ||φ− ψ||E0 + 1

32 ||φ− ψ||E0 + ||Tψ − ψ(c)||E≤ 1

2M(φ, ψ) + 132M(φ, ψ) +M(φ, ψ)

= [12 + 132 + 1]M(φ, ψ)

= 4932M(φ, ψ) = 49

64 η(M(φ, ψ)).Therefore

η(||Tφ− Tψ||E) <49

64η(M(φ, ψ)). (67)

We choose λ = max 49256 ,

2564 ,

161256 ,

4964. Clearly λ ∈ (0, 1).

From (64), (65), (66) and (67) we getη(||Tφ− Tψ||E) < λ η(M(φ, ψ)).This implies thatλ η(M(φ, ψ))− η(||Tφ− Tψ||E) > 0 and hence

ζ(η(||Tφ− Tψ||E), η(M(φ, ψ))) > 0. (68)

Therefore T is a generalized Suzuki type ZG,η−contraction with respect to ζ.

G. V. R. Babu, M. Vinod Kumar, Adv. Theory Nonlinear Anal. Appl. 3 (2019), 121–140. 139

For any n ∈ R, we define φn : I → E by

φn(x) =

nx2 if x ∈ [12 , 1]nx2

if x ∈ [1, 2].

Clearly φn ∈ E0, ||φn||E0 = ||φn(c)||E and hence φn ∈ Rc for any n ∈ R.Let F0 = φn | n ∈ R. Then F0 ⊆ Rc and F0 is algebraically closed with respect to the difference.

Therefore T satisfies all the hypotheses of Corollary 4.2 and hence φ0 ∈ Rc is a PPF dependent fixedpoint of T .

5. Acknowledgements

The authors would like to thank the honorable referee for his/her valuable suggestions which helped usto improve the presentation of the paper.

References

[1] A.H. Ansari, Note on φ−ψ− contractive type mappings and related fixed point, The 2nd Regional Conference on Mathe-matics and Applications, Payame Noor University Tehran, (2014), 377-380.

[2] A.H. Ansari, J. Kaewcharoen, C− class functions and fixed point theorems for generalized α− η−ψ− φ−F−contractiontype mappings in α− η complete metric spaces, J. Nonlinear Sci. Appl., 9 (6) (2016), 4177-4190.

[3] G.V.R. Babu, P.D. Sailaja , A fixed point theorem of Generalized Weakly contractive maps in Orbitally Complete Metricspace, Thai J. Math., 9 (1) (2011), 1-10.

[4] G.V.R. Babu, G. Satyanarayana, M. Vinod Kumar, Properties of Razumikhin class of functions and PPF dependent fixedpoints of Weakly contractive type mappings, Bull. Int. Math. Virtual Institute, 9 (1) (2019), 65-72.

[5] G.V.R. Babu, M. Vinod Kumar, PPF dependent coupled fixed points via C−class functions, J. Fixed Point Theory, 2019(2019), Article ID 7.

[6] B.C. Dhage, On some common fixed point theorems with PPF dependence in Banach spaces, J. Nonlinear Sci. Appl., 5(2012), 220-232.

[7] S.R. Bernfeld, V. Lakshmikantham, Y.M. Reddy, Fixed point theorems of operators with PPF dependence in Banachspaces, Appl. Anal., 6 (4) (1977), 271-280.

[8] L. Ćirić, S.M. Alsulami, P. Salimi, P. Vetro, PPF dependent fixed point results for triangular αc−admissible mappings,Hindawi Publishing corporation, (2014), Article ID 673647, 10 pages.

[9] Z. Dirci, F.A. McRae, J. Vasundharadevi, Fixed point theorems in partially ordered metric spaces for operators with PPFdependence, Nonlinear Anal., 67 (2007), 641-647.

[10] A. Farajzadeh, A. Kaewcharoen, S. Plubtieng, PPF dependent fixed point theorems for multivalued mappings in Banachspaces, Bull. Iranian Math. Soc., 42 (6) (2016), 1583-1595.

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[13] J. Harjani, B. Lopez, K. Sadarangani, Fixed point theorems for mixed monotone operators and applications to integralequations, Nonlinear Anal. 74 (2011), 1749-1760.

[14] N. Hussain, S. Khaleghizadeh, P. Salimi, F. Akbar, New Fixed Point Results with PPF dependence in Banach SpacesEndowed with a Graph, Abstr. Appl. Anal., (2013), Article ID 827205.

[15] E. Karapınar, Fixed points results via simulation functions, Filomat, 30 (8) (2016), 2343 - 2350.[16] E. Karapınar, P. Kumam, P. Salimi, On a α − ψ−Meir-Keeler contractive mappings, Fixed point theory Appl., (2013),

Article Number 94 (2013).[17] P. Kumum, D. Gopal, L. Budhia, A new fixed point theorem under Suzuki type Z−contraction mappings, J. Math. Anal.,

8 (1) (2017), 113-119.[18] M.A. Kutbi, W. Sintunavarat, On sufficient coniditons for the existence of Past-Present-Future dependent fixed point in

Razumikhin class and application, Abstr. Appl Anal., (2014), Article ID 342684.[19] X.L. Liu, A.H. Ansari, S. Chandok. S. Radenović, On some results in metric spaces using auxiliary simulation functions

via new functions, J. Comput. Anal. Appl., 24 (6) (2018).[20] M. Mursaleen, S.A. Mohiuddine, R.P. Agarwal, Coupled fixed point theorems for α − ψ−contractive type mappings in

partially ordered metric spaces, Fixed Point Theory Appl., (2012), Arcticle Number 228 (2012).[21] A. Padcharoen, P. Kumum, P. Saipara, P. Chaipunya, Generalized Suzuki type Z−contraction in complete metric spaces,

Kragujevac J. Math., 42 (3) (2018), 419-430.[22] O. Popescu, Some new fixed point theorems for α−Geraght contractive type maps in metric spaces, Fixed Point Theory

Appl., (2014), Article Number 190 (2014).

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[23] S. Radenović, F. Vetro, J. Vujaković, An alternative and easy approach to fixed point results via simulation functions,Demonstr. Math., 50 (1) (2017), 223-230.

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[26] B. Samet, C. Vetro, P. Vetro, Fixed point theorems for α− ψ−contractive type mappings, Nonlinear Anal., 75 (4) (2012),2154-2165.

ORIENTAL JOURNAL OF CHEMISTRY

www.orientjchem.org

An International Open Access, Peer Reviewed Research Journal

ISSN: 0970-020 XCODEN: OJCHEG

2019, Vol. 35, No.(1): Pg. 485-486

This is an Open Access article licensed under a Creative Commons license: Attribution 4.0 International (CC- BY).

Published by Oriental Scientific Publishing Company © 2018

Vibrational Spectra of C2Cl4 Using Lie Algebraic Technique

J. VIJAYASEkHAR1, S. UMA DEVI2 and T. SREENIVAS3*

1Department of Mathematics, GITAM University, Hyderabad, India.2Department of Engineering Mathematics, Andhra University, India.

3Department of Mathematics, ANITS, Visakhapatnam, India.*Corresponding author E-mail: sreenivas.teppala@gmail.com

http://dx.doi.org/10.13005/ojc/350165

(Received: January 09, 2019; Accepted: February 07, 2019)

AbSTRACT

In this paper, we have calculated the vibrational frequencies of medium-size molecule (5-100 atoms), Tetrachloroethylene (C2Cl4) in fundamental mode by one dimensional unitary Lie algebraic technique. In this method, bonds C-Cl and C-C of the Tetrachloroethylene are replaced with a unitary Lie algebras and Hamiltonian expressed in terms of Casimir and Majorana invariant operators and parameters. This Hamiltonian operator describes stretching vibrations of molecule. The determined vibrational frequencies by this theoretical method are compared with experimental results. The obtained results are consistent with experimental results.

keywords: Vibrational spectra, Lie algebraic technique, Tetrachloroethylene.

INTRODUCTION

Spectroscopy today, becomes the most accurate source of information of the spectral structure and energy calculation of molecules. There is considerable present research interest in the analysis of vibrational frequencies of molecules. The emergence of new experimental methods to detect higher vibrational excitations in polyatomic molecules require theoretical methods for their interpretation. In 1991, Iachello and Oss introduced a theoretical method, Lie algebraic method as a computational tool for the analysis and interpretation of experimental and ro-vibrational spectra of molecules. This method is based on the mathematical theory of unitary Lie algebras1,2.

Lie algebraic method The Hamiltonian operator associated with Lie algebraic method3 for n interacting bonds of molecule is

Where, Ci and Cij are invariant operators of the uncoupled and coupled bonds respectively and are given by

( ) ( )( )4

4= + − + +

= − −

2

2

, v ; ,v ,v ; ,v v v v v

( v v ),

,

i

i i j j ij i i j j i j i j

i

j

i

i

iN

N N C N N N

C

N

brief Communication

486SREENIVAS et al., Orient. J. Chem., Vol. 35(1), 485-486 (2019)

and the Majorana operator Mij is used to describe local mode interactions in pairs and contains both diagonal and non-diagonal matrix elements given by

( )( )( )( )( )( )( )

v v v v v v v v

v v v v v v 1 v v 1

v v v v v v v

1 1

1 .v 11

i i j j i i j j i j j i i j

1/ 2

i i j j i i j j j i i i j j

1/ 2

i i j j i i j j i

i

j j j i i

j

ij

ij

N , ; N , N , ; N , = N + N – 2

N , ; N , N , ; N , = - + N – N – +

N , ; N , N , ; N , = - +1 N – N –

M

+ - M

- M + +

Where, vi,vj are vibrational quantum numbers of bonds i and j respectively. The Vibron number Ni is calculated by

...(we and wexe are the spectroscopic constants6).

We consider a numerical fitting procedure to adjust the parameters Ai, lij and Aij over an experimental results of vibrational energy levels4,5. The starting guess for the parameter Ai will be

obtained by using the energy equation for the single-

oscillator fundamental mode as

( ) ( )v 1 4 1 .i iE A N= = − −

The parameter lij can be calculated from

the relation,

lij .2i j

ij

E Eë

N−

=

Structure of Tetrachloroethylene

Tetrachloroethylene containing two carbon

and four chlorine atoms with one (C-C) and four (C-Cl) bonds. Symmetry species of C2Cl4 are Ag, Au, B1g, B1u, B2g, B2u, B3u (Point group, D2h).

Fig.1.Tetrachloroethylene

Table 1. Algebraic parameters

Stretching parameters

N1= 174 (C-C), N2= 132 (C-Cl)A1 = 9.52, A2 = 8.27(cm-1)

A12 = 2.321, A13=-1.35 (cm-1)l12 = 2.321, l23=-1.35 (cm-1)

Table 2: Experimental and calculated vibrational frequencies (in cm-1) of C2Cl4

Symmetry Mode Experimental7,8 Calculated Species

Ag C-C Stretch 1571 1571.38 Ag C-Cl2 Symmetric Stretch 447 446.82 B1g C-Cl2 Asymmetric Stretch 1000 1000.32 B2u C-Cl2 Asymmetric Stretch 908 997.62 B3u C-Cl2 Symmetric Stretch 777 776.61

RESULTS

CONCLUSION

In this paper we have calculated the vibrational spectra of C2Cl4 in fundamental mode by Lie algebraic method and also compared with available experimental results. These calculations shows that this technique is a reliable appoarch of other theoretical techniques like Ab initio methods.

ACkNOwLEDgMENT

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

REFERENCES

1. Iachello, F.; Levine, R. D. Oxford University Press, Oxford., 1995.

2. Oss, S. Adv. Chem. Phys., 1996, 93, 455-649. 3. Iachello, F.; Oss S. J. Mol. Spectrosc., 1991,

146, 56-78.4. Karumuri, S. R.; Vijayasekhar, J.; Sreeram, V.;

Uma Maheswara Rao, V.; Basaveswara Rao, M. V. J. Mol. Spectrosc., 2011, 269, 119-123.

5. Karumuri, S. R.; Girija Sravani, K.; Vijayasekhar,

J.; Reddy, L. S. S. Acta Phys. Pol. A., 2012, 122(1), 1111-1114.

6. K. Nakamoto. Wiley-interscience, New York., 1978.

7. Shimanouchi, T. Tables of Molecular Vibrational Frequencies Consolidated Volume I, National Bureau of Standards., 1972.

8. Karl K. Irikura, J. Phys. Chem. Ref. Data., 2007, 36(2), 389-397.

Journal of Physics: Conference Series

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Locally Rotationally Symmetric Bianchi Type-I Cosmological Model inf(R,T) GravityTo cite this article: M.Vijaya Santhi et al 2019 J. Phys.: Conf. Ser. 1344 012004

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1

Locally Rotationally Symmetric Bianchi Type-I Cosmological Model in f (R,T ) Gravity

M.Vijaya Santhi1, Daba Meshesha Gusu1, V.U.M.Rao1, G.Suryanarayana2

Department of Applied Mathematics, Andhra University, Visakhapatnam-530003, India1

Department of Mathematics, ANITS Engineering College (A), Andra Pradesh, India2

E-mail: gv.santhi@live.com, dabam7@gmail.com, umrao57@hotmail.com,

gsuryanaraya.maths@anits.edu.in

Abstract. In this paper, we have investigated a spatially homogeneous locally rotationally symmetricBianchi type-I space-time with cosmological term Λ in presence of perfect fluid distribution in f (R,T )gravity theory. We have derived explicitly the field equations of the theory and obtained the exact solutionof field equations by employing a periodic varying deceleration parameter, which is a unique feature of themodel. We have also performed the analysis of the model such as the equation of state parameter, pressure,energy density, density parameter and jerk parameter which are significant in the discussion of cosmology.Some physical and geometrical properties of the model have also been discussed along with the graphicalrepresentation of various parameters. We obtained the presence of quintessence and phantom regions basedon chosen parameters. It is observed that the deceleration parameter exhibits a smooth transition from earlydeceleration to late time acceleration of the universe and oscillate based on chosen parameters. We haveobserved that the presented model is compatible with the recent cosmological observations.

1. IntroductionIn cosmology, the late-time accelerated expansion of the universe has been a major subject of

investigation. Modified gravity approach is one of the best ways to explain the cosmic accelerationand ultimate fate of universe. It seems attractive to explain the phenomena of dark energy and late-time acceleration. Hence, the modified theories of gravity is attracting currently several researchers toinvestigate dark energy (DE) models. Among these geometrically modified theories, f (R,T ) theory hasattracted a lot of attention of many cosmologists and astrophysicists in recent times because of its abilityto explain several issues in cosmology and astrophysics [1, 2]. The evolution of the universe from earlydeceleration to late time acceleration is effectively described by f (R,T ) theory of gravity. The f (R,T )modified theory of gravity developed [3], where the gravitational Lagrangian is given by an arbitraryfunction of the Ricci scalar R and the trace T of the energy-momentum tensor. It is to be noted thatthe dependence of T may be induced by exotic imperfect fluid or quantum effects. They have obtainedthe gravitational field equations in the metric formalism, as well as, the equations of motion of testparticles, which follow from the covariant divergence of the stress-energy tensor. They have derivedsome particular models corresponding to specific choices of the function f (R,T ).

In this theory, the interactions of matter with space-time curvature become a well motivation toconsider cosmological consequence with different matter components [4]. Some cosmological modelsin f (R,T ) gravity were reconstructed [5] where it was proved that the dust fluid reproduced ΛCDM.Friedmann-Lemaitre-Robertson-Walker (FLRW) space-time cosmological models have explored [6, 7].FRW cosmological model in f (R,T ) gravity have been investigated along with perfect fluid matter and

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linearly varying deceleration parameter with magnetized strange quark matter and Λ [8, 9]. Moreover, aperiodic time varying deceleration parameter (PVDP) has been introduced [10] in order to account for anoscillating cosmological models with quintom matter. Since these models are very natural to resolve thecoincidence problem due to periods of acceleration [11]. Some Bianchi space-time have studied [12,13].Cosmological models and solar system consequences of the theory have reconstructed [14]. The Bianchitype-I cosmological model have studied in [15, 16]. Dark energy Bianchi type-I cosmological modelshave been explored [17]. Aspects of anisotropic cosmological models were studied [18]. Recently, theinvestigation from the transition of deceleration to acceleration [19].

Various researchers have studied locally rotational symmetric (LRS) Bianchi-type models. Aninhomogeneous LRS model investigated by [20, 21], which was later continued [22–28]. In this study,we have explored the LRS Bianchi type-I space-time in f (R,T ) theory of gravity. On the other hand,the cosmological term Λ has an important role in the study of the accelerating universe, and which isalso a candidate for dark energy. The cosmological constant in the gravitational Lagrangian is a functionof the trace of the stress-energy tensor, and consequently the model was denoted Λ(T ) gravity. It wasargued that recent cosmological data favor a variable cosmological constant, which are consistent withΛ(T ) gravity, without the need to specify an exact form of the function Λ(T ) [29, 30].

The investigation of Bianchi-type models in modified or alternative theories of gravity is anotherinteresting topic of discussion. Perfect fluid solutions using a Bianchi type-I space-time in scalar tensortheory have been explored [31]. With the above motivation, we have investigated a class of LRSBianchi type-I model with variable Λ term within the framework of f (R,T ) gravity theory by choosingf (R,T ) = R+ 2 f (T ), where f (T ) = λT , and λ is an arbitrary constant. The paper is organized asfollows. The field equations in f (R,T ) gravity are derived in section 2. In section 3, we present themetric and field equations. The solution of the field equations has been explored in section 4. In section5, some physical and geometrical properties of the model are also investigated. Finally, conclusions aregiven in section 6.

2. Field equations in f (R,T ) theory of gravityThe f (R,T ) theory of gravity is one of the important modifications of general theory of gravity proposed[3]. Here in this theory, the gravitational Lagrangian is described by an arbitrary function of the Ricciscalar R and the trace T of the energy-momentum tensor Ti j. Following [3], let us consider the action ofthe form in the units 8πG = 1 = c

S =12

∫f (R,T )

√−gd4x+

∫Lm√−gd4x, (1)

where g is the determinant of the metric tensor gi j, f (R,T ) is the function of Ricci scalar, R and trace ofenergy-momentum tensor, T and Lm represents the matter Lagrangian density. The energy-momentumtensor of the matter is defined as

Ti j =−2√−g

δ (√−gLm)

δgi j , (2)

so that trace T = gi jTi j.Considering Lagrangian density Lm of matter depends only on the metric tensor components gi j,

equation (2) becomes

Ti j = gi jLm−2∂Lm

∂gi j . (3)

Varying the action S mention in equation (1) with respect to the metric tensor components gi j, the fieldequations of f (R,T ) gravity can be written by [3] as

fR(R,T )Ri j−12

f (R,T )gi j +(gi j−∇i∇ j) fR(R,T ) = Ti j− fT (R,T )Ti j− fT (R,T )Θi j, (4)

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where ≡ gi j∇ j∇i ≡ ∇i∇i is d’Alembert operator, fR(R,T ) =∂ f (R,T )

∂R , fT (R,T ) =∂ f (R,T )

∂T and ∇idenotes the covariant derivative.

The expansion tensor Θi j is given by

Θi j =−2Ti j +gi jLm−2gαβ ∂ 2Lm

∂gi j∂gαβ. (5)

For the perfect fluid, the energy-momentum tensor Ti j is given by

Ti j = (ρ + p)uiu j− pgi j, (6)

where ui = (0,0,0,1) is the four velocity in co-moving coordinates which satisfies the conditionsuiui = 1 and ui∇ jui = 0. Here ρ is the energy density and p the pressure of the fluid. Moreover,the matter Lagrangian is not uniquely specified. So, the source term is described as a function of theLagrangian matter through different choices of it. Here we choose the matter Lagrangian as Lm = −p,so that equation (5) becomes

Θi j =−2Ti j− pgi j. (7)

Since the field equations in f (R,T ) gravity also depend on the physical nature of the matter field (throughthe tensor Θi j), for each choice of f , we obtain several theoretical models. Among these, we assumef (R,T ) gravity as suggested by [3]

f (R,T ) = R+2 f (T ), (8)

where f (T ) is an arbitrary function of trace T . Using equations (7) and (8) into (4), we have obtained

Ri j−12

Rgi j = Ti j +2 f ′(T )Ti j +[2 f ′(T )p+ f (T )

]gi j, (9)

where a prime denotes derivative with respect to the argument.We also wish to consider the following choice of f (T )

f (T ) = λT, (10)

where λ is an arbitrary constant.Using equation (10) in (9) and re-arranging, the field equations become

Ri j−12

Rgi j = (1+2λ )Ti j +(2p+T )λgi j. (11)

Let us recall Einstein’s equations with cosmological constant on the right side,

Ri j−12

Rgi j = Ti j−Λgi j. (12)

By comparing equations (11) and (12), and taking the coupling parameter λ to be small, we see that aneffective cosmological parameter as a function of T may be defined in f (R,T ) as

Λ = Λ(T ) =−(2p+T )λ = (p−ρ)λ . (13)

For this correspondence further details of it given [29]. Thus, we can also regard this form of f (R,T )theory for the case f (R,T ) = R+ 2λT for a perfect fluid as equivalent to general relativity with aneffective cosmological parameter.

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3. Metric and field equationsWe consider the spatially homogeneous and anisotropic LRS Bianchi type-I spacetime as

ds2 = dt2−A2dx2−B2 (dy2 +dz2) , (14)

where A and B are functions of cosmic time t only.The energy-momentum tensor for a perfect fluid is taken as:

Ti j = (ρ + p)uiu j− pgi j. (15)

Now assuming the co-moving coordinate system, the field equations (11) for the metric (14) with thehelp of (15) can be written as

AA+

BB+

ABAB

= ρλ − (1+3λ ) p, (16)

2BB+

B2

B2 = ρλ − (1+3λ ) p, (17)

B2

B2 +2ABAB

= (1+3λ )ρ−λ p, (18)

where dot denotes ordinary differentiation with respect to cosmic time t.From equation (16) and (17) we get

AA− B

B+

ABAB− B2

B2 = 0. (19)

Integrating equation (19), we obtainAA− B

B=

c1

AB2 , (20)

where c1 is a constant of integration.We define the following physical parameters for the LRS Bianchi type-I model: The average scale

factor a and the volume scale factor V are defined as

a =3√

AB2, V = a3 = AB2. (21)

The average Hubble parameter H is given in the form

H =aa=

13(H1 +H2 +H3) =

13

(AA+2

BB

), (22)

where H1 = AA , H2 = H3 = B

B are the directional Hubble parameters along the respective axes. Thephysical quantities of observational interest in cosmology, which are the expansion scalar θ , the shearscalar σ2 and the average anisotropy parameter Am, are defined as

θ = ui;i = 3H =

AA+

2BB, (23)

σ2 =

12

(3

∑i=1

H2i −3H2

)=

13

(AA− B

B

)2

, (24)

Am =13

3

∑i=1

(∆Hi

H

)2

=23

σ2

H2 , (25)

where ∆Hi = Hi−H and Hi for i = 1,2,3 are directional Hubble’s parameters in the directions of x, yand z respectively.

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4. Solutions of the field equationsWe can observe that the field equations (16)–(18) are a system of three independent differential equationswith four unknowns, namely A, B, p and ρ . Hence, in order to solve this inconsistent system we requiresome additional conditions to get viable cosmological model with determinate solutions.

Periodically varying deceleration parameterThe deceleration parameter (q) in cosmology is a measure of the cosmic acceleration of the universe’sexpansion and is defined as

q =−1− HH2 , (26)

where the overhead dots denote derivatives with respect to cosmic time. It is the geometrical parametersthrough which the dynamics of the universe can be quantified. Many researchers have used a constantdeceleration parameter to obtain the solutions of the model which gives a power law for the metricpotentials [32–35].

Based on the late time cosmic speed up phenomena with a cosmic transit from a phase of decelerationto acceleration at some redshift (z). It can be a speculate signature flipping of the deceleration parameter.Geometrical parameter such as jerk parameter is usually extracted from observation of high z supernova.However, the exact time dependence of these parameter is not known to a satisfactory extent. In theabsence of any explicit form of parameters, many authors have used parametrized forms especiallythat of the deceleration parameter to address different cosmological issues. Many parametrized formsof deceleration parameters such as linearly and quadratic varying deceleration parameter are studiedby [36]. A special law of variation of Hubble parameter in FLRW-spacetime, which yields a constantform of deceleration parameter [37–39]. This law of variation for Hubble’s parameter is valid for slowlyvarying deceleration parameter models [32, 40].

Linear parametrization of the deceleration parameter shows quite natural phenomena toward thefuture evolution of the universe whether it expands forever or ends up with Big rip in finite future. Sucha parametrization has been used frequently by [41, 42]. It is to mention here that the general dynamicalbehavior can be assessed through the values of the deceleration parameter in the negative domain.While de-Sitter expansion occurs for q = −1, accelerating power-law expansion can be achieved for−1 < q < 0. A super-exponential expansion of the universe occurs for q <−1. There is an apprehensionin determining the deceleration parameter but from observational data most of the studies in recent timesconstrain this parameter in the range −1≤ q < 0 [43–45].

Considering in view the signature flipping nature of q, we assume a periodic time varying decelerationparameter [10].

q = h1 cosk1t−1, (27)

where h1 and k1 are positive constants. Here k1 decides the periodicity of the periodic varyingdeceleration parameter and can be considered as a cosmic frequency parameter. h1 is an enhancementfactor that enhances the peak of the periodic varying deceleration parameter. This model simulates apositive deceleration parameter q = h1− 1 (for h1 > 1) at an initial epoch and evolves into a negativepeak of q =−h1−1. After the negative peak, it again increases and comes back to the initial states. Theevolutionary behavior of q is periodically repeated. In other words, the universe in the model starts witha decelerating phase and evolves into a phase of super-exponential expansion in a cyclic history.

Integration of equation (27) and assuming constant of integration equal to zero yields the Hubblefunction becomes

H =k1

h1 sink1t. (28)

Using equation (28) and a = aH, we get H =−h1H2 cosk1t. The scale factor a is obtained by integrating

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the Hubble function in equation (28) as

a = a0

[tan(

12

k1t)] 1

h1. (29)

where a0 is the scale factor at the present epoch and taking a0 = 1.From the above equations (28) and (29), we obtain volume of the scale factor V and expansion scalar

θ as

V =

[tan(

12

k1t)] 3

h1, (30)

θ =3k1

h1 sink1t. (31)

The redshift z is given by

z =1a−1 =

[tan(

12

k1t)]− 1

h1−1. (32)

From equation (32), we obtain

t =2tan−1

[(z+1)−h1

]k1

. (33)

Integrating equation (20)

AB= c2 exp

[c1

∫ dtAB2

]= c2 exp

[c1

∫ [tan(

12

k1t)]− 3

h1dt

], (34)

where c2 is integration constant.From equations (20), (21) and (34), the values of metric potentials are

A = c232

[tan(

12

k1t)] 1

h1exp

[2c1

3

∫ [tan(

12

k1t)]− 3

h1dt

], (35)

and

B = c− 1

32

[tan(

12

k1t)] 1

h1exp

[−c1

3

∫ [tan(

12

k1t)]− 3

h1dt

]. (36)

where c1 and c2 are integrating constants.The metric (14) can now be written as

ds2 = dt2− c432 a2 exp

[4c1

3

∫a−3dt

]dx2− c

− 23

2 a2 exp[−2c1

3

∫a−3dt

](dy2 +dz2) , (37)

where a =[tan(1

2 k1t)] 1

h1 .By using equations (28), (35) and (36), we obtain the value of directional Hubble parameters for our

model as

H1 =k1

h1 sink1t+

2c1

3

[tan(

12

k1t)]− 3

h1, (38)

H2 =k1

h1 sink1t− c1

3

[tan(

12

k1t)]− 3

h1. (39)

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The shear scalar (σ2) and anisotropy parameter (Am) become

σ2 =

c1

3

[tan(

12

k1t)]−6

h1, (40)

Am =2c1

9k21[h1 sin(k1t)]2

[tan(

12

k1t)]−6

h1. (41)

5. The physical and geometrical properties of the modelIn this section, we will discuss the physical and geometrical properties of the model which are importantfor descriptions of cosmology. From the above equations (28), (31), (38) and (39), it can be noticed thatHubble parameter, scalar expansion and directional Hubble parameters diverge at t = nπ

k1, where n is a

positive integer including zero and they all tend to constants as t→∞. The directional Hubble parametersdiffer from H by certain dynamical parameters. It can be mentioned here that the anisotropy condition,i.e., σ2

θ 2 6= 0 as t→ ∞, when c1 6= 0. If c1 = 0, our model becomes isotropic and shear scalar vanishes.It can be observed from equation (30) that the spatial volume is zero at t = 2nπ

k1for n is a positive

integer including zero. It suggests that the universe starts evolving with zero volume at t = 2nπ

k1, i.e. it has

the big bang scenario. It can be observed that the average scale factor is zero at the epoch t = 2nπ

k1. Within

the time frame, the scale factor increases with cosmic time whereas the Hubble parameter decreases withcosmic time. However, the evolutionary behavior of the scale factor is governed by a tangent functionand that of the Hubble parameter is governed by a sine function. Hence the model has a point typesingularity [46]. As t→ ∞, both the metric potentials A and B tend to infinity. It shows that the universeexpands constantly at later times.

An equivalent present epoch can be derived from redshift relation given in equation (32) as t =(8n+1

k1

)π

2 , where n is a positive integer including zero. Therefore, it is possible to express the decelerationparameter of equation (27) in terms of redshift. In figure 1, we have shown the evolutionary aspect of the

cosmic time t (Gyr)

5 10 15

Dec

eler

atio

n p

aram

eter

(q)

-2.5

-2

-1.5

-1

-0.5

0

0.5

h1=0.5, k

1=0.5

h1=1, k

1=0.5

h1=1.5, k

1=0.5

Figure 1: Plot of deceleration parameter q versus cosmic time t for h1 = 0.5, h1 = 1, h1 = 1.5 andk1 = 0.5.

deceleration parameter as a function of cosmic time for three different domain of the parameter h1 namely

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h1 = 0.5, h1 = 1 and h1 = 1.5. The periodic nature of the periodically varying deceleration parameteris clearly depicted in this figure. Figure 2 shows the plot of deceleration parameter q versus redshift z.The evolutionary behavior of the periodically varying deceleration parameter is affected by the choiceof the parameter h1. Hence, the deceleration parameter oscillates in between h1− 1 and −h1− 1. Forh1 = 0, deceleration parameter becomes a constant quantity with a value of−1 and can lead to a de-Sitterkind of expansion. For 0 < h1 ≤ 1, it varies periodically in the negative domain and provides acceleratedmodels. However, for h1 > 1, q evolves from a positive region to a negative region showing a signatureflipping at some redshift z. It is interesting to mention here that, the transition redshift depends on thechoice of the parameter h1. This can be constrained from the cosmic transit behavior and transit redshiftz. Figure 2 shows the behavior of deceleration parameter versus redshift z for different values of h1. It

redshift (z)-1 0 1 2 3

Dec

eler

atio

n pa

ram

eter

(q)

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

h1=0.5, k

1=0.5

h1=1, k

1=0.5

h1=1.5, k

1=0.5

Decelerated phase

Accelerated phase

Figure 2: Plot of deceleration parameter q versus redshift z for h1 = 0.5, h1 = 1, h1 = 1.5 and k1 = 0.5.

may be noted that for h1 = 0.5 and 1 the model exhibits completely accelerating universe while h1 = 1.5exhibits a smooth transition from decelerated phase to the accelerated phase of the universe. It may beseen that the model enters the accelerated phase for h1 = 1.5 at z ≈ 0.71. This is quite in accordancewith recent cosmological observations [19, 47–49]. In the event of non-availability of any observationaldata regarding cosmic oscillation and corresponding frequency, we consider k1 as a free parameter. Herein this work, we are interested for a time varying deceleration parameter that oscillates in between thedecelerating and accelerating phase to simulate the cosmic transit phenomenon. In order to assess thedynamical features of the model through numerical plots, we assume a small value for k1, say k1 = 0.5.

The physical properties of the model from the assumed dynamics of the universe with a periodicvarying deceleration parameter helps us to study the energy density and pressure of the universe. Fromequations (16), (17) and (18), we can get the energy density ρ and pressure p of the fluid as

ρ =(5λ +2) B2

B2 +(4+11λ ) AA

BB −λ

(AA +3 B

B

)2((1+3λ )2−λ 2

) , (42)

p =(λ +1) B2

B2 +(1−λ ) AA

BB +(1+3λ )

(AA +3 B

B

)2(

λ 2− (1+3λ )2) . (43)

Applying the corresponding metric potentials and their derivatives, for a periodic varying deceleration

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parameter as defined in equation (27) we get the density ρ and pressure p of fluids as

ρ =k2

1 (3+2λ (3+h1))(tan(1

2 k1t))−2 (sec

(12 k1t

))4

4h21

[(1+3λ )2−λ 2

]−

λk21(sec(1

2 k1t))2

2h1

[(1+3λ )2−λ 2

] − (1+4λ )c21(tan(1

2 k1t))− 6

h1

3[(1+3λ )2−λ 2

] ,

(44)

p =−k2

1 (1+2(1+3λ )(1−h1))(tan(1

2 k1t))−2 (sec

(12 k1t

))4

4h21

((1+3λ )2−λ 2

)−

(1+3λ )k21(sec(1

2 k1t))2

2h1

((1+3λ )2−λ 2

) −(1+4λ )c2

1(tan(1

2 k1t))−6

h1

3((1+3λ )2−λ 2

) .

(45)

The cosmological parameter Λ obtained from equation (13) as

Λ =

[3 B2

B2 +5 ABAB + A

A +3 BB

](1+2λ )λ

2(

λ 2− (1+3λ )2) . (46)

Substituting corresponding metric potentials A and B with their respect derivatives, we get

Λ =

[k2

1 (3−h1)(tan(1

2 k1t))−2 sec4

(12 k1t

)+ k2

1h1 sec2(1

2 k1t)]

(1+λ )λ

2h21

(λ 2− (1+3λ )2

) . (47)

cosmic time t (Gyr)7 8 9 10

ρ

0

5

10

15

20

25

30

h1=0.5, λ=0.4

h1=1, λ=1

h1=1.5, λ=1.6

Figure 3: Plot of energy density ρ versus cosmic time t for k1 = 0.5 and c1 = 1.

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cosmic time t (Gyr)1 2 3 4 5

p

-5

-4

-3

-2

-1

0

h1=0.5, λ=0.4

h1=1, λ=1

h1=1.5, λ=1.6

Figure 4: Plot of pressure p versus cosmic time t for k1 = 0.5 and c1 = 1.

cosmic time t (Gyr)

0 1 2 3 4 5

Λ

-5

-4

-3

-2

-1

0

h1=0.5, λ=0.4

h1=1, λ=1

h1=1.5, λ=1.6

Figure 5: Plot of cosmological constant (Λ) versus cosmic time t for k1 = 0.5 and c1 = 1.

A signature flipping behavior of the deceleration parameter fixes h1 to be greater than 1 (see figure 2).In view of this, one may take λ as a free parameter with positive values only. Here, we have consideredthree moderate values, λ = 0.4, 1 and 1.6 for numerical calculations of the dynamical parameters.

Figure 3 represents the behavior of energy density ρ versus cosmic time t. It can be seen from thegraph that it decreases as the cosmic time increases. In this case we take the values of the parametersh1 = 0.5, 1, 1.5, λ = 0.4, 1, 1.6, k1 = 0.5 and c1 = 1. For the above choice of parameters the energydensities are positive throughout the evolution of the model. It is observed that the energy densities arealways positive and decrease with increasing cosmic time in the model. The evolutionary trend of theenergy density is not changed by a variation of λ , rather an increase in λ simply decreases the value ofρ at a given time.

Figure 4 describes the behaviour of pressure versus cosmic time t. It shows that the pressure p of theuniverse is an increasing function of cosmic time t, which begins from a large negative value and tends tozero at present epoch. As per the observation, the negative pressure is due to dark energy in the contextof accelerated expansion of the universe. Hence, the behavior of pressure in our model is agreed with

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cosmic time t (Gyr)

4 4.5 5

EoS

par

amet

er (ω

)

-4

-3

-2

-1

0

h1=0.5, λ=0.4

h1=1, λ=1

h1=1.5, λ=1.6

quintessence region

phantom

region

Figure 6: Plot of equation of state parameter (ω) versus cosmic time t for k1 = 0.5 and c1 = 1.

this observation. Here, pressure is a negative quantity at the present epoch in a given model. The choiceof the parameter h1 and λ has some effects on the evolutionary trend. In general, lower value of λ resultsin a pressure curve that lies to more negative values.

The equation of state (EoS) parameter can be obtained in a straightforward manner from Eqs. (44)and (45) as

ω =pρ=−3k2

1 (1+2(1+3λ )(1−h1))X−2Y 4−6k21h1 (1+3λ )Y 2−4c2

1h21 (1+4λ )X

−6h1

3k21 (3+2λ (3+h1))X−2Y 4−6k2

1λh1Y 2−4h21c2

1 (1+4λ )X−6h1

, (48)

where X = tan(1

2 k1t)

and Y = sec(1

2 k1t).

The behavior of EoS parameter (ω) versus cosmic time t for our model is depicted in figure 6 for thechosen constant parameters h1 and λ . It may be observed that the model starts in quintessence regionsfor h1 = 0.5, 1, and λ = 1.4, 1 which varies in the same region. In this case, the EoS parameter remainswithin the quintessence region with a value close to ΛCDM model in late times. However, the modelfor parameters h1 = 1.5 and λ = 1.6 starts in high phantom region and lies in the same region which adark energy-driven accelerated phase (ω <−1) which is consistent with the current observational dataof the universe [50]. It may be noted that the equation of state parameter became influenced by theparameters of h1 and λ . Moreover, the EoS parameter exhibits an oscillatory behavior in both regions.One interesting feature of the equation of state parameter is that, it does not acquire any singular valuesduring the cosmic cycle within time frame. Since the periodic varying deceleration parameter does nothave singularity, the same thing also occurs in the EoS parameter. In these constructed model, the EoSevolves with cosmic time which is more evident in the reconstruction history of the dynamical darkenergy based on recent data sets [51, 52].

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We can obtain the density parameter (Ω) for the present model as

Ω =ρ

3H2 =(3+2λ (3+h1))

3[(1+3λ )2−λ 2

] − 2λh1 sin2 (12 k1t

)3[(1+3λ )2−λ 2

]−

(1+4λ )c21h2

1(tan(1

2 k1t))−6

h1 sin2 (k1t)

9k21

[(1+3λ )2−λ 2

] . (49)

It can be expressed as a function of redshift (z) as

Ω =(3+2λ (3+h1))

3[(1+3λ )2−λ 2

] − 2λh1

[(z+1)2h1 +1

]−1

3[(1+3λ )2−λ 2

]− 4(1+4λ )c2

1h21 (z+1)6+2h1

9k21

(1+(z+1)2h1

)2 [(1+3λ )2−λ 2

] . (50)

z-1 0 1 2 3

dens

ity p

aram

eter

(Ω

)

0

0.2

0.4

0.6

0.8

1

h1=0.5, λ=0.4

h1=1, λ=1

h1=1.5, λ=1.6

Figure 7: Plot of density parameter (Ω) versus redshift (z) for k1 = 0.5 and c1 = 1.

Figure 7 represents the behavior ofΩ versus redshift z. It can be seen that it increases as the universeevolves. Here we have chosen the constant values (c1 = 1, k1 = 0.5, h1 = 0.5, 1, 1.5 and λ = 0.4, 1, 1.6)such that we arrive at Ω approaching 1 for small values of h1 and λ . Hence, the density parameter showsthat in agreement with the observational data of the universe. Moreover, with the cosmic evolution, Ω

decreases with cosmic time. The density parameter, at a given redshift, is observed to have lower valuefor higher values of λ .

Model of the universe close to ΛCDM can be described using the cosmic jerk parameter j, adimensionless third derivative of the scale factor with respect to the cosmic time [53]. The value ofthe jerk parameter is constant for a flat ΛCDM model. The jerk parameter j which shows the deviationof a model from the ΛCDM in our case is given by

j =

...aaH3

= q(1+2q)− qH

= h21(cos2 (k1t)+1

)−3h1 cos(k1t)+1. (51)

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For small value(s) parameters h1 near to zero, then the jerk parameter becomes approach to 1 whichexplains the time evolution of the ΛCDM model.

6. ConclusionsIn this paper, we investigated a spatially homogeneous locally rotationally symmetric Bianchi type-Ispace-time in the presence of perfect fluid cosmological model within the framework of f (R,T ) theoryof gravity by following the works of [3], choosing f (R,T ) = R+2 f (T ), where f (T ) = λT . To obtain anexact solution of the model, we assumed periodically varying deceleration parameter. We examined themodel by looking at cosmological parameters as the followings: the deceleration parameter is assumedto be periodically varying declaration parameter and its graphical representation with respect to cosmictime is shown in figure 1. It can be observed from figure 2 that for h1 = 1.5 the model describes asmooth transition from early deceleration to the present accelerated phase of the universe. It may also beseen that the model exhibits transition at z≈ 0.71 which is quite in accordance with recent cosmologicalobservations [54–56]. But from the figure 2, it can be observed that the universe completely lays inthe accelerating phase for h1 = 0.5 and 1. The energy density (ρ) of universe is positive decreasingfunctions of cosmic time (figure 3). The energy density of the model positive throughout the evolutionof the Universe and approaches to zero for large values of cosmic time t. The evolution of the universein model for the pressure p versus cosmic time t is shown in figure 4 with different values of parametersh1 and λ , with constants k1 and c1. The pressure has negative values for the model which shows that theuniverse is accelerated expanding for late times. Similarly figure 5 shows the cosmological parameter Λ

versus cosmic time becomes approaching to zero at late times. Hence our model is in excellent agreementwith observational constraints providing that the present value of Λ is chosen [57]. The dynamics of theuniverse is studied through the equation of state parameter. As it is shown in figure 6, for the values ofh1 = 0.5, 1 and λ = 0.4, 1 the equation of state parameter ω lies in quintessence region which an EoSparameter that more likely approaches to ΛCDM model at late times, while for h1 = 1.5 and λ = 1.6larger values of the phantom behavior is attained in the near future. An interesting consequence of thepresent model is that it allows both quintessence and the phantom like behaviour for free parameters.From figure 7, we can conclude that for smaller values of the parameters h1 = 0.5 and λ = 0.4, thedensity parameter approaches to 1 which describes the flatness of universe which confirms the presentcosmological data of the universe. It can be seen that the density parameter Ω increases with respectto redshift as the universe evolves. We have seen that for small value(s) of h1 approaching to zero, thejerk parameter becomes approximately equal to 1 which indicates a flat ΛCDM model. Therefore, it isconcluded that the findings support the current accelerating expansion of the universe.

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650 Int. J. Computer Aided Engineering and Technology, Vol. 10, No. 6, 2018

Copyright © 2018 Inderscience Enterprises Ltd.

Dynamic analysis of composite propeller of ship using FEA

Roopsandeep Bammidi* Department of Mechanical Engineering, Anil Neerukonda Institute of Technology and Sciences, Bheemunipatnam, Sanghivalasa, Visakhapatnam, Andhra Pradesh 531162, India Email: roopsandeepbammidi@gmail.com *Corresponding author

K. Spandana Veronika Department of Mechanical Engineering, Centurion University, Parlakhemundi, India Email: spandy371@gmail.com

Abstract: Ships and underwater vehicles use propeller for propulsion. In general, propellers are used as propulsors and they are also used to develop significant thrust to propel the vehicle at its operational speed and RPM. The blade geometry and design are more complex involving many controlling parameters. In current years the increased need for light weight structural element with composite materials has led to use of S2 glass fabric/epoxy to propeller. The present research work is to carry out the model and static analysis of aluminium, composite material which is a combination of glass fibre reinforced plastics (GFRP) materials. The research work deals with modelling and analysing the propeller blade of an underwater vehicle for their strength. A propeller is a complex geometry which requires high end modelling software. The solid model of propeller is developed in CATIA V5 R21. Static model analysis of the propellers made of aluminium and composite materials are carried out in advanced numerical simulation systems (ANSYS). We applied thrust force at blade blend section and centrifugal force at the centre of gravity and found out Von Mises stresses, total deformation, directional deformation, principal and shear stresses of aluminium and composite propeller.

Keywords: propeller; glass fibre reinforced plastics; GFRP; composites; CATIA; advanced numerical simulation systems; ANSYS.

Reference to this paper should be made as follows: Bammidi, R. and Veronika, K.S. (2018) ‘Dynamic analysis of composite propeller of ship using FEA’, Int. J. Computer Aided Engineering and Technology, Vol. 10, No. 6, pp.650–672.

Biographical notes: Roopsandeep Bammidi is an Assistant Professor in Department of Mechanical Engineering at Anil Neerukonda Institute of Technology and Sciences, Visakhapatnam and his research interests are finite element methods, road vehicle aerodynamics and CFD.

Dynamic analysis of composite propeller of ship using FEA 651

K. Spandana Veronika is an MTech student in Department of Mechanical Engneerng at Centurion University and research interests are FEA, design and manufacturing.

This paper is a revised and expanded version of a paper entitled ‘Static and dynamic analysis of composite ship propeller using FEA’ presented at International Conference on Mechanical and Industrial Engineering (ICMIE), Pune, 15 July 2012.

1 Introduction

Ships and under water vehicles like submarines, torpedoes and submersibles etc., uses propeller as propulsion. The blade geometry ant its design is more complex involving many controlling parameters. The strength analysis of such complex 3D blades with conventional formulas will give less accurate values. In such cases numerical analysis (finite element analysis) gives comparable results with experimental values. In the present work the propeller blade material is converted from aluminium metal to fibre reinforced composite material for underwater vehicle propeller. Such complex analysis can be easily solved by finite element method techniques. The force needed to propel a ship is obtained from the reaction against the water, causing a stream of water to move in the opposite direction. The devices like oars, paddle wheel, jets etc are used to propel ship. From this basic knowledge propellers came into existence.

The experiments that are normally carried out with ship models and model propellers are resistance experiments, open water experiments, self-propulsion experiments, wake measurements and cavitations experiments. Propeller being an important component for propulsion, more emphasis is done on design of the propeller. It has to withstand to the high pressure acting over on it. The metal propellers generally used cause vibration during its operation. In order to avoid it, conventional isotropic materials are replaced with composite materials. Glass fibre reinforced plastics (GFRP) materials are woven with fibre orientation angles 45, –45. Strength analysis is carried out for composite propeller by using different number of layers for composite materials and inter laminar shear stresses are found out.

2 Literature review

A literature survey was taken up to review present status of research in the field of theoretical analysis of stresses and deflection on propeller blades and identify great areas requiring focused attention focused specifically relevant to the project topic. The papers collected could be broadly classified into theoretical study on propeller strength and experiential studies on propeller strength and a few on composite materials and their fem treatment. Many investigators discussed with the strength of the propeller blade and only a few of them are fitted in this report. To find out the stresses and deflections of a propeller blade subjected to hydrodynamic loading.

652 R. Bammidi and K.S. Veronika

2.1 Strength of propeller blades

The strength requirements of propellers dictate that not only the blades be sufficiently robust to withstand long periods of arduous service without suffering failure or permanent distortion, but also that the elastic deflection under load should not alter the geometrical shape to such an extent as to modify the designed distribution of loading .A first approach to strength problem was made by Taylor (1933) who considered a propeller blade as a cantilever rigidly fixed at the boss. The stresses were evaluated following the theory of simple bending using section of the blade by a cylinder. Such sections are having straight faces and curved backs. The greatest tensile stress was calculated to occur at the trailing edge and the greatest compressive stress at the centre of the back. This method being the simplest of all is still widely used for simple and conventional propeller geometries, with narrow blades. But the method is suspect when used for propellers with wide blades and width comparable ton length.

Conolly (1960) addresses the problem of wide blades, tried to combine both theoretical and experimental investigations. The author carried out the measurements of deflection and stresses on model blades subjected to simulated loads with an aim to develop a theoretical model calibrated against the laboratory experiments. This model was validated by measurements of pressure and stress distribution on the blades of a full scale ship propeller at sea based on the experimental results it was concluded that wide blades are subjected to tensile stress on the face and compressive stress of similar magnitude on back side. It was pointed out the accuracy of the predication from the model depends on the accuracy of working load determined. Sonntvedt (1974) studied the application of finite element methods for frequency response and improve to the frozen type of hydrodynamic loading. The thin shell element of the triangular type and the super parametric shell element are used in the finite element model it presents the realistic and dynamic stresses in marine propeller blades. Stresses and deformation calculated for ordinary geometry and highly skewed propellers are compared with experimental results.

2.2 Propeller blade failure

Lee et al. (2003) investigated the main sources of propeller blade failures and resolved the problem systematically. An FEM analysis is carried out to determine the blade strength in model and full scale condition and the range of safety factor for the propeller under study is determined. Jourdian and Armand (1978) recognised that the failure of un-numerous blades is due to fatigue, which cannot be taken into account in a conventional static strength calculation. When comparing to Conolly (1960), improvements were taken on the structural model and also loading is taken into account the wake pattern. The feasibility of a dynamic analysis combined with an improved knowledge of fatigue resistance of the material will result in a reliable cure for this situation. General three dimensional solid elements of the results directly compared with the measured values. Correlation was made between model and full scale results. The radial stresses have been chosen as typical for the stress situation at each point.

Dynamic analysis of composite propeller of ship using FEA 653

2.3 Hub-blade interaction in propeller strength

The finite element method is generally used for calculating the stresses in propeller blade and hub separately. In the blade stress calculation, the hub is assumed to be rigid. On other hand the blade is completely ignored if hub strength is considered. Beek and Drunen (1978) the interference between the stresses conditions in both parts. Strong tools are available to shift disturbing boundary conditions from the blade root to the hub-shaft interface and obtain reliable information about the blade root loading an its resolution in the hub. Detailed experimental data, obtained in strain gauge measurements on a full scale blade, proved validity of the chosen element type and for application on propeller blades. Finite element calculations of propeller blade stresses for a blade clamped to an assumed rigid hub give reliable results over the whole blade except for the very close vicinity of the clamped section. It is noticed that special attention must be paid in hub blade transition for highly skewed blades.

2.4 Computer technique for propeller blade section design

Dekanski et al. (1993) developed a numerical procedure and computer program for the lifting surface design of sub-cavitations propellers. The procedure includes careful treatment of the effect of radical-induced velocities on blade-section design and provides for independent specification of the pitch angle of trailing vortices. Numerical results demonstrate the importance of including radial-induced velocities for propeller blades having significant skew, rake and/or a non-uniform radial pitch distribution. Blak et al. (1990) generate a geometry for given operating conditions by using the partial differential equation (PDE) method approach. Standard techniques of surface representation, such as B-spline interpolation, require a large number of control points to achieve this. The PDE method approaches the representative of the blade as a boundary valued problem which ensures that a fair surface is generated and secondly that a small set of design parameters are needed. The small parameter set is of importance since it can firstly manipulate the propeller design with ease and secondly, use it to great advantage in the task of functional design.

3 Modelling of propeller

Modelling of the propeller is done using CATIA V5 R 19. In order to model the blade, it is necessary to have sections of the propeller at various radii. These sections are drawn and rotated through their respective pitch angles. Then all rotated sections are projected onto right circular cylinders of respective radii as shown. As the above process is very complicated we model the propeller blade by using single section surface option.

654 R. Bammidi and K.S. Veronika

3.1 Design parameters

Figure 1 Design parameters of a ship propeller

3.2 Modelling of propeller

Figure 2 Initial stage of design (see online version for colours)

Dynamic analysis of composite propeller of ship using FEA 655

Figure 3 Design of single blade (see online version for colours)

Figure 4 Design view of blade (see online version for colours)

Figure 5 Final stage of design (see online version for colours)

656 R. Bammidi and K.S. Veronika

Figure 6 Final stage of design (see online version for colours)

Figure 7 Aluminium alloy model (see online version for colours)

4 Aluminium propeller analysis in ANSYS

4.1 Properties of aluminium

Casting condition: chill cast.

Proof stress: 230 N/sq.mm.

Tensile strength: 280 N/sq.mm.

Young’s modulus: 7.00 × 1 e4 N/sq.mm.

Rigidity modulus: 2.71 × 1 e4 N/sq.mm.

Poisson’s ratio: 0.29.

Density: 2.7 g/cc.

Dynamic analysis of composite propeller of ship using FEA 657

% elongation: 2.

Hardness: 105 BHN.

Melting point: 650 C.

Figure 8 Propeller meshing (see online version for colours)

Figure 9 ANSYS analysis of a propeller (see online version for colours)

Figure 10 Nodal solution of propeller (see online version for colours)

658 R. Bammidi and K.S. Veronika

Figure 11 Nodal analysis of propeller (see online version for colours)

5 Calculations of stresses in a propeller

The calculation of the stresses in a propeller is extremely complicated owing to a number of reasons: the loading fluctuates, its distribution over the propeller blade surface is difficult to calculate, and the geometry of the propeller is rather complex. It is therefore usual to use simplifies methods to calculate the stresses in the propeller blades and to adopt a large factor of safety based on experience. The simple method described here is based on the following principal assumptions:

1 The propeller blade is assumed to be a cantilever fixed to the boss at the root. The critical radius is just outside the root fillets.

2 The propeller thrust and torque, which arise from the hydrodynamic pressure distribution over the propeller blade surface, are replaced by single forces each acting at a point on the propeller blade.

3 The centrifugal force on the propeller blade is assumed to act through the centroid of the blade, and the moment of the centrifugal force on the critical section can be obtained by multiplying the centrifugal force by the distance of the centroid of the critical section from the line of action of the centrifugal force.

4 The geometrical properties of the radial section (expanded) at the critical radius may be used instead of a plane section of the propeller blade at that radius, and the neutral axes may be taken parallel and perpendicular to the base line of the expended section.

Dynamic analysis of composite propeller of ship using FEA 659

The following notation is adopted:

2

1

2

m239.36 mm 29,000 watts 13.49s

100 mm 15 mm 0.402 9,810 mm/sec60

60 32

A P A

R r r RπN Pω T g

V

Q P ZπN

Actual thrust acting 0.85 T

Actual torque transmitted to propeller 0.75 Q

1Thrust force per blade, Ft TZ

2

2 2

Centrifugal force, (2 )

4

wFc r πng

wπ n rg

For aluminium propeller:

Frequency, 36 Hzf

Speed, 36 60 2,160 rpmN

29,000Thrust, 2,149.74 N-m13.49

PTV

Actual thrust 0.85 0.85 2,149.74 1,827.28 N-mT

1,827.28Actual thrust per blade 609.1 N-m3

6029,000 128.152 2,160

Qπ

2 2

22

Centrifugal force 4

2 2,1604 2.12 169 N

9.81

wπ n rg

ππ

2Angular speed, 226.1960πN radω

s

Weight, 2.12 kgW

660 R. Bammidi and K.S. Veronika

5.1 Stress analysis for aluminium propeller in ANSYS

When an elastic system free from external forces is disturbed from its equilibrium position it vibrates under the influence of inherent forces and is said to be in the state of free vibration. It will vibrate at its natural frequency and its amplitude will gradually become smaller with time due to energy being dissipated by motion. The main parameters of interest in free vibration are natural frequency and the amplitude. The natural frequencies and the mode shapes are important parameters in the design of a structure for dynamic loading conditions. Modal analysis is used to determine the vibration characteristics such as natural frequencies and mode shapes of a structure or a machine component while it is being designed. Modal analysis is used to determine the natural frequencies and mode shapes of a structure or a machine component.

The rotational speed is limited by lateral stability considerations. Most designs are sub critical, i.e. rotational speed must be lower than the first natural bending frequency of the propeller.

Figure 12 Thrust force applied on the blade (see online version for colours)

Figure 13 Deflection of aluminium propeller in X direction (see online version for colours)

Dynamic analysis of composite propeller of ship using FEA 661

Figure 14 Deflection of aluminium propeller in Y direction (see online version for colours)

Figure 15 Deflection of aluminium propeller in Z direction (see online version for colours)

Figure 16 Normal stress of aluminium propeller in X direction (see online version for colours)

662 R. Bammidi and K.S. Veronika

Figure 17 Normal stress of aluminium propeller in Y direction (see online version for colours)

Figure 18 Normal stress of aluminium propeller in Z direction (see online version for colours)

Figure 19 Von Mises stresses of aluminium propeller (see online version for colours)

Dynamic analysis of composite propeller of ship using FEA 663

5.2 Composite propeller analysis in ANSYS

Properties of S2 glass fabric/epoxy:

Ex = 22.925 GPa

Ey = 22.925 GPa

Ez = 22.925 GPa

NUxy = 0.12

NUyz = 0.2

NUzx = 0.2

Gxy = 4.7 GPa

Gyz = 4.2 GPa

Gzx = 4.2 GPa

Density = 1.8 gm/cc.

5.4 Composite material layer stacking

Figure 20 Composite material layer stacking (see online version for colours)

Figure 21 Meshed model of composite propeller (see online version for colours)

664 R. Bammidi and K.S. Veronika

Figure 22 Modal analysis on blade 1 (see online version for colours)

Figure 23 Modal analysis on blade 2 (see online version for colours)

Figure 24 Modal analysis on blade 3 (see online version for colours)

Dynamic analysis of composite propeller of ship using FEA 665

Figure 25 Deflection of a composite propeller with eight layers in X direction (see online version for colours)

5.3 Calculations of stresses in a propeller

The following notation is adopted: For composite propeller:

Frequency, 81 Hzf

Speed, 81 60 4,860 rpm N

29,000Thrust, 2,149.74 N-m13.49

PTV

Actual Thrust = 0.85 0.85 2,149.74 1,827.28 N-mT

1,827.28Actual thrust per blade 609.1 N-m3

6029,000 56.952 4,860

Qπ

2 2

22

Centrifugal force 4

2 4,8604 2.12 867 N

9.81

wπ n rg

ππ

2Angular speed, 509.1460πN radω

s

Weight, 1.2 kgW

666 R. Bammidi and K.S. Veronika

5.4 Static analysis of composite propeller

Figure 26 Deflection of a composite propeller with eight layers in Y direction (see online version for colours)

Figure 27 Deflection of a composite propeller with eight layers in Z direction (see online version for colours)

Figure 28 Normal stress in composite propeller in X direction (see online version for colours)

Dynamic analysis of composite propeller of ship using FEA 667

Figure 29 Normal stress in composite propeller in Y direction (see online version for colours)

Figure 30 Normal stress in composite propeller in Z direction (see online version for colours)

Figure 31 Von Mises stress of a composite propeller (see online version for colours)

668 R. Bammidi and K.S. Veronika

6 Results and discussion

6.1 Linear static analysis

Linear static analysis is concerned with the behaviour of elastic continua under prescribed boundary conditions and statically applied loads. The applied load in this case is thrust acting on blades. Under water vehicle with contra rotating (aft) propeller is chosen for FE analysis. The FE analysis is carried out using advanced numerical simulation systems (ANSYS). The deformations and stresses are calculated for aluminium (isotropic) and composite propeller (orthotropic material). In composite propeller number of layers is taken as eight.

6.2 Static analysis of aluminium propeller

The thrust of 2,150 N is applied on face side of the blade in the region between 0.7R and 0.75R. The intersection of hub and shaft point’s deformations in all directions are fixed. The thrust is produced because of the pressure difference between the face and back sides of propeller blades. This pressure difference also causes rolling movement of the underwater vehicle. This rolling movement is nullified by the forward propeller which rotates in other direction (reverse direction of aft propeller). The propeller blade is considered as cantilever beam i.e. fixed at one end and free at other end. The deformation pattern for aluminium propeller and the maximum deflection was found as 3.62 mm in Y-direction. Similar to the cantilever beam the deflection is maximum at free end. Maximum principal stress value for the aluminium propeller and Von Mises stress on the basis of shear distortion energy theory also calculated in the present analysis. The maximum Von Mises stress induced for aluminium blade is 124.34 N/mm2 and the stresses are greatest near to the mid chord of the blade-hub intersection with smaller stress magnitude toward the tip and edges of the blade. Table 1 Static analysis of aluminium propeller

Results Aluminium propeller

Deflection in mm 3.62 Max normal stress N/mm2 93.718 Von Mises N/mm2 124.34 1st principal stress N/mm2 57.039 2nd principal stress N/mm2 32.09 3rd principal stress N/mm2 37.115 Frequency in Hz 36

Dynamic analysis of composite propeller of ship using FEA 669

Figure 32 Fundamental frequency of aluminium propeller (see online version for colours)

Figure 33 Max normal stress of aluminium propeller (see online version for colours)

Figure 34 Von Mises stresses of aluminium propeller (see online version for colours)

670 R. Bammidi and K.S. Veronika

Figure 35 Frequency of composite propeller (see online version for colours)

Table 2 Static analysis of composite propeller with eight layers

Results Composite propeller

Deflection in mm 0.7 Max normal stress N/mm2 50.12 Von Mises N/mm2 80.28 1st principal stress N/mm2 30.784 2nd principal stress N/mm2 30.025 3rd principal stress N/mm2 26.106 Frequency in Hz 81

Figure 36 Max normal stress in composite propeller (see online version for colours)

Dynamic analysis of composite propeller of ship using FEA 671

Figure 37 Von Mises stress of a composite propeller (see online version for colours)

7 Conclusions and future scope of work

7.1 Conclusions

1 The deflection for composite propeller blade was found to be around 0.7 mm for all layers is much less than that of aluminium propeller i.e. 3.62 mm, and shows composite materials is much stiffer than aluminium propeller.

2 Modal analysis results showed that the natural frequencies of composite propeller were 150% more than aluminium propeller, indicates that the operation range of frequency is higher for composite propeller.

3 Static analysis results showed that the max normal stresses of aluminium propeller are 87% higher than the composite propeller and Von Mises stresses are 50% higher in aluminium propeller than that of the composite propeller.

4 Aluminium propeller can rotate at maximum speed of 2,160 rpm without failing while composite propeller can rotate at a maximum speed of 4,860 rpm. The weight of the propeller can also be significantly reduced by using composite materials without sacrificing the mechanical properties.

7.2 Future scope of work

1 The present work only consists of static and modal analysis, which can be extended for Eigen value, harmonic, transient and spectrum analysis in case of both aluminium and composite materials.

2 There is also a scope of future work to be carried out for different types of materials. For present purpose only modelling and analysis of a propeller blade is carried only for GFRP materials.

672 R. Bammidi and K.S. Veronika

References Beek, G.H.M. and Drunen, B.V. (1978) ‘Hub-blade interaction in propeller strength’, The Society

of Naval Architects and Marine Engineers, May 24–25, Vol. 2, No. 5, pp.19-1–19-14. Blak, W.K., Kerwin, J.E., Weitendorf, E. and Friesch, J. (1990) ‘Deign of Aplc-10 propeller with

full scale measurements and observations under service conditions’, SNAME Transitions, Technical Resource Library, Vol. 98, No. 1, pp.77–111.

Conolly, J.E. (1960) Strength of Propellers, Reads in London at a Meeting of the Royal Intuition of Naval Architects on Dec. 1, 1960, pp.139–160.

Dekanski, C.W., Blor, M.L.G. and Wilson, M.J. (1993) ‘The generation of propeller blade geometries using the PDE method’, Journal of Ship Research, Vol. 39, No. 2, pp.108–116.

Jourdian, M. and Armand, J.L. (1978) ‘Strength of propeller blades – a numerical approach’, 18Symposlum, The Society of Naval Architects and Marine Engineers, Virginia Beach, Virginia, 24–25 May 1978.

Lee, C-S., Kim, Y-J., Kim, G-D. and Nho, I-S. (2003) Case Study on the Structural Failure of Marine Propeller Blades, Vol. 6, No. 3, The Society of Naval Architects of Korea.

Sonntvedt, T. (1974) ‘Propeller blade stresses, application of finite element methods’, Computers & Structures, Vol. 4, pp.193–204, Pergamon Press, Chat Britain.

Taylor, D.W (1933) The Speed and Power and Ships, Press of Ransdell Incorporated, Washington DC.

http://www.iaeme.com/IJMET/index.asp 516 editor@iaeme.com

International Journal of Mechanical Engineering and Technology (IJMET) Volume 10, Issue 01, January 2019, pp. 516–526, Article ID: IJMET_10_01_053

Available online at http://www.iaeme.com/ijmet/issues.asp?JType=IJMET&VType=10&IType=01

ISSN Print: 0976-6340 and ISSN Online: 0976-6359

© IAEME Publication Scopus Indexed

A REVIEW ON TRIBOLOGICAL

PERFORMANCE CHARACTERISTICS OF

PLASTIC GEARS

S.Phani Kumar*

Assistant Professor,Department of Mechanical Engineering,

Anil Neerukonda Institute of Technology & Sciences, Visakhapatanam, India

Dr.K.N.S.Suman

Assistant Professor,Department of Mechanical Engineering,Andhra

University,Visakhapatanam, India

S.Ramanjaneyulu

Assistant Professor,Department of Mechanical Engineering,

Anil Neerukonda Institute of Technology & Sciences, Visakhapatanam, India

*corresponding author

ABSTRACT

Polymers are presently broadly utilized as substitute material for steel gear in low

load devices. Its malfunction differs from gears made of steel, thus it is imperitive to sort

out the failures shown by polymer gears. Numerous earlier studies noted that wear

recognition, microstructure surface condition monitoring, weight loss and temperature

detection can be used in the analysis for the failure of polymer gear. The principle

objective of the current work is to conduct an analysis for failure detection methods as

defined above. Other researcher works were studied and their findings were extracted in

order to identify the methods they used. The most common method used was wear

detection and it was supplemented by other methods such as microstructure surface

condition monitoring. Failures shown by polymer can be concluded to be tooth breakage,

tooth deformation, material removal and surface fatigue.

Key words: Polymer gears, Acetal copolymer, Nylon 6/66, Microstructure surface,

Thermal damage

Cite this Article: S.Phani Kumar, Dr.K.N.S.Suman and S.Ramanjaneyulu, a Review on

Tribological Performance Characteristics of Plastic Gears, International Journal of

Mechanical Engineering and Technology, 10(01), 2019, pp.516–526

http://www.iaeme.com/IJMET/issues.asp?JType=IJMET&VType=10&Type=01

S.Phani Kumar, Dr.K.N.S.Suman and S.Ramanjaneyulu

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1 INTRODUCTION

Polymers are chosen to substitute steel gear in low load devices owing to the technical advantage

and economic gains. The advantages of polymer gear is ability to operate with limited amount of

lubrication, light in weight material and low operating noise [1] compared to metal gear.

However, considerations such as; working environment, strength, weight, material properties and

cost [2] must be cautiously considered when deciding to supplant steel with polymer. Standards

for polymer gear like AGMA [3] and British Standard [4] can be referred when designing the

gears.

Polymer gear failures are different from steel for the reason that the material properties of

polymer are totally different. An example of polymer gear failure is melting of material which

does not occur for steel gears. These types of failure can be categorized under thermal damage.

This article reviews four types of failure characteristic wear, microstructure surface condition

monitoring, weight loss and thermal damage which is shown in Fig. 1. These characteristics are

often studied together as they complement one another. The other two characteristics are not

discussed in this article. Wear detection is the most commonly used technique [5] to determine

failure as it covers a diversity of conditions such as cracks, breakage and debris formation [6].

Microstructure surface condition monitoring provides a ready and quick method to inspect the

surface failure [7] which are not visible during wear detection. Weight loss failure characteristic

is only suitable for gears made from pure polymer as reinforcements may affect the weight of

gear. In thermal damage characteristic, temperature detection has a vital role to determine as to

when the damage occurs for a defined temperature.

Figure 1. Types of failure characteristics for polymer gear

2 FAILURE CHARACTERISTICS

2.1. Wear

There are many types of failure that can be categorized under wear, such as crack or breaking,

tooth thickness reduction and debris formation. Each review will include all known wear

formation of polymer gear.

2.1.1. Wear debris formation

Acetal gear are found to have different failure compared to Nylon gear as reported by K. Mao et

al [8]. Polymer gear wear can be divided into three stages; running in, linear and final rapid wear

as shown in Table 1. The wear debris size increases as the gear approaches final wear period.

When Acetal gears were tested in the high range load, 10 – 16.1 N.m, the wear debris formed

immediately after the test started. When Nylon gears were tested at high load, 10 N.m, it fractures

after going through running in and linear wear period. The gear made from Acetal failed by

melting and in Nylon by fracture as shown in Fig. 2.

The same result was also obtained by W. Li et al [9] in their research where the testgears were

paired with different materials. Acetal gear started to melt at load torques higher than 9 N.m and

fracture occurs when load is 10 N.m and above. However, the wear performance improved when

A Review on Tribological Performance Characteristics of Plastic Gears

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it is paired with dissimilar material where Acetal as the driver gear and Nylon as the driven gear,

this pair showed the highest performance from other pairs which can be seen in Fig. 3.

Table 1.Stages of polymer gear wear.

Phase Explanation

Running in

wear Occurs for a short time but the amount of wear is high

Linear wear

Low amount of wear can be seen but is progressive

Final rapid wear

High wear rate but small amount of debris, indicating debris is due to deformation

undergone by the polymer gear caused by thermal effects

Figure 2.Wear on polymer gears with module 2 mm and 30 gear tooth [8].

Figure 3.Results on Nylon/Acetal (N/A), Acetal/Nylon (A/N), Acetal/Acetal (A/A) and Nylon/Nyon

(N/N) gear pairs [9].

2.1.2. Tooth thickness reduction

Wear rates for Acetal gear produced by machine cut and injection mould in gear independent to

the manufacturing process as reported by K. Mao et al [10]. The gears undergo testing at load 6

– 9 Nm at 100 rpm and undergo wear in three phases. The running-in and linear phase produced

little wear debris, but in the rapid wear period, wear debris increased and so does the wear rate.

After about 33% of tooth thickness removed, the gear started to fail. An incremental step loading

(load is incrementally added without changing the test gears) was used and it was compared to

the conventional procedure (test gears is changed for every load value). The result obtained

S.Phani Kumar, Dr.K.N.S.Suman and S.Ramanjaneyulu

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showed that the incremental step loading test produces adequate result and can be completed

within hours compared to conventional testing which takes up weeks. It was also noted that

bending occurred when the material is momentarily melted, causing it to jump out of mesh. The

repeated motion of sliding at addendum and dedendum region produced heat caused by the

friction of the tooth surface leading to adhesive wear.

The various types of failure in unreinforced and reinforced Nylon 66 gear was studied by S.

Senthilvelan and R. Gnanamoorthy [11] using tooth thickness and weight loss measurement

technique. Un-reinforced and reinforced Nylon 66 gears were meshed with a stainless steel

(SS316) gear. Fig. 4 shows the deformation of teeth region undergone by the glass reinforced

Nylon 66 gear. Reinforced gears showed a uniform material loss compared to un-reinforced gears

because glass fiber have better adhesion to the matrix compared to carbon fibre. Wear of tooth

flank region in glass reinforced fibre is caused by softening of material and scraping by opposing

stainless steel gear tooth. The wear occurred is due to the low thermal resistance of the material.

In the case of carbon fibre reinforced gear, no appreciable tooth deformation was present due to

high stiffness and good thermal resistance of the material. This result was obtained at the test

parameter of 1000 rpm rotational speed and loads ranging from 1.5 N.m to 3 N.m.

Figure 4.Tooth thickness reduction due to scraping of steel gear tooth [11].

2.1.3. Cracks

Cracks often occur at the root of the tooth and will propagate causing tooth breakage. The effect

of rotational speed on the performance of unreinforced and glass reinforced Nylon 6 was studied

by S. Senthilevan and R. Gnanamoorthy [12]. The glass fibre reinforced gear showed

improvement in mechanical strength and thermal deformation. They noted that the performance

of gears was influenced by the load applied. The performance was only influenced by speed at

the higher load condition. Gear root tooth crack and tooth wear were observed occurring at lower

load, 8 MPa for both materials. When the load is higher, plastic deformation occurs on the

unreinforced gear and at 15 MPa deformation starts to occur in glass fibre reinforced gear. At

low stress levels, gear tooth root cracking and tooth wear was the main factor of failure, and in

the higher stress level, deformation of material at high temperature causes failure. Modification

on gear tooth made from Nylon 6 was reported by H. Imrek [13] and the failure for each design

was studied. The tooth was modified as seen in Fig. 5 so that the single mesh area was increased

thus reducing the load and temperature of the area. This reduces the wear rate and improves the

overall teeth temperature. The unmodified gear showed cracking at the pitch area in Fig. 5 while

in modified gear, cracking occurred at tooth roots.

Modification of gear tooth was also studied by H. Duzcukoglu [14] where holes are

introduced to the tooth body of the gear. This serves as a cooling mechanism and to improve the

heat distribution. Gears with modification shown smooth wear transition compared to unmodified

gears. As the tooth load increases, the tooth profile wear becomes more noticeable at the tooth

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root and tooth tip region. It was concluded that wear occurs due to softening and detachment of

material from the contact area and with the modification, the damage is delayed with the help of

increased heat transfer from the gear.The effect of different hub type on the spur gear performance

was studied by S.

Senthilvelan and R. Gnanamoorthy [15] using gears made from Nylon 66 and reinforced with

carbon fibre. The hub was made from Nylon 66 and in cylindrical or spline shape. At 15 MPa of

bending stress, both unreinforced and reinforced gears showed wear at tooth surface and flank.

When the bending stress is at 20 MPa, the gear fitted with cylindrical hub failed at the gear and

hub joint at 2 x 105 cycle and the gear with spline hub showed wear characteristics such as cracks

at tooth root region. The failure between circular hub and gear is caused by the joint failure

meanwhile in spline hub and gear the failure is caused by the gear tooth. The failure for

unreinforced and reinforced gear were the same in the spline hub,

Unmodified Modified

Figure 5.Gear profile models, arrow indicates crack propagation [13].

2.2. Microstructure surface condition monitoring

This method is used to detect micro crack or deformation on the gear surface which are notvisible

with naked eye. With the introduction of composite polymer, this method becomes more

important as it is capable to inspect the fibre structure and alignment.

Scanning electron micrograph (SEM) was used by A. R. Breeds et al [16] in order tostudy the

surface condition. Three major interest region were dedendum, addendum, and pitch line. With

the help of SEM, they were able to detect large pits or scoops of material were removed at the

dedendum, smooth surface due to wear at addendum caused by sliding and rolling motion of

gears and the formation of a ridge at pitch line caused by rolling. An SEM examination around

the gear tooth pitch and root areas were conducted by K. Mao et al [10] to determine whether

wear occurs at that region. From the gear mesh theory, there is nearly zero friction around the

pitch point, however the images from SEM showed otherwise. This shows that SEM can also be

a reliable method to detect failure in polymer gears. Fig. 6 shows the difference of wear occurring

at the tip and pitch point of the gear.

The effect of fibre orientation was analyzed using SEM by S. Sentilvelan and

R.Gnanamoorthy [11]. A perpendicular aligned fibre orientation showed better performance

where it helps slowing the crack growth, thus improving the gear life. In the glass reinforced gear,

matrix nylon material was found adhered to the protruded glass fibre on the fracture surface. The

cracked surface showed few cavities and nearly flat. This is due to the better adhesion of glass

fibre and nylon matrix. Molten smeared layers were also seen on the surface. A high number of

cavities were observed in the carbon reinforced gear cracked surface due to the poor adhesion

S.Phani Kumar, Dr.K.N.S.Suman and S.Ramanjaneyulu

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between carbon fibresand nylon matrixes. Molten material was absent due to better thermal

properties of the carbonfibre reinforced Nylon 66. The fibre orientation is influenced by the gate

location and tooth geometry. Both glass and carbon reinforced show poor wear resistance

compared to unreinforced gear at the investigated condition. Molten material can be seen on the

unreinforced Nylon 66 in Fig. 7and on glass fibre reinforced Nylon 66 in Fig. 8. There was no

molten material present on carbon fibre reinforced Nylon 66. However, cavities are present on

both glass fibre and carbon fibre reinforced Nylon 66 in Fig. 8.

The surface of loading tooth was inspected by H. Duzcukoglu et al [17] to determine the

presence of transverse crack. It was found that the transverse cracked occurred due to thermal

softening caused by accumulated heat. These cracks shall merge and grow resulting removal of

material in the shape of flakes. The possibility of controlling wear by applying coatings on tooth

flanks were studied by K. Dearn et al [18]. Five types of coating were used; PTFE, boron nitride,

molybdenum disulphide and graphite to protect the gear. SEM was used to study the surface of

each gear with different coatings. PTFE and graphiteprovide most optimum protection as it

lowers the friction between gear teeth, reduces running temperature and subsequently the wear

of gears. However, it is possible that the coating will lose its effectiveness as the protection film

wears over time.

Wear at tip Wear at pitch

Figure 6.SEM image of wear on polymer gear occurring at tip and pitch [10].

Figure 7.Surface condition of unreinforced Nylon 66 [11].

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Glass Fibre Reinforced Nylon 66 Carbon Fibre Reinforced Nylon 66

Figure. 8.Surface condition of polymer gear [11].

2.3. Weight loss

This characteristic was found to be acceptable if gears were made using pure polymer.However,

if it is made from composite, it become less reliable as the weight is affected by the composition

of fibre and moisture or water presence. This was shown by N. A. Wright and S. N. Kurenka

where they introduced a pair of control gear in their research [19]. They stated that the weight

loss from running test can be considered as one of the method to determine failure, if only the

material used does not have a high affinity for water. They noted that Polyamide 66 absorbs a

significant amount of water, therefore the introduction of control gears and it was placed on the

drive gearbox. They calculated the weight loss by subtracting the weight loss by control gear

from the total measure weight loss by the test gear.

The weight loss method was used by C. H. Kim [20] to determine the wear volume of both

Nylon and Acetal pinion. The pinion had three design, a solid gear tooth body, a drilled hole on

the gear tooth body and a hole inserted with steel pin. The hole type and insert type showed less

wear rate than the solid one. In the Nylon gear, hysteric heat loss was decreased by the hole in

the tooth, while in the steel pin type, heat is absorbed and distributed by the pin. Both design led

to decrease in wear rate and degradation of Nylon material. In the case of Acetal pinion, the

variation of cross section increased the specific wear rate. The decrease of cross section area led

to deformation and plastic flow on the Acetal pinion. This will lead to severe wear due to

interference and severe contact between the Acetal pinion and steel driver gear. The wear rate in

Nylon pinion decreased by over 30% and an increase in service life by 415%.While the Acetal

pinion, it causes increase in wear, therefore this method can only be applied to visco-elastic

material only.

The wear resistance of carbon nanotube reinforced Acetal gear was studied by S.Youseff [21]

by determining the weight loss of the gear. It was then compared to results from previous research

[22]. The results showed that the average wear resistance of Acetalreinforced with carbon

nanotube compared to Acetal improved significantly. Spur gear improved by 28%, helical gear

by 35%, bevel gear by 44% and lastly worm gear up to 47%.

S.Phani Kumar, Dr.K.N.S.Suman and S.Ramanjaneyulu

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2.4. Thermal damage and temperature detection using thermal camera

ortemperature sensor

In this method, temperature of the gears is taken during or after they were tested. Some researcher

also used data acquisition system to record the operating temperature. This failure detection

method is essential as different loads will influence the running temperature and affect the

material properties when it reaches the glass transition Temperature or the melting temperature.

The failure mode of polymer and polymer composite was found to be different asshown by

S. Sentilvelan and R. Gnanamoorthy [11]. The gears were made from Nylon and reinforced with

carbon or glass fibre. They also found that the surface temperature of unreinforced gear was

higher compared to reinforced gear. In the reinforced Nylon, carbon reinforced had a lower

temperature than glass reinforced. The reinforced gears lower temperature was contributed by a

better tooth stiffness, lower friction and good thermal properties. A high tooth stiffness prevents

tooth deflection which contribute to less unwanted contact between tooth surfaces which causes

heat. The improved heat dissipation ability increased the gears life considerably. The introduction

of cooling holes was reported by H. Duzcukoglu [17] in order to decrease thermal damage. Three

design of gears were studied, first is unmodified, the second gear had a hole drilled at the pitch

point of the gear tooth and the third design have holes at the pitch point and on the body of the

tooth as seen in Fig. 9. The temperature was detected using a non-contact infrared temperature

sensor and recorded on a PC by using data acquisition system. The first design failed at the

vicinity of the pitch diameter, caused by softening. This was due to the gear inability to emit heat

which was accumulated during the running process. As the load increases, the thermal damage

also increased. This causes the material to soften and severe tooth deformation occurs. In the

second design, partial thermal softening at the pitch region and tooth root region was observed.

The amount of thermal damage was reduced by using this design, however, there is still damage

on the surface of the loading tooth. For the third design, only thermal damage initiation was

observed at the high load, 18.1 N.m. The heat produced in each design is from the friction between

the driver and driven gear. The result from heat produced affecting the gear tooth can be seen in

Fig. 9.

First Second Third

Figure 9.Tooth condition for each design when the load is at 6.1 N/mm [17].

A design in which an internal hole or steel pin inserts are introduced to the tooth body was

presented by C. H. Kim [20] to improve heat transfer process and stress concentration.

Temperature of the tooth surface was measured and investigated using a non-contact type

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temperature sensor. Three load value was used, which are 9.8N/mm, 19.6 N/mm and 29.4 N/mm.

At first load value, the hole type pinion had the lowest temperature value but have a higher

fluctuation. The steel insert pinion has a slightly higher temperature, but the temperature

maintained. When the load is 19.6 N/mm, the insert type pinion showed better performance than

the others. At the highest test load, fracture can be seen from all types of design. the insert type

took the longest time before failing followed by hole type and lastly solid type. It can be noted

that the decrease in tooth temperature will result in better life and Reduction of wear.

Polymer gear can fail in two typical ways, fatigue or sudden melting as reported in the

research by A. Pogacnik and J. Tavcar [23]. A new multilevel accelerated testing procedure was

proposed by the authors and the results which are life span and gear temperature were compared

with a calculation procedure. The temperature was recorded using a thermal camera and the

materials were PA 6, PA 6 with 30% glass fibres and Polyacetal.

The maximum gear temperatures and load levels are different for every pair of materials.

PA6/PA6 pair generated the highest temperature due to the high coefficient of friction. POM/PA6

pair gives the lower temperature due to lower coefficient of friction. The melting of gears was a

consequence of overload and an increase in temperature. By avoiding problematic material

combination, the failure due to thermal characteristic can be avoided.

The effect of different surface roughness was studied by J. Mertens and S. Sentilvelan[24]

where three different value of coefficient of friction studied. Three stainless steel gearwith

coefficient of friction 3.8-4.1μm, 2.5-2.8μm and 1.9-2.2 μm was mated with polypropylene gear.

The surface temperature of test gear was measured using a non-contact infra-red sensor. The

frictional values of the surface are influenced by the hardness and micro geometry of the stainless

steel gear. When a polymer slides on steel, adhesion and deformation occurs, contributing to the

friction between those two surfaces. At a higher load, the surface interaction will increase,

causing the friction, wear and temperature to increase which can be seen in Fig. 10. Polymer gear

will generate more heat when meshed with surfaces having a high friction coefficient thus

affecting the performance of the polymer gear. It can be seen that Gear A have the highest friction

followed by B and C which relates to the higher temperature produced by A and followed by B

and C at each load.

Figure 10.Comparison of surface temperature for test gears [25].

S.Phani Kumar, Dr.K.N.S.Suman and S.Ramanjaneyulu

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3. CONCLUSIONS

From the reviewed articles, it can be concluded that detection of wear, microstructure surface

condition monitoring, weight loss and thermal damage and temperature detection are the most

prominent method alongside vibration and meshing displacement in order to detect failure in

polymer gears. Most of the failures occurred due to the limitation of material, such as the load

handling capability and thermal properties. In order to optimize the usage of polymer gear in

applications, the operating parameters such as load and running temperature must be calculated

beforehand so that the working environment of the polymer gear is the most optimum.

REFERENCES

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(2006).

[2] J. L. Elmquist. Deciding When to Go Plastic.Gear Technology. 46 - 47 (2014).

[3] A. G. M. Association, "Standard 1106-A97. Tooth Proportions for Plastic Gears," ed.

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[4] B. S. Institution, "BS 6168. Specification for non-metallic spur gears," ed. London, (1987).

[5] K. M. Marshek and P. K. C. Chan, "Qualitative analysis of plastic worm and worm gear

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surface microcracks produced in gear fatigue," Journal of Materials ScienceLetters, 13, pp.

1551-1554, (1994).

[8] K. Mao, W. Li, C. J. Hooke, and D. Walton, "Friction and wear behaviour of acetaland nylon

gears," Wear, 267, pp. 639-645, 6/15/ (2009).

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dissimilar polymer gear engagements," Wear,271, pp. 2176-2183, Jul 29 (2011).

[10] K. Mao, P. Langlois, Z. Hu, K. Alharbi, X. Xu, M. Milson, et al., "The wear andthermal

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5// (2015).

[11] S. Senthilvelan and R. Gnanamoorthy, "Damage Mechanisms in Injection Molded

Unreinforced, Glass and Carbon Reinforced Nylon 66 Spur Gears," Applied

CompositeMaterials, 11, pp. 377-397, (2004).

[12] S. Senthilvelan and R. Gnanamoorthy, "Effect of rotational speed on the performance of

unreinforced and glass fiber reinforced Nylon 6 spur gears," Materials & Design,28, pp. 765-

772, // (2007).

[13] H. İmrek, "Performance improvement method for Nylon 6 spur gears," Tribology

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[15] S. Senthilvelan and R. Gnanamoorthy, "Selective Carbon Fiber Reinforced Nylon 66 Spur

Gears: Development and Performance," Applied Composite Materials, 13, pp. 43- 56,( 2006).

[16] A. R. Breeds, S. N. Kukureka, K. Mao, D. Walton, and C. J. Hooke, "Wear behavior of acetal

gear pairs," Wear,166, pp. 85-91, 1993/06/15 (1993).

[17] H. Duzcukoğlu, R. Yakut, and E. Uysal, "The Use of Cooling Holes to Decrease the Amount

of Thermal Damage on a Plastic Gear Tooth," Journal of Failure Analysis andPrevention, 10,

pp. 545-555, (2010).

[18] K. D. Dearn, T. J. Hoskins, D. G. Petrov, S. C. Reynolds, and R. Banks, "Applicationsof dry

film lubricants for polymer gears," Wear,.298–299, pp. 99-108, 2/15/ 2013.

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[19] N. A. Wright and S. N. Kukureka, "Wear testing and measurement techniques for polymer

composite gears," Wear, 251, pp. 1567-1578, 10//( 2001).

[20] C. H. Kim, "Durability improvement method for plastic spur gears," Tribology International,

39, pp. 1454-1461, 11// (2006).

[21] S. Yousef, T. A. Osman, A. Abdalla, and G. Zohdy, "Wear Characterization of Carbon

Nanotubes Reinforced Acetal Spur, Helical, Bevel and Worm Gears Using a TS Universal

Test Rig," JOM, pp. 1-8, 2014/12/23( 2014).

[22] S. Yousef, A. Khattab, M. Zaki, and T. A. Osman, "Wear Characterization of Carbon

Nanotubes Reinforced Polymer Gears," Nanotechnology, IEEE Transactions on, 12, pp. 616-

620, (2013).

[23] A. Pogačnik and J. Tavčar, "An accelerated multilevel test and design procedure for polymer

gears," Materials & Design, 65, pp. 961-973, 1// 2015.

[24] A. J. Mertens and S. Senthilvelan, "Effect of Mating Metal Gear Surface Texture on the

Polymer Gear Surface Temperature," Materials Today: Proceedings,.2, pp. 1763- 1769, //

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[25] A. J. Mertens and S. Senthilvelan, "Effect of mating metal gear surface texture on the polymer

gear surface temperature," Materials Today-Proceedings. 2, pp. 1763-1769, (2015).

Prespacetime Journal| December 2019 | Volume 10 | Issue 8 | pp. 1094-1103

Reddy, D. R. K., & Raju, K. D., Dark Energy Model in a Five-Dimensional Space-Time with a Massive Scalar Field

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Article

Dark Energy Model in a Five-Dimensional

Space-Time with a Massive Scalar Field

D. R. K. Reddy*1

&K. Deniel Raju1,2

1Department of Applied Mathematics, Andhra University, Visakhapatnam-530003, India

3Department of Mathematics, ANITS (A), Visakhapatnam-531162, India

Abstract The main aim of this investigation is to study the dynamical aspects of a cosmological model

with dark energy (DE) and an attractive massive scalar field as source in a five dimensional

spherically symmetric space-time. An exact solution of the field equations of general relativity is

obtained which represents a massive scalar field DE model in five dimensions. The cosmological

parameters of the model are computed and their dynamical behavior is studied. It is noted that

our model is a quintessence model which represents accelerated expansion of the universe and is

in good agreement with the recent cosmological observation.

Keywords: Massive scalar field, cosmological model, dark energy, five dimensional, space time.

1.Introduction

We live in this universe. Hence it is necessary to know about our universe. There have been

several modern cosmological observations to study the nature of the universe. The recent

experimental study of type 1a supernova [1-2] and cosmic microwave background data [3]

suggests that the universe is spatially flat and accelerating. It is supposed that this is because of

the hither to unknown fluid with high negative pressure known as 'dark energy' (DE). Even today,

it has been a challenging problem to explain DE which is driving the acceleration of the universe.

The simplest candidate for DE is the cosmological constant. But this suffers from coincidence

problem. Hence several DE models like quintessence [4] phantom [5] quintom [6], tachyon [7]

and holographic models [8] have been considered.

In recent years, scalar field DE models have been attracting several authors, in particular, in the

presence of Brans-Dicke [9], Saez-Ballester [10] and Barber's [11] scalar fields. The following

are some of the quintessence models obtained in Bianchi type space-times. Rao et al. [12]

obtained a Bianchi type-II modified holographic DE model in Barber's second Self-Creation

theory of gravity while Reddy [13] discussed Bianchi type-V modified holographic Ricci DE

models in Saez-Ballester scalar tensor theory of gravitation. Reddy et al. [14] investigated

*Correspondence: D.R.K. Reddy. Department Applied Mathematics, Andhra University, Visakhapatnam, India.

Email: reddy_einstein@yahoo.com

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minimally interacting Kaluza-Klein holographic DE model in the presence of Brans-Dicke scalar

field while Kiran et al. [15,16] have presented Bianchi type minimally interacting holographic DE

models in Brans-Dicke and Saez-Ballester theories of gravitation. Also Rao et al. [17] obtained

five dimensional FRW modified holographic Ricci DE model in Saez-Ballester scalar tensor

theory. Very recently Aditya and Reddy [18] have investigated FRW type Kaluza-Klein modified

holographic DE models in Brans-Dicke theory.

Very recently, investigation of DE models in the presence of scalar-meson fields is attracting

many researchers. It is well known that the scalar meson fields are of two categories, namely,

zero mass and massive scalar fields. Mass less scalar fields describe long range interactions while

massive scalar fields represent short range interactions. Massive and mass scalar fields along

with different physical sources have been investigated by various authors [19-23]. Attractive

massive scalar fields play a vital role in the discussion of scalar field DE cosmological models.

Hence, Reddy [24] obtained Bianchi type-V DE model with zero mass scalar fields. Naidu [25]

presented Bianchi type-II modified holographic Ricci DE model in the presence of massive scalar

field. Aditya and Reddy [26] have investigated Bianchi type-III massive scalar field DE model

while Reddy et al. [27] discussed Kantowski-Sachs DE model in the presence of anisotropic DE

fluid. Also, five dimensional Kaluza-Klein DE model with zero mass scalar fields has been

presented by Reddy and Ramesh [28]. Reddy and Ramesh [29] discussed inflationary

cosmological model with flat potential in the presence of mass less scalar field. Recently, Naidu

et al. [30] have obtained a DE model with massive scalar field in Bianchi type-V space-time.

Here our main interest is to explore higher dimensional DE models with massive scalar fields as

source. The significance of the higher dimensional space-time in the discussion of early stage of

evolution of the universe is well known. This is because of the fact that the cosmos in its early

stages might have had higher dimensional era before the universe has undergone compactification

transition. Several authors, therefore, have been attracted to this subject and obtained five

dimensional cosmological models [31-35]. In particular, Samantha and Dhal [36] have obtained a

new class of higher dimensional cosmological models in f(R,T) gravity [37]. Rao and Jayasudha

and Raju et al. [38-40] have discussed five dimensional spherically symmetric cosmological

models in Brans-Dicke and Saez-Ballester theories of gravitation.

Inspired by the above investigations and the discussion, we propose to explore five dimensional

anisotropic spherically symmetric cosmological model in the presence of DE fluid and massive

scalar field. Following is the scheme of this paper: Section 2 deals with the metric and

corresponding field equations. In sec.3 we obtain the solution of the field equations and present

the cosmological DE model in five dimensions. Sec.4 is devoted to the discussion of

cosmological parameters. The last section contains conclusions.

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2. Metric and Field Equations

Five-dimensional spherically symmetric anisotropic space-time is described by the metric

2 2 2 2 2 2 2 2 2( sin )ds dt e dr r d r d e d (1)

Here and are functions of time t and the fifth coordinate is considered space-like [41]

The non-vanishing components of Einstein tensor for the metric (1) are

0

0

2 2

1 2 3

1 2 3

2

4

4

3

4

3 1 1 1

4 2 4 2

3

2

G

G G G

G

(2)

where an overhead dot indicates differentiation with respect to t .

Einstein's field equations in the presence of DE fluid and massive scalar field are written as

(de) (s)

ij ij ij ij

1R g R (T T )

2 (3)

Here ( )de

ijT is the energy momentum tensor of DE given by

(de)

ij i j ijT ( p )u u p g (4)

and

(s ) ,k 2 2

ij ,i , j ,k

1T ( M )

2 (5)

where is the massive scalar field which satisfies the Klein-Gordon equation.

ij 2

;ijg M 0 (6)

M is the mass of the scalar field , p and are DE pressure and DE density respectively.

Also, comma and semicolon denote ordinary and covariant differentiation respectively.

(Gravitational units are used so that 8 G c 1 ).

The energy momentum tensor of DE given by eq. (4) can be parameterized as

(de)

ij deT diag 1, , ( ), ( ) ( ) (7)

where

Prespacetime Journal| December 2019 | Volume 10 | Issue 8 | pp. 1094-1103

Reddy, D. R. K., & Raju, K. D., Dark Energy Model in a Five-Dimensional Space-Time with a Massive Scalar Field

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p

(8)

is the equation of state (EoS) parameter of DE and , , are the skewness parameters which are

the deviations from along y, z and axes respectively.

Now using co moving coordinates and equations (2),(3),(5) and (7) the Einstein field equations

for the metric (1) can be written as

22 2 23

4 2 2

M

(9)

2 2 22 23

4 2 4 2 2 2

M

(10)

2 2 2

2 23( )

4 2 4 2 2 2

M

(11)

2 2 2

2 23( )

4 2 4 2 2 2

M

(12)

22 2 23

( )2 2 2

M

(13)

The energy conservation equation, ;0ij

jT gives

31 0

2

(14)

The Klein-Gordon equation (6) becomes

23M 0

2

(15)

where an overhead dot denotes differentiation with respect to time t

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Reddy, D. R. K., & Raju, K. D., Dark Energy Model in a Five-Dimensional Space-Time with a Massive Scalar Field

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For the space-time given by Eq. (1), we define the following cosmological parameters which will

be useful in solving the field equations (9) - (15)

The average scale factor a(t) and the spatial volume V are given by

3

4 2V a (t) e

(16)

Generalized Hubble parameter is

1 3 1

H4 2 2

(17)

The scalar expansion , shear scalar 2 and the average anisotropy parameter are

i

;i

3 1u 4H

2 2

(18)

2 2 2

2 ij

ij

1 3 1 1 1 2

2 8 8 2 6 9

(19)

24i

h

i 1

H H1A

4 H

(20)

where iH (i 1,2,3,4) represents directional Hubble parameters.

The deceleration parameter is

d 1

q 1dt H

(21)

3. Solution of field equations and DE model

From Eqs.(10), (11) and (12) we, immediately, obtain

0 (22)

Using Eq. (22), Eqs. (9) - (15) reduce to the following independent equations:

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Reddy, D. R. K., & Raju, K. D., Dark Energy Model in a Five-Dimensional Space-Time with a Massive Scalar Field

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22 2 23

4 2 2

M

(23)

2 2 22 23

4 2 4 2 2 2

M

(24)

22 2 23

( )2 2 2

M

(25)

23M 0

2

(26)

[Eq. (14) being conservation equation].

Now, the field equations (23) - (26) are a system of four independent equations in five unknowns

, , , and . To obtain a determinate solution we use the relation between metric potentials

given by [36,38]

k (27)

where k 0 is a constant.

Using Eq. (27) in Eq. (26) we have

23k 1

M 02

(28)

To reduce the mathematical complexity we have [22,23,29]

3k 1

2

(29)

which amounts to a power law between average scale factor and the scalar field .

Now using Eq. (29) in Eq. (28) we get

2 2

0 1

M texp t

2

(30)

Prespacetime Journal| December 2019 | Volume 10 | Issue 8 | pp. 1094-1103

Reddy, D. R. K., & Raju, K. D., Dark Energy Model in a Five-Dimensional Space-Time with a Massive Scalar Field

ISSN: 2153-8301 Prespacetime Journal

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1100

From Eqs. (29) and (30) we obtain

2 2

0 12 2

3 1

M t t

k

(31)

From Eqs. (27) and (31) we have

2 2

0 12 23 1

kM t t

k

(32)

Here 0 and 1

are constants of integration.

Now using Eqs. (31) and (32),in Eq. (1) the DE modelcan be written as

2 2 2 2 2 2 2 2 2 2

0 1

2 220 1

exp 2 2 sin3 1

2 2exp

3 1

kds dt M t t dr r d r d

k

M t td

k

(33)

and the scalar field in this model is given by Eq. (30).

4. Discussion of Cosmological parameters

In this section we construct the cosmological parameters corresponding to the model (33) and

discuss their physical significance.

Spatial volume of the model is

2 2

0 1exp2

M tV t

(34)

The average Hubble parameter is

2

0

1

4H M t (35)

The scalar expansion is

2

0M t (36)

The shear scalar is

2

22 2 2

0 0

3 1 3 2 2

2(3 1) 6(3 1) 9

k kM t M t

k k

(37)

The average anisotropy parameter is

21

33 1

h

kA

k

(38)

Now using Eqs. (30) - (32) in Eq. (23) we obtain DE density as

Prespacetime Journal| December 2019 | Volume 10 | Issue 8 | pp. 1094-1103

Reddy, D. R. K., & Raju, K. D., Dark Energy Model in a Five-Dimensional Space-Time with a Massive Scalar Field

ISSN: 2153-8301 Prespacetime Journal

Published by QuantumDream, Inc.

www.prespacetime.com

1101

22 2

2 02 2 2

0 0 12

( 1)exp 2 2

(3 1) 2

M t Mk kM t t M t

k

(39)

Now using Eqs. (30) - (32) in Eq. (24) we get the EoS parameter of DE as

22 2

0 2

2

22 2

02 2

0 1

(3 2 4)1 2 1

(3 1) 3 1

exp 2 22

M t k k kM

k k

M t Mt M t

(40)

where is given by Eq. (39).

From Eq. (24), (25) and (30) - (32) we obtain the skewness parameter as

2

2 2

0

1 1

3 1

kM t M

k

(41)

where is given by Eq. (39).

The deceleration parameter is

2

22

0

41

Mq

M t

(42)

The jerk parameter is given by[42]

5

0

2

2

4

0

2

2

2

0

2

2

2 )(

16

)(

32

)(

12121)(

tM

M

tM

M

tM

M

H

qq

aH

atj (43)

The model given by Eq. (33) describes five dimensional DE model with massive scalar field with

the above cosmological parameters and the massive scalar field given by Eq. (30). It can be

observed that the spatial volume of the universe increases exponentially from a finite value so

that we have exponential expansion of the universe. The parameters 2

, ,H are finite at 0t

and tend to infinity as .t Also DE density is always positive and increases with time. It

can be seen that the EoS parameter is function of cosmic time 1 which implies that the

model lies in the quintessence region. The skewness parameter also increases with time. The

Prespacetime Journal| December 2019 | Volume 10 | Issue 8 | pp. 1094-1103

Reddy, D. R. K., & Raju, K. D., Dark Energy Model in a Five-Dimensional Space-Time with a Massive Scalar Field

ISSN: 2153-8301 Prespacetime Journal

Published by QuantumDream, Inc.

www.prespacetime.com

1102

average anisotropy parameter is constant and is anisotropic throughout the evolution except when

1.k It is observed that the deceleration parameter 1q at late times showing that our

universe accelerates. In cosmology jerk parameter plays an important role in describing the

models close to CDM and it is given by Eq.(43) It is believed that a transition of the universe

from deceleration to acceleration occurs for models with negative values of q and positive values

of j. At late times our model has q=-I and j=1. Thus our model represents accelerated expansion

of the universe which is in accordance with the modern cosmological observations.

5. Conclusions

Five-dimensional anisotropic DE models are significant in the discussion of early stages of

evolution of the universe. Scalar fields are useful in describing DE cosmological model. Hence,

in this article, we have discussed an anisotropic DE model in the frame work of five dimensional

spherically symmetric space-time. It is observed that the model non-singular and there is an

exponential expansion of volume from a finite value. The model represents a quintessence

universe. It is also observed that our model represents an accelerating expansion of the universe.

Thus all the results obtained are in good agreement with the recent exponential data.

Received November 3, 2019; Accepted December 10, 2019

References [1] A. G. Riess et al. Astron. J. 116, 1009 (1998). [2] S. Perlmutter et al.Astrophys.J. 517, 567(1999).

[3] C. L. Bennett et al. Astrophys. J. Suppl. Ser. 148, 1 (2003).

[4] P. J. Steinhardt et al. Phys. Rev. D. 59,123504 (1999).

[5] R. R. Caldwell Phys. Lett. B 545, 23 (2002). [6] S. Nojiri et al. Phys. Rev. D. 71,063004 (2005).

[7] T. Padmanabham Phys. Rev. D. 66, 021301 (2002). [8] M. R. Setare Phys. Lett. B, 654, 1 (2007).

[9] C. H. Brans and R. H. Dicke Phys. Rev. 124,925 (1961). [10] D. Saez and V. J. Ballester Phys. Lett. A 113, 467 (1986).

[11] G. A. Barber Gen. Relativ. Gravit. 14, 117 (1982).

[12] M. P. V. V. BhaskaraRao et al. Can. J. Phys. 94, 1314 (2016). [13] D. R. K. Reddy Can. J. Phys. 95, 145 (2017).

[14] D. R. K. Reddy et al. Astrophys. Space Sci. 361, 386 (2016).

[15] M. Kiran et al. Astrophys. Space Sci. 354, 577 (2014).

[16] M. Kiran et al. Astrophys. Space Sci. 356, 407 (2015). [17] M. P. V. V. BhaskaraRao et al. Prespace time Journal 7, 1749 (2016).

[18] Y. Aditya, D. R. K. Reddy Eur. Phys. J. C. 78, 619 (2018).

[19] G. F. R. Ellis General Relativity and Cosmology Academic Press, Academic Press, New York (1971).

[20] G. Mohanty, B. D. Pradhan Int. J. Theor. Phys. 31, 151 (1992).

Prespacetime Journal| December 2019 | Volume 10 | Issue 8 | pp. 1094-1103

Reddy, D. R. K., & Raju, K. D., Dark Energy Model in a Five-Dimensional Space-Time with a Massive Scalar Field

ISSN: 2153-8301 Prespacetime Journal

Published by QuantumDream, Inc.

www.prespacetime.com

1103

[21] J. K. Singh, S. Ram Astrophys. Space Sci.236 (1996).

[22] J. K Singh Nuovocimento B 120, 1259 (2005). [23] J. K. Singh, S. Rani Int. J. Theor. Phys. 54, 545 (2015).

[24] D. R. K. Reddy DJ J. Engg. App. Math. 4, 13 (2018).

[25] R. L. Naidu Can. J. Phys. 97, 330 (2018).

[26] Y. Aditya, D. R. K. Reddy Astrophys. Space Sci. 363, 207 (2018). [27] D. R. K. Reddy et al. J. Dyna. Syst. Geom. Theor. 17, 1 (2019).

[28] D. R. K. Reddy, G. Ramesh Int. J. Cosmology. Astron. Astrophys. 1, 67 (2019).

[29] D. R. K. Reddy, G. Ramesh Prespace time Journal 10, 301 (2019). [30] R. L. Naidu et al. J. Heliyondoi. org/ 10.1016/ j. heliyon2019. e01645.

[31] A. Chodos and S. Detweller Phys. Rev. D 21, 2167 (1980).

[32] T. Appelqunist, A. Chodos Phys. Rev. Lett. 50,141 (1983). [33] D. R. K. Reddy et al. Astrophys. Space Sci. 342, 245 (2012).

[34] D. R. K. Reddy, R. L. Naidu Astrophys. Space Sci. 307, 395 (2007).

[35] D. R. K. Reddy et al. Int. J. Theor. Phys. 47, 2966 (2008).

[36] G. C. Samantha, S. N. Dhal Int. J. Theor. Phys. 52, 1334 (2013). [37] T. Harko et al. Phys. Rev. D, 84, 024020 (2011).

[38] V. U. M. Rao, V. JayasudhaAstrophys. Space Sci. 358, 29 (2015).

[39] V. U. M. Rao, V. JayasudhaAstrophys. Space Sci. 358, 8 (2015). [40] P. Raju et al. Astrophys. Space Sci. 361, 77 (2016).

[41] P. S. Wesson Astron. Astrophys. 119, 1 (1983).

[42]T.Chiba T.Nakamura.Prog.Theor.Phys.100,1077(1998)

Vol.:(0123456789)1 3

Evolutionary Intelligence https://doi.org/10.1007/s12065-018-0170-4

SPECIAL ISSUE

Velocity adaptation based PSO for localization in wireless sensor networks

Vyshnavi Nagireddy1 · Pritee Parwekar1 · Tusar Kanti Mishra1

Received: 18 April 2018 / Revised: 31 July 2018 / Accepted: 7 September 2018 © Springer-Verlag GmbH Germany, part of Springer Nature 2018

AbstractWireless sensor networks are a network of sensors interconnected through a wireless medium. Wireless sensor networks are utilized for many array of applications where determining precise location of the sensors are treated to be the crucial task. The prime job of localization is to determine the exact location of sensors placed at particular area as it makes the reference of anchor nodes to determine the location of remaining nodes in the network. Position information of sensor node in an area is useful for routing techniques and some application specific tasks. The localization accuracy is affected due to the estimations in anchor node placements. Localization information is not always easy as it varies with respect to the environment in which the sensors are deployed. Ranging errors occur in hostile environments and accuracy effects as there are signal attenuations in sensors when deployed underwater, underground etc. Efficiency can be enhanced by reducing the error using localization algorithms. Particle swarm optimization is one approach to overcome the localization problem. Results are considered for localization algorithms like Particle swarm optimization, Social group optimization and Velocity adaptation based Particle swarm optimization. The goal of this work is to implement a velocity adaptation based particle swarm optimization for localization method to achieve minimum error. The results reveal that the proposed approach works better for obtaining improved location accuracy.

Keywords Wireless sensor network (WSN) · Localization · Global positioning system (GPS) · Anchor nodes · Sensors · Particle swarm optimization (PSO)

1 Introduction

A wireless sensor network as shown in Fig. 1 includes nodes which have the sensing capabilities to monitor diverse changes revised in environment like temperature, humid-ity, light. Data in the network is communicated among the sensors and distributed further using wireless links, data is transmitted from the nodes to the base station also called as gateway. These nodes can be either randomly distributed or specifically placed according to the application. They are often low cost, small in size and are battery powered

devices due to these features nodes are rapidly deployed and utilized for numerous applications. Real time sensor deploy-ment applications include tracing target, defense and mili-tary, monitoring environmental aspects, resources inspection and other surveillance and monitoring [1]. Wireless sensor network (WSN) has some constraints which affects the per-formance of the network. They are hardware limitations, deployed atmosphere, topology management, transmission channel effects, compatibility issues, and localization [2].

The crucial part of WSNs is that when there is any anom-alous situation, position information is needed to locate the issue and communicate it to the base station [3]. Localiza-tion is about determining the location of the sensors. When many nodes are present, uncertainties arise regarding the position. GPS can be used to avoid ranging errors. GPS can be affixed to nodes to get the exact location information of whereabouts of the sensor node which improves the locali-zation accuracy. Providing GPS facility to nodes is not cost effective and not always feasible. For some applications, they require sensor node size to be small fixing GPS to sensor

* Vyshnavi Nagireddy nvyshnavi6@gmail.com

Pritee Parwekar pritee.cse@anits.edu.in

Tusar Kanti Mishra tusar.cse@anits.edu.in

1 Anil Neerukonda Institute of Technology and Sciences, Vishakhapatnam, India

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nodes increases the size and GPS consumes more energy which results in energy depletion and this effect the network lifetime. Deploying GPS on a massive scale undergoes some performance issues like weak signals due to dense environ-ments and changes in climate like snowfall, moisture, rain and fog. Environmental constraints like underground and underwater sensors cannot use GPS [4].

Localization is classified into range based versus range-free, centralized versus distributed, anchor free versus anchor based [5]. Range based is a distance based strategy and Range free is a connectivity based strategy. In distance based localization various algorithms are prevalent like the Time of arrival (TOA), the Angle of arrival (AOA), the Received signal strength indication (RSSI) [6]. Range free techniques include hop counting techniques and it doesn’t include estimations based on distances or the signal strength that is received or the angle measurement [7].

In general optimum means either to maximize or to mini-mize. Optimization is used to improve the performance of the network [8]. In networks optimization is applied to mini-mize the energy consumption and maximize the network lifetime. The optimization problems include fault tolerance and connectivity, node deployment and coverage, mobility and scheduling [9, 10]. Coverage and connectivity are radi-cal problem in wireless sensor network as they are directly associated with resources in sensing optimizing environ-ment. Hence the right choice of the optimization algorithm is a crucial element as that yields better solutions. The opti-mization algorithm used in this paper is Particle Swarm opti-mization (PSO) which comes under bio-mimetic optimiza-tion algorithms. Bio-mimic algorithms are nature inspired algorithms which are suited for global optimization. Diverse meta-heuristic algorithms are genetic algorithm (GA), par-ticle swarm optimization (PSO), cuckoo search algorithm

(CS), Ant colony optimization algorithm (ACO) [11].Opti-mization is emphasized to obtain location accuracy.

This paper proposes an objective function for localization with a modified Particle swarm optimization with respect to velocity.PSO is applied to mitigate the localization prob-lem. The paper is organized as follows: Sect. 2 examines the related work; Sect. 3 includes proposed approach; Sect. 4 describes simulation and results; Conclusion is reviewed in Sect. 5.

2 Related work

In WSNs, localization is a vital aspect that developed a con-siderable research interest in academia and industry through-out the world. Technological advances in WSN introduced micro electro mechanical systems (MEMS). MEMS are beneficial with respect to cost and size as they are micro-machined and the circuit integration in it combines multiple components in one chip [12]. These smaller units with high functionality and greater connectivity lead to build effi-cient network which are cost efficient. When the sensors are deployed randomly on a large scale, the major challenge is to identify the location of the sensors. Localization locates the autonomous sensors in a network which has a great role in development of WSNs. It is essential to trace the sensors in most of the geographical tracking, patient monitoring, envi-ronmental monitoring, Health care and home intelligence applications. Significant application of PSO is observed in overcoming optimization problem in WSN. In medical appli-cations it is important to have accuracy in classifiers as it leads to further diagnosis [13]. Major applications include cancer classification, identifying life span of the patient by making use of various classifiers, pattern recognition and in bio metrics etc. Clustering is a technique used in WSN to minimize the energy consumption and thereby leads to minimal battery consumption and this helps in prolonging lifetime of the network [14]. PSO in clustering has its ben-efits in pattern recognition, image segmentation and to sim-plify complex data. Hence major improvements are done in PSO-Clustering as it is aggregating many disciplines within it and improves the performance in respective applications with utmost accuracy. Apart from these it has its extended applications in various Robotic and Electric set-ups. The benefit of PSO compared to other algorithms is that it does not involve complex operators. Hence it is subjected to vari-ous modifications to improve the convergence. VAPSO is one of such improvement to existing PSO which results in improving the convergence and minimizing localization error which helps to attain location accuracy. VAPSO is considered to solve the localization problem and improve the network lifetime. Therefore VAPSO can be implemented

Fig. 1 Wireless sensor network communication architecture

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to all the applications areas mentioned in above to improve the performance accordingly.

2.1 Particle swarm optimization

Particle swarm optimization (PSO) algorithm is a nature driven population based metaheuristic algorithm proposed by James Kennedy, Russell C. Eberhart in 1995. Swarm optimization techniques are inspired by observing behaviors of animals like bird flocking, fish schooling. The concept behind this algorithm is generating swarm of particles move around a space searching for their food or a place that suits their needs. Position of every particle changes according to the current position and current velocity, distance between current position and personal best, distance between their current position and global best. It thrives to optimize the fitness. This search mechanism combines self-experiences with social experiences and the search area includes possible solutions. Each particle holds its best, also called as per-sonal best then according to the position and velocity, itera-tive update in the objective function is found in the particle neighbourhood. Best among all the particles considered that is global best. The optimum solution can be achieved in PSO algorithm by the collaboration of each individual. PSO is effective and accessible with limited parameters and calcu-lations when compared to other optimization approaches. It achieves high-quality results within briefer calculation time and constant convergence tendency than other methods [15].

In [15], position based PSO approach is utilized to reduce the channel interface errors and RSSI based ranging errors thereby to improve the location accuracy. Localization prob-lem includes range based distance calculations by using anchor nodes and unknown nodes. A localization algorithm with PSO and Quasi-Newton approach is introduced to solve the non-linear optimization problems. Quasi-Newton method is a Newton based method which resolves the drawbacks in the Newton system like faster computation and cost effec-tive [16]. Newton system has precise convergence where Quasi-Newton method has a less precise convergence when compared to Newton system [16]. Consequently, a hybrid PSO-QN based algorithm is proposed where selection of primary location is done using PSO and then results are obtained through iterative approach in QN method so does the location accuracy improves through the desired results [15]. In [17], Object recovery problem is used which makes use of PSO algorithm. Object tracking is a method which tracks the sensor nodes and updates the modified position of the node in particular area with respect to time. When nodes in the network are not sensed for particular time, they are considered to be missing. In such case the PSO is utilized to estimate the position of nodes. Localization error and aver-age localization error are considered to calculate the location of the object which is moving. Three methods are explored

in this paper like centroid method, weighted centroid method and PSO method for object recovery. Results state that level of accuracy and minimal localization error is encountered by PSO compared to other object recovery methods.

In [18], a target tracking algorithm which integrates learning regression tree approach is presented with filtering methods which makes use of a RSSI metric. Indoor environ-ment based target tracking application of WSN is applied. For position estimations based on RSSI, Regression tree algorithm is explored.

Various approaches are proposed based on coordinate estimation of the sensor nodes in the network. Estimating coordinates of the nodes is a predominant task in localiza-tion as majority of applications are organized based on it.

In [19], a self positioning algorithm called Matrix trans-form based self positioning algorithm (MSPA) is presented. This algorithm doesn’t rely on a GPS for position estimation. This GPS-free MSPA approach is divided into two phases constructing local coordinate system and converging local coordinate to global coordinate system. The MSPA algo-rithm proposed includes a standardized clustering based scheme to construct local coordinate system and a transfor-mation matrix based scheme to combine them into a global coordinate system. Efficiency of the proposed algorithm has been analyzed based on the energy consumption and parameter setting guide lines reached with reference to exist-ing MSPA. Apart from this work, a set of parameter-setting strategies for the presented algorithm based on a probability model is developed and energy requirements are explored. In [20], improved distance vector hop (IDVHop) algorithm is presented. In this work, a traditional DV-Hop algorithm and Teaching learning based optimization (TLBO) are reviewed. Then IDV-Hop algorithm using TLBO is proposed which concentrates on four stages like hop size of anchor nodes are customized by adding up the correction factor, localization errors induced by the anchor nodes are reduced using the concept of degree of collinearity, Upgrading target nodes to assistant anchor nodes and application of TLBO algorithm on IDV-Hop. Results show that the IDV-Hop using TLBO locates the target nodes precisely and achieves higher con-vergence rate. In [21], a distributed localization algorithm is proposed which focuses on position uncertainties. Modi-fied Spring Mass Method (SMM) is used in estimating node position which is fault- tolerant. Algorithm called Localiza-tion considering position uncertainty (LCPU) is presented which uses SMM as it estimates target node position and thereby corrects the reference nodes’ position.

Majority of the optimization algorithms’ motive is to improve the lifetime of the sensor network. Clustering and routing are such mechanisms that are to be highlighted when we come across optimization. In [22], protocols related to energy issues in WSN are discussed. Low energy adaptive clustering hierarchy (LEACH) is ad-hoc low energy routing

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protocol in WSN. It is a cluster based protocol where it mini-mizes the energy consumption by forming clusters. Various improvements are made to this existing LEACH due to its quality and energy savage. Other protocol is the power-effi-cient gathering in sensor information systems (PEGASIS) which is determined as the advancement of existing LEACH protocol. Improved performance is observed in PEGASIS when compared to LEACH [22]. Apart from these, there are many other protocols like TEEN (threshold sensitive energy efficient sensor network protocol) [22], APTEEN (Adaptive Threshold sensitive Energy Efficient sensor Network) [22] to improve the network life time for respective network topolo-gies. Table 1 explores various optimization algorithms that are prevailing, their advantages and limitations.

Considering this summary, PSO is considered for improvement as the implementation of it is facile compared to other approaches. The presented algorithm is a distance based method which achieves accuracy with minimum num-ber of anchor nodes.

3 Proposed system

The paper considers a localization problem which uses PSO. The concept of velocity is targeted in PSO as particle moves with a displacement and direction for a given amount of time. The aim is to reduce the localization error. This can be achieved by proposing the modified velocity parameters in the PSO algorithm. In the existing system location estima-tion is made such that the unknown locations of sensors are computed. For the experimental setup there are two types of nodes the sensors and the anchors. The sensors are randomly deployed and the position of the anchor nodes is fixed so the coordinates of the anchor nodes are known.

3.1 Problem statement

The aim of the proposed method is to reduce the localization error in locating a particular sensor which will be helpful for the identification of the randomly deployed sensors. For this a velocity adaptive concept is proposed in particle swarm optimization which improves the location accuracy.

The existing work done in [28] which talks about a PSO based localization has a goal for localization in a network of n nodes out of total m sensor nodes based on the a-pri-ori information about locations of m − n anchor nodes. Accordingly for a 2D localization problem, a total of 2n unknown coordinates, l= [lx, ly]; where lx = x1, x2,…,xn, ly=y1,y2,…,yn, are to be predicted using the coordinates of anchor node xn+1,…,xn andyn+1,...,ym. Assuming coordinates of ath anchor node as (xa, ya) and the coordinates of the unknown nodes are (xm, ym). So the distance estimate da is Euclidian distance calculated by considering the refer-ence node ‘a’ locating in (xa, ya) is:

From Eq. (1), where a = 1, 2, 3…n. Here, the difference considered is computed between actual and measured dis-tances. The objective function proposed to resolve the locali-zation problem is formulated as:

From Eq. (2), M is the number of anchor nodes, da is men-tioned above Eq. (1), da denotes noisy measurements calcu-lated from da. This objective function is to be minimized as it represents the error from inaccurate measurements. To

(1)da =

√

(xa − xm)2 + (ya − ym)

2.

(2)f(x, y) =1

M

(

M∑

=1

da − da

)

.

Table 1 Summary on various optimization techniques in WSN

Algorithm/technique PROs CONs Category

QUASI-NEWTON [23] Improves the result with good performance in optimization as it is simple to apply. Converges fast

Increases computational complexity, requires derivatives

Variable-metric method

LEACH [24] Clusters and cluster heads are formed such that energy utilized is minimum

As it is single hopped, it cannot be sug-gested for networks at a large scale

Hierarchical protocol

TLBO [25] It does not require specific parameters to be considered unlike other population based algorithms so it can be easily computed

Combining TLBO with other optimization techniques increases the complexity in computing it

Population based algorithm

PSO [26] Easy to implement with very few param-eters, optimizes iteratively by utilizing minimum time and an efficient global search algorithm

Convergence is week Nature inspired algorithm

PEGASIS [27] Improves lifetime of the network due to minimum transmission distance and uniform energy distribution

When the size of network is large, there is a lengthy delay in transmission

Hierarchical protocol

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reduce the inaccuracy, the existing equation is modified to Eq. (3).

From Eq. (3), β is a constant value between 0 and 1, a minimal value is considered such that makes the conver-gence faster. M is the number of anchor nodes, da is the distance estimate, da denotes noisy measurements calculated from da. The further development made to reduce the loca-tion error by using a velocity adaptation algorithm. Veloc-ity and position update equation are considered by assum-ing an n dimension search space with p particles. Position and velocity of ith particle be xid and vid respectively where 1 ≤ i ≤ p, 1 ≤ d ≤ n. pbestid is the personal best, gbestd is the global best. At s iteration, particles evolve according to the Eqs. (4), (5) for velocity and position:

where W is the inertia weight, c1 and c2 are cognitive and social constants; rand1 and rand2 are random numbers distributed uniformly in range [0, 1]. These mathematical expressions suggest the action of each particle toward its best position and the swarm’s best position with velocity and position updates. To improve the convergence, the Eq. (3) can be modified using velocity improvement.

3.2 Velocity adaptation algorithm (VAA)

Velocity is a vector quantity and can be defined as rate of change of the position. Motion of a body includes speed and direction.

Hence the proposed velocity update is presented in Eq. (6) as,

where =vid(s)∑M

j=1vij

+ vid(s).

Vid is the velocity of ith particle, vij is the average velocity of all particles, partial derivative of Personal best (pid) and Global best (gid) are considered with respect to time (t) where it can be done using command called ‘diff’ in MATLAB i.e, abs(diff(personal best-global best)/ time interval).The notation is difference of pid and gid from previous co-ordinate numeri-cally. Introducing α to this term is to affirm the value quite closer to vid value. When there is a rapid growth in velocity,

(3)f(x, y) = 𝛽 ∗1

M

(

M∑

i=1

da − da

)

.

(4)

Vid(s + 1) = wi × vid(s) + c1 × rand1 ×(

pbestid − xid)

+ c2 × rand2 ×(

gbestd − xid)

(5)Xid(s + 1) = Xid(s) + vid(s + 1),

Velocity = displacement∕time.

(6)Vid(s + l) = +

t(pid − gd),

it considerably computes fast and there by reduces the error in localization. Algorithm 1, explains the VA based PSO approach.

Algorithm 1. -VAA (Swarm_of_particles)

1: Begin2: initialize swarm_of_particles3: repeat4: for each particle,5: Update the pbest6: Iff(x)<f(pbesti) then7: Pbesti=x8: End if9: Iff(pbesti)< f(gbest) then10: Gbest=pbesti

11.End if12.end for13. For each particle do14. Vid= + ( − )

15. Xid(s+1) =Xid(s) +vid(s+1)16. End for17. End for18. it=it+119. Untilit>Max_iterations

4 Simulation and results

This work, particle swarm optimization (PSO) and social group optimization (SGO) are implemented along with the proposed Velocity Adaptation Algorithm (VAA) for localization problem in WSN. Comparisons of PSO, SGO and VAA are done such that the error obtained is visual-ized. The implementations of these works are processed using MATLAB. In a sensor area of size 100*100, results are observed for a fixed count of anchor nodes and remain-ing nodes for maximum iterations. It is observed that the localization error is minimized for the proposed Velocity adaptation based PSO when compared to existing PSO and SGO. Maximum convergence is observed in the proposed work as shown in Fig. 4 for VAA rather Figs. 2 and 3 (PSO and SGO). Normalized plot is represented for both proposed and existing works for 50,000 iterations.

Comparative analysis of all the three algorithms are presented in Table 2, for a fixed grid dimension, using minimum number of anchor nodes it is shown that veloc-ity adaptation algorithm converges faster when compared to PSO and SGO. It is observed that Velocity adaptation

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algorithm outperforms PSO and SGO with respect to time, and min–max errors.

From Figs. 5 and 6, it is observed that there is a consistent decrease in the error for values obtained for VAA when com-pared with SGO and PSO. From the above plots and tables, it is clear that the proposed velocity adaptation based PSO is performing better compared to the existing algorithms.

4.1 Case study

Quasi-Newton method is adopted when optimization of the result is required. Because of its performance, conveni-ence and efficiency it is considered for solving optimization problems. To maximize the lifetime of the network, it is significant to consider optimization as the vital objective. Algorithms discussed in WSN must focus on performance optimization. In this work, performance is optimized by

minimizing the localization error in network.PSO is used to locate the initial position and then final outcome is driven with the Quasi-Newton method. This kind of aggregation of these two methods is put forward to improve the location accuracy in the network. Figures 7 and 8 display the mean square error using minimum number of anchor nodes and the regular nodes. The difference between estimated location and true location are shown in the plots Figs. 7, 8.

From Table 3, it is clear that in PSO approach, error is getting minimized upon the increase in the percentage of anchor nodes; apart from this location accuracy is improved using this approach. On associating this case study of the traditional PSO with the proposed approach (VAPSO), the results reveal that anchor node based location estimations which has a location based error is getting minimized in a network.

Fig. 2 Plot for PSO

Fig. 3 Plot for SGO

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5 Conclusion

This paper focuses on a swarm intelligence based Particle swarm optimization algorithm using Velocity adaptation to minimize the localization error in wireless sensor networks.

Considering the nature inspired algorithm perspective veloc-ity and position factors are improved for a better conver-gence of the results. The performance of three algorithms are analyzed and compared for a finite number of iterations with unknown nodes and minimum number of anchor nodes. Comparative analysis states that Velocity adaptive PSO algorithm is efficient in improving the location accuracy due to the optimal results rather SGO and PSO. From the

Fig. 4 Plot for VA based PSO

Table 2 Comparative analysis of PSO, SGO and VA based PSO

Algorithm name Number of sensors

Percentage of anchors

Grid dimension Iterations Elapsed time (s) Absolute error

Min Max

PSO 500 2 100*100 2500 9.49675 0.0327 122.88PSO 500 10 100*100 5000 27.4676 0.0108 119.0939PSO 1000 10 100*100 10,000 98.85207 0.0064 104.7993 SGO 500 2 100*100 2500 9.30566 0.0312 120.4235 SGO 500 10 100*100 5000 27.420 0.0160 119.961 SGO 1000 10 100*100 10,000 97.9633 0.0049 111.174

VAA 500 2 100*100 2500 9.29074 0.0156 119.2140VAA 500 10 100*100 5000 27.3478 0.0078 106.3856VAA 1000 10 100*100 10,000 97.9186 0.0045 94.4935

0

0.01

0.02

0.03

0.04

2500 5000 10000

Erro

r

Iterations

Minimum Error

PSO

SGO

VAPSO

Fig. 5 Minimum localization error representation for PSO, SGO and VA based PSO

0

50

100

150

2500 5000 10000

Erro

r

Iterations

Maximim error

PSO

SGO

VAPSO

Fig. 6 Maximum localization error representation for PSO, SGO and VA based PSO

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obtained results, it is concluded that the Velocity Adapta-tion based PSO performance is considerably efficient due to the observed minimum values of error upon the increase of anchor node percentage. Future work can be extended to

overcome the slow convergence in PSO computation and optimize the complexity of computations. Proposed scheme can be further improved and can be applied in environments like Mobile sensor networks.

References

1. Ding C, Yang L, Meng Wu (2017) Localization-free detection of replica node attacks in wireless sensor networks using similarity estimation with group deployment knowledge. Sensors 17(1):160

2. Alrajeh N, Ali M, Bashir, Shams B (2013) Localization techniques in wireless sensor networks. Int J Distrib Sens Netw 9(6):304628

3. X Han et al (2017) An RSSI based DV-hop algorithm for wireless sensor networks. In: Communications, computers and signal pro-cessing (PACRIM), 2017 IEEE Pacific Rim conference on IEEE

4. Karim L et al (2017) Localization in terrestrial and underwater sensor-based m2m communication networks: architecture, clas-sification and challenges. Int J Commun Syst 30:4

5. Kumar M (2014) Localization by decreasing the impact of obsta-cles in wireless sensor networks (Doctoral dissertation)

6. Wu W et al (2018) Efficient range-free localization using elliptical distance correction in heterogeneous wireless sensor networks. Int J Distrib Sens Netw 14(1):1550147718756274

7. Han G et al (2013) Localization algorithms of wireless sensor networks: a survey. Telecommun Syst 52(4):2419–2436

8. Hemalatha P, Gnanambigai Dr J (2015) A survey on optimization techniques in wireless sensor networks. Int J Adv Res Comput Eng Technol (IJARCET) 4(12)

9. Gogu A et al (2012) Review of optimization problems in wireless sensor networks. Telecommunications networks-current status and future trends. InTech

10. Gogu A et al (2011) Optimization problems in wireless sensor networks. Complex, intelligent and software intensive systems (CISIS), 2011 International Conference on IEEE

11. Adnan M et al (2013) Bio-mimic optimization strategies in wire-less sensor networks: a survey. Sensors 14.1:299–345

12. Tuna G, Gungor VC, Dursun B (2017) Wireless MEMS for smart grids. Wirel MEMS Netw Appl. https ://doi.org/10.1016/B978-0-08-10044 9-4.00011 -7

13. Inbarani H, Hannah AT, Azar Jothi G (2014) Supervised hybrid feature selection based on PSO and rough sets for medical diag-nosis. Comput Methods Prog Biomed 113.1: 175–185

14. Parwekar P, Rodda S, Kalla N (2018) A study of the optimization techniques for wireless sensor networks (WSNs).” Information systems design and intelligent applications. Springer, Singapore, pp 909–915

15. Cao J (2015) A localization algorithm based on particle swarm optimization and quasi-newton algorithm for wireless sensor net-works. J Commun Comput 12:85–90

16. Tatsumi K, Tetsuzo T (2017) A perturbation based chaotic system exploiting the quasi-newton method for global optimization. Int J Bifur Chaos 27(04):1750047

17. Pavalarajan S, Krishna Moorthy R (2014) Swarm intelligence based location estimation for wireless sensor network. Appl Mech Mater, vol 573. Trans Tech Publications

18. Ahmadi H, Viani F, Ridha B (2018) An accurate prediction method for moving target localization and tracking in wireless sensor networks. Ad Hoc Netw 70:14–22

19. Wang L, Xu Q (2010) GPS-free localization algorithm for wireless sensor networks. Sensors 10(6):5899–5926

Fig. 7 Mean Estimation error for 100*100

Fig. 8 Mean Estimation Error 200*200

Table 3 Mean square error using Quasi Newton approach

Anchor nodes Mobile nodes Network size Mean square error

4 200 100 24.9644 200 200 52.6595 200 100 23.7945 200 200 49.67910 200 100 16.12310 200 200 32.92715 200 100 11.70015 200 200 24.873

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20. Sharma G, Kumar A (2018) Improved DV-hop localization algo-rithm using teaching learning based optimization for wireless sen-sor networks. Telecommun Syst 67(2):163–178

21. Kim Y et al (2018) Localization technique considering position uncertainty of reference nodes in wireless sensor networks. IEEE Sens J 18(3):1324–1332

22. Pantazis NA, Stefanos A, Nikolidakis Vergados DD (2013) Energy-efficient routing protocols in wireless sensor networks: a survey. IEEE Commun Surv Tutor 15(2):551–591

23. Xu Y, Zhuang Y, Jing-jing Gu (2015) An improved 3D localiza-tion algorithm for the wireless sensor network. Int J Distrib Sens Netw 11(6):315714

24. Al-Baz A, El-Sayed A (2018) A new algorithm for cluster head selection in LEACH protocol for wireless sensor networks. Int J Commun Syst 31(1):e3407

25. Sharma G,, Kumar A (2018) Modified energy-efficient range-free localization using teaching–learning-based optimization for wire-less sensor networks. IETE J Res 64(1):124–138

26. Lee KY, Jong-Bae P (2006) Application of particle swarm opti-mization to economic dispatch problem: advantages and disad-vantages. In: Power systems conference and exposition, (2006) PSCE’06. 2006 IEEE PES. IEEE

27. Sujata B (2017) Energy efficient PEGASIS routing protocol in wireless sensor network. Int Res J Eng Tech 4(7)

28. Lavanya D, Udgata SK (2011) Swarm intelligence based localiza-tion in wireless sensor networks. In: International workshop on multi-disciplinary trends in artificial intelligence. Springer, Berlin, Heidelberg, pp 317–328

Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Received: 5 March 2018 Revised: 11 May 2018 Accepted: 6 June 2018

RE S EARCH ART I C L E

DOI: 10.1002/dac.3763

Provable secure lightweight hyper elliptic curve‐basedcommunication system for wireless sensor networks

Vankamamidi S. Naresh1 | Reddi Sivaranjani2 | Nistala V.E.S. Murthy3

1Department of Computer Science andEngineering, Sri Vasavi EngineeringCollege, Tadepalligudeam, AndhraPradesh 534101, India2Department of Computer Science andEngineering, Anil Neerukonda Institute ofTechnology and Sciences, Visakhapatnam,Andhra Pradesh 530003, India3Department of Mathematics, AndhraUniversity, Visakhapatnam, AndhraPradesh 530003, India

CorrespondenceVankamamidi S. Naresh, Department ofComputer Science and Engineering, SriVasavi Engineering College,Tadepalligudeam, Andhra Pradesh534101, India.Email: vsnaresh111@gmail.com

Int J Commun Syst. 2018;e3763.https://doi.org/10.1002/dac.3763

Summary

It is widely believed that hyper elliptic curve cryptosystems (HECCs) are not

attractive for wireless sensor network because of their complexity compared

with systems based on lower genera, especially elliptic curves. Our contribu-

tion shows that for low cost security applications HECs cryptosystems can out-

perform elliptic curve cryptosystems. The aim of this paper is to propose a

discrete logarithm problem‐based lightweight secure communication system

using HEC. We propose this for different genus curves over varied prime fields

performing a full scale study of their adaptability to various types of

constrained networks. Also, we propose to evaluate the performance of the pro-

tocol for computational times with respect to different genus for main opera-

tions like Jacobian, Divisor identifications, key generation, signature

generation/verification, message encryption, and decryption by changing the

size of the field. A formal security model was established based on the hardness

of HEC‐Decision Diffie‐Hellman (HEC‐DDH). Finally, a comparative analysis

with ECC‐based cryptosystems was made, and satisfactory results were

obtained.

KEYWORDS

Diffie‐Hellman, elliptic curve, genus, hyper elliptic curve, Jacobian, wireless sensor networks

1 | INTRODUCTION

In modern world, most of the wireless systems require resource constrained devices such as RFID tags, sensors, smartcards, small processors, PDA's, and smart phones. These devices play a major role in providing security for satellite com-munication, internet security, e‐banking, e‐commerce, Internet Of Things (IOT) applications, and embedded systems.Implementing security for wireless communication system using these devices is the most challenging problem. Manycryptographic algorithms were developed to accomplish their requirements for secure data communication in wirelesssystems. These algorithms have many limitations, which include increased power consumption, communication, andcomputational complexity with increased processing time. Thus, an efficient cryptographic algorithm that overcomesthese limitations is the need of the hour.

Public key cryptography (PKC)1 offers a solution to the above limitations by using 2 different keys known as thepublic and private keys. The secret (private) key is chosen by the user and is well known only to him. The publickey is computed from the private key by using a reversible mathematical process and is made open to all. Both the keysare interoperable on each other and are used for the decryption and encryption processes. As the private key is neverrevealed, PKC is highly secured unlike symmetric key cryptography. Based on the arithmetic operations, PKC is broadly

© 2018 John Wiley & Sons, Ltd.wileyonlinelibrary.com/journal/dac 1 of 16

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classified as Rivet‐Shamir‐Adeleman (RSA) and elliptic curve cryptosystem (ECC). RSA scheme is built on large primenumber factorization problem. Security threats on RSA can be reduced by improving key size, which made the key gen-eration, encryption, and decryption scheme more difficult with higher storage requirements as well as processing times.The weaknesses of RSA are overcome by ECC,2 which is a famous PKC‐based system for wireless systems. ECC offersbetter security with reduced key sizes and require less processing time and involves less computational complexity.

In 1988, Koeblitz Neal proposed a novel curves with greater genus for cryptographic implementations acknowledgedas hyper elliptic curve cryptosystem (HECC),3 which emerged as an alternative for ECC in the era of PKC. Hyper ellipticcurve is a generalization of EC of genus “g” greater than or equal to 2. Elliptic curves are HECs having genus equal to 1.Hyper elliptic curve cryptosystem has more advantages than ECC like smaller key size, low computational load, highspeed, bandwidth savings, and low power consumption. Hyper elliptic curve cryptosystem can be used efficiently tosecure engineering applications like cryptocurrency, web servers, chip cards, electronic commerce, and cellular tele-phones. These advantages make HECC to implement both in software and hardware simply and make HEC a goodchoice for lightweight cryptosystems.

1.1 | Related work

Modern cryptography replaces the multiplicative groups in Diffie‐Hellman (DH) with elliptic curve groups, as proposedby Miller4 and independently by Koblitz.3 This loses an important constant factor in the number of field operationsrequired for a group operation, but it gains much more from avoiding index‐calculus attacks. Particularly, to achievea security level around 2128, EC‐groups use base fields of size around 2256, while multiplicative groups need base fieldsof size around 23000 (see, for example, Giry5).

The latest paper6 by Bos et al shows that for high‐security DH one obtains even better performance from a differentoption: Jacobian groups of hyperelliptic curves of genus 2. The main advantage of genus 2 over genus 1 is that a muchsmaller base field, specifically a field of size around 2128, produces a group of size around 2256 and a security levelaround 2128. Reducing the number of bits in the field by a factor of 2 typically produces a speed‐up factor around 3,depending on various details of field arithmetic. The disadvantage of genus 2 is that each group operation requires manymore field operations; but for Gaudry7 and Gaudry et al8 Kummer‐surface formulas, this loss factor is only slightlyabove 2. Even better, 24% of Gaudry's field multiplications are multiplications by curve parameters that can be chosento be small; a secure small‐parameter genus‐2 curve was announced by Gaudry et al8 after a massive point‐countingcomputation. A further advantage of genus 2, exploited in a very recent paper9 by Bernstein, Chuengsatiansup, Lange,and Schwabe, is a synergy between the structure of Gaudry's formulas and the availability of vector operations in mod-ern CPUs.

Hyper elliptic curve cryptosystem uses higher genus curves in which variable and divisor coefficients are all boundedto members of a finite field. Hyper elliptic curve cryptosystem of varying genus is broadly classified as HECC over primefield and HECC over binary field. To begin with, the performance analysis of HECC over prime field of different genuscurves has been performed.

The security of HECC relies on the hardness of solving HEC‐discrete logarithm problem (DLP). In 2000, Gaudry,Hess, and Smart8 showed how the ECDLP for an elliptic curve over a characteristic 2 finite field F2

n can be reducedto the DLP in the Jacobian of a hyper elliptic curve C defined over a subfield of F2

n. For some elliptic curves, the genusof C is small, and the Gaudry, Hess, and Smart reduction yields an algorithm for solving the ECDLP instance that isfaster than Pollard's rho algorithm. Therefore, efficient algorithms for performing the group law in the Jacobian of hyperelliptic curves and for solving the HEC‐DLP are also of interest because they can be used to attack elliptic curvecryptosystems.

In HECC systems, we form a group with respect to addition of divisors of Jacobian of the HEC.10 The procedure byCantor11 was the best for adding divisors of Jacobian up to 2000. Afterwards, there was a lot of progress and numerousinvestigations took place in the area of HECC. A comparison with reference to the key size of RSA/DH, ECC, and anHEC‐based is shown in the Table 1. Regrettably, in HECC, the approaches for group operation are not as speed as infinite fields (other studies9,12-14) or for elliptic curves. For genus g ≥ 2 curves,15 the approaches for group operation2

can be optimized to a superior level (other studies7,16-18) leading to build much faster cryptosystems making the studyof HECC and the addition operation therein important. So in this work, we propose a secure lightweight HEC‐DH withdigital signature, and the analysis of HECC for curves of different genus over varying prime fields is performed. Further,from the performance metrics of processing time and key size, we observe that different types of networks are actuallyin need of the same protocol but with different combinations of curves of different genus over different prime fields.

TABLE 1 Key size comparison in bits needed to attain equivalent level of security

Field RSA and DH EC‐based HEC‐based

F (280) 1024 160 50‐80

F (2112) 2048 224 112

F (2128) 3072 256 128

F (2192) 7680 384 192

F (2256) 15360 512 256

Abbreviations: DH, Diffie‐Hellman; EC, elliptic curve; HEC, hyper elliptic curve.

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From the performance metrics of processing time and key size, it is inferred that genera19 2 and 3 are used for resourceconstrained devices while genera 4 and 6 can be used for large scale networks. We studied various HECC‐based existingprotocols (other studies20-24) and compared with the proposed to obtain satisfactory results.

1.2 | Our contribution

The foremost goal of this work is to build “Lightweight Secure Communication System based on HEC‐DDH assumptionfor Wireless Sensor Networks.” The basic idea behind the proposed construction is “HEC Cryptosystem with an operandsize, only a fractional amount of that of the operand size of EC” and based on the hardness of HEC‐DLP.

We made a concrete and comprehensive attempt to implement the proposed protocol for curves of different genusover different prime fields to infer their adaptability to various types of constraint networks.

As part of security analysis, we build a formal security model along with formal proofs for the security of proposedprotocols.

Our comparative assesses and measures the efficiency of the proposed protocol and compares it with some otherlightweight schemes with respect to energy cost for communication and computation and show that the proposed pro-tocol is optimal.

Outline of rest of the paper: Section 2 describes the prerequisite for the proposed protocols. The proposed schemesare described in Section 3. Section 4 talks about the security analysis. Section 5 provides experimental results and acomparative analysis with the current prevalent lightweight schemes. Lastly, Section 6 concludes with certainobservations.

2 | BACK GROUND OF HEC

We can see HECs3 as generalizations of ECs.2

Definition (HEC). Let the algebraic closure of a given field K be K . A HEC “C” of genus g ≥ 1 over K is defined by,

“C:y2 þ h xð Þy ¼ f xð Þ in K x; y½ ; ” (1)

where h (x) ∈ K [x] ∋deg (h(x)) ≤ g and f (x), a monic polynomial with deg ( f ) = 2 g + 1 and there are no singular points

on “C,” ie, there exists no point (x, y) ∈ K2, which concurrently satisfy Equation (1) and the partial derivatives “2y + h(x) = 0 and h1(x) y − f 1(x) = 0.”

Definition (Rational points, points at infinity, and finite points). Let an extension field of K be Lsuch that K⊆L⊆K: The set of L − rational points on C are indicated as C (L) = (x, y) ∈ L × L/Equation (1)is to be satisfied∪∞ where “∞” is a special point at infinite. The C(KÞ is merely indicated by C.

Definition (Opposite, special, and ordinary points). The opposite point of a finite point P = (x, y) on C

is the point eP = (x, −y − h (x)).We define e∞ = ∞. A point Q≠ ∞ is said to be a special point.If eQ = Q, elseQ is an ordinary point.

Definition (Coordinate ring and polynomial functions). The coordinate ring of C over K is given by “

K [C] = K [x, y]/(y2 + h (x) y − f (x)).” The members of K[C] are termed as polynomial functions.

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Definition (Function field and rational functions). The function field of C over K is the field of

fractions K(C) =GH

/G, H ∈ K [C], deg (G) = deg (H). A member of K [C] is termed as a rational

function.

Definition (Divisor). “A divisor D is a formal sum of points P ∈ CD ¼ ∑P∈CmPP where mP∈ Z andalmost all mP are zero; the degree of D is given by deg Dð Þ ¼ ∑

P∈CmP ∈Z. The order of D at P is ordP

(D) = mP ∈ Z.”

Definition (support of a divisor). “The support (supp) of D ¼ ∑P∈CmPP is supp(D) = P∈C: mP≠0”

Definition (Divisors of degree 0). Is the subgroup Div0 (C) of Div (C) with divisors of degree 0.

Definition (Addition of divisors and GCD of divisors). Let D1 = ∑P∈CmPP and D2 = ∑P∈CnPP , thenthe addition25 and GCD of divisors are respectively given byD1 + D2 = ∑P∈CmPP+ ∑P∈CnPP =∑P∈C mP þ nPð ÞP and GCD (D1, D2) = ∑P∈C min mP; nPð ÞP:Definition (Semireduced divisors). A divisor is said to be semireduced divisor if it is of the form:D = ∑i mi Pi − (∑i mi) ∞, ∀mi ≥ 0 and all Pi′s are finite points ∋ if P∈ supp (D) then eP∉ supp (D) unlessP is special in which case mi = 1.∀ D ∈D0 ∃ a semireduced divisor D1 (D1 ∈D0) ∋ D ∼ D1.

Definition (Reduced divisor). A semireduced divisor D = ∑i mi Pi − (∑i mi) ∞ is said be a reduceddivisor if ∑imi ≤ g. ∀ D ∈D0, ∃ a unique reduced divisor D1 ∋ D ∼ D1.

Definition (Principal divisor). For G, H∈K [C], the divisor of a rational function R = G/H∈K (C) is calleda PD and defined as, Div Rð Þ ¼ ∑

P∈CordP Rð ÞP. We have Div (R) = div (G) − div (H) and Div (R) ∈D0.

Definition (Group of principal divisors (Jacobian)). “The set of divisors of rational functions formthe principal divisors, P = Div (R)|R ∈K (C) with P ⊂ Div0 (C) ⊂ Div (C) then the Group of PrincipalDivisor (Jacobian)10 of the curve C is defined as the quotient group: J = J (C) = Div0 (C)/P.For D1, D2 ∈ Div(C) the relation on Div(C) defined by D1 ∼ D2 ⇔D1 − D2 ∈P(ie, D1 ∼ D2 ⇒ ∃R ∈K(C): D1 = D2 + div (R)) is an equivalence relation. Every D∈ J can be uniquely represented asD = P1 + P2 + · · · + Pr – r. ∞, where r ≤ g and Pi is not symmetric of Pj”

Representations of divisors:For the purpose of computations, D =∑i mi Pi is not suggestible. Some shortcoming of this representation is that Pi's

coordinate values ∈ K the closure of the field K on which our curve is defined.Mumford representationLet D = ∑i mi Pi − (∑i mi) ∞ a semireduced divisor, which can be characterized by 2 polynomials

1. “A monic polynomial of degree mi, U(x) = Π(x − xi)mi, having root which has equal x‐coordinate of the points in the

support of the divisor. The multiplicities of the roots are same as order of the corresponding P on it.”2. At this juncture there are 2 cases3. If every Pi are distinct.

V(x) = ∑i yi∏j≠i x−xj

∏j≠i xi−xj

!, the unique polynomial ∋deg (V) ≤ deg (U) – 1 and

V(xi) = yi, ∀xi.

4. If all Pi are not distinct

We need to compute V (x) the unique polynomial ∋deg (V) ≤∑imi − 1 that fulfills the subsequent condition along

with the V (xi) = yi and if multiplicity of Pi = mi ∋ ddx

jV xð Þ 2þ V xð Þh xð Þ− f xð Þ½ x¼xi

¼ 0; for 0 ≤ j ≤ mi – 1,

ie, ∃ a unique V (x) polynomial ∋ x−xið Þmi |(V (x)2 + V(x) h(x) − f (x)).

NARESH ET AL. 5 of 16

3 | SECURE LIGHTWEIGHT HYPER ELLIPTIC CURVE CRYPTOSYSTEM

3.1 | Proposed authenticated HEC‐DH cryptosystem

3.1.1 | HEC Diffie‐Hellman with digital signature

To design a cryptosystem based on the DH key exchange using the Jacobian of a HEC:Standard and public parameters: These parameters are publicly known to all users.Let p be a large prime and choose an HEC over finite field Fp be C(Fp) and let the Jacobian of C(Fp) be J (Fp).

Choose a designated member D ∈ J (Fp) as a reduced divisor, with an order equal to a very large prime “n.” Theparameters (p, Fp,C (Fp), D, n) and let m be the message.

Party A: public key and signature generation:Randomly party A, chooses a large integer a∈[1,n‐1] as private key,computes the public key X = a D and transmits to party B where a D is a reduced divisor in the group J (Fp).Further chooses an integer k∈ [1, n‐1] ∋(k, n) = 1 and computes R = k D.The function θ: J (Fp)→Fp∋θ(R) = r′, ie, Map R to r′ via the Mumford's representation for the points of J (Fq), the

function θ can be defined by

θ Rð Þ ¼u1; if D ¼ x2 þ u1x þ u0; v xð Þ

u0; if D ¼ x þ u0; v xð Þ½ 0; if D ¼ 1; 0½ :

8><>: Calculate r′ ¼ θ Rð Þmod n:

s = k−1(H (m)+ar′) mod nA transmits the public key with signature (X,(R, s)) to party B.Party B: public key and signature verification:After receiving the message party B Computes r′ = θ(R) mod nCompute v1 ≡ s −1 H (m) mod n andv2 ≡ s −1r′mod n.Signature is verified based on whether the equation v1D + v2X = R is satisfied or notIf the signature is valid thenThe party B randomly chooses a private key, large integer b, then computes the reduced divisor Y = b D and

transmits to party A.Generation of shared key:Party A computes the shared key as reduced divisor S.K = a.Y = a (bD), which is equal to (ab)D.Party B computes the shared key as reduced divisor S.K = b.X = b (aD), which also is equal to (ab)D.Both parties now have the same key, which is an member of the Jacobian J (Fp), and so is a reduced divisor, which

consists of a set of points of the HEC.This reduced divisor is a secret known to both users, the key could be composed of the x‐coordinate of the points of (ab)D.

3.1.2 | Encryption and decryption of HEC‐DH

Encryption:Party A:HEC EncryptionInput: Domain parameters, Public Key Y of Party‐B, message mOutput: cipher text (C)

1. choose a random integer a ∈ [1, n− 1]2. compute a.Y3. return the cipher text as C= m+ a Y.

Decryption:Party B:HEC DecryptionInput: Domain parameters, Public Key “X” of Party‐A, Private Key “b” of Party‐B, cipher text C

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Output: Plaintext (m)

1. compute b X2. return the message m=C‐b X

4 | SECURITY ANALYSIS OF HEC ‐DH

4.1 | HEC‐DDH

“Let D ∈ JC (F p) be a member of prime order n. The DLP in J (F p) is: given another member D′ ∈ < D>, to find theinteger l ∋D′ = l D. We define the HEC‐DH function as DHJD(aD, bD) = abD where a, b are in F q. The HEC‐DHproblem is to compute DHJD (D1, D2) given (C, J (F p), D, D1, D2).”

Theorem 4.1. Let C be an HEC with genus 2 over F p. Let D be a member of J (F p) of prime order n. Given(C, J (F p), D, aD, bD), ∃no proficient algorithm for calculating any coordinate of abD.

Proof The cryptanalyst only has reduced divisors D, aD, and bD from which to compute (ab)D. Because it isbelieved that this task is as difficult as the discrete log problem in the Jacobian of a HEC, the cryptanalystwill fail to find (ab)D. ◻

4.2 | Security model

Before entering into security model first we define HEC‐DDH using distinguishers as follows:HEC‐DDH:HEC‐DDH problem on G = J (F p) is sketched as under:“Instance: (aD, bD, cD) for a, b, c ∈ Fq.Output: yes if c = ab, else, no.

ΔReal ¼ a; b←Fq;U ¼ aD;V ¼ bD;W ¼ abD: U;V;Wð Þ ΔRand ¼ a; b; c←Fq;U ¼ aD;V ¼ bD;W ¼ cD: U;V;Wð Þ

The benefit of each probabilistic polynomial time (ppt) 0/1‐ valued distinguisher D in solving HEC‐DDH on G isstated as follows:

“ AdvDDHD;G ¼ ∣Prob U;V ;Wð Þ←ΔReal:D U;V ;Wð Þð Þ ¼ 1½ −Prob U;V ;Wð Þ←ΔRand:D U;V ;Wð Þð Þ ¼ 1½ ;|”where the probability is nearby the choice of D's coin tosses (and random choice of a, b).

D is called a (t, ϵ)‐DDH distinguisher for G if AdvDDHD;G ≥ϵ, where D runs in time at most t.

Now define AdvDDHG ¼ MaxD AdvDDHD;G

n owhere the max is taken over all D with time complexity t.”

DDH assumption: “no (t, ϵ)‐DDH distinguisher for G, ie, ∀ ppt 0/1‐ valued distinguisher D and for every adequately

small ϵ > 0, AdvDDHD;G ≤ϵ.”

4.2.1 | Security of UP (HEC‐DH)

Theorem 4.2. The Unauthenticated Protocol (UP) HEC‐DH is secure from passive adversary based on HEC‐

DDH supposition established as follows and attains secrecy:

AdvHEC−DHUP t; qeð Þ≤2 AdvHECDDHG t” þ 2qe=∣G∣;where t’ = t + O (2qe tsm), tsm is the time needed to execute scalar

multiplications over G = J (F p) and qe is the number of queries that an adversary may ask.

Proof Let A be a passive adversary for HEC‐DH. Given A, we construct an algorithm D which solves HEC‐DDH problem with non‐negligible advantage. A has access to Reveal, Execute, and Test queries. Since long‐term private key is not used in the UP, HEC‐DH consequently query corrupt may be just ignored and the

NARESH ET AL. 7 of 16

protocol accomplishes forward secrecy routinely. Assume that A sends a single Execute query (E.q). Let ussketch Real and Fake distributions for (transcript key, session key) pairs (T, sk) as below:

Real≔x1; x2←Fq;

y1 ¼ x1D; y2 ¼ x2D

T ¼ y1; y2ð Þ; sk ¼ x1x2D

: T; skð Þ

8><>:9>=>;

Fake≔x1; x2←Fq;

y1 ¼ x1D; y2 ¼ x2D

T ¼ y1; y2ð Þ; sk←G

: T; skð Þ

8><>:9>=>;

First we prove the following 2 lemmas that relates to the Reveal and Execute queries.

Lemma 4.1. For every adversary running in time t, the following holds:

|Prob[(T, sk) ← Real: A (T, sk) = 1]‐Prob[(T, sk) ← Fake: A (T, sk) = 1]|≤AdvHECDDHG (t”) + 1/|G|, where t” ≤ t + 2.tsm

Proof. Let (A, B, C) ∈ G3 be any instance for solving HEC‐DDH problem. Using A, we form a distinguisherD for solving HEC‐DDH problem with input (A, B, C) and produce whatever A produces. Let Dist besketched as under with (A, B, C):

Dist≔x1; x2←Fq;

y1 ¼ x1A; y2 ¼ x2B

T ¼ y1; y2ð Þ; sk ¼ x1x2C

: T; skð Þ

8><>:9>=>;

Now, we examine the output of the Real Dist and the Dist: a, b ϵ Fq, A = aD, B = bD, C = abD; (T, sk)←Dist: (T, sk)are statistically equivalent when the multiples a, b used are random in Dist. On the other hand, the Fake distributionand the distribution:a, b ∈Fq, c∈ Fq − ab, A = aD, B = bD, C = cD; (T,sk) ← Dist:(T, sk) are statistically closer withina factor of 1/|G|. The unique diversity is in Fake the value of sk picked arbitrarily from G, but in Dist this value is chosenrandomly from G −abP, else, these 2 distributions are statistically equivalent by the definition of random self‐reducibility of HEC‐DDH. Subsequently, established as under:

∣Prob T; skð Þ←Real: A T; skð Þ ¼ 1½ −Prob T; skð Þ←Fake: A T; skð Þ ¼ 1½ ∣

≤∣Prob a; b∈Fq: D aP; bP; abPð Þ ¼ 1½ −Prob a; b∈Fq; c∈Fq− abf g: D aP; bP; cPð Þ ¼ 1½ þ 1=∣G∣;

≤AdvHECDDHG t” þ 1=∣G∣;where t”≤tþ 2:tsm:◻

Lemma 4.2. For any adversary A who is computationally unbounded, the following holds: Prob[(T,sk0) ← Fake;sk1 ← G;b ← 0,1:A(T, skb) = b] = 1/2.

Proof. This lemma is to deal with the Test query, the proof of which follows from the fact that in Fake thesession key sk is completely independent of the transcript T generated there and Using the same techniquesas in lemma 2 of Naresh.26 ◻

8 of 16 NARESH ET AL.

Equipped with above 2 lemmas, we show the theorem as under:

AdvHECDHUP t; 1ð Þ ¼ ∣Prob Succ½ −1∣¼ 2∣Prob T; sk0ð Þ←Real; sk1←G; b← 0; 1f g:A T; skbð Þ ¼ b½ −1=2∣¼ 2∣Prob T; sk0ð Þ←Real; sk1←G; b← 0; 1f g:A T; skbð Þ ¼ b½ −Prob T; sk0ð Þ←Fake;½sk1←G; b← 0; 1f g:A T; skbð Þ ¼ b∣ by;Lemma4:2:ð Þ≤2AdvHECDDHG t

þ 2=∣G∣ by;Lemma4:1:ð Þ

Then continuing as above we may define distributions “Realqe, Fakeqe and Distqe, we use the respective tuple (Ai, Bi,Ci) for the ith copy. We use (T, sk) to represent the (transcript, session key) pair created by these distributions.” Thensimilar to the lemma 4.1 and 4.2, we can get the lemmas as under.

Lemma 4.3. For any adversary running in time t, holds as under:

|Prob[(T, sk) ← Realqe: A(T, sk) = 1]‐prob[(T, sk) ← Fakeqe:A(T, sk) = 1|≤AdvHECDDHG (t”) + 1/|G|, where t’ = t +

O (2 qe tsm). ◻

Lemma 4.4. For any adversaryA who is computationally unbounded holds as under Prob[(T, sk0)← Fakeqe;sk1 ← Gqe; b ← 0,1:A (T, skb) = b] = 1/2. ◻Armed with lemma 4.3 and 4.4 we can simply prove the statement of the theorem. ◻

4.2.2 | Security of AP (AHEC‐DH)

Theorem 4.3. The AP (AHEC‐DH) is protected against active adversary based on HEC‐DDH assumption,establishing as under:

AdvAHEC−DHAP (t, qe,qs)≤ AdvHEC−DHUP (t,(qe,+qs)/2)+2 AdvDsig(t′), where qs, qe are the maximum amount of Send and

Execute queries respectively that an adversary can make.

Proof. “Let A' be an adversary attacking the AP. Herewith we form an adversary A attacking the UP. Ini-tially We bound the probability of the Forge event, that A' outputs a valid forge with respect to public keyPK for Party A beforehand production of the corrupt query (A).” ◻

Lemma 4.5. Let the event Forge be that a signature of Dsig is forged by A', then

Prob Forge½ ≤2 AdvDsig t′

Proof. A'made a signature forgerF to counterattack Dsig‐scheme. The goal ofF is that once a public key PK isspecified as input,F has access to a signing oracle linkedwith this key which produces a valid forgery (m, σ) wrtPK, ie, γPK (m, σ)= 1∋σwas not previously generated by the signing oracle as a signature onmessagem. TheFstarts choosing a party arbitrarily, and sets PKƒ to PK. For other Party, F genuinely yields a pair (public key,private key) by implementing theHEC‐DH protocol. Next,F brings out the approach necessary for initializationof UP. Afterwards F executes A' as a subprogram ∋ queries from A' are simulated as in Dutta and Barua27:

ProbA′;AP SuccjForge½ ¼ 1=2:

AdvA;UP ¼ 2 ProbA;UP Succ½ −1=2 ¼ 2 ProbA′;AP Succ∧Forge

þ ProbA′;AP Succ∧Forge½ −1=2

¼ 2 ProbA′;AP Succ∧Forge þ ProbA′;AP SuccjForge½ ProbA′;AP Forge½ −1=2

¼ 2 ProbA′;AP Succ∧Forge

þ 12ProbA′;AP Forge½ −1=2

¼ 2 ProbA′;AP Succ½ −ProbA′;AP Succ∧Forge½

þ 1=2ð ÞProbA′;AP Forge½ −1=2≥ 2ProbA′;AP Succ½ −1j−jProbA′;AP Forge½ −2ProbA′;AP Succ∧Forge½

≥AdvA′;UP−Prob Forge½ ”

NARESH ET AL. 9 of 16

“A poses an E.q in accordance with each E.q of A'. Likewise, A poses an E.q for every session initiated by A' usingqueries. As a session contains not below 2 instances, such an E.q is handled consequently after a minimum of 2 sendqueries of A'. The quantity of such queries is at most qs/2, where qs is the quantity of send queries made by A'. Thenumber of E.qs' handled by A is at most qe + qs/2, where qe is the amount of E.q's handled by A'.”

Already, we have AdvAHEC−DHA;UP (t, qe,qs)≤ AdvHEC−DHUP (t,(qe,+qs)/2) by supposition.

We get,

AdvAHEC−DHAP t; qe; qsð Þ≤

AdvHEC−DHUP t; qe;þqsð Þ=2ð Þ þ Prob Forge½ ;

AdvAHEC−DHAP t; qe; qsð Þ≤

AdvHEC−DHUP t; qe;þqsð Þ=2ð Þ þ 2 AdvDsig t′

;

Which proves the statement of the theorem

4.3 | Security attributes of HEC‐DH

Now we present various important security attributes for HEC‐DH28

4.3.1 | Confidentiality

To be confidential, information should be only intangible to unauthorized access and non‐intangible to eaves dropper/interceptor. If an adversary wants to get session key k; he/she needs to calculate b from Y = b or a and k from X = aDand r = k which is equivalent to solving HEC‐DLP.

4.3.2 | Integrity

Integrity check insures that the data have not been changed and is that one sent by the sender. Due to the property ofRandom Oracle Model “it is not feasible that two different messages have equal digest/hash value.” In our schemes, thereceiver verifies the signature based on hash of the message to check the integrity.

4.3.3 | Authenticity

The property that we associate with entity from where it came is called authenticity. In our proposed schemeauthenticity is based on signature and verification.

4.3.4 | Unforgeability

It means it is infeasible for an attacker to create valid signature without secret key. The proposed scheme is unforgeableas it is based on unforgeable HECDSA.

4.3.5 | Nonrepudiation

Non repudiation restricts Alice from denying the signcrypted text she sent. Unforgeability insures non repudiation. Ifsender denies, recipient send signcrypted text to judge, by using verification technique judge can decide that themessage is sent by Alice.

4.3.6 | Forward secrecy

“Infers that session key used in communication would not be disclosed even though a long‐term private key revealed. Inthe proposed system if an adversary obtain da for computing session key “k” need “r.” Computing “r” is equivalent tosolve computationally infeasible HEC‐DLP.”

10 of 16 NARESH ET AL.

4.3.7 | Public verifiability

“The property, when Alice denies his sign the recipient Bob can prove in a secure way that just Alice has signed themessage. In the proposed schemes one for public verification the receiver needs to engage with the signcrypted textis directly verified.”

5 | RESULTS AND COMPARATIVE ANALYSIS

In this section, we presented the cost analysis of the proposed scheme with respect to computation cost and communi-cation overhead and its comparison with existing schemes.

5.1 | Proposed protocol's cost analysis

Hyper elliptic curve cryptosystem is the natural generalization of ECC systems that are appropriate for realizing highsecurity in resource constrained environment. The HECC can provide the same security with lesser key sizes as inTable 2. The proposed scheme being HECC with lesser key sizes will have less computational expenses than ECC‐basedschemes.

In the theoretical part of the results, the major and most expensive operations in the existing20-24 and proposedschemes are EC point multiplication/addition (ECPM/ECPA) and HEC divisor scalar multiplication/addition(HECDM/HECDA). Comparative computational costs analyses of existing and proposed schemes are presented inTable. 2 are based on these major operations. As per results in Nizamuddin and Amin,21 it is clear that computationtime of one scalar multiplication for EC is nearly double of that of one scalar multiplication for HEC. Since our protocoldeals with HECDM, it is optimal with respect to time complexity.

5.2 | Experimental results and computation

Proposed HECC Algorithm for HE curves of genera 2, 3, 4, 5, and 6 were implemented using “magma software packagedesigned for computations in algebra, number theory, algebraic geometry, and algebraic combinatorics on intel® core™i5‐6500 CPU @ 3.20GHz, 4GB RAM with 64 bit windows operating system.” Table 3 shows the HE curves used in gen-era 2, 3, 4, 5, and 6 over the finite field GF (p).

First, we focused on comparative analysis for computational times with respect to different genus for mainoperations like Jacobian operation, divisor identifications, key generation, signature generation/verification,message encryption, and decryption by changing the size of the field such as 2^3, 2^30, and 2^127,respectively. Further for a particular genus, we compared proposed protocol with respect to HEC over finite fieldsof various sizes.

5.2.1 | Cost analysis of proposed schemes

The experimental comparative analysis various functions of proposed protocol for various length finite fields separatelywith respect to different genus HEC:

We implemented the proposed protocol for HE curves of different genus over finite fields of various lengths, findingthe computation time for various functions such as Jacobian computation time, key computation time, signature gen-eration/verification time, and encryption/decryption time (the computation time for encryption and decryption is same,due to that only encryption computation time is facilitated) and the results are indicated in Figures 1 to 4, respectively.From the figures, we can observe that the computation time is increasing with the increasing of the genus and with thefield sizes, ie, with lesser field sizes and with less genus we are getting optimal results. Since “HEC with an operand sizeis only a fractional amount of that of the EC operand size,” the proposed scheme is appropriate for devices, whichrequire lesser key sizes and storage requirements.

TABLE2

Com

parision

ofnum

berof

operationsof

variou

sschem

es

Schem

esParticipan

tECPM

ECPA

HECDM

HECDA

Division

Multiplica

tion

Addition/Subtraction

Hash

Prop

osed

Alice

Bob

… …

… …

3 4… 1

1 …

… 11 …

2 2

Nizam

udd

inan

dAmin

21Alice

Bob

3 4… …

… …

… 11 …

1 11 …

2 2

Zhen

gan

dIm

ai24

Alice

Bob

3 4… 1

… …

… …

1 …

1 11 …

2 2

Hwan

get

al22

Alice

Bob

4 5… 1

… …

… …

… …

1 …

1 …

1 1

Too

ranian

dBeh

eshtiSh

irazi20

Alice

Bob

4 51 2

… …

… …

… …

1 …

3 12 2

Moh

apatra

andMajhi23

Alice

Bob

5 4… 1

… …

… …

1 …

… 11 …

3 3

NARESH ET AL. 11 of 16

FIGURE 1 Jacobian computation time

TABLE 3 Hyper elliptic curves used in genera 2, 3, 4, 5, and 6 over the finite field GF (p)

Genus Hyper elliptic curve (HEC)

2 y2 = x5 + f 3x3 + f 2x

2 + f 1x + f 0

3 y2 = x7 + f 5x5 + f 4x

4 + f 3x3 + f 2x

2 + f 1x + f 0

4 y2 = x9 + f 7x7 + f 6x

6 + f 5x5 + f 4x

4 + f 3x3 + f 2x

2 + f 1x + f 0

5 y2 = x11 + f 9x9 + f 8x

8 + f 7x7 + f 6x

6 + f 5x5 + f 4x

4 + f 3x3 + f 2x

2 + f 1x + f 0

6 y2 = x13 + f 11x11 + f 10x

10 + f 9x9 + f 8x

8 + f 7x7 + f 6x

6 + f 5x5 + f 4x

4 + f 3x3 + f 2x

2 + f 1x + f 0

Wheref11:= 132713617209345335075125059444256188021;f10:= 90907655901711006083734360528442376758f9:= 34744234758245218589390329770704207149;f8:= 132713617209345335075125059444256188021;f7:= 90907655901711006083734360528442376758f6:= 34744234758245218589390329770704207149;f5:= 132713617209345335075125059444256188021;f4:= 90907655901711006083734360528442376758f3:= 34744234758245218589390329770704207149;f2:= 132713617209345335075125059444256188021;f1:= 90907655901711006083734360528442376758;f0:= 6667986622173728337823560857179992816;

FIGURE 2 Key computation time

12 of 16 NARESH ET AL.

FIGURE 3 Signature generation/verification time

FIGURE 4 Encryption/decryption Time

FIGURE 5 Computation time of HECC over Genus‐2

FIGURE 6 Computation time of HECC over Genus‐3

NARESH ET AL. 13 of 16

14 of 16 NARESH ET AL.

5.2.2 | The experimental comparative analysis of proposed protocol for HE curves ofdifferent genus separately with respect to finite fields of various lengths

We presented the comparative analysis of proposed protocol for finite fields of various lengths and of HE curves ofdifferent genus separately with respect to computation time for various functions such as Jacobian computation time,key computation time, signature generation/verification time, and encryption/decryption time (the computation timefor encryption and decryption is same, due to that only encryption computation time is facilitated), and the resultsare shown in Figures 5 to 9, respectively.

FIGURE 7 Computation time of HECC over Genus‐4

FIGURE 8 Computation time of HECC over Genus‐5

FIGURE 9 Computation time of HECC over Genus‐6

NARESH ET AL. 15 of 16

From the figures, we can observe that the computation time is increasing with the increase of the size of the fieldfor all the operations.

6 | CONCLUSIONS

Hyper elliptic curve cryptosystem is apt for secure communication in wireless networks for constrained devices as“Operand size of HEC is only a fractional amount of that of the operand size of EC” and almost all the standardDLP‐based schemes such as the DH and EIGamal can be realized using HEC. In this paper, we proposed a provablesecure lightweight HEC cryptosystem for WSN, which permits both the entities to verify each other's authenticity.It is shown that the proposed scheme is proficient as the timings of various operations such as signaturegeneration/verification, key generation, message encryption, and decryption compares favorably with the timings ofECC Systems available in existing literature. As HECC (genus 2) of 80‐bit operand lengths provide the same securitylevel with ECC of 160‐bit, and from the results of the present work, HECC is more appropriate for implementationin the constrained platforms in wireless network.

As part of finding proper genus for practical development, from the performance metrics such as processing timeand key size of implementation results it is inferred that HEC of genera 2 and 3 can be used for resource constraineddevices. Hyper elliptic curve of genera 4, 5, and 6 is well suited for large scale network. Hence, HECC over genus 2 withan appropriate key length can be implemented for power and memory constrained devices to improve the secrecy ofdata and efficiency of the system. The performance of higher genus HECC gets degraded in terms of divisor generation,key generation, and encryption and decryption process. However, it is observed that the implementation of genus 4HECC on an embedded microprocessor that this cryptosystem is better suited for the use in embedded environments,than to general purpose processors.

ORCID

Vankamamidi S. Naresh http://orcid.org/0000-0002-6273-9041Reddi Sivaranjani http://orcid.org/0000-0001-6260-8909Nistala V.E.S. Murthy http://orcid.org/0000-0003-2738-9362

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How to cite this article: Naresh VS, Sivaranjani R, V.E.S. Murthy N. Provable secure lightweight hyper ellipticcurve‐based communication system for wireless sensor networks. Int J Commun Syst. 2018;e3763. https://doi.org/10.1002/dac.3763

Uppalapati.Jyothi et al iJ. Pharm. Sci. & Res. Vol. 1O(9), 2018, 2201-2204

ISSN:O075-14i9Jr:rui aa i:tl Fh;t tlrt.*rrtir;tiSt:i+itttr,; irrrd Rtrli:lt'r:ii

ii lr .ii:lr lirr:lr:urt.;,.iri

).1 f - l4,4

A bstract:

Stability Indic atrng RP-HPLC Method for the SimultaneousEstimation of Sacubitril and Valsartan in Drug product

IDepartmenrofChemistryl,on,,rn,,,,oo,Wi;!iil2,;",i,ir!,ea|t)2sa,tlisa*napa,tnanl,AndhraPradesh531l622Depqrtment of Chemistry, GITAM Instinrte of Science, jITAM Universiry, Uisakhdpatnam, Anclhra pradesh 530045

The airn of the method lvas to develop and validate a rapid. sensitive and accurate trethocl tbr sinrLrltaneous esti6atron ol' \/alsartan andSacubitril in drug product by liquid chrotratography. The chrorratographic separation rvas achieved on C8 colunrn (t_Lrna Cij l -<(t*-1 6. 3utn) atalnbienttelnperaturc.Theseparationachievedernplofinga:nobilephaseconsrstsol0.i%v/v'l rilluoroaceticacidrn\\at!.r \lerhanol (2j:75)

S.,.uLrtill$ords: Valsartatt. Sacubitril. Isocratic. HPLC. I.,una, Tntluoro acetic acid. Acetonilrile. Methanol and validatron

1. INTRoDUCTToN\ alsartan

Irig. l. Cherllical structure: Valsartan

Yalsartan is an angiotensin Il receptor commonly calledangiotcnsin reccptor blocker. Valsartan is nrainly used lortreiltrnent o1'high blood pressure. congestive hcart lailure. and tuincrease the chances o1'living longcr after a heart attacl<.\ialsartan blocl<s the actions olangiotensin [1. rvhich includeconstricting blood vessels and actir.,ating aldosterone. to rcduceblooci pressure. The drug bincls to angiotensin type I receptors(,1 l). rvorking as an antagonist. l'his ntechanisr.r.r of action isdillirent than the ACI: inhibitor.drugs. *'hich block thccorrrcrsion ol'angiotensin I to angiotensin IL\':rlsnrtan is chemicallv designatecl as (S)-3-methyl-2-(N-t2,-(2H- 1.2.3.4-tetrazol-5- yl)biphenyl-4-lllnrethlllpentanamido)butanoic acid. Its rnolcoular lirmula isC21H29N503. and its molccular rvcight is 435.5 l9 g/mol.Sacu bitrilSacubitril is an antihypertensive dnrg . Sacubitril is a prodrug thatis activated to Sacubitrilat by de-ethllation via esterase, SacLrbitrilinhibits the enzyrneneprillsin. rvhich is responsible tbr thedegrirclltion o1'atrial and brain natriuretic peptidc. trvo blooclpressure-lor.r,ering peptides that rvork nrairrly by reducing bloodrolLrnrc. In addition. nepril,rsin dcgrades a variclr ol'peptidcsinclrrding brady'kinin. an inflantntatorl. rlediator exerling ptrtentr asodilirtory action.Sacubitril is chemically dcsrgnated as 4-([(2S.1R)-l-(4-Biphcnl'lyl)-5-ethoxy-4-rnethyl-5-oxo-2-perrranyl larnino ) -4-oxobulanoic acid. Its molecular fbrntula is C24H29NO5. and itsn:olecuiar rveight is 4l 1.49 g/rnol.

2. !lA rERrALs AND NI ITHoDS2.1 Equipments: The chromatographic technique perfbrrlecl on a*aters 2695 rvith 2,187 detector ancl lntpoue12 soft*arc.reversed phase C8 oolurln (Luna C8 150x4.6.3pm) as sri.rtionar\phase .LJltrtrsonic cleaner. Scaletech anall tical balance anclVaccunr nricro flltration unit riith 0.45pt rrembrane tjlter.

2,2 Naterials: Phanlaceutically pure sanrplc ol'Valsartan/Sacr.rbitril rvere obtained as gili san-rplcs fiom Forluncpharma training instirutc. Sri Sai nagar colonr.. KpHB.Hl derabad. lndia.HPLC-grade Nlethanol and ,Acetonitrile *,ere obtainecl fionrqualigens reagents pvt ltd. '['riflouro acetic acid (.,\R gracle) l,asfiom sd flnc chern.

2.3 (lhrontatographic conditions'l'lre santple separati()n \\asachieved on a ([.una CU 150*11.6.3 pnr) C8 column. aicled b1mobilc phase ntixturc ol0. lTor'/r, f riflr-roro acetic acid itr tl ater :

Mcthanol (25:75). The flou. rate \\,as 1.0 ml/ ntinute ancl r-rltraviolet detector at 2(r7nm. that \\,as filtercd anci ciegasseci prior touse. In je ction r,olume is l0 prl and anrbicnt ter.nperatltres.PreparaIion o1' mobile phase:Buller Preparation: laken acourately lntl o1"l'riflr-rorr.l acctic acicjin l000mL o1-rvaterMobile phase: l'hen addcd 25 r,olr.rntes ol'bLr1l'er and 75 r,olunrcsof Methanol ntixed r.vell and sonicated fbr 5 rnin.Diluent: Water: Acetonitrile: 50:50 v\v

2,,1 Preparation of solutions2..1.1 Standard solution:51.5 mg olpure Valsartan ancl 48.5 mg ol'Sacubitril ucrc rr,e ighed and transt'clrcd to _50 ntl ol rolr:rletricllasl< and clissolved in clilucnt.'l'he tlask uas slraken ancl volur.ncl,as made up to rrark rvith diluent to git,e a prirnarl stocksolution. Front thc above solution lrnl of solution is pipctru outinto a l0 nrl volr.rrnetric flask and volurne u,as rladc up to ntarlrwith water to give a solution containing l03pg/ml ol'Valsartanand 97 pg/rrl Sacubitril .

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. ..,._t.,.._ 1Ji r./ ,./1 ,..( (r( iittrt l.tllc,.

Graphical Abstract

INTRODUCTIOI

N-Bromosuccinimide (NBS) is a versatile analytical reagent in organic reactions. In most of thereactions' NBS acts as oxidizing as well as brominating;g.;t^rr-nl. rt ir.rr"Jur"one of anaryicarreagent because of its sensitivity, accuracy, simplicity"rni iow cost. Suren dra et a/., reported thecaralyic nature of NBS in organic transformatiorr. ;q. Nns h*; significant role in the estimation ofd*rgs bv its oxidation ts-lSl'Jn uq".ou. ,orutions ,il *id;ri;; properties of NBS were attributed by€Csxr

403

2019, B (1):403_411(International Peer Reviewed Journal)

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M. Kalyana chakravarthyl*, K. Ramakrishna2 and p. v.Subba Rao3l Department of chemistry, Anil Neerukonda Institute ofrechnology and Sciences, visakhapatnam, INDIA2 Deparhnent of ch^emistry,.college of Science, Gitam uni"riersity, vlsat t apatnam, tNfilA3. School of Chemistry, Andhra University, IIDIA

Email : kalyanamutnuru @ gmail. com

ABSTRACT

Indigo Carmine(IC), Sodium dodecyl sulphate (SDS), Cetyl

:.ra

F..:;'_Lij,

2018-19

2018-19

Astrophys Space Sci (2019) 364:190https://doi.org/10.1007/s10509-019-3681-2

ORIGINAL ARTICLE

Kaluza-Klein dark energy model in Lyra manifold in the presenceof massive scalar field

Y. Aditya1 · K. Deniel Raju2,3 · V.U.M. Rao3 · D.R.K. Reddy3

Received: 31 August 2019 / Accepted: 30 October 2019 / Published online: 5 November 2019© Springer Nature B.V. 2019

Abstract In this investigation we intend to study the dy-namics of an anisotropic dark energy cosmological modelin the presence of a massive scalar field in a modified Rie-mannian manifold proposed by Lyra (Math. Z. 54:52, 1951)in the background of a five dimensional Kaluza-Klein spacetime. We solve the Einstein field equations using some phys-ically significant conditions and present a deterministic darkenergy cosmological model. We use here the time depen-dent displacement vector field of the Lyra manifold. All thedynamical parameters of the model, namely, average Hub-ble parameter, anisotropy parameter, equation of state pa-rameter, dark energy density, deceleration parameter andstatefinders are evaluated for our model and their physicalrelevance to modern cosmology is discussed in detail.

Keywords Kaluza-Klein model · DE model · Lyramanifold · Massive scalar meson field

1 Introduction

The subject that is attracting several researchers, in mod-ern cosmology, is the accelerated expansion of the universe(Riess et al. 1998; Perlmutter et al. 1999). It has been said

B Y. Adityayaditya2@gmail.com

D.R.K. Reddyreddy_einstein@yahoo.com

1 Department of Mathematics, GMR Institute of Technology,Rajam 532127, India

2 Department of Mathematics, ANITS (A), Visakhapatnam531162, India

3 Department of Applied Mathematics, Andhra University,Visakhapatnam 530003, India

that this is caused by an exotic negative pressure which isknown as dark energy (DE). Several DE models have beenproposed to explain this phenomenon which even today re-mains as mystery. The cosmological constant is supposedto account for this DE. But it has some serious problems.Hence two approaches have been suggested to describe thismysterious concept. One method is to construct DE modeland study their significance in relation to this cosmic infla-tion. Another way is to modify Einstein’s theory of gravi-tation and to construct DE models with a special referenceto the observations of modern cosmology which throws abetter light to explain this scenario.

For this purpose, there have been several modifications ofEinstein’s theory of gravitation by modifying the Einstein-Hilbert action of general relativity and introducing ScalarFields (SFs) into Einstein theory. Introduction of SFs leadsto the well known quintessence models which help to ex-plain the accelerated expansion of the universe. Thus themodified theories of gravitation are f (R) and f (R,T ) the-ories (Nojiri and Odintsov 2003; Harko et al. 2011) andscalar-tensor theories proposed by Brans and Dicke (1961)and Saez and Ballester (1986). Anisotropic DE modelsin the above modified theories of gravitation were inves-tigated by numerous researchers (Copeland et al. 2006;Nojiri et al. 2005; Kiran et al. 2014; Reddy et al. 2014;Aditya et al. 2016; Rao et al. 2018; Aditya and Reddy2018a, 2019).

Here we are interested in the interacting scalar mesonfields. Scalar meson fields are of two types—zero mass SFsand massive SFs. Zero mass SFs describe long range inter-actions while massive SFs represent short range interaction.In fact, this physical significance of SFs leads to immensestudy of SFs. Also, SFs are very important since they repre-sent matter fields with spin less quanta and describe gravita-tional fields. In the literature, there are several investigations

190 Page 2 of 8 Y. Aditya et al.

of cosmological models in the presence of mass less andmassive SFs coupled with different physical sources. Herewe are mainly concerned with models in the presence ofmassive scalar fields. Some note worthy models are obtainedby Naidu (2018), Aditya and Reddy (2018a) and Reddy andRamesh (2019) in the presence of massive SFs.

In an attempt to unify gravity and electromagnetic fieldsseveral modifications of Riemannian geometry have beenproposed. Significant among them is Lyra (1951) geometry.In this geometry a gauge function has been introduced intothe structureless manifold so that displacement field arisesnaturally. The energy conservation is not valid in this the-ory. The displacement field, in this theory plays the samerole as the cosmological constant in general relativity. Sev-eral cosmological models in this particular theory have beendiscussed extensively. The following are relevant and signif-icant to our investigation: Singh and Rani (2015) have dis-cussed Bianchi type-III cosmological models into coupledperfect fluid and attractive massive scalar field as physicalsource in Lyra geometry. Very recently, Reddy et al. (2019)investigated Bianchi type-III DE cosmological model in thepresence of massive scalar field in this geometry.

In order to discuss the early stages of evolution of theuniverse, immediately after big bang, higher dimensionalcosmology plays a vital role. Subsequently, the universehas undergone compactification and we have the presentfour dimensional universe. Witten (1984) and Appelquistet al. (1987) are some of the authors who have studiedhigher dimensional cosmology. In particular, in Kaluza-Klein (Kaluza 1921; Klein 1926) five dimensional geometrythe extra dimension is used to couple the gravity and electro-magnetism. Hence, Kaluza-Klein models gain importance.Kaluza-Klein cosmological models have been discussed byseveral authors in modified theories of gravity (Reddy andLakshmi 2014; Sahoo et al. 2016; Santhi et al. 2016a;Naidu et al. 2018a; Reddy and Aditya 2018; Aditya andReddy 2018b).

The above discussion motivates us to investigate Kaluza-Klein cosmological model in the presence of anisotropicDE fluid coupled with an attractive massive scalar field.The plan of this paper is the following: In Sect. 2, theKaluza-Klein model and the field equations in the presenceof anisotropic DE fluid and massive scalar field are derived.Section 3 presents the solution of the field equations and themodel. Section 4 is devoted to compute all the dynamicalparameters and to present physical discussion. In Sect. 5 theresults are summarized with conclusions.

2 Basic field equations

Here we derive the basic field equations with the help of theKaluza-Klein (KK) metric which is defined as

ds2 = dt2 − A2(dx2 + dy2 + dz2) − B2dψ2 (1)

where A, B are functions of cosmic time t and fifth coordi-nate ψ is space-like. Unlike Wesson (1983), here, the spatialcurvature has been taken as zero (Gron 1988).

We consider the field equations in the normal gauge inLyra manifold as

Rij − 1

2gijR + 3

2

(didj − 1

2gij dkd

k

)= −Tij (2)

where di is the displacement vector field of the manifold(function of time t) defines as

di = [β(t),0,0,0

](3)

here we assume gravitational units so that 8πG = c = 1.The other symbols have their usual meaning. Tij is theenergy-momentum tensor given by

Tij = T deij + T s

ij (4)

where T deij is the energy-momentum of DE given by

T deij = (ρΛ + pΛ)uiuj − pΛgij , uiu

i = 1 (5)

which can also be written as

T deij = diag[ρΛ,−pΛ,−pΛ,−pΛ]. (6)

We assume the anisotropic distribution of DE to ensurethe present acceleration of Universe. Hence the energy-momentum tensor T de

ij can be parameterized as

T deij = [1,−wx,−wy,−wz,−wψ ]ρΛ

= [1,−wΛ,−(wΛ + α),−(wΛ + γ ),−(wΛ + δ)

]ρΛ

(7)

where ωx = ωΛ, ωy = ωΛ + α, ωz = ωΛ + γ and ωψ =ωΛ + δ are the directional equations of equation of state(EoS) parameters on x, y, z and ψ axes respectively. Here,α, γ and δ are the deviations from ωΛ on y, z and ψ axes re-spectively. pΛ and ρΛ being the energy density and pressureof DE fluid, wΛ = pΛ

ρΛis the EoS parameter of DE.

Also

T(s)ij = φ,iφ,j − 1

2

(φ,kφ

′k − M2φ2) (8)

where φ is the massive scalar field, M is the mass of thescalar field (SF). This scalar field satisfies the Klein-Gordonequation, which is given by

gijφ;ij + M2φ = 0. (9)

With the use of Eqs. (3)–(9), the Lyra manifold fieldequations (2) for the KK metric (1), explicitly, can be de-rived as (we use co-moving coordinates)

3

(A2

A+ AB

AB

)− ρΛ − φ2

2− M2φ2

2− 3

4β2 = 0 (10)

Kaluza-Klein dark energy model in Lyra manifold in the presence of massive scalar field Page 3 of 8 190

2A

A+ A2

A2+ 2

AB

AB+ B

B+ wΛρΛ + φ2

2

−M2φ2

2+ 3

4β2 = 0 (11)

2A

A+ A2

A2+ 2

AB

AB+ B

B+ (wΛ + α)ρΛ + φ2

2

−M2φ2

2+ 3

4β2 = 0 (12)

2A

A+ A2

A2+ 2

AB

AB+ B

B+ (wΛ + γ )ρΛ + φ2

2

−M2φ2

2+ 3

4β2 = 0 (13)

3

(A

A+ A2

A2

)+ (wΛ + δ)ρΛ + φ2

2− M2φ2

2+ 3

4β2 = 0

(14)

φ + φ

(3A

A+ B

B

)+ M2φ = 0. (15)

Here an overhead dot indicates differentiation with respectto time t .

The following cosmological parameters are useful tosolve our field equations:

Spatial volume (V ), average scale factor (a(t)), meanHubble parameter (H ) and scalar expansion (θ ) are givenby

V = a3(t) = A3B (16)

H = a

a= 1

4

(3A

A+ B

B

)(17)

θ = 4H =(

3A

A+ B

B

). (18)

Shear scalar σ 2, average anisotropy parameter () and de-celeration parameter (DP) q are given by

σ 2 = 1

2σ ijσij = 1

2

( 4∑

i=1

H 2i − 1

3θ2

)(19)

= 1

4

4∑

i=1

(Hi − H

H

)2

(20)

q = −1 + d

dt

(1

H

). (21)

The nature of expansion of the model can be explained us-ing the DP. For positive value of DP, the model deceleratesin the standard way. If DP vanishes then the model expandswith constant rate. For −1 ≤ q < 0, we get accelerated ex-pansion of the universe. The model exhibits an exponentialexpansion for q = −1 and super exponential expansion forq < −1.

3 Kaluza-Klein DE model

Here, we solve the field equations (10)–(15) and presentKaluza-Klein DE model within the framework of Lyra man-ifold in the presence of massive scalar field.

From Eqs. (11) and (12) we have

α = 0. (22)

From Eqs. (12) and (13) we obtain

α = γ (23)

consequently from Eqs. (22) and (23), we obtain

α = γ = 0. (24)

This is because of the fact that the universe is isotropic in x,y and z directions and hence the deviations from EoS of DEvanished.

Using Eq. (24) in Eqs. (10)–(15) reduce to the followingindependent equations

3

(A2

A+ AB

AB

)− ρΛ − φ2

2− M2φ2

2− 3

4β2 = 0 (25)

2A

A+ A2

A2+ 2

AB

AB+ B

B+ wΛρΛ + φ2

2

−M2φ2

2+ 3

4β2 = 0 (26)

3

(A

A+ A2

A2

)+ (wΛ + δ)ρΛ + φ2

2− M2φ2

2+ 3

4β2 = 0

(27)

φ + φ

(3A

A+ B

B

)+ M2φ = 0. (28)

Now Eqs. (25)–(28) are a system of four independent dif-ferential equations in seven unknowns (A,B,φ,ρΛ,wΛ, δ

and β). Hence, we are free to choose three more mathemati-cal or physical conditions to find a deterministic model. Wechoose the following conditions:

(i) We use the fact that expansion scalar θ is proportionalto shear scalar σ 2, so that we have (Collins et al. 1980)a relation between the metric potentials as follows:

A = Bn (29)

where n is a positive constant which retains theanisotropy of the space-time. The motivation behindconsidering this relation is explained by Thorne (1967).Observations from the velocity-red-shift relation forextragalactic sources suggest that Hubble expansion ofthe universe is isotropic at present within ≈ 30 per cent(Kantowski and Sachs 1966; Kristian and Sachs 1966).

190 Page 4 of 8 Y. Aditya et al.

In particular, the studies of red-shift survey place thelimit as

σ

H≤ 0.3, (30)

in the neighborhood of our present day Galaxy. Collinset al. (1980) have shown that the normal congruence tothe homogeneous expansion satisfies the condition σ

H

is constant.(ii) In recent years, it is quite natural to use a power-law

relation between scalar field φ and average scale factora(t) of the form (Johri and Sudharsan 1989; Johri andDesikan 1994)

φ ∝ [a(t)

]m (31)

where m is a power index. Several researchers havestudied different aspects of this form of scalar field φ

(Rao et al. 2015; Santhi et al. 2016b; Aditya and Reddy2018b). In view of the physical importance of aboverelation, here we assume the following assumption toreduce the mathematical complexity of the system

φ

φ= −(3n + 1)

B

B. (32)

This is a consequence of Eq. (31). This relation (32)has been already taken by many authors and have con-structed cosmological models using this relation. Singh(2005), Singh and Rani (2015), Aditya and Reddy(2018a, 2019) and Naidu et al. (2019) have studiedBianchi type cosmological models with massive scalarfields using the above relation (32).

(iii) In addition to the above, we have taken a power law re-lation between β(t), the displacement vector field andaverage scale factor a(t) given by

β(t) = β0[a(t)

]k (33)

where β0 = 0 and k are positive constants.

Now from Eqs. (28)–(32) we get

φ = exp

(φ0t − M2t2

2+ φ1

)(34)

where φ0 and φ1 are constants of integration.Equations (32) and (33) together yield

A = exp

(n(M2t2 − 2φ0t − 2φ1)

2(3n + 1)

),

B = exp

(M2t2 − 2φ0t − 2φ1

2(3n + 1)

).

(35)

Using Eq. (35) in Eq. (1), the Kaluza-Klein model in thepresence of massive scalar field is given by

ds2 = dt2 − exp

(n(M2t2 − 2φ0t − 2φ1)

3n + 1

)

× (dx2 + dy2 + dz2)

− exp

(M2t2 − 2φ0t − 2φ1

3n + 1

)dψ2 (36)

and the massive scalar field in the model is given by Eq. (34).

4 Dynamical parameters of the model

Dynamical or cosmological parameters (16)–(21) have a sig-nificant role in the discussion of the cosmological models ofthe universe. Hence we evaluate them and present here

V = exp

(M2t2 − 2φ0t − 2φ1

2

)(37)

H =(

M2t − φ0

3

)(38)

θ = M2t − φ0 (39)

σ 2 = 1

2

(M2t − φ0

3n + 1

)2

(n − 1)2 (40)

= 9n2 − 12n + 7

4(3n + 1)2(41)

q = −(

1 + 3M2

(M2t − φ0)2

). (42)

Now from Eqs. (16), (34) and (35) we get

β(t) = β0 exp

(k(M2t2 − 2φ0t − 2φ1)

6

). (43)

Now from Eqs. (25)–(27), (33), (35) and (43) we obtainenergy density ρΛ, EoS parameter ωΛ of DE and skewnessparameter δ as

ρΛ = n(4n + 1)

(3n + 1)2

(M2t − φ0

)2

−[(M2t − φ0)

2 + M2

2

]exp

(2φ0t − M2t2 + 2φ1

)

− 3

4β2

0 exp

(k(M2t2 − 2φ0t − 2φ1)

6

). (44)

wΛ = − 1

ρΛ

2n(n + 1)(M2t − φ0)

2

(3n + 1)2+ nM2

3n + 1

+[(M2t − φ0)

2 − M2

2

]exp

(2φ0t − M2t2 + 2φ1

)

Kaluza-Klein dark energy model in Lyra manifold in the presence of massive scalar field Page 5 of 8 190

+ 3

4β2

0 exp

(k(M2t2 − 2φ0t − 2φ1)

6

)(45)

δ = n(1 − 3n)

ρΛ(3n + 1)2

[M2 − (

M2t − φ0)2]

, (46)

where ρΛ is given by Eq. (44).Several DE models have been formulated for analyzing

the DE phenomenon in the accelerated expansion of the uni-verse. Hence there is a need to distinguish these DE modelsso that one can decide which DE model provides good ex-planation for the present status of the universe. Because ofthe fact that various DE models give almost the same presentvalues of the deceleration and Hubble parameters. Hence,these parameters can not differentiate the DE models com-pletely. For this reason, Sahni et al. (2003) have introducedtwo new dimensionless parameters known as statefinders de-fined as follows:

r =...a

aH 3, s = r − 1

3(q − 12 )

.

For our model the above parameters are obtained as

r = 1 + 27M2

(M2t − φ0)2(47)

s = −6M2

(M2t − φ0)2 + M2. (48)

For (r, s) = (1,0), (1,1) we obtain ΛCDM and CDM

limits, respectively. However, s > 0 and r < 1 shows theDE regions such as phantom and quintessence-like, s < 0and r > 1 indicate the Chaplygin gas. Recently, many au-thors have investigated the statefinders analysis with dif-ferent geometries (Jawad 2014; Singh and Kumar 2016;Santhi et al. 2017; Naidu et al. 2018b; Sharma and Pradhan2019).

5 Physical discussion

Here, we have obtained Kaluza-Klein DE universe (Eq. (36))in the presence of attractive massive scalar field in the frame-work of Lyra manifold. It can be seen that the volume ofour model is non-zero at t = 0, i.e., the model is free fromthe initial singularity. It is clear from Eq. (25) that the vol-ume of the model is exponential function, hence the modelexhibits an exponential expansion from a finite volume ast increases. Also, it is observed that the physical parame-ters H,θ,σ 2 are finite at t = 0 and they tend to infinity ast → ∞. We observed that the anisotropy parameter is in-dependent of the time t . Hence the universe is uniform andspatially homogeneous. In Fig. 1, we have plotted the behav-ior of displacement vector β versus redshift z for different

Fig. 1 Plot of β(t) versus redshift z for M = 1.5, k = 0.18, φ1 = 10,n = 0.9 and β0 = 0.01

Fig. 2 Plot of scalar field versus redshift z for M = 1.5, k = 0.18,φ1 = 10, n = 0.9 and β0 = 0.01

values of φ0. It can be seen form Fig. 1 that β is a decreas-ing and positive function. The function β(t) decreases withincrease in φ0.

In order to study the behavior of physical parameters wehave plotted them in terms of cosmological redshift z. Weused the relation between the redshift z and the average scalefactor a(t) as 1 + z = a0

a. We consider the present value of

average scale factor a0 which has been normalized to one.

Scalar field: In Fig. 2, we have plotted the behavior of mas-sive scalar field φ versus redshift z for different values of φ0.We observed that φ is positive and increasing function forall the three values of φ0. We, also, observe that the scalarfield increases as φ0 increases. It can be observed that thescalar field shows increasing behavior and hence we canconclude that the corresponding kinetic energy decreases.The massive scalar field shows rapid increase from very lowvalues and approaches maximum value. This behavior isquite similar to the behavior of exponential potential which

190 Page 6 of 8 Y. Aditya et al.

Fig. 3 Plot of energy density ρΛ versus redshift z for M = 1.5,k = 0.18, φ1 = 10, n = 0.9 and β0 = 0.01

Fig. 4 Plot of EoS parameter versus redshift z for M = 1.5, k = 0.18,φ1 = 10, n = 0.9 and β0 = 0.01

correspond to cosmological scaling solutions obtained byCopeland et al. (2006) and interacting modified ghost SFmodels of DE constructed by Jawad (2015).

Energy density: Fig. 3 depicts the behavior of energy den-sity of DE ρΛ versus redshift z. It can be observed that ρΛ ispositive and decreasing function. Also, ρΛ increases as theSF increases.

EoS parameter: The EoS parameter of fluid relates its pres-sure p and energy density ρ by the relation, w = p

ρ. Differ-

ent values of EoS parameter correspond to various epochs ofthe universe from early decelerating to present acceleratingexpansion phases. It includes stiff fluid, radiation and mat-ter dominated (dust) for w = 1, w = 1

3 and w = 0 (deceler-ating phases) respectively. Also, it represents quintessencefor −1 < w < −1/3, cosmological constant (vacuum) forw = −1 and phantom for w < −1. Figure 4 describes thebehavior of EoS parameter of DE versus redshift for various

Fig. 5 Plot of skewness parameter versus redshift z for M = 1.5,k = 0.18, φ1 = 10, n = 0.9 and β0 = 0.01

values of φ0. It is observed that for all the three values of φ0

the model starts in quintessence region −1 < wΛ < −1/3,crosses the phantom divided line wΛ = −1 at late times andapproaches the aggressive phantom region wΛ −1. Also,as scalar field increases the EoS parameter of our DE modelapproaches the quintessence region. The trajectories of EoSparameter of DE model coincide with the Planks collabora-tion (Ade et al. 2014) and WMAP nine years observationaldata (Hinshaw et al. 2013) which give the ranges for EoSparameter as

−0.92 ≤ wΛ ≤ −1.26 (Planck + WP + Union 2.1),

−0.89 ≤ wΛ ≤ −1.38 (Planck + WP + BAO),

−0.983 ≤ wΛ ≤ −1.162 (WMAP+ eCMB+BAO+H0).

Skewness parameter: The physical significance of skew-ness parameters is that the amount of anisotropy in the DEfluid. Here the surviving skewness parameter δ is depicted inFig. 5 for various values of φ0. We observed that the skew-ness parameter is positive in the initial epoch and attains anegative value at late times. We can conclude that the DEin our model is anisotropic throughout the evolution of theuniverse and hence it helps to study the anisotropies at smallangular scales which play a key role in the formation of largescale structures of the universe.

Deceleration parameter: The nature of expansion of themodel can be explained using the deceleration parameter(DP). For example, the model decelerates in the standardway for positive value of DP and the model expands withconstant rate as DP vanishes. The model exhibits acceler-ated expansion for −1 ≤ q < 0, an exponential expansionfor q = −1 and super exponential expansion for q < −1.Figure 6 describes the behavior of DP versus redshift z forvarious values of φ0. We observe that DP remains less than

Kaluza-Klein dark energy model in Lyra manifold in the presence of massive scalar field Page 7 of 8 190

Fig. 6 Plot of deceleration parameter versus redshift z for M = 1.5

Fig. 7 Plot of r versus s for M = 1.5 and φ0 = 0.5

−1 and hence we obtain a universe with exponential expan-sion. Also, it can be seen that the model approaches superexponential expansion for q < −1. It can be seen that asEoS parameter of our model attains aggressive phantom re-gion (wde −1) hence we get super exponential expansion.Also, as q < −1 the model expands with super exponentialexpansion. The same phenomenon occurred in our model.

Satefinders: In order to verify the viability of various DEmodels statefinder parameters (r, s) are proposed. Theserepresent well-known DE regions which are given as fol-lows: (r, s) = (1,0), (1,1) represent the ΛCDM and CDM

limit, respectively. However, s > 0 and r < 1 shows the DEregions such as phantom and quintessence-like, s < 0 andr > 1 indicate the Chaplygin gas. In the present study, wedevelop r − s plane for φ0 = 0.5 is shown in Fig. 7. It canbe seen that our DE model corresponds to ΛCDM limit((r, s) = (1,0)) at late times which is in accordance withthe recent observational data. Also, it can be observed thatthe r-s plane correspond to Chaplygin gas model.

6 Conclusion

In this work, we have constructed Kaluza-Klein DE modelwith massive scalar field within the framework of Lyra man-ifold. In order to obtain a deterministic solution of the fieldequations we have used various physically valid conditions.We have computed all the cosmological and kinematicalparameters and discussed their physical significance in thelight of the present cosmological scenario and observations.We summarize our results as follows:

Our Kaluza-Klein DE model with massive scalar field isnon-singular and from a finite volume the model exhibits anexponential expansion leading to early inflation. The decel-eration parameter also confirms that our model starts withexponential expansion (inflation) and attains a super expo-nential expansion at late times. The average anisotropy pa-rameter is constant, the model is uniform throughout and ho-mogeneous. Due to the exponential expansion of the model,all the physical quantities of the model are finite initially andapproach to infinity at late times. The massive scalar field ofour model is positive throughout the evolution of the uni-verse and increases rapidly at present epoch. The behaviorof massive scalar field in our DE model is quite similar tothe behavior of exponential potential which correspond tocosmological scaling solutions obtained by Copeland et al.(2006) and interacting modified ghost SF models of DE con-structed by Jawad (2015). Statefinders plane (r-s plane) anal-ysis shows that the model finally approaches to ΛCDM

limit which is in accordance with the recent observationsand also our DE model corresponds to Chaplygin gas model.The energy density ρΛ of our model is always positive anddecreasing function. It can be seen from the analysis of EoSparameter that the model starts in the quintessence region(−1 < wΛ < −1/3), crosses the phantom divided line andfinally approaches to aggressive phantom region. We ob-served that the skewness parameter is positive in the initialepoch and attains a negative value at late times. We can con-clude that the DE in our model is anisotropic throughoutthe evolution of the universe and hence it helps to study theanisotropies at small angular scales which are play a key rolein the formation of large scale structures of the universe. Inour model, it is observed that the massive scalar field influ-ences all the physical parameters of the model at minimumscale. We hope and believe that the higher dimensional mas-sive scalar field model in Lyra manifold will help to have abetter insight into the understanding of DE which is respon-sible for cosmic acceleration.

Acknowledgements The authors are very much grateful to the re-viewer for constructive comments which certainly improved the qual-ity and presentation of the paper.

Compliance with ethical standards The authors declare that theyhave no potential conflict and will abide by the ethical standards of thisjournal.

190 Page 8 of 8 Y. Aditya et al.

Publisher’s Note Springer Nature remains neutral with regard to ju-risdictional claims in published maps and institutional affiliations.

References

Ade, P.A.R., et al.: Astron. Astrophys. 571, A16 (2014)Aditya, Y., Reddy, D.R.K.: Astrophys. Space Sci. 363, 207 (2018a)Aditya, Y., Reddy, D.R.K.: Eur. Phys. J. C 78, 619 (2018b)Aditya, Y., Reddy, D.R.K.: Astrophys. Space Sci. 364, 3 (2019)Aditya, Y., et al.: Astrophys. Space Sci. 361, 56 (2016)Appelquist, T., et al.: Modern Kaluza-Klein Theories. Addison-Wesley,

Reading (1987)Brans, C., Dicke, R.H.: Phys. Rev. 124, 925 (1961)Collins, C.B., et al.: Gen. Relativ. Gravit. 12, 805 (1980)Copeland, E.J., et al.: Int. J. Mod. Phys. D 15, 1753 (2006)Gron, O.: Astron. Astrophys. 193, 1 (1988)Harko, T., et al.: Phys. Rev. D 84, 024020 (2011)Hinshaw, G.F., et al.: Astrophys. J. Suppl. Ser. 208, 19 (2013)Jawad, A.: Astrophys. Space Sci. 353, 691 (2014)Jawad, A.: Astrophys. Space Sci. 357, 19 (2015)Johri, V.B., Desikan, K.: Gen. Relativ. Gravit. 26, 1217 (1994)Johri, V.B., Sudharsan, R.: Aust. J. Phys. 42, 215 (1989)Kaluza, T.: Sitz. Preuss. Akad. Wiss. Phys.-Math. Kl. 1, 966 (1921)Kantowski, R., Sachs, R.K.: J. Math. Phys. 7, 433 (1966)Kiran, M., et al.: Astrophys. Space Sci. 354, 577 (2014)Klein, O.: Z. Phys. 37, 895 (1926)Kristian, J., Sachs, R.K.: Astrophys. J. 143, 379 (1966)Lyra, G.: Math. Z. 54, 52 (1951)Naidu, R.L.: Can. J. Phys. 97, 330 (2018)

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International Journal of Recent Technology and Engineering (IJRTE)

ISSN: 2277-3878, Volume-8 Issue-6, March 2020

5622

Published By:

Blue Eyes Intelligence Engineering & Sciences Publication

Retrieval Number: F9965038620/2020©BEIESP

DOI:10.35940/ijrte.F9965.038620

Extractive Text Summarization for Sports Articles

using Statistical Method

Sai Teja Polisetty, K Selvani Deepthi, Shaik Ameen, Ravivarma G, M Mounisha

Abstract: The past decade has endorsed a great rise in

Artificial Intelligence. Text summarization which comes under

AI has been an important research area that identifies the

relevant sentences from a piece of text. By Text Summarization,

we can get short and precise information by preserving the

contents of the text. This paper presents an approach for

generating a short and precise extractive summary for the given

document of text. A statistical method for extractive text

summarization of sports articles using extraction of various

features is discussed in this paper. The features taken are TF-

ISF, Sentence Length, Sentence Position, Sentence to Sentence

cohesion, Proper noun, Pronoun. Each sentence is given a score

known as the predictive score is calculated and the summary for

the given document of text is given based on the predictive score

or also known as the rank of the sentence. The accuracy is

checked using the BBC Sports Article dataset and sports articles

of various newspapers like the New York Times, CNN. The

precision of 73% is acquired when compared with System

Generated Summary (SGS) and manual summary, on an

average.

Keywords: Artificial Intelligence, Cosine similarity, Natural

Language Processing, System Generated Summary (SGS), Term

frequency inverse sentence frequency.

I. INTRODUCTION

Text mining can also be referred as Text analysis, which

uses different Artificial Intelligence technologies. Text

mining can be referred to as deriving high-quality

information by drawing patterns and identifying the

important keywords from the unstructured data. Text mining

tasks include text classification which is the classification of

the text for example genre, the sentimental analysis which

tells about how the author wrote the sentences that is in

which tone, document clustering.

Text mining tasks include text classification [2] which is

the classification of the text for example genre, the

sentimental analysis which tells about how the author wrote

the sentences that is in which tone, document clustering

refers to as clustering the documents using unsupervised

learning, text summarization for obtaining a short and

precise text. Each task is different from each other and has

its own probe and methodologies. Natural Language

Processing (NLP)is all about computer and human

Revised Manuscript Received on February 01, 2020. *Correspondence Author

*Sai Teja Polisetty, pursuing B. Tech final year, Department of CSE,

Anil Neerukonda Institute of Technology and Sciences, India, Dr. K. Selvani Deepthi, Associate Professor, Department of CSE, Anil

Neerukonda Institute of Technology and Sciences, India,

Shaik Ameen, pursuing B. Tech final year, Department of CSE, Anil Neerukonda Institute of Technology and Sciences, India,

Ravivarma G, pursuing B. Tech final year, Department of CSE, Anil

Neerukonda Institute of Technology and Sciences, India, M Mounisha, pursuing B. Tech final year, Department of CSE, Anil

Neerukonda Institute of Technology and Sciences, India,

languages interactions. A finest procedure that helps the

system in reading and understanding the given text. With

the rapid increase in data the text summarization manually

consumes a lot of time. That resulted in development of

various summarization methods. Using them, the system

should deliver a précis with the main essence of the

document in reduced size. Text summarization is of two

types: i) extractive ii) abstractive. Firstly, the extractive text

summarization where important sentences and words from

the given text document are identified and those are

combined into the summary in a meaningful way. Finally, in

abstractive text summarization, new sentences are

fabricated, and the summary is given without the loss of

information which is understandable. This can be done

using different methods that include statistical measures,

deep learning techniques with supervised or unsupervised

learning and word vector embedding [10]. The literature

survey is as follows:

Fabio Bif Goularte et.al (2018) [1] proposed a method

using fuzzy rules to retrieve the most important sentences in

a document. Fuzzy logic is applied because it can be used

for sentences having ambiguity. The summaries are trained

for Brazilian Portuguese texts and the summary generated is

tested by the summary given by the domain experts. The

author has cross-checked with other methods as well as

using ROUGE measures. The f-measure obtained is 0.95

which is huge than obtained in any other method. This

method works accurately with only unigrams and it doesn’t

work for semantic verification of the information.

Taeho Jo et.al (2017) [2] proposed a method that uses

the feature vector of certain features and obtains the

correlation between the vectors. In previous works only the

feature value is considered but, in this work, the author

considers both features and its value for better performance

for this KNN (K Nearest Neighbour) is used to check the

correlation between the vectors. The author has used a

binary classification, KNN is used which tells whether a

sentence is important or not. This is way different from text

categorization which uses the previous knowledge for

classifying.

Deepa Anand et.al (2019) [3] proposed a method for

summarizing the Indian legal judgment documents for this

the author has used a neural network approach that is semi-

supervised. The author used two different LSTM

architectures for both word embedding and sentence

embedding. The author used Glove vectors and co-selection

measures are used for analysis. The recall of 0.7 to 0.8

approximately is obtained which is moderate. This approach

cannot be used for long sentences so sentence simplification

methods should be used.

Extractive Text Summarization for Sports Articles using Statistical Method

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Retrieval Number: F9965038620/2020©BEIESP DOI:10.35940/ijrte.F9965.038620

Krishnaveni et.al (2017) [4] proposed a heading-based

text summarization since the context can be known from the

heading. Each sentence is given a rank on considering how

much relevant the sentences are with the heading and the

top sentences are retrieved based on the compression ratio.

Since the summary is generated based on the heading there

is not irrelevancy in context. The summaries are compared

with that of the summaries generated from the main

summariser, Ms-word summarizer.

Jingqiang Chen et.al (2018) [5] proposed a multi-model

neural network-based extractive summarizer. A multi-model

architecture is used on is bidirectional RNN and the other is

CNN. This method encodes the text and images using this

architecture and the probability of the sentence is calculated

using a logistic classifier that has text coverage and text

redundancy as features. The dataset used the Daily mail

corpus by gathering the images from the internet. Encoding

of the images is done using bidirectional RNN.

Jorge V. Tohalino et.al (2018) [6] has proposed a multi-

layered neural network model to generate the extractive

summary based on the sentences having high importance in

the context for multi-documents. A graph which consists of

nodes and edges is built with sentences and words

respectively. The datasets used are CST News for

Portuguese multi- document summarization and DUC-2002,

DUC-2004 for English multi-document summarization. The

word embedding feature is lacking in this work for

generating a better summary.

J.N.Madhuri et.al (2019) [7] has proposed a method to

generate an extractive summary using sentence ranking

technique using the term frequency after the removal of stop

words. It works for any kind of text but cannot semantically

distinguish sentences. The evaluation is done using the

MSWord summarizer and human summarized summary.

The summaries are converted to mp3 format so that it is

easy to evaluate or know the summary.

Nikhil S. Shirwandkar et.al (2018) [8] has proposed an

approach using both Restricted Boltzmann machine and

Fuzzy logic to identify the important sentences. Two

summaries for the same input document are obtained and

then the final summary is obtained by considering both the

summarization get meaningful and no loss of information.

The author did summarization for only a single document.

The f-measure obtained is 84%.

Mahmood Yousefi-Azar et.al (2016) [9] has proposed a

text summarizer for a query-oriented system using an

unsupervised deep learning network which is a deep auto-

encoder to calculate feature space from term frequency.

This is a stochastic machine. The experiments are done on

SKE and BC3 email datasets. A semi-supervised learning

approach works better because it can be used for unlabelled

data.

Aditya Jain et.al (2017) [10] has proposed a neural

network-based text summarization, the input of the nodes is

the values of different features [11] and the dataset used is

DUC-2002. The summary is compared with four online

summarizers. The accuracy of summarization can be exalted

using a large training dataset for a neural network.

II. PROPOSED METHOD

The author used statistical methods for generating

precis. There are seven features which are chosen to precis

the information. All the seven features that are calculated

for each sentence and the sentences are given a ranking

from which the threshold known as the compression ratio

[6] is chosen which tells how many sentences should be

present in the summary. Suppose ‘k’ sentences should be

included in the summary then ‘k’ sentences with the highest

rank are added to the summary. The sentences with a certain

compression ratio should be in ascending order so that the

context of the article will not change.

A. Architecture

1) Input Document: To generate a summary some input

should be given in the file format “.txt”. The text

documents of sports articles are considered for

summarization and are valid since the author has

proposed a method for summarizing sports articles.

2) Pre-processing: Stop word removal and stemming

are done to the text considered as input after lower

casing. Fig2 shows various techniques of pre-

processing the author used. In the lower casing, the

whole document is converted into lowercase

alphabetical letters. This is done to remove the

ambiguity of the words with different casing. The

stop words in the document are removed from the

document because these are the most frequent words

like articles, prepositions, conjunctions which don’t

add any value in defining the importance of a

sentence. The stop words are removed using nltk

(natural language tool kit) library. Stemming [4] is

referred as getting the words reduced to their root

form. It removes the different forms of the same

word present in different sentences. For example,

eating, ate and eat are identified as eat.

Fig.1 Architecture of the proposed method

Fig.2 Steps included in Pre-processing

International Journal of Recent Technology and Engineering (IJRTE)

ISSN: 2277-3878, Volume-8 Issue-6, March 2020

5624

Published By:

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Retrieval Number: F9965038620/2020©BEIESP

DOI:10.35940/ijrte.F9965.038620

3) Feature Extraction – After pre-processing the text,

sentence features are extracted to calculate the predictive

scores.

TF-ISF: Term frequency and inverse document

frequency (TF-IDF) [4] are used for information

retrieval systems. In this paper, the summarization is

done for a single document of sports article and Term

frequency inverse sentence frequency (TF-ISF) is

calculated as follows.

(1)

(2)

Where TFj indicates the term frequency of a word in a

sentence, ISFj is the inverse sentence frequency of a jth

word in a sentence, TF-ISFi(2) is the Term frequency and

inverse sentence frequency of ith

sentence in a document, n

cites to the number of words in a sentence, N cites to the

number of sentences in a document.

Sentence length: This feature provides the less weightage

to short sentences as short sentences are relatively less

important when compared to long sentences [2]. It is

measured by calculating the ratio of no of words in the

sentence to the no of words in the longest sentence of the

document. This feature’s value is calculated using eq. (3).

(3)

Sentence position: In a document the numerical value of

the sentence's position is gauged(4). The position is

evaluated as the normalized percentile score in the range of

0 to 1.

(4)

p is the position of the current sentence and n is the total

number of sentences.

Cosine similarity: Analogy between sentences is calculated

using this property. This briefs how much similar this

sentence with other sentences in a document is [7]. A

vector for each sentence is assessed and (5) is practiced

obtaining the score of the feature.

(5)

A is the first vector and B is the second vector with which

it is compared

Existence of proper noun: Proper names broaching to

places and people might be useful to decide the relevance

of a sentence. This binary feature, with value ‘1’ if a

sentence contains these proper names and 0 otherwise.

Existence of pronoun: Pronouns cites to the persons which

are described in the previous sentences. This binary

feature, with value ‘1’ if a sentence contains these proper

names and 0 otherwise.

Sentence containing scores: Since in sports the score of the

team or the individual participant is important the score is

taken into consideration. This binary feature, with value ‘1’

if a sentence contains these proper names and 0 otherwise.

4) Predictive score of each sentence: The predictive

score is calculated by considering the mean of cosine

similarity feature and TF-ISF feature and taking the

sentence length and sentence position and addition of the

remaining features and the score for each sentence is given

known as the predictive score.

5) Calculate the threshold number of sentences in a

summary: The threshold value known as the

compression ratio tells us how much percent of the

total document is presented as a summary.

6) Precis of the document: This is the output of the text

document which is the summary. Suppose ‘k’ be the

threshold and the ‘k’ sentences with the highest

predictive score should be in ascending order so that

the context of the article will not change.

B. Algorithm

Step1: Take text file as an input

Step 2: Split the entire document into sentences.

Step 3: Convert all the sentences into lowercase.

Step 4: Split the sentences into words and remove the

stop words from the obtained words.

Step 5: Stemming for all the words after stop word

removal is done using nltk toolkit.

Step 6: For all the sentences

Step 7: Calculate the TF-ISF for each word after

stemming and take the mean of all the values of a

sentence and one value is obtained which is the TF-

ISF for one sentence.

Step 8: Repeat step 7.

Step 9: Calculate the sentence length for each

sentence.

Step 10: Check for proper noun in a sentence if

present update to 1 else 0.

Step 11: Check for pronoun in a sentence if present

update to 1 else 0.

Step 12: Calculate the sentence position using (4)

Step 13: Check if a value is present in sentence if

present update to 1 else 0.

Step 14: Convert the sentences into vectors.

Step 15: For all the sentence vector

Step 16: For all the next sentence vector

Step 17: Calculate the cosine similarity. Repeat 16

Step 18: Repeat 15

Step 19: Calculate the mean of all the cosine

similarity with respect to other sentences and obtain a

value for each sentence.

Step 20: Calculate the mean of TF-ISF and cosine

similarity values, obtain predictive score.

Step 21: Sort the predictive scores and list the top

sentences based on the compression ratio.

Step 22: Add the sentences to the final summary.

Step 23: Print the final summary.

C. Pseudo Code

Pre-processing

f=open(“input.txt”,’r’)

m= [] ([] : Empty list)

for line in f:

Extractive Text Summarization for Sports Articles using Statistical Method

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Published By:

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Retrieval Number: F9965038620/2020©BEIESP DOI:10.35940/ijrte.F9965.038620

m.append(line.split(‘.’)

sentence= []

for i in range(len(m)):

sentence.append(m[i].lower())

words_after_stemming=[]

for i in range(len(sentence)):

if(sentence[i] not a stop word):

words_after_stemming.append(PorterSt

emmer().stem(sentence[i]))

TF-ISF

tfisf=[]

for i in range(len(words_after_stemming):

for j in range(len(words_after_stemming[i]):

tf=words_after_stemming[i].count(words_after_ste

mming[i][j])

isf=math.log(N/c); tfisf[i]=tf*isf

Sentence length

maximum_length=max(words_after_stemming.cou

nt())

sentence_length=[]

for i in range(len(words_after_stemming):

sentence_length.append(words_after_stemming.co

unt()/maximum_length)

Proper Noun

propernoun=[]

for i in range(len(words_after_stemming)):

if(pos_tag(words_after_stemming[i].split())[

0][1]==’NN’)

propernoun.append(1)

else

propernoun.append(0)

Pronoun

pronoun=[]

for i in range(len(words_after_stemming)):

if(pos_tag(words_after_stemming[i].

split())[0][1]==’PRP’)

pronoun.append(1)

else

pronoun.append(0)

Sentence position

sentenceposition=[]

for i in range(1,len(words_after_stemming)+1):

sentenceposition[i-

1]=(i/len(words_after_stemming))

Sentence containing scores

score=[]

for i in range(len(words_after_stemming)):

if(sords_after_stemming[i] in

‘0123456789’):

score.append(1)

else

score.append(0)

Cosine similarity

cosine_similarity=[]

generate vectors for words

apply (5)

Mean of TF-ISF and cosine similarity

score=[]

for i in range(len(cosine_similarity)):

score=mean(cosine_similarity[i],tfisf[i])

Sort predictive score

sort.score()

summary=[]

summary.append(score(i,compression_ratio))

Print summary

print(summary)

III. RESULTS

The author considered BBC sports article data set since

it is the worldwide famous for its news. It consists of 737

articles for five different types of sports namely athletics,

cricket, football, rugby, tennis and 101, 124, 265, 147, 100

articles respectively. The model is tested for these articles

and performed with a good amount of accuracy by

generating the relevant context leaving the unnecessary

sentences. The author also considered different sports

articles from New York Times, CNN during the testing of

model for accuracy.

A. Sample Input

Fig.3 Sample input from dataset

B. Sample Output

Fig.4 Sample Output for compression ratio (α=50%)

International Journal of Recent Technology and Engineering (IJRTE)

ISSN: 2277-3878, Volume-8 Issue-6, March 2020

5626

Published By:

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Retrieval Number: F9965038620/2020©BEIESP

DOI:10.35940/ijrte.F9965.038620

IV. DISCUSSION

The evaluation measures used are for context evaluation

which comes under co-selection. In co-selection there are

three types of measures they are precision, recall, f-measure.

Precision (P) can be defined as the fraction of the

number of sentences common in manual and automated

summary to the number of sentences in the automated

summary. Precision (P) can be procured using equation (6).

(6)

Recall (R) can be defined as the ratio of the number of

sentences common in manual and automated summary to

the manual summary. The recall can be calculated using

equation (7).

(7)

F-measure (F) can be defined as the harmonic mean of

precision and recall. F-measure (F) can be procured using

equation (8).

(8)

The System generated summary (SGS) is tested for

accuracy with a manual summary and an online summarizer

tool. The manual summary is given by one of the literates

who have secured a degree in Literature. The online tool

used for generating the summary is Text Compactor. The

author has considered the compression ratio half of the

original document.

Firstly, a manual summary is compared with SGS and a

manual summary is compared with online summarizer tool

for precision, recall, and f-measure. The documents tested

are the sports articles from various newspapers including the

New York Times, CNN. For analysing the accuracy, the

author also checked it with BBC Sports Articles Dataset.

The Table-1 shows the number of sentences in each

document taken from the news articles. The data of 10

documents is shown.

Title Number

of

sentences

in

document

Number

of

sentences

in SGS

(α=50%)

Doc-1 11 5

Doc-2 16 7

Doc-3 15 7

Doc-4 8 4

Doc-5 36 19

Doc-6 15 7

Doc-7 39 20

Doc-8 27 14

Doc-9 19 10

Doc-10 32 16

The evaluation measures for some of the documents are

given below.

Table.1 Shows number of sentences in document as well

as SGS.

Title Manual w.r.t SGS

P R F

Doc-1 0.8 0.8 0.8

Doc-2 0.7142 0.625 0.6662

Doc-3 0.7142 0.8333 0.7689

Doc-4 0.75 0.75 0.75

Doc-5 0.6842 0.7647 0.7219

Doc-6 0.7142 0.7142 0.7142

Doc-7 0.7 0.7368 0.7245

Doc-8 0.7142 0.7692 0.7401

Doc-9 0.8 0.889 0.8336

Doc-10 0.75 0.8 0.7741

Table.2 Comparison of performance between manual

w.r.t SGS

Title Manual w.r.t online summarizer tool

P R F

Doc-1 0.667 0.8 0.7274

Doc-2 0.6667 0.5 0.5714

Doc-3 0.5 0.667 0.5714

Doc-4 0.667 0.5 0.5714

Doc-5 0.7334 0.6470 0.6874

Doc-6 0.5714 0.5714 0.5714

Doc-7 0.834 0.834 0.834

Doc-8 0.5834 0.5384 0.56

Doc-9 0.667 0.667 0.667

Doc-10 0.5625 0.6 0.58

Table.3 Comparison of performance between manual

w.r.t online summarizer tool.

Average

P R F

Manual w.r.t SGS 0.7341 0.7681 0.7493

Manual w.r.t online

summarizer tool

0.6452 0.6328 0.6341

Table. 4 The average of the Precision(P), recall(R), f-

measure(F) for all the 10 documents

Fig.5 Comparison between SGS and Online

summarizer tool

V. CONCLUSION AND FUTURE SCOPE

The extractive text summarization has many tasks

involved in generating the précis. The author proposed a

statistical approach for generating an extractive précis of a

sports articles that uses sentence ranking based on the

selected features for the sentences. The most critical part is

identifying the important sentences without loss of meaning

from the given document. The manual summary is

generated for all the documents and a summary from online

summarizer tool and SGS is

compared with them.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Pre

cisi

on

val

ue

s o

f d

ocu

me

nts

Documents

SGS Online Summarizer Tool

Extractive Text Summarization for Sports Articles using Statistical Method

5627

Published By:

Blue Eyes Intelligence Engineering & Sciences Publication

Retrieval Number: F9965038620/2020©BEIESP DOI:10.35940/ijrte.F9965.038620

Summing up using the proposed method on an average,

73% precision,

76% recall, and 74% f-measure is obtained which when

compared with a manual generated summary and an online

summarizer tool.

In nearly future, the proposed model can also be

developed to a neural network model that takes the values of

different features as input for generating an accurate precis

by considering the contextual meaning. To increase the

accuracy of the precis, various features can also be added.

REFERENCES

1. Goularte, Fábio & Nassar, Silvia & Fileto, Renato & Saggion,

Horacio. (2018). A Text Summarization Method based on Fuzzy Rules and applicable to Automated Assessment. Expert Systems with

Applications. 115. 10.1016/j.eswa.2018.07.047.

2. Jo, Duke Taeho. (2017). K nearest neighbor for text summarization using feature similarity. 1-5. 10.1109/ICCCCEE.2017.7866705.

3. Anand, Deepa & Wagh, Rupali. (2019). Effective Deep Learning

Approaches for Summarization of Legal Texts. Journal of King Saud University - Computer and Information Sciences.

10.1016/j.jksuci.2019.11.015.

4. P. Krishnaveni and S. R. Balasundaram, "Automatic text summarization by local scoring and ranking for improving

coherence," 2017 International Conference on Computing

Methodologies and Communication (ICCMC), Erode, 2017, pp. 59-64. doi: 10.1109/ICCMC.2017.8282539

5. J. Chen and H. Zhuge, "Extractive Text-Image Summarization Using

Multi-Modal RNN," 2018 14th International Conference on Semantics, Knowledge and Grids (SKG), Guangzhou, China, 2018, pp. 245-248.

doi: 10.1109/SKG.2018.00033

6. Valverde Tohalino, Jorge & Amancio, Diego. (2017). Extractive Multi-document Summarization Using Multilayer Networks. Physica A:

Statistical Mechanics and its Applications. 503.

10.1016/j.physa.2018.03.013. 7. J. N. Madhuri and R. Ganesh Kumar, "Extractive Text Summarization

Using Sentence Ranking," 2019 International Conference on Data Science and Communication (IconDSC), Bangalore, India, 2019, pp.1-

3.doi: 10.1109/IconDSC.2019.8817040

8. N. S. Shirwandkar and S. Kulkarni, "Extractive Text Summarization Using Deep Learning," 2018 Fourth International Conference on

Computing Communication Control and Automation (ICCUBEA),

Pune, India, 2018, pp. 1-5. doi: 10.1109/ICCUBEA.2018.8697465

9. Azar, Mahmood & Hamey, Len. (2016). Text Summarization Using

Unsupervised Deep Learning. Expert Systems with Applications. 68. 10.1016/j.eswa.2016.10.017.

10. Jain, Aditya & Bhatia, Divij & Thakur, Manish. (2017). Extractive

Text Summarization Using Word Vector Embedding. 51-55.

10.1109/MLDS.2017.12. 11. K.Selvani Deepthi, Dr.Radhika Y(2015),“Extractive Text

Summarization using Modified Weighing and Sentence Symmetric

Feature Methods”, in an International Journal of Modern Education and Computer Science, ISSN: 2075-0161 Volume 7 No-10 pp: 33-39,

October 2015.

AUTHORS PROFILE

Sai Teja Polisetty is currently pursuing final year in

Computer Science and Engineering, Anil Neerukonda

Institute of Technology and Sciences. His area of

Interest is Machine Learning, Natural Language

Processing, Big Data Analytics.

K. Selvani Deepthi is doctorate in computer science

and engineering. She is working as Associate Professor, Department of CSE, Anil Neerukonda Institute of

Technology. Her area of Interest is Machine Learning,

Natural Language Processing, Text mining.

Shaik Ameen is currently pursuing final year in

Computer Science and Engineering, Anil Neerukonda Institute of Technology and Sciences. His area of

Interest is Machine Learning, Business Intelligence.

Ravivarma G is currently pursuing final year in Computer Science and Engineering, Anil Neerukonda

Institute of Technology and Sciences. His area of

Interest is Deep Learning, Natural Language

Processing.

Mounisha M is currently pursuing final year in

Computer Science and Engineering, Anil Neerukonda Institute of Technology and Sciences. Her area of

Interest is Natural Language Processing.

lnternational Journal of Advanced Science and TechnotogyVot.28, No. I6, (2019), pp. ss4_slrExperimental rnvestigation on performance of Multiwalled

carbon Nano Tubes Grafted carbopFibeiReinforced FrictionMaterial

1 xNaresh KyTul Konada, 2 K.N. S. Suman, 35. Siva Kumar| * As st profess or, M""hani"ai-iep artment, Anits Engineerin g co ilege,Visakhapatnam, India, Email: nareshkonoda@gmoil.Zom

- o-,2Associate profess.li, Departmeniiryecnyl"at nigin;)ring, Andhra (lniversity,

Vis akhap atnam, Indi a, Email : Sumanklka@yaho o. com3As* professir, clr"irrrry i"p*1*ent, Anits Engineering corege,vis akhapatnam, India, Emaii : sivaratmar. chemitryqpnits. edu. in

AbstractThe present research wo1\wa.s prompted

lo sludy the actual performance of carbon fiberreinforced friction materiar using a two wheelerirrk";;il[. n ,rri, worrg carbon fiber(cF) content after truee surface keatment T:th9e^ *idrt ; (-aFr), nitric acid (cF2), andmulti walled carbon nano tubes functionariz.o rnrwCNi-_r)grated on cF (cF3), ismixed with phenol polymer matrix and other remaining ingredients. Friction materials aredeveloped by considering surface treated gT *:",

"r:6 ilt;) using injection mouldingmethod by using a standard die. After fabrication, tiction'nraterials are machined toremove unwanted edges to meet with the standard dimensions oiti.tioo

"rt"riri, n irgouter radius 75mn! inner radius 55 mm and thickness o mm. Three slotted frictionmaterials are obtained at equal distance of 20mm for eacl J"i.ro with a depth of 4mm.The performance of surfaie treated friction rrrut.riur, *iirr three slot condition areevaluated using a test rig ut ,?ryrlg speeds.and pressures ,ppii"o on the friction materiar.These materials are compared wit[ tire existing standard iJbestos fiber friction material(AF)' The test was performed based on.r"*-usu*ptiorr"-ail friction materials arecharactetized for their physical, mechanical, chemic'al uia iriuorogical performance.scanning electron microscope images (sEM); Atomic ror".-*iooscope images (AFM)and energy dispersiveX-Ray spectr-oscopy fBosl *" ou.*"J-for all the sanrples beforeand after performing th3.jest, to identiff uniform distribution of all the ingredients andworn surface analysis of the materials. riellsult-s.1ev"ut ttat,itvrwcNT_F grafted on cFtbree slotted condition friction material),-F3 exhibits goJ li"*u performance intermsof fade, recovery, weight loss and unirorm diskibution ir urr irr" irrgredients compared toother formulations of friction materials.

Keywords: Asbestos fiber, carbonfiber, Fade, Recovery, Two wheerer test rig

1. Introduction

Friction material present in an automobile is mainly responsible for controlling thevehicle at various operating speeds and. pressures during running condition of theautomobile' The main role of friition material present i;;;"rt;*obile is to produce highstable coeffrcient of friction and le'ss wear rateind should sustain variable loads acting onit' The other important characteristics of friction materiar incruJe ability to operate overvarious atmospheric conditions, fade resistan:., g:oq recoverybehavior, less squealingaction, high reliability and less cost of fabricatio, [o "huo

et alioL4).All these importantcharacteristics of a friction material can be achieved by profo selection of fiber, fillermaterials and polymer. Trtrir. Many of the research"r, iruri p".formed trreir work-uyaltering the fiber materials and filler materials. Few of them "vuiuut"o

the performance of

ISSN: 2005-4238 IJASTCopyrisht @ 2019 SERSC

5s4

2019-20

2019-20

Dynamics of Continuous, Discrete and Impulsive SystemsSeries A: Mathematical Analysis 27 (2020) 181-194Copyright c©2020 Watam Press http://www.watam.org

PPF DEPENDENT FIXED POINTS OFCONTRACTIVE TYPE MAPPINGS

G.V.R. Babu1 and M. Vinod Kumar2

1,2 Department of Mathematics, Andhra University, Visakhapatnam-530 003, India.

2 Permanent Address : Department of Mathematics, Anil Neerukonda Institute ofTechnology and Sciences, Sangivalasa, Visakhapatnam-531 162, India.

1E-mail : gvr babu@hotmail.com 2E-mail : dravinodvivek@gmail.com

Abstract. We introduce generalized Kannan, generalized Chatterjea, and generalized

Kannan and Chatterjea type mappings and weakly Chatterjea contractive type mappings

which are non-self mappings defined on a Banach space of all continuous functions with the

range space is a Banach space and prove the existence and uniqueness of PPF dependent

fixed points of these mappings. We provide examples in support of our results.

Keywords. PPF dependent fixed point, Razumikhin class, generalized Kannan type

mapping, generalized Chatterjea type mapping, generalized Kannan and Chatterjea type

mapping, weakly Chatterjea contractive type mapping.

AMS (MOS) Subject Classification: 47H10, 54H25.

1 Introduction

Fixed point theory plays an important role in nonlinear analysis and itsapplications. In 1922, the Polish mathematician Banach[5] established theexistence and uniqueness of a fixed point of a contraction map in completemetric spaces and this result is known as Banach contraction principle. TheBanach contraction principle is one of the basic and important tool whichprovides an idea for proving the existence of solutions of ordinary differentialequations and integral equations and for solving various problems in mathe-matical science and engineering. Thereafter, Chatterjea[8], Kannan[12] andReich[17] proved different types of fixed point theorems in complete metricspaces.

Several authors extended, generalized and improved Banach fixed pointtheorem in different ways. In 1968, Kannan[12] defined a mapping known asKannan mapping and he proved the existence and uniqueness of fixed pointin a complete metric space. Several years later, in 1972, Chatterjea[8] alsodefined a mapping known as Chatterjea mapping and he proved the existenceand uniqueness of fixed point in a complete metric space, for more details werefer [1, 3, 15, 16].

182 G.V.R. Babu and M.V. Kumar

On the other hand, some authors proved the existence and uniqueness offixed point of non-self mapppings. One of these type of fixed points is a PPFdependent fixed point. In 1977, Bernfeld, Lakshmikantham and Reddy[7]introduced the concept of fixed point for mappings that have different do-mains and ranges which is called PPF (Past, Present and Future) dependentfixed point. Furthermore, they introduced the notation of Banach type con-traction for a non-self mapping and proved the existence of PPF dependentfixed points of Banach type contractive mappings in the Razumikhin class,for more details we refer [2, 4, 6, 10, 11, 14].

In 2009, Choudhury[9] introduced C-weakly contractive self mappingsand proved the existence and uniqueness of fixed points. Kasamsuk Ung-chittrakool [13] introduced non-self Kannan-Chatterjea type mappings andproved the existence of best proximity points in complete metric spaces.

Based on these concepts, in this paper we introduce generalized Kannan,generalized Chatterjea, and generalized Kannan and Chatterjea type map-pings and weakly Chatterjea contractive type mappings which are non-selfmappings defined on a Banach space of all continuous functions with therange space is a Banach space and prove the existence and uniqueness ofPPF dependent fixed points of these mappings.

Throughout this paper, we denote the real line by R, R+ = [0,∞), andthe set of all natural numbers by N. Suppose that (E, ||.||E) is a Banachspace, I = [a, b] ⊆ R and E0 = C(I, E) denotes the set of all continuousfunctions on I equipped with the supremum norm ||.||E0

and we define it by||φ||E0

= supa6t6b

||φ(t)||E for any φ ∈ E0.

2 Razumikhin class of functions and its prop-erties

For a fixed c ∈ I, the Razumikhin class Rc of functions in E0 is definedby Rc =

φ ∈ E0/ ||φ||E0

= ||φ(c)||E

. Clearly every constant function fromI to E belongs to Rc so that Rc is a non-empty subset of E0 .

Definition 2.1. Let Rc be the Razumikhin class of continuous functions inE0. Then we say that

i) the class Rc is algebraically closed with respect to the differenceif φ− ψ ∈ Rc whenever φ, ψ ∈ Rc.

ii) the class Rc is algebraically closed if φ+ ψ ∈ Rc whenever φ, ψ ∈ Rc.iii) the class Rc is topologically closed if it is closed with respect to the

topology on E0 by the norm ||.||E0.

iv) the class Rc is a convex set if for any λ ∈ [0, 1], λRc + (1− λ)Rc ⊆ Rc.

The Razumikhin class of functions Rc has the following properties.

PPF Dependent Fixed Points of Contractive Type Mappings 183

Theorem 2.2. Let Rc be the Razumikhin class of functions in E0. Theni) E0 = ∪Rc

c∈[a,b].

ii) for any φ ∈ Rc and α ∈ R, we have αφ ∈ Rc.iii) the Razumikhin class Rc is topologically closed with respect to the norm

defined on E0.iv) ∩Rc

c∈[a,b]= φ ∈ E0/φ : I → E is constant .

v) if the Razumikhin class Rc is algebraically closed with respect to thedifference then Rc is a convex set.

Proof. i), ii) and iii) follow from Theorem 2.1 of [4].iv) Let φ ∈ ∩Rc

c∈[a,b]. Then φ ∈ Rc for any c ∈ I.

By the definition of Rc, we have ||φ||E0= ||φ(c)||E for any c ∈ I.

Therefore φ is a constant function and hence∩Rcc∈[a,b]

= φ ∈ E0/φ : I → E is constant .

v) Suppose that Rc is algebraically closed with respect to the difference.Then by the definition Rc −Rc ⊆ Rc.From (ii), we have λRc ⊆ Rc for all λ ∈ [0, 1].Therefore λRc+(1−λ)RC ⊆ Rc for any λ ∈ [0, 1] and hence Rc is a convexset.

The following example shows that, for a fixed c ∈ [a, b] there exist twofunctions in Rc such that their difference is in Rc but not their sum.

Example 2.3. Let I = [0, 1], E = R. Fix c = 12 ∈ [0, 1]. Let E0 = C(I,R).

We define φ : I → E by

φ(x) =

x if x ∈ [0, 12 ]2x(1− x) if x ∈ [ 12 , 1].

Clearly φ ∈ Rc.We define ψ : I → E by

ψ(x) =

−2x2 if x ∈ [0, 12 ]x− 1 if x ∈ [ 12 , 1].

Clearly ψ ∈ Rc.Clearly

(φ+ψ)(x) =

x− 2x2 if x ∈ [0, 12 ]3x− 2x2 − 1 if x ∈ [ 12 , 1],

and

(φ−ψ)(x) =

x+ 2x2 if x ∈ [0, 12 ]x− 2x2 + 1 if x ∈ [ 12 , 1].

184 G.V.R. Babu and M.V. Kumar

Clearly ||φ+ ψ||E06=∣∣∣∣φ( 1

2 ) + ψ( 12 )∣∣∣∣E

and ||φ− ψ||E0=∣∣∣∣φ( 1

2 )− ψ( 12 )∣∣∣∣E.

Therefore φ+ ψ /∈ Rc and φ− ψ ∈ Rc .

The following example shows that, there exists a subset F0 of E0 whichis algebraically closed with respect to the difference.

Example 2.4. Let I = [0, 2], E = R, E0 = C(I,R) and n ∈ Z. Fix c = 1.We define φn : I → E by

φn(x) =

nx2 if x ∈ [0, 1]nx2 if x ∈ [1, 2].

Clearly φn is continuous on [0, 2] and ||φn||E0 = ||φn(c)||E so that φn ∈ Rc.Let F0 = φn/n ∈ Z. Then F0 is algebraically closed with respect to thedifference and F0 ⊆ Rc ⊆ E0.

Lemma 2.5. Let c ∈ I be a fixed element. Let Rc be algebraically closedwith respect to the difference. We define a relation 6 on E0 by φ 6 ψ if andonly if φ− ψ ∈ Rc for any φ, ψ ∈ E0. Theni) the relation 6 is an equivalence relation on E0.ii) the equivalence class of φ contains the elements of the form φ+ η

for some η ∈ Rc.iii) for any φ, ψ ∈ E0, φ 6 ψ if and only if the equivalence class of φ and ψ

are equal.iv) any two equivalence classes are either equal or disjoint.

Proof. Let c ∈ I be a fixed element.Let Rc be algebraically closed with respect to the difference.Then by definition φ− ψ ∈ Rc whenever φ, ψ ∈ Rc.We define a relation 6 on E0 by φ 6 ψ if and only if φ− ψ ∈ Rcfor any φ, ψ ∈ E0.i) Since Rc is algebraically closed with respect to the difference,

we have 6 is reflexive, symmetric and transitive.Therefore 6 is an equivalence relation on E0.

ii) We denote the equivalence class of φ by [φ] and defined by[φ] = ψ ∈ E0/ψ 6 φ .Since φ ∈ [φ], we have [φ] is a non-empty subset of E0.We consider

[φ] = ψ ∈ E0/ψ 6 φ= ψ ∈ E0/ψ − φ ∈ Rc= ψ ∈ E0/ψ = φ+ η for some η ∈ Rc.

i.e., [φ] = φ+ η/for some η ∈ Rc.iii) Since Rc is algebraically closed with respect to the difference, we have

φ 6 ψ if and only if [φ] = [ψ].iv) Suppose that [φ] and [ψ] are not disjoint for φ, ψ ∈ E0.

Let η ∈ [φ] ∩ [ψ]. Then η = φ+ ζ1 and η = ψ + ζ2 for some ζ1, ζ2 ∈ Rc.Since Rc is algebraically closed with respect to the difference, we have

PPF Dependent Fixed Points of Contractive Type Mappings 185

φ− ψ ∈ Rc. Therefore φ 6 ψ and hence [φ] = [ψ].Hence any two equivalence classes are either equal or disjoint.

Theorem 2.6. Let c ∈ I be a fixed element. Let Rc be algebraically closedwith respect to the difference. Then the set E0 can be partitioned into disjointequivalence classes.

Proof. Let Rc be algebraically closed with respect to the difference.By definition, [φ] = ψ ∈ E0/ψ 6 φ .Clearly [φ] ⊆ E0 for any φ ∈ E0.Therefore ∪

φ∈E0

[φ] ⊆ E0. (2.6.1)

Let φ ∈ E0. Then φ ∈ [φ] ⊆ ∪φ∈E0

[φ].

This is true for every φ ∈ E0. Therefore E0 ⊆ ∪φ∈E0

[φ]. (2.6.2)

From (2.6.1) and (2.6.2), we have E0 = ∪φ∈E0

[φ]. (2.6.3)

By Lemma 2.5 of (iv), [φ] = [ψ] or [φ]∩[ψ] = ∅ for any φ, ψ ∈ E0. (2.6.4)From (2.6.3) and (2.6.4), the set E0 is partitioned into disjoint equivalenceclasses under the equivalence relation defined by φ 6 ψ if and only ifφ− ψ ∈ Rc for any φ, ψ ∈ E0.

The following lemma is useful to prove our main results in Section 3.

Lemma 2.7. Let (E, ||.||E) be a Banach space, I = [a, b] ⊆ R andE0 = C(I, E) be the set of all continuous functions on I equipped with thesupremum norm ||.||E0

and we define it by ||φ||E0= sup

a6t6b||φ(t)||E for any

φ ∈ E0. Let φn be a sequence in E0 such that ||φn − φn+1||E0→ 0 as

n → ∞. If φn is not a Cauchy sequence then there exist an ε > 0 andsequences of positive integers m(k) and n(k) with n(k) > m(k) > k suchthat

∣∣∣∣φm(k) − φn(k)∣∣∣∣E0

> ε,∣∣∣∣φm(k) − φn(k)−1

∣∣∣∣E0

< ε and

i) limk→∞

∣∣∣∣φm(k) − φn(k)∣∣∣∣E0

= ε ii) limk→∞

∣∣∣∣φm(k) − φn(k)−1∣∣∣∣E0

= ε

iii) limk→∞

∣∣∣∣φm(k)−1 − φn(k)∣∣∣∣E0

= ε iv) limk→∞

∣∣∣∣φm(k)−1 − φn(k)−1∣∣∣∣E0

= ε.

Proof. Runs as that of Lemma 1.4 of [3].

Definition 2.8. [7] Let T : E0 → E be a mapping. A function φ ∈ E0 issaid to be a PPF dependent fixed point of T if T (φ) = φ(c) for some c ∈ I.

Definition 2.9. [7] Let T : E0 → E be a mapping. Then T is called aBanach type contraction if there exists k ∈ [0, 1) such that for any φ, ψ ∈ E0

||Tφ− Tψ||E 6 k ||φ− ψ||E0.

Theorem 2.10. [7] Let T : E0 → E be a Banach type contraction. Let Rc bean algebraically closed with respect to the difference and topologically closed.Then T has a unique PPF dependent fixed point in Rc.

186 G.V.R. Babu and M.V. Kumar

Remark 2.11. By Theorem 2.2 of (iii), Rc is topologically closed with respectto the norm topology on E0 and hence Theorem 2.10 is still valid if we dropthe condition ‘Rc is topologically closed’.

Definition 2.12. Let c ∈ I be a fixed element. A mapping T : E0 → E issaid to be a Kannan type mapping if there exists a constant k ∈ [0, 12 ) suchthat for any φ, ψ ∈ E0

||Tφ− Tψ||E 6 k[||φ(c)− Tφ||E + ||ψ(c)− Tψ||E ]. (2.12.1)

Definition 2.13. Let c ∈ I be a fixed element. A mapping T : E0 → E issaid to be a Chatterjea type mapping if there exists a constant k ∈ [0, 12 ) suchthat for any φ, ψ ∈ E0

||Tφ− Tψ||E 6 k[||φ(c)− Tψ||E + ||ψ(c)− Tφ||E ]. (2.13.1)

Definition 2.14. Let c ∈ I be a fixed element. A mapping T : E0 → E issaid to be a weakly Chatterjea contractive type mapping if for any φ, ψ ∈ E0

||Tφ− Tψ||E 6 12 [||φ(c)− Tψ||E + ||ψ(c)− Tφ||E ]

− f(||φ(c)− Tψ||E , ||ψ(c)− Tφ||E) (2.14.1)where f : R+ × R+ → R+ is a continuous function such that f(x, y) = 0 ifand only if x = 0 = y.

Definition 2.15. Let c ∈ I be a fixed element. A mapping T : E0 → Eis said to be a generalized Kannan type mapping if there exist non-negativeconstants k1 and k2 such that k1 + 2k2 < 1 and for any φ, ψ ∈ E0

||Tφ−Tψ||E 6 k1[||φ−ψ||E0]+k2[||φ(c)−Tφ||E+||ψ(c)−Tψ||E ]. (2.15.1)

Definition 2.16. Let c ∈ I be a fixed element. A mapping T : E0 → E issaid to be a generalized Chatterjea type mapping if there exist non-negativeconstants k1 and k2 such that k1 + 2k2 < 1 and for any φ, ψ ∈ E0

||Tφ−Tψ||E 6 k1[||φ−ψ||E0]+k2[||φ(c)−Tψ||E+||ψ(c)−Tφ||E ]. (2.16.1)

Definition 2.17. Let c ∈ I be a fixed element. A mapping T : E0 → E issaid to be a generalized Kannan and Chatterjea type mapping if there existnon-negative constants k1, k2, k3 such that k1 + 2k2 + 2k3 < 1 and for anyφ, ψ ∈ E0

||Tφ− Tψ||E 6 k1[||φ− ψ||E0] + k2[||φ(c)− Tφ||E + ||ψ(c)− Tψ||E ]+ k3[||φ(c)− Tψ||E + ||ψ(c)− Tφ||E ]. (2.17.1)

Remark 2.18. From the above defintions we observe the following.i) Let f(x, y) = [ 12 − k][x+ y] (where k ∈ [0, 12 )). Then (2.13.1) is a special

case of (2.14.1).ii) Let k2 = k3 = 0. Then (2.17.1) is a Banach type contraction.

iii) Let k1 = k2 = 0. Then (2.17.1) is a Chatterjea type mapping.iv) Let k1 = k3 = 0. Then (2.17.1) is a Kannan type mapping.v) Let k3 = 0. Then (2.17.1) is a generalized Kannan type mapping.

vi) Let k2 = 0. Then (2.17.1) is a generalized Chatterjea type mapping.

PPF Dependent Fixed Points of Contractive Type Mappings 187

3 Main Results

Theorem 3.1. Let (E, ||.||E) be a Banach space, I = [a, b] ⊆ R,E0 = C(I, E). Fix c ∈ I. Assume that Rc is algebraically closed with respectto the difference. If T : E0 → E is a Chatterjea type mapping then T has aunique PPF dependent fixed point in Rc.

Proof. Let φ0 ∈ Rc ⊆ E0. Clearly Tφ0 ∈ E.Let x1 = Tφ0. We choose φ1 ∈ Rc such that x1 = φ1(c).Then Tφ0 = φ1(c). Since Rc is algebraically closed with respect to thedifference, we have ||φ1 − φ0||E0 = ||φ1(c)− φ0(c)||E .Clearly Tφ1 ∈ E. Let x2 = Tφ1. We choose φ2 ∈ Rc such that x2 = φ2(c).Then Tφ1 = φ2(c). Since Rc is algebraically closed with respect to thedifference, we have ||φ2 − φ1||E0

= ||φ2(c)− φ1(c)||E .On continuing this process, we define a sequence φn inductively byTφn = φn+1(c) and ||φn+1 − φn||E0

= ||φn+1(c)− φn(c)||E (3.1.1)for any n ∈ N ∪ 0. If φn+1 = φn for some n ∈ N ∪ 0, thenTφn = φn+1(c) = φn(c) so that T has a PPF dependent fixed point in Rc.Suppose that φn+1 6= φn for any n ∈ N ∪ 0.We consider||φn+1 − φn||E0

= ||φn+1(c)− φn(c)||E = ||Tφn − Tφn−1||E6 k[||φn(c)− Tφn−1||E + ||φn−1(c)− Tφn||E ]= k[||φn(c)− φn(c)||E + ||φn−1(c)− φn+1(c)||E ]= k[||φn−1 − φn+1||E0

]6 k[||φn−1 − φn||E0

+ ||φn − φn+1||E0].

Hence||φn+1 − φn||E0 6 ( k

1−k )||φn − φn−1||E0 .

6 ( k1−k )2||φn−1 − φn−2||E0

...6 ( k

1−k )n||φ1 − φ0||E0 .Therefore||φn+1 − φn||E0

6 ( k1−k )n||φ1 − φ0||E0

. (3.1.2)Now for any m > n||φm − φn||E0

6 ||φm − φm−1||E0+ ||φm−1 − φm−2||E0

+ ...+ ||φn+1 − φn||E0

6 ( k1−k )m−1||φ1 − φ0||E0

+ ( k1−k )m−2||φ1 − φ0||E0

+ ...

+( k1−k )n||φ1−φ0||E0

= ( k1−k )n[1 + ( k

1−k ) + ( k1−k )2 + ..+ ( k

1−k )m−n−1]||φ1 − φ0||E0

6 ( k1−k )n 1

1− k1−k

||φ1 − φ0||E0.

Since 0 6 k < 12 , we have k

1−k < 1 .

Therefore as n→∞, ( k1−k )n → 0 and hence φn is a Cauchy sequence in E0.

Since E0 is complete, we have the sequence φn converges and limn→∞

φn = φ∗,

where φ∗ ∈ E0. Since Rc is topologically closed, we have φ∗ ∈ Rc.We now show that φ∗ is a PPF dependent fixed point of T.

188 G.V.R. Babu and M.V. Kumar

We consider||Tφ∗ − φ∗(c)||E 6 ||Tφ∗ − Tφn||E + ||Tφn − φ∗(c)||E

6 k[||φ∗(c)− Tφn||E + ||φn(c)− Tφ∗||E ] + ||Tφn − φ∗(c)||E= k[||φ∗ − φn+1||E0

+ ||φn(c)− Tφ∗||E ] + ||φn+1 − φ∗||E0

6 k[||φ∗−φn+1||E0]+k[||φn(c)−φ∗(c)||E+ ||φ∗(c)−Tφ∗||E ]

+ ||φn+1 − φ∗||E0.

Hence [1− k]||Tφ∗ − φ∗(c)||E 6 k[||φ∗ − φn+1||E0 + ||φn − φ∗||E0 ]+ ||φn+1 − φ∗||E0 .

On applying limits as n→∞, we get [1− k]||Tφ∗ − φ∗(c)||E 6 0.Therefore Tφ∗ = φ∗(c) and hence φ∗ is a PPF dependent fixed point of T .We now show that φ∗ is a unique PPF dependent fixed point of T .Let φ∗, ψ∗ be any two PPF dependent fixed points of T in Rc.By definition we get Tφ∗ = φ∗(c) and Tψ∗ = ψ∗(c).We consider||φ∗ − ψ∗||E0

= ||φ∗(c)− ψ∗(c)||E = ||Tφ∗ − Tψ∗||E6 k[||φ∗(c)− Tψ∗||E + ||ψ∗(c)− Tφ∗||E ]6 k[||φ∗(c)− ψ∗(c)||E + ||ψ∗(c)− φ∗(c)||E ] = 2k||φ∗ − ψ∗||E0

and hence [1− 2k]||φ∗ − ψ∗||E0 6 0. Therefore φ∗ = ψ∗.Hence T has a unique PPF dependent fixed point in Rc.

Example 3.2. Let I = [0, 1], E = R. Fix c = 12 ∈ [0, 1]. Let E0 = C(I,R).

We define T : E0 → E by T (φ) = 14φ( 1

2 ) + 3128 , φ ∈ E0.

For any φ, ψ ∈ E0

||Tφ− Tψ||E = 14 ||φ( 1

2 )− ψ( 12 )||E

6 14 [||φ( 1

2 )− Tψ]E + ||Tψ − Tφ||E + ||Tφ− ψ( 12 )||E ].

Therefore 34 ||Tφ− Tψ||E 6 1

4 [||φ( 12 )− Tψ||E + ||ψ( 1

2 )− Tφ||E ] and hence||Tφ− Tψ||E 6 1

3 [||φ( 12 )− Tψ||E + ||ψ( 1

2 )− Tφ||E ].Therefore T is a Chatterjea type mapping.Therefore T satisfies all the hypotheses of Theorem 3.1 and hence T has aPPF dependent fixed point. We now compute this PPF dependent fixed point.We define φ : I → E by

φ(x) =

x2

8 if x ∈ [0, 12 ]132 if x ∈ [ 12 , 1].

Clearly, ||φ||E0= 1

32 =∣∣∣∣φ( 1

2 )∣∣∣∣E

and T (φ) = φ( 12 ). Therefore φ is a PPF

dependent fixed point of T in Rc.

The following example shows that the Chatterjea type mapping T mayhave more than one PPF dependent fixed point in Rc when Rc is notalgebraically closed with respect to the difference.

Example 3.3. Let I, E,E0, c and T be defined as in the Example 3.2.We define ψ : I → E by

ψ(x) =

116

√x2 if x ∈ [0, 12 ]

116

√1−x2 if x ∈ [ 12 , 1].

PPF Dependent Fixed Points of Contractive Type Mappings 189

Clearly, ||ψ||E0= 1

32 =∣∣∣∣ψ( 1

2 )∣∣∣∣E

and T (ψ) = ψ( 12 ).

Therefore ψ is a PPF dependent fixed point of T in Rc.Now

(φ−ψ)(x) =

x2

8 −116

√x2 if x ∈ [0, 12 ]

132 −

116

√1−x2 if x ∈ [ 12 , 1].

Clearly ||φ− ψ||E06= ||φ(c)− ψ(c)||E , so that φ− ψ /∈ Rc.

Therefore Rc is not algebraically closed with respect to the difference withc = 1

2 .

Theorem 3.4. Under the hypotheses of Theorem 3.1, if T : E0 → E is aweakly Chatterjea contractive type mapping then T has a unique PPF depen-dent fixed point in Rc.

Proof. Let φ0 ∈ Rc ⊆ E0. Clearly Tφ0 ∈ E.Let x1 = Tφ0.We choose φ1 ∈ Rc such that x1 = φ1(c).Then Tφ0 = φ1(c). Since Rc is algebraically closed with respect to thedifference, we have ||φ1 − φ0||E0

= ||φ1(c)− φ0(c)||E .As in the proof of the Theorem 3.1, here also we define a sequence φn suchthat Tφn = φn+1(c) and ||φn+1 − φn||E0

= ||φn+1(c)− φn(c)||E (3.4.1)for any n ∈ N ∪ 0.Without loss of generality, we suppose that φn+1 6= φn for any n ∈ N ∪ 0.We consider||φn+1 − φn||E0

= ||φn+1(c)− φn(c)||E= ||Tφn − Tφn−1||E6 1

2 [||φn(c)− Tφn−1||E + ||φn−1(c)− Tφn||E ]− f(||φn(c)− Tφn−1||E , ||φn−1(c)− Tφn||E ])

= 12 [||φn(c)− φn(c)||E + ||φn−1(c)− φn+1(c)||E ]

− f(||φn(c)− φn(c)||E , ||φn−1(c)− φn+1(c)||E)= 1

2 ||φn−1−φn+1||E0−f(0, ||φn−1−φn+1||E0

) (3.4.2)6 1

2 ||φn−1 − φn+1||E0

6 12 [||φn−1−φn||E0

+ ||φn−φn+1||E0]. (3.4.3)

Therefore ||φn+1 − φn||E0 6 ||φn − φn−1||E0 so that the sequence||φn+1 − φn||E0 is a decreasing sequnce in R+, and hence it is convergent.Let lim

n→∞||φn+1− φn||E0

= r (say). (3.4.4)

We now show that r = 0.From (3.4.3) we have||φn+1 − φn||E0

6 12 ||φn−1−φn+1||E0

6 12 [||φn−1−φn||E0

+ ||φn−φn+1||E0].

On applying limits as n→∞, we getr 6 1

2 [ limn→∞

||φn−1 − φn+1||E0] 6 1

2 [r + r].

Therefore limn→∞

||φn−1−φn+1||E0= 2r. (3.4.5)

From (3.4.2)||φn+1 − φn||E0 6 1

2 ||φn−1 − φn+1||E0 − f(0, ||φn−1 − φn+1||E0).On applying limits as n→∞, we get

190 G.V.R. Babu and M.V. Kumar

r 6 12 [2r]− f(0, 2r), which implies that f(0, 2r) 6 0.

Therefore f(0, 2r) = 0 and hence r = 0.From (3.4.4), we havelimn→∞

||φn+1−φn||E0= 0. (3.4.6)

We now show that φn is a Cauchy sequnce. Suppose that φn is not aCauchy sequence. Then there exist an ε > 0 and sequence of positive integersm(k) and n(k) with n(k) > m(k) > k such that||φm(k)−φn(k)||E0 > ε, ||φm(k)−φn(k)−1||E0 < ε. (3.4.7)We consider

ε 6 ||φm(k) − φn(k)||E0= ||φm(k)(c)− φn(k)(c)||E

= ||Tφm(k)−1 − Tφn(k)−1||E6 1

2 [||φm(k)−1(c)− Tφn(k)−1||E + ||φn(k)−1(c)− Tφm(k)−1||E ]− f(||φm(k)−1(c)− Tφn(k)−1||E , ||φn(k)−1(c)− Tφm(k)−1||E)

= 12 [||φm(k)−1 − φn(k)||E0

+ ||φn(k)−1 − φm(k)||E0]

− f(||φm(k)−1 − φn(k)||E0, ||φn(k)−1 − φm(k)||E0

]).On applying limits as k →∞, by Lemma 2.7 we getε 6 1

2 [ε+ ε]− f(ε, ε) and hence ε = 0, a contradiction.Therefore φn is a Cauchy sequence in E0.Since E0 is complete, we have the sequence φn converges and lim

n→∞φn = φ∗,

where φ∗ ∈ E0. Since Rc is topologically closed, we have φ∗ ∈ Rc.We now show that φ∗ is a PPF dependent fixed point of T .We consider||Tφ∗ − φ∗(c)||E 6 ||Tφ∗ − Tφn||E + ||Tφn − φ∗(c)||E

6 12 [||φ∗(c)− Tφn||E + ||φn(c)− Tφ∗||E ]−f(||φ∗(c)−Tφn||E , ||φn(c)−Tφ∗||E)+||Tφn−φ∗(c)||E

= 12 [||φ∗(c)− φn+1(c)||E + ||φn(c)− Tφ∗||E ]− f(||φ∗(c)− φn+1(c)||E , ||φn(c)− Tφ∗||E)+ ||φn+1(c)− φ∗(c)||E .

On applying limits as n→∞, we get||Tφ∗ − φ∗(c)||E 6 1

2 [0 + ||φ∗(c)− Tφ∗||E ]− f(0, ||φ∗(c)− Tφ∗||E) + 06 1

2 ||φ∗(c)− Tφ∗||E .

Hence||Tφ∗ − φ∗(c)||E 6 0.Therefore Tφ∗ = φ∗(c) and hence φ∗ is a PPF dependent fixed point of T .We now show that φ∗ is a unique PPF dependent fixed point of T .Let φ∗, ψ∗ be any two PPF dependent fixed points of T in Rc.Then by definiton Tφ∗ = φ∗(c) and Tψ∗ = ψ∗(c).We consider||φ∗ − ψ∗||E0 = ||φ∗(c)− ψ∗(c)||E = ||Tφ∗ − Tψ∗||E

6 12 [||φ∗(c)− Tψ∗||E + ||ψ∗(c)− Tφ∗||E ]

− f(||φ∗(c)− Tψ∗||E , ||ψ∗(c)− Tφ∗||E)= 1

2 [||φ∗−ψ∗||E0+ ||ψ∗−φ∗||E0

]− f(||φ∗−ψ∗||E0, ||ψ∗−φ∗||E0

)and hence f(||φ∗ − ψ∗||E0

, ||φ∗ − ψ∗||E0) 6 0 so that ||φ∗ − ψ∗||E0

= 0.Therefore φ∗ = ψ∗ and hence T has a unique PPF dependent fixed point inRc.

PPF Dependent Fixed Points of Contractive Type Mappings 191

Example 3.5. Let I = [0, 2], E = R. Fix c = 1 ∈ [0, 2]. Let E0 = C(I,R).Let f : R+ × R+ → R+ be a function defined by f(x, y) = x+y

3 for anyx, y ∈ R+. Then f is a continous function such thatf(x, y) = 0 ⇐⇒ x = 0 = y.We define T : E0 → E by T (φ) = 1

7φ(1) + 187 , φ ∈ E0.

For any φ, ψ ∈ E0, we consider||Tφ− Tψ||E = 1

7 ||φ(1)− ψ(1)||E6 1

7 [||φ(1)− Tψ||E + ||Tψ − Tφ||E + ||Tφ− ψ(1)||E ].Hence it follows that||Tφ− Tψ||E 6 1

6 [||φ(1)− Tψ||E + ||ψ(1)− Tφ||E ]= 1

2 [||φ(1)− Tψ||E + ||ψ(1)− Tφ||E ]− 1

3 [||φ(1)− Tψ||E + ||ψ(1)− Tφ||E ]= 1

2 [||φ(1)− Tψ||E + ||ψ(1)− Tφ||E ]− f(||φ(1)− Tψ||E , ||ψ(1)− Tφ||E).

Therefore T is a weakly Chatterjea contractive type mapping and hence sat-isfies the hypothesis of Theorem 3.4, so that T has a PPF dependent fixedpoint in Rc.We now compute this PPF dependent fixed point of T in Rc.We define φ : I → E by

φ(x) =

3√x if x ∈ [0, 1]

4− x2 if x ∈ [1, 2].

Clearly, ||φ||E0= 3 = ||φ(1)||E . Therefore φ ∈ Rc and T (φ) = φ(1) so that

φ is a PPF dependent fixed point of T in Rc.

Theorem 3.6. Under the hypotheses of Theorem 3.1, if T : E0 → E is ageneralized Kannan and Chatterjea type mapping then T has a unique PPFdependent fixed point in Rc.

Proof. Let φ0 ∈ Rc ⊆ E0. Clearly Tφ0 ∈ E.Let x1 = Tφ0.We choose φ1 ∈ Rc such that x1 = φ1(c).Then Tφ0 = φ1(c). Since Rc is algebraically closed with respect to thedifference, we have ||φ1 − φ0||E0 = ||φ1(c)− φ0(c)||E .As in the proof of the Theorem 3.1, here also we define a sequence φnsuch that Tφn = φn+1(c) and ||φn+1 − φn||E0

= ||φn+1(c)− φn(c)||E for anyn ∈ N ∪ 0.Without loss of generality we suppose that φn+1 6= φn for any n ∈ N ∪ 0.We consider||φn+1 − φn||E0

= ||φn+1(c)− φn(c)||E= ||Tφn − Tφn−1||E6 k1[||φn − φn−1||E0

]+ k2[||φn(c)− Tφn||E + ||φn−1(c)− Tφn−1||E ]+ k3[||φn(c)− Tφn−1||E + ||φn−1(c)− Tφn||E ]

= k1[||φn − φn−1||E0 ]+ k2[||φn − φn+1||E0

+ ||φn−1 − φn||E0]

192 G.V.R. Babu and M.V. Kumar

+ k3[||φn − φn||E0+ ||φn−1 − φn+1||E0

]= k1[||φn − φn−1||E0 ]

+ k2[||φn − φn+1||E0 + ||φn−1 − φn||E0 ]+ k3[||φn−1 − φn+1||E0

]6 k1[||φn − φn−1||E0

]+ k2[||φn − φn+1||E0

+ ||φn−1 − φn||E0]

+ k3[||φn−1 − φn||E0 + ||φn − φn+1||E0 ]= [k1 + k2 + k3]||φn − φn−1||E0 + [k2 + k3]||φn − φn+1||E0 .

Therefore [1 − k2 − k3]||φn+1 − φn||E06 [k1 + k2 + k3]||φn − φn−1||E0

and

hence ||φn+1 − φn||E06 [k1+k2+k3]

[1−k2−k3] ||φn − φn−1||E0. (3.6.1)

Since [k1 + 2k2 + 2k3] < 1 we have k = [k1+k2+k3][1−k2−k3] < 1.

From (3.6.1)||φn+1 − φn||E0 6 k||φn − φn−1||E0

6 k.k||φn−1 − φn−2||E0= k2||φn−1 − φn−2||E0

...6 kn||φ1 − φ0||E0

.Therefore ||φn+1−φn||E0 6 kn||φ1−φ0||E0 . (3.6.2)Now for any m > n, we have||φm − φn||E0

6 ||φm − φm−1||E0+ ||φm−1 − φm−2||E0

+ ...+ ||φn+1 − φn||E0

6 km−1||φ1 − φ0||E0+ km−2||φ1 − φ0||E0

+ ..+ kn||φ1 − φ0||E0

= kn[1 + k + k2 + ..+ km−n−1] ||φ1 − φ0||E0

6 kn 11−k ||φ1 − φ0||E0

.On applying limits as n→∞, we get kn → 0.Therefore φn is a Cauchy sequence in E0.Since E0 is complete, we have the sequence φn converges and lim

n→∞φn = φ∗,

where φ∗ ∈ E0. Since Rc is topologically closed, we have φ∗ ∈ Rc.We now show that φ∗ is a PPF dependent fixed point of T.We consider||Tφ∗ − φ∗(c)||E 6 ||Tφ∗ − Tφn||E + ||Tφn − φ∗(c)||E

6 k1[||φ∗−φn||E0] + k2[||φ∗(c)−Tφ∗||E + ||φn(c)−Tφn||E ]

+k3[||φ∗(c)−Tφn||E+||φn(c)−Tφ∗||E ]+||Tφn−φ∗(c)||E= k1[||φ∗ − φn||E0 ] + k2[||φ∗(c)− Tφ∗||E + ||φn − φn+1||E0 ]

+k3[||φ∗−φn+1||E0+ ||φn(c)−Tφ∗||E ]+ ||φn+1−φ∗||E0

.Hence [1− k2]||Tφ∗ − φ∗(c)||E 6 k1[||φ∗ − φn||E0

] + k2[||φn − φn+1||E0]

+ k3[||φ∗ − φn+1||E0+ ||φn(c)− Tφ∗||E ]

+ ||φn+1 − φ∗||E0

6 k1[||φ∗ − φn||E0 ]+ k2[||φn − φ∗||E0

+ ||φ∗ − φn+1||E0]

+ k3[||φ∗ − φn+1||E0+ ||φn(c)− Tφ∗||E ]

+ ||φn+1 − φ∗||E0.

On applying limits as n→∞, we get[1− k2]||Tφ∗ − φ∗(c)||E 6 k3||φ∗(c)− Tφ∗||E and hence[1− k2 − k3]||Tφ∗ − φ∗(c)||E 6 0.

PPF Dependent Fixed Points of Contractive Type Mappings 193

Therefore Tφ∗ = φ∗(c) and hence φ∗ is a PPF dependent fixed point of T .We now show that φ∗ is a unique PPF dependent fixed point of T .Let φ∗, ψ∗ be any two PPF dependent fixed points of T in Rc.Then by definition Tφ∗ = φ∗(c) and Tψ∗ = ψ∗(c).We consider||φ∗ − ψ∗||E0

= ||φ∗(c)− ψ∗(c)||E= ||Tφ∗ − Tψ∗||E6 k1[||φ∗ − ψ∗||E0 ] + k2[||φ∗(c)− Tφ∗||E + ||ψ∗(c)− Tψ∗||E ]

+ k3[||φ∗(c)− Tψ∗||E + ||ψ∗(c)− Tφ∗||E ]= k1[||φ∗ − ψ∗||E0

] + k3[||φ∗ − ψ∗||E0+ ||ψ∗ − φ∗||E0

]= [k1 + 2k3]||φ∗ − ψ∗||E0

.Hence [1− k1 − 2k3]||φ∗ − ψ∗||E0

6 0 so that ||φ∗ − ψ∗||E0= 0.

Therefore φ∗ = ψ∗and hence T has a unique PPF dependent fixed point inRc.

Corollary 3.7. Under the hypotheses of Theorem 3.1, if T : E0 → Eis either a generalized Kannan type mapping or generalized Chatterjea typemapping then T has a unique PPF dependent fixed point in Rc.

4 Acknowledgements

The authors would like to thank the honorable editor and referee for theirvaluable suggestions.

References[1] Y. I. Alber and S. Guerre-Delabriere. Principle of weakly contractive maps in Hilbert

spaces. In: Gohberg I., Lyubich Y. (Eds) New results in operator theory and itsapplications. Operator Theory: Advances and Applications, Birkhuser, Basel, 98,(1997) 7-22.

[2] Ali Farajzadeh and Anchalee kaewcharoen, On fixed point theorems for mappingswith PPF dependence, J. Inequal. Appl., 2014, (2014) 372.

[3] G.V.R. Babu and P. D. Sailaja, A fixed point theorem of Generalized Weakly con-tractive maps in Orbitally Complete Metric space, Thai J. Math., 9(1), (2011) 1-10.

[4] G. V. R. Babu, G. Satyanarayana, and M. Vinod Kumar, Properties of Razumikhinclass of functions and PPF dependent fixed points of weakly contractive type maps,Bull. Int. Math. Virtual Institute, 9, (2019) 65-72.

[5] Banach S., : Sur les operations dans les ensembles abstraits et leur application auxequations integrales, Fund. Math., 3, (1922) 133-181.

[6] Bapurao C. Dhage, On some common fixed point theorems with PPF dependencein Banach spaces, J. Nonlinear Sci. and Appl., 5(3), (2012) 220-232.

[7] S. R. Bernfeld, V. Lakshmikantham and Y. M. Reddy, Fixed point theorems ofoperators with PPF dependence in Banach spaces, Appl. Anal., 6(4), (1977) 271-280.

[8] Chatterjea S.K., Fixed point theorems, C.R.Acad. Bulgare Sci., 25, (1972) 727-730.

[9] B. S. Choudhury, Unique fixed point theorems for weackly C-Contractive mappings,Khatmandu University J. Sci. Tech., 5(1), (2009) 6-13.

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[10] Z. Dirci, F. A. McRae and J. Vasundharadevi, Fixed point theorems in partiallyordered metric spaces for operators with PPF dependence, Nonlinear Anal. The-ory,Methods and Appl., 67(2), (2007) 641-647.

[11] N. Hussain, S. Khaleghizadeh, P. Salimi and F. Akbar, New Fixed Point Resultswith PPF dependence in Banach Spaces Endowed with a Graph, Abstr. Appl. Anal.,2013, (2013) 9 pages, Article ID 827205.

[12] Kannan R., Some results on fixed points, Bull. Calcutta Math. Soc., 60, (1968)71-76.

[13] Kasamsuk Ungchittrakool, A Best proximity point theorem for Generalized Non-selfKannan-type and Chatterjea-type mappings and Lipschitzian mappings in completemetric spaces, J. Funct. Spaces, 2016, (2016) 11 pages, Article ID 9321082.

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[15] B. E. Rhoades, A comparison of various definitions of contractive mappings, Trans.Amer. Math. Soc., 226, (1977) 257-290.

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Received June 2018; revised May 2019.

email: journal@monotone.uwaterloo.cahttp://monotone.uwaterloo.ca/∼journal/

Novi Sad J. Math. Vol. 50, No. 1, 2020, 45-66https://doi.org/10.30755/NSJOM.08210

PPF dependent fixed points of generalized contractivetype mappings using C−class functions with an

application

Gutti Venkata Ravindranadh Babu1 and Madugula Vinod Kumar23

Abstract. In this paper, we prove the existence of PPF dependentfixed points of single-valued generalized α− η − ψ − φ− F−contractiontype mappings and extend it to multi-valued α∗−ψ−φ−F−contractiontype mappings in Banach spaces. Also, we introduce the concept of anf − α∗−admissible mapping and prove the existence of PPF dependentcoincidence points of a pair of single-valued and multi-valued mappings.A fixed point result in a Banach space endowed with a graph is obtainedas an application of PPF dependent fixed point result of a single-valuedmapping.

AMS Mathematics Subject Classification (2010): 47H10; 54H25

Key words and phrases: PPF dependent fixed point; Razumikhin class;multi-valued mapping; α−admissible mapping; α∗−admissible mapping;C−class function; graph; G-contraction

1. Introduction

Banach contraction principle is one of the most important result in anal-ysis and it is the main source of metric fixed point theory. The significanceof the proof of the Banach fixed point theorem is that it not only providesthe existence and uniqueness of fixed point, but also furnishes a method forconstructing the fixed point. Several mathematicians generalized Banach’scontraction condition by changing either the domain space or extending asingle-valued mapping to a multi-valued mapping, for more details we referto [1, 6, 8, 13, 14, 20, 21, 22, 25, 11, 12, 26, 27, 30]. In 2012, Samet, Vetroand Vetro [29] introduced the concept of α−admissible self mappings and theyproved the existence of fixed points by using contractive type conditions involv-ing an α−admissible mapping in complete metric spaces, for more details werefer to [18, 23, 28]. In 2012, Asl, Rezapour and Shahzad [5] extended these no-tions to multi-functions by introducing the notions of α∗ − ψ−contractive andα∗−admissible mappings and obtained some fixed points theorems, for moredetails we refer to [3]. In 2013, Ali and Kamran [2] extended the notion ofα∗−ψ−contractive mappings to multi-valued functions and proved some fixed

1Department of Mathematics, Andhra University, Visakhapatnam - 530 003, India.e-mail: gvr babu@hotmail.com

2Department of Mathematics, Andhra University, Visakhapatnam - 530 003, India.Permanent Address : Department of Mathematics, ANITS, Sangivalasa, Visakhapatnam -531 162, India. e-mail: dravinodvivek@gmail.com

3Corresponding author

46 Gutti Venkata Ravindranadh Babu, Madugula Vinod Kumar

point theorems. In 2016, Ansari, Kaewcharoen [4] introduced a new type con-traction, namely the generalized α− η−ψ− φ−F−contraction type mappingand proved the existence of fixed points of such mappings.

In 1977, Bernfeld, Lakshmikantham and Reddy [10] introduced the conceptof a fixed point for mappings that have different domains and ranges which iscalled PPF (Past, Present and Future) dependent fixed point. Furthermore,they introduced a notion of Banach type contraction for non-self mapping andproved the existence of PPF dependent fixed points in the Razumikhin classfor Banach type contraction mappings. The PPF dependent fixed point theo-rems are useful for proving the solutions of nonlinear functional differential andintegral equations which may depend upon the past history, present data andfuture consideration. Several mathematicians proved the existence of a PPFdependent fixed point of single-valued and multi-valued mappings, for moredetails we refer to [7, 9, 16, 17, 19, 24].

In 2014, Ciric, Alsulami, Salimi and Vetro [15] introduced the concept oftriangular αc−admissible mappings with respect to ηc non-self mappings andestablished the existence of PPF dependent fixed points for contraction map-pings involving triangular αc−admissible mappings with respect to ηc non-selfmappings in the Razumikhin class.

In this paper, we denote the real line by R, R+ = [0,∞), and N is theset of all natural numbers. Let (E, ||.||E) be a Banach space and we denote itsimply by E. Let I = [a, b] ⊆ R and E0 = C(I, E) be the set of all continuousfunctions on I equipped with the supremum norm ||.||E0

and we define it by||φ||E0

= supa≤t≤b

||φ(t)||E for any φ ∈ E0. We use the following proposition in

proving our results.

Proposition 1.1. If an and bn are two real sequences, bn is bounded,then lim inf(an + bn) ≤ lim inf an + lim sup bn.

In Section 2, we present basic definitions, lemmas, and preliminaries thatare needed to develop the paper. Also we extend the concept of generalizedα− η−ψ− φ−F−contraction type mapping from the metric space setting toE0 and based on this we define multi-valued α∗ −ψ− φ−F−contraction typemapping on E0 and also introduce the concept of a f−α∗−admissible mappingon E0. In Section 3, we prove the existence of PPF dependent fixed points ofa single-valued generalized α− η − ψ − φ− F−contraction type mapping anddraw some corollaries. In Section 4, we prove the existence of PPF dependentfixed points of multi-valued α∗ − ψ − φ − F−contraction type mappings andPPF dependent coincidence points of a pair (f, T ) where f is a single-valuedfunction and T is a multi-valued function. In Section 5, a fixed point resultin a Banach space endowed with a graph is drawn as an application of PPFdependent fixed point result of a single-valued map.

2. Preliminaries

In this section we present some basic definitions and lemmas for single andmulti-valued mappings in a metric space and then we present the corresponding

PPF dependent fixed points of generalized contractive type mappings ... 47

definitions that are related to PPF dependent fixed points.

Definition 2.1. ([29]) Let T : X → X and α : X×X → R+ be two functions.We say that T is an α−admissible mapping if for any x, y ∈ X withα(x, y) ≥ 1 =⇒ α(Tx, Ty) ≥ 1.

Definition 2.2. ([28]) Let T : X → X and α, η : X × X → R+ be threefunctions. We say that T is an α−admissible mapping with respect to η if forany x, y ∈ X with α(x, y) ≥ η(x, y) =⇒ α(Tx, Ty) ≥ η(Tx, Ty).

Note that if we take η(x, y) = 1 for any x, y ∈ X, then Definition 2.2 reducesto Definition 2.1. Also, if we take α(x, y) = 1 for any x, y ∈ X, then we saythat T is an η−subadmissible mapping.

In 2013, Karapinar, Kumam and Salimi [23] introduced the notion of tri-angular α−admissible mappings as follows.

Definition 2.3. ([23]) Let T : X → X and α : X×X → R+ be two functions.Then T is said to be a triangular α−admissible mapping if for any x, y, z ∈ X,

α(x, y) ≥ 1 =⇒ α(Tx, Ty) ≥ 1 andα(x, z) ≥ 1, α(z, y) ≥ 1 =⇒ α(x, y) ≥ 1.

Example 2.4. Let X = R. We define T : X → X by T (x) = x2, x ∈ X andα : X ×X → R+ by

α(x, y) =

√x2 + y2 if x ≥ 1 and y ≥ 1

0 otherwise.

Then T is a triangular α−admissible mapping.

Definition 2.5. Let T : X → X and α, η : X ×X → R+ be three functions.Then T is said to be a triangular α−admissible mapping with respect to η iffor any x, y, z ∈ X,

α(x, y) ≥ η(x, y) =⇒ α(Tx, Ty) ≥ η(Tx, Ty) andα(x, z) ≥ η(x, z), α(z, y) ≥ η(z, y) =⇒ α(x, y) ≥ η(x, y).

Example 2.6. Let X = R. We define T : X → X by T (x) = x2, x ∈ X andα, η : X ×X → R+ by

α(x, y) =

x− y + 2 if x ≥ y14 otherwise,

and

η(x, y) =

x− y + 1 if x ≥ y12 otherwise.

Then T is a triangular α−admissible mapping with respect to η.

In 2014, Ansari [3] introduced the concept of C−class functions as follows.

48 Gutti Venkata Ravindranadh Babu, Madugula Vinod Kumar

Definition 2.7. ([3]) A mapping F : R+×R+ → R is called a C−class functionif it is continuous and for any s, t ∈ R+ the function F satisfies the followingconditions :i) F (s, t) ≤ s and

ii) F (s, t) = s implies that either s = 0 or t = 0.The family of all C−class functions is denoted by ζ.

Example 2.8. ([3]) The following functions belong to ζ.i) F (s, t) = s− t for all s, t ∈ R+.ii) F (s, t) = ks for all s, t ∈ R+ where 0 < k < 1.

iii) F (s, t) = s(1+t)r for all s, t ∈ R+ where r ∈ (0,∞).

iv) F (s, t) = sβ(s) for all s, t ∈ R+ where β : R+ → [0, 1) is continuous.v) F (s, t) = s− φ(s) for all s, t ∈ R+ where φ : R+ → R+ is continuous

and φ(t) = 0 ⇐⇒ t = 0.vi) F (s, t) = sh(s, t) for all s, t ∈ R+ where h : R+ × R+ → R+ is continuous

such that h(s, t) < 1 for all s, t ∈ R+.

Definition 2.9. ([4]) Let (X, d) be a metric space and α, η : X ×X → R+ betwo functions. A mapping T : X → X is said to be a generalized α− η − ψ −ϕ− F−contraction type mapping if for any x, y ∈ X,

α(x, y) ≥ η(x, y) =⇒ ψ(d(Tx, Ty)) ≤ F (ψ(M(x, y)), ϕ(M(x, y))),

where M(x, y) = maxd(x, y), d(x, Tx), d(y, Ty), F ∈ ζ, ψ, ϕ : R+ → R+ areboth continuous such that ψ(t) = 0 ⇐⇒ t = 0, ψ is a nondecreasing functionand ϕ(t) > 0 for t ∈ (0,∞).

Definition 2.10. ([20]) Let (X, d) be a metric space and α, η : X×X → R+ betwo functions. Then X is said to be an α − η−complete metric space if everyCauchy sequence xn in X with α(xn, xn+1) ≥ η(xn, xn+1) for any n ∈ Nconverges in X.

Definition 2.11. ([20]) Let (X, d) be a metric space and α, η : X ×X → R+

be two functions. A mapping T : X → X is said to be an α − η−continuousmapping if each sequence xn in X with xn → x as n→∞ and α(xn, xn+1) ≥η(xn, xn+1) for all n ∈ N =⇒ Txn → Tx as n→∞.

Theorem 2.12. ([4]) Let (X, d) be a metric space. Assume that α, η : X×X →R+ and T : X → X. Suppose that the following conditions are satisfied:

i) (X, d) is an α− η−complete metric space,ii) T is a generalized α− η − ψ − φ− F−contraction type mapping,

iii) T is a triangular α−orbital admissible mapping with respect to η,iv) there exists x1 ∈ X such that α(x1, Tx1) ≥ η(x1, Tx1) andv) T is an α− η−continuous mapping.

Then Tnx1 converges to x∗ in X and x∗ is a fixed point of T .

For a fixed c ∈ I, the Razumikhin class Rc of functions in E0 is defined byRc =

φ ∈ E0 | ||φ||E0

= ||φ(c)||E

. Clearly, every constant function from Ito E belongs to Rc and thus Rc is a non-empty subset of E0 .

PPF dependent fixed points of generalized contractive type mappings ... 49

Definition 2.13. Let Rc be the Razumikhin class of continuous functions inE0. Then we say thati) the class Rc is algebraically closed with respect to the difference if φ−ψ ∈ Rc

whenever φ, ψ ∈ Rc.ii) the class Rc is topologically closed if it is closed with respect to the topology

on E0 by the norm ||.||E0.

The Razumikhin class of functions Rc has the following properties.

Theorem 2.14. ([7]) Let Rc be the Razumikhin class of functions in E0. Theni) for any φ ∈ Rc and α ∈ R, we have αφ ∈ Rc.

ii) the Razumikhin class Rc is topologically closed with respect to the normdefined on E0.

iii) ∩Rcc∈[a,b]

= φ ∈ E0 | φ : I → E is constant .

Definition 2.15. ([10]) Let T : E0 → E be a mapping. A function φ ∈ E0 issaid to be a PPF dependent fixed point of T if T (φ) = φ(c) for some c ∈ I.

Definition 2.16. ([10]) Let T : E0 → E be a mapping. Then T is called aBanach type contraction if there exists k ∈ [0, 1) such that||Tφ− Tψ||E ≤ k ||φ− ψ||E0

for any φ, ψ ∈ E0.

Theorem 2.17. ([10]) Let T : E0 → E be a Banach type contraction. LetRc be algebraically closed with respect to the difference and topologically closed.Then T has a unique PPF dependent fixed point in Rc.

Definition 2.18. Let c ∈ I. Let T : E0 → E and α : E × E → R+ be twofunctions. Then T is said to be an αc−admissible mapping if for any f, g ∈ E0

(2.1) α(f(c), g(c)) ≥ 1 =⇒ α(Tf, Tg) ≥ 1.

Definition 2.19. Let c ∈ I. Let T : E0 → E, α, η : E × E → R+ be threefunctions. Then T is said to be an αc−admissible mapping with respect to ηcif for any f, g ∈ E0,

(2.2) α(f(c), g(c)) ≥ η(f(c), g(c)) =⇒ α(Tf, Tg) ≥ η(Tf, Tg).

Definition 2.20. ([15]) Let c ∈ I. Let T : E0 → E and α, η : E × E → R+

be three functions. Then T is said to be a triangular αc−admissible mappingwith respect to ηc if for any f, g, h ∈ E0

(2.3)(i) α(f(c), g(c)) ≥ η(f(c), g(c)) =⇒ α(Tf, Tg) ≥ η(Tf, Tg) and

(ii) α(f(c), g(c)) ≥ η(f(c), g(c)), α(g(c), h(c)) ≥ η(g(c), h(c))=⇒ α(f(c), h(c)) ≥ η(f(c), h(c)).

Note that if η(x, y) = 1 for any x, y ∈ E then we say that T is a triangularαc−admissible mapping and if α(x, y) = 1 for any x, y ∈ E then we say that Tis a triangular ηc−subadmissible mapping.

We use the following lemma in our main results.

50 Gutti Venkata Ravindranadh Babu, Madugula Vinod Kumar

Lemma 2.21. ([15]) Let T be a triangular αc−admissible mapping with respectto ηc. We define the sequence φn by Tφn = φn+1(c) for any n ∈ N∪0, whereφ0 ∈ Rc is such that α(φ0(c), Tφ0) ≥ η(φ0(c), Tφ0). Then α(φm(c), φn(c)) ≥η(φm(c), φn(c)) for any m,n ∈ N with m < n.

We denote Ψ = ψ : R+ → R+ | ψ is continuous and ψ(t) = 0 ⇐⇒ t = 0.Now, motivated by the results of Ansari and Kaewcharoen [4] we introduce

the following.

Definition 2.22. Let c ∈ I. Let T : E0 → E, α, η : E × E → R+ be threefunctions. If there exist ψ, φ ∈ Ψ, with ψ strictly monotonically increasing,functions such that

(2.4)α(f(c), g(c)) ≥ η(f(c), g(c)) =⇒

ψ(||Tf − Tg||E) ≤ F (ψ(M(f, g)), φ(M(f, g))),

where M(f, g) = max||f − g||E0, ||f(c)− Tf ||E , ||g(c)− Tg||E ,12 [||f(c)− Tg||E + ||g(c)− Tf ||E ]

for any f, g ∈ E0, then we say that T is a generalized α − η − ψ − φ −F−contraction type mapping.

If we take η(x, y) = 1 for any x, y ∈ E, then T is said to be a generalizedα− ψ − φ− F−contraction type mapping.

Let K(E) be the collection of all non-empty compact subsets of E. Thenthe Hausdorff metric induced by the norm ||.||E is defined by

HE(A,B) = max supa∈A

d(a,B), supb∈B

d(A, b),

where d(a,B) = infb∈B||a−b||E and d(A, b) = inf

a∈A||a−b||E for any A,B ∈ K(E).

Nadler[25] proved the following lemma in metric spaces.

Lemma 2.23. ([25]) Let A and B be two non-empty compact subsets of ametric space X. If a ∈ A then there exists b ∈ B such that d(a, b) ≤ H(A,B).

In 2016, Farajzadeh, Kaewcharoen and Plubtieng [17] introduced the con-cept of a PPF dependent fixed point and PPF dependent coincidence point ofmulti-valued mappings as follows.

Definition 2.24. ([17]) Let T : E0 → K(E) be a multi-valued mapping. Apoint f ∈ E0 is said to be a PPF dependent fixed point of T if f(c) ∈ Tf forsome c ∈ I.

Definition 2.25. ([17]) Let f : E0 → E0 be a single-valued mapping andT : E0 → K(E) be a multi-valued mapping. A point g ∈ E0 is said to be aPPF dependent coincidence point of f and T if fg(c) ∈ Tg for some c ∈ I.

Notice that if f is the identity mapping then clearly g is a PPF dependentfixed point of the multi-valued mapping T .

Definition 2.26. Let c ∈ I. Let T : E0 → K(E), α : E × E → R+ andα∗ : K(E) × K(E) → R+ be three mappings. Then T is said to be anα∗−admissible mapping if for any f, g ∈ E0

α(f(c), g(c)) ≥ 1 =⇒ α∗(Tf, Tg) ≥ 1,

PPF dependent fixed points of generalized contractive type mappings ... 51

where α∗(Tf, Tg) = infα(a, b) | a ∈ Tf, b ∈ Tg.

Based on the generalized α−ψ−φ−F−contraction type mapping of single-valued functions, we define the generalized α∗ − ψ − φ − F−contraction typemapping for multi-valued functions as follows.

Definition 2.27. Let c ∈ I. Let T : E0 → K(E), α : E × E → R+ andα∗ : K(E)×K(E)→ R+ be three functions. If there exist functions ψ, φ ∈ Ψ,with ψ strictly monotonically increasing, such that

(2.5) α∗(Tf, Tg) ≥ 1 =⇒ ψ(HE(Tf, Tg)) ≤ F (ψ(M(f, g)), φ(M(f, g))),

where M(f, g) = max||f − g||E0, d(f(c), Tf), d(g(c), T g),12 [d(f(c), T g) + d(g(c), Tf)]

for any f, g ∈ E0, then we say that T is a generalized α∗−ψ−φ−F−contractiontype mapping.

Based on the concept of α∗−admissible mappings, we define an f−α∗−admissiblemapping as follows.

Definition 2.28. Let c ∈ I. Let T : E0 → K(E), α : E × E → R+,α∗ : K(E)×K(E)→ R+ and f : E0 → E0 be four mappings. Then T is saidto be an f − α∗−admissible mapping if for any φ, ψ ∈ E0

(2.6) α(fφ(c), fψ(c)) ≥ 1 =⇒ α∗(Tφ, Tψ) ≥ 1.

We observe that T is an α∗−admissible mapping if f is the identity mapping.

Example 2.29. Let E0 = R and c ∈ [a, b] ⊆ R. Let E0 = C(I, E).We define T : E0 → K(E) by

Tφ =

[||φ(c)||E + 1, 3] if ||φ(c)||E ≤ 1[1, ||φ(c)||E ] if ||φ(c)||E > 1,

f : E0 → E0 by f(φ) = kφ, k ≥ 1 and φ ∈ E0,α : E × E → R+ by

α(x, y) =

y − x+ 2 if x ≤ y,both x and y non-negative, or

both x and y negative, orx is negative and y is positive,

2 if x ≥ y, both x and y non-negative ,0 otherwise,

and α∗ : K(E)×K(E)→ R+ byα∗(A,B) = infα(a, b)/a ∈ A and b ∈ B for any A,B ∈ K(E).

Let φ, ψ ∈ E0 be such that α(fφ(c), fψ(c)) ≥ 1.Case (i): Suppose that both fφ(c), fψ(c) are non-negative and fφ(c) ≤ fψ(c).Since k ≥ 1, we have both φ(c), ψ(c) are non-negative and φ(c) ≤ ψ(c) andhence ||φ(c)||E ≤ ||ψ(c)||E .Subcase (i): Suppose that ||φ(c)||E , ||ψ(c)||E ∈ [0, 1].

52 Gutti Venkata Ravindranadh Babu, Madugula Vinod Kumar

We have α∗(Tφ, Tψ) = infα(a, b)/a ∈ Tφ and b ∈ Tψ= infα(a, b)/a ∈ [||φ(c)||E + 1, 3] and

b ∈ [||ψ(c)||E + 1, 3].Therefore α∗(Tφ, Tψ) = 2 > 1.Subcase (ii): Suppose that ||φ(c)||E , ||ψ(c)||E ∈ (1,∞).In this case, α∗(Tφ, Tψ) = infα(a, b)/a ∈ [1, ||φ(c)||E ] and

b ∈ [1, ||ψ(c)||E ].Therefore α∗(Tφ, Tψ) = 2 > 1.Subcase (iii): Suppose that ||φ(c)||E ∈ (1,∞) and ||ψ(c)||E ∈ [0, 1].Here, α∗(Tφ, Tψ) = infα(a, b)/a ∈ [1, ||φ(c)||E ] and

b ∈ [||ψ(c)||E + 1, 3].Therefore α∗(Tφ, Tψ) = 2 > 1.Subcase (iv): Suppose that ||ψ(c)||E ∈ (1,∞) and ||φ(c)||E ∈ [0, 1].Here, α∗(Tφ, Tψ) = infα(a, b)/a ∈ [||φ(c)||E + 1, 3] and

b ∈ [1, ||ψ(c)||E ].Therefore α∗(Tφ, Tψ) = 2 > 1.Case (ii): Suppose that both fφ(c), fψ(c) are negative and fφ(c) ≤ fψ(c).Since k ≥ 1, we have both φ(c), ψ(c) are negative and φ(c) ≤ ψ(c) and hence||φ(c)||E ≥ ||ψ(c)||E .As in Case (i), we get α∗(Tφ, Tψ) = 2 > 1.Case (iii): Suppose that both fφ(c) is negative and fψ(c) is positive andfφ(c) ≤ fψ(c).Since k ≥ 1, we have φ(c) is negative and ψ(c) is positive and φ(c) ≤ ψ(c).As in Case (i), we get α∗(Tφ, Tψ) = 2 > 1.Subcase (i): Suppose that ||φ(c)||E ≥ ||ψ(c)||E .As in Case (i), we get α∗(Tφ, Tψ) = 2 > 1.Subcase (ii): Suppose that ||φ(c)||E ≤ ||ψ(c)||E .As in Case (i), we get α∗(Tφ, Tψ) = 2 > 1.Case (iv): Suppose that both fφ(c), fψ(c) are non-negative and fφ(c) ≥ fψ(c).Since k ≥ 1, we have both φ(c), ψ(c) are non-negative and φ(c) ≥ ψ(c) andhence ||φ(c)||E ≥ ||ψ(c)||E .As in Case (i), we get α∗(Tφ, Tψ) = 2 > 1.Hence from all the above cases, we get T is f − α∗−admissible mapping.

We use the following lemma in our main results.

Lemma 2.30. Let φn be a sequence in E0 such that ||φn − φn+1||E0→ 0 as

n → ∞. If φn is not a Cauchy sequence then there exist an ε > 0 and twosubsequences φm(k) and φn(k) of φn with m(k) > n(k) > k such that∣∣∣∣φn(k) − φm(k)

∣∣∣∣E0≥ ε,

∣∣∣∣φn(k) − φm(k)−1∣∣∣∣E0

< ε and

i) limk→∞

∣∣∣∣φn(k) − φm(k)

∣∣∣∣E0

= ε, ii) limk→∞

∣∣∣∣φn(k) − φm(k)−1∣∣∣∣E0

= ε,

iii) limk→∞

∣∣∣∣φn(k)−1 − φm(k)

∣∣∣∣E0

= ε, iv) limk→∞

∣∣∣∣φn(k)−1 − φm(k)−1∣∣∣∣E0

= ε.

Proof. Similar to the proof of Lemma 1.4 of [6].

PPF dependent fixed points of generalized contractive type mappings ... 53

3. PPF dependent fixed points of a single-valued map-pings

Theorem 3.1. Let c ∈ I. Let T : E0 → E and α, η : E × E → R+ be threefunctions satisfying the following conditions:

i) T is a generalized α− η − ψ − φ− F−contraction type mapping,ii) T is a triangular αc−admissible mapping with respect to ηc,

iii) Rc is algebraically closed with respect to the difference,iv) if φn is a sequence in E0 such that φn → φ as n→∞ and

α(φn(c), φn+1(c)) ≥ η(φn(c), φn+1(c)) for any n ∈ N ∪ 0, thenα(φn(c), φ(c)) ≥ η(φn(c), φ(c)) for any n ∈ N ∪ 0, and

v) there exists φ0 ∈ Rc such that α(φ0(c), Tφ0) ≥ η(φ0(c), Tφ0).Then T has a PPF dependent fixed point in Rc.

Proof. Let φ0 ∈ Rc be such that α(φ0(c), Tφ0) ≥ η(φ0(c), Tφ0). Since Tφ0 ∈ E,there exists x1 ∈ E such that Tφ0 = x1. We choose φ1 ∈ Rc such thatx1 = φ1(c). Then Tφ0 = φ1(c). Since Tφ1 ∈ E, there exists x2 ∈ E such thatTφ1 = x2. We choose φ2 ∈ Rc such that x2 = φ2(c). Then Tφ1 = φ2(c).Continuing this process, we can define a sequence φn in Rc inductively byTφn = φn+1(c) and ||φn+1 − φn||E0

= ||φn+1(c)− φn(c)||E for any n ∈ N∪0.If φn+1 = φn for some n ∈ N ∪ 0, then Tφn = φn+1(c) = φn(c) so that φn isa PPF dependent fixed point of T in Rc.Suppose that φn+1 6= φn for any n ∈ N ∪ 0.Since Tφn = φn+1(c) for any n ∈ N ∪ 0 and α(φ0(c), Tφ0) ≥ η(φ0(c), Tφ0),from Lemma 2.21, we have α(φm(c), φn(c)) ≥ η(φm(c), φn(c)) for any m,n ∈ Nwith m < n.We considerψ(||φn+1 − φn+2||E0

) = ψ(||Tφn − Tφn+1||E)

(3.1) ≤ F (ψ(M(φn, φn+1)), φ(M(φn, φn+1)))

(3.2) ≤ ψ(M(φn, φn+1)).

We considerM(φn, φn+1) = max||φn − φn+1||E0

, ||φn(c)− Tφn||E , ||φn+1(c)− Tφn+1||E ,12 [||φn(c)− Tφn+1||E + ||φn+1(c)− Tφn||E ]

= max||φn − φn+1||E0, ||φn+1 − φn+2||E0

,12 [||φn − φn+2||E0 ]

≤ max||φn − φn+1||E0 , ||φn+1 − φn+2||E0 ,12 [||φn − φn+1||E0

+ ||φn+1 − φn+2||E0]

= max||φn − φn+1||E0, ||φn+1 − φn+2||E0

≤M(φn, φn+1).

Hence M(φn, φn+1) = max||φn − φn+1||E0, ||φn+1 − φn+2||E0

.Suppose that max||φn − φn+1||E0 , ||φn+1 − φn+2||E0 = ||φn+1 − φn+2||E0 .Then M(φn, φn+1) = ||φn+1 − φn+2||E0 .From (3.1), ψ(||φn+1−φn+2||E0

) ≤ F (ψ(||φn+1−φn+2||E0), φ(||φn+1−φn+2||E0

))≤ ψ(||φn+1 − φn+2||E0

)

54 Gutti Venkata Ravindranadh Babu, Madugula Vinod Kumar

and hence F (ψ(||φn+1−φn+2||E0), φ(||φn+1−φn+2||E0)) = ψ(||φn+1−φn+2||E0).Therefore either ψ(||φn+1 − φn+2||E0

) = 0 or φ(||φn+1 − φn+2||E0) = 0 and

hence φn+1 = φn+2, a contradiction. Therefore M(φn, φn+1) = ||φn−φn+1||E0.

From (3.2), ψ(||φn+1 − φn+2||E0) ≤ ψ(||φn − φn+1||E0

).Since ψ is strictly monotonically increasing, we have||φn+1 − φn+2||E0 ≤ ||φn − φn+1||E0 .Therefore the sequence ||φn − φn+1||E0

is a decreasing sequence in R+ andhence it is convergent.Let lim

n→∞||φn − φn+1||E0

= r. We now show that r = 0.

From (3.1), ψ(||φn+1 − φn+2||E0) ≤ F (ψ(||φn − φn+1||E0

), φ(||φn − φn+1||E0)).

On applying limits as n→∞, we get ψ(r) ≤ F (ψ(r), φ(r))) ≤ ψ(r) andhence F (ψ(r), φ(r)) = ψ(r). Therefore either ψ(r) = 0 or φ(r) = 0 andhence r = 0. Therefore

(3.3) limn→∞

||φn − φn+1||E0 = 0.

We now show that the sequence φn is a Cauchy sequence in Rc.Suppose that the sequence φn is not a Cauchy sequence.By Lemma 2.30, there exist an ε > 0 and two subsequences φmk

and φnk

of φn with mk > nk > k such that ||φnk−φmk

||E0≥ ε , ||φnk

−φmk−1||E0< ε

and

(3.4) limk→∞

||φnk− φmk

||E0= ε.

By the triangular inequality, we have||φnk+1−φmk+1||E0

≤ ||φnk+1−φnk||E0

+ ||φnk−φmk

||E0+ ||φmk

−φmk+1||E0.

On applying limit superior as k →∞ on both sides we get

(3.5) lim supk→∞

||φnk+1 − φmk+1||E0≤ ε.

By the triangular inequality, we have||φnk

−φmk||E0 ≤ ||φnk

−φnk+1||E0 + ||φnk+1−φmk+1||E0 + ||φmk+1−φmk||E0 .

Now by applying Proposition 1.1 with ak = ||φnk+1 − φmk+1||E0 andbk = ||φnk

− φnk+1||E0+ ||φmk+1 − φmk

||E0we have

(3.6) ε ≤ lim infk→∞

||φnk+1 − φmk+1||E0 .

From (3.5) and (3.6), we get

(3.7) limk→∞

||φnk+1 − φmk+1||E0= ε.

From (3.4) and (3.7) we have

(3.8) limk→∞

||φnk− φmk+1||E0

= ε = limk→∞

||φmk− φnk+1||E0

.

We considerM(φnk

, φmk) = max||φnk

− φmk||E0 , ||φnk

(c)− Tφnk||E , ||φmk

(c)− Tφmk||E ,

PPF dependent fixed points of generalized contractive type mappings ... 55

12 [||φnk

(c)− Tφmk||E + ||φmk

(c)− Tφnk||E ]

= max||φnk− φmk

||E0, ||φnk

− φnk+1||E0, ||φmk

− φmk+1||E0,

12 [||φnk

− φmk+1||E0+ ||φmk

− φnk+1||E0].

On applying limits as k →∞, we get

(3.9) limk→∞

M(φnk, φmk

) = maxε, 0, 0, 1

2[ε+ ε] = ε.

We considerψ(||φnk+1 − φmk+1||E0

) = ψ(||Tφnk− Tφmk

||E),≤ F (ψ(M(φnk

, φmk)), φ(M(φnk

, φmk))).

On applying limits as k →∞, we get ψ(ε) ≤ F (ψ(ε), φ(ε)) ≤ ψ(ε) and henceF (ψ(ε), φ(ε)) = ψ(ε). Therefore either ψ(ε) = 0 or φ(ε) = 0 and hence ε = 0, acontradiction. Therefore the sequence φn is a Cauchy sequence in Rc ⊆ E0.Since E0 is complete, we have φn → φ∗ as n→∞ for some φ∗ ∈ E0.Since Rc is topologically closed, we have φ∗ ∈ Rc.From (iv), we have α(φn(c), φ∗(c)) ≥ η(φn(c), φ∗(c)) for any n ∈ N ∪ 0.Since T is a generalized α− η − ψ − φ− F− contraction type mapping,we haveψ(||φn+1(c)− Tφ∗||E) = ψ(||Tφn − Tφ∗||E)

(3.10) ≤ F (ψ(M(φn, φ∗)), φ(M(φn, φ

∗))),

where||φ∗(c)− Tφ∗||E ≤M(φn, φ

∗)= max||φn − φ∗||E0

, ||φn(c)− Tφn||E , ||φ∗(c)− Tφ∗||E ,12 [||φn(c)− Tφ∗||E + ||φ∗(c)− Tφn||E ]

≤ max||φn − φ∗||E0 , ||φn − φn+1||E0 , ||φ∗(c)− Tφ∗||E ,12 [||φn(c)−Tφ∗||E + ||φ∗−φn||E0 + ||φn−φn+1||E0 ].

On applying limits as n→∞, we get||φ∗(c)− Tφ∗||E ≤ lim

n→∞M(φn, φ

∗) ≤ max0, 0, ||φ∗(c)− Tφ∗||E ,12 [||φ∗(c)− Tφ∗||E ]

= ||φ∗(c)− Tφ∗||E .Hence lim

n→∞M(φn, φ

∗) = ||φ∗(c)− Tφ∗||E .On applying limits as n→∞ to inequality (3.10), we getψ(||φ∗(c)− Tφ∗||E) ≤ F (ψ(||φ∗(c)− Tφ∗||E), φ(||φ∗(c)− Tφ∗||E))

≤ ψ(||φ∗(c)− Tφ∗||E)and henceF (ψ(||φ∗(c)− Tφ∗||E), φ(||φ∗(c)− Tφ∗||E)) = ψ(||φ∗(c)− Tφ∗||E).Therefore ψ(||φ∗(c)− Tφ∗||E) = 0 or φ(||φ∗(c)− Tφ∗||E) = 0 andhence Tφ∗ = φ∗(c). Therefore φ∗ ∈ Rc is a PPF dependent fixed point of T .

Corollary 3.2. Let c ∈ I. Let T : E0 → E and α : E × E → R+ be twofunctions satisfying the following conditions:

i) T is a generalized α− ψ − φ− F−contraction type mapping,ii) T is a triangular αc−admissible mapping,

iii) Rc is algebraically closed with respect to the difference,

56 Gutti Venkata Ravindranadh Babu, Madugula Vinod Kumar

iv) if φn is a sequence in E0 such that φn → φ as n→∞and α(φn(c), φn+1(c)) ≥ 1 for any n ∈ N ∪ 0, then α(φn(c), φ(c)) ≥ 1for any n ∈ N ∪ 0, and

v) there exists φ0 ∈ Rc such that α(φ0(c), Tφ0) ≥ 1.Then T has a PPF dependent fixed point in Rc.

Proof. Follows by choosing η(φ(c), ψ(c)) = 1 for any φ, ψ ∈ E0 in Theorem3.1.

Corollary 3.3. Let c ∈ I. Let T : E0 → E and α, η : E × E → R+ be threefunctions satisfying the following conditions:

i) T satisfies the inequalityα(f(c), g(c)) ≥ η(f(c), g(c)) =⇒

||Tf − Tg||E ≤ k ·max||f − g||E0, ||f(c)− Tf ||E , ||g(c)− Tg||E ,

12 [||f(c)− Tg||E + ||g(c)− Tf ||E ]

for any f, g ∈ E0, where 0 < k < 1,ii) T is a triangular αc−admissible mapping with respect to ηc,

iii) Rc is algebraically closed with respect to the difference,iv) if φn is a sequence in E0 such that φn → φ∗ as n→∞

and α(φn(c), φn+1(c)) ≥ η(φn(c), φn+1(c)) for any n ∈ N ∪ 0, thenα(φn(c), φ∗(c)) ≥ η(φn(c), φ∗(c)) for any n ∈ N ∪ 0, and

v) there exists φ0 ∈ Rc such that α(φ0(c), Tφ0) ≥ η(φ0(c), Tφ0).Then T has a PPF dependent fixed point in Rc.

Proof. Follows by choosing F (s, t) = ks where 0 < k < 1 and ψ(t) = t for anys, t ∈ R+ in Theorem 3.1.

Corollary 3.4. Let c ∈ I. Let T : E0 → E and α : E × E → R+ be twofunctions satisfying the following conditions:

i) T satisfies the inequalityα(f(c), g(c)) ≥ 1 =⇒

||Tf − Tg||E ≤ kmax||f − g||E0, ||f(c)− Tf ||E , ||g(c)− Tg||E ,

12 [||f(c)− Tg||E + ||g(c)− Tf ||E ]

for any f, g ∈ E0, where 0 < k < 1,ii) T is a triangular αc−admissible mapping,

iii) Rc is algebraically closed with respect to the difference,iv) if φn is a sequence in E0 such that φn → φ∗ as n→∞ and

α(φn(c), φn+1(c)) ≥ 1 for any n ∈ N ∪ 0, then α(φn(c), φ∗(c)) ≥ 1for any n ∈ N ∪ 0, and

v) there exists φ0 ∈ Rc such that α(φ0(c), Tφ0) ≥ 1.Then T has a PPF dependent fixed point in Rc.

Proof. Follows by choosing η(φ(c), ψ(c)) = 1 for any φ, ψ ∈ E0 in Corollary3.3.

PPF dependent fixed points of generalized contractive type mappings ... 57

4. PPF dependent fixed points and coincidence points ofmulti-valued mappings

Theorem 4.1. Let c ∈ I. Let T : E0 → K(E), α : E × E → R+ andα∗ : K(E)×K(E)→ R+ be three functions satisfying the following conditions:

i) T is a generalized α∗ − ψ − φ− F−contraction type mapping,ii) T is an α∗−admissible mapping,

iii) Rc is algebraically closed with respect to the difference,iv) Tφ ⊆ Rc(c) for any φ ∈ E0,v) if φn is a sequence in E0 such that φn → φ∗ as n→∞

and α(φn(c), φn+1(c)) ≥ 1 for any n ∈ N ∪ 0, then α(φn(c), φ∗(c)) ≥ 1for any n ∈ N ∪ 0, and

vi) there exist φ0 ∈ Rc and φ1(c) ∈ Tφ0 such that α(φ0(c), φ1(c)) ≥ 1.Then T has a PPF dependent fixed point in Rc.

Proof. Let φ0 ∈ Rc and φ1(c) ∈ Tφ0 be such that α(φ0(c), φ1(c)) ≥ 1.If φ0 = φ1 then φ0 is a PPF dependent fixed point of T . Suppose that φ0 6= φ1.Since T is an α∗−admissible mapping, we have α∗(Tφ0, Tφ1) ≥ 1.Since T is a generalized α∗ − ψ − φ− F− contraction type mapping, we haveψ(HE(Tφ0, Tφ1)) ≤ F (ψ(M(φ0, φ1)), φ(M(φ0, φ1))).Since x1 ∈ Tφ0, by Lemma 2.23 there exists x2 ∈ Tφ1 such that||x1−x2||E ≤ HE(Tφ0, Tφ1). Since x2 ∈ Tφ1 and Tφ1 ⊆ Rc(c), we choose φ2 ∈Rc such that x2 = φ2(c) ∈ Tφ1. If φ1 = φ2 then φ1 is a PPF dependent fixedpoint of T . Suppose that φ1 6= φ2. Clearly α(φ1(c), φ2(c)) ≥ α∗(Tφ0, Tφ1) ≥ 1and hence α(φ1(c), φ2(c)) ≥ 1. Since T is an α∗−admissible mapping, we haveα∗(Tφ1, Tφ2) ≥ 1. Since T is a generalized α∗ − ψ − φ − F−contraction typemapping, we have ψ(HE(Tφ1, Tφ2)) ≤ F (ψ(M(φ1, φ2)), φ(M(φ1, φ2))).Since x2 ∈ Tφ1, by Lemma 2.23 there exists x3 ∈ Tφ2 such that||x2 − x3||E ≤ HE(Tφ1, Tφ2). On continuing this process, we get a sequenceφn in Rc satisfying the following:for any n ∈ N,

(4.1)

φn−1 6= φn,xn = φn(c) ∈ Tφn−1,||φn − φn+1||E0 = ||φn(c)− φn+1(c)||E

= ||xn − xn+1||E ≤ HE(Tφn−1, Tφn),α∗(Tφn−1, Tφn) ≥ 1 and henceψ(HE(Tφn−1, Tφn)) ≤ F (ψ(M(φn−1, φn)), φ(M(φn−1, φn))).

From (4.1) we have||φn − φn+1||E0

≤ HE(Tφn−1, Tφn), which implies thatψ(||φn − φn+1||E0

) ≤ ψ(HE(Tφn−1, Tφn))

(4.2) ≤ F (ψ(M(φn−1, φn)), φ(M(φn−1, φn))).

Now we considerM(φn−1, φn) = max||φn−1 − φn||E0

, d(φn−1(c), Tφn−1), d(φn(c), Tφn),12 [d(φn−1(c), Tφn) + d(φn(c), Tφn−1)]

58 Gutti Venkata Ravindranadh Babu, Madugula Vinod Kumar

= max||φn−1 − φn||E0 , d(φn(c), Tφn).Suppose that M(φn−1, φn) = d(φn(c), Tφn).From (4.2) we haveψ(||φn − φn+1||E0

) ≤ F (ψ(d(φn(c), Tφn)), φ(d(φn(c), Tφn)))≤ ψ(d(φn(c), Tφn))

and hence||φn − φn+1||E0

= ||φn(c)− φn+1(c)||E ≤ d(φn(c), Tφn), a contradiction.Therefore M(φn−1, φn) = ||φn−1 − φn||E0

.From (4.2) ψ(||φn − φn+1||E0

) ≤ F (ψ(||φn−1 − φn||E0), φ(||φn−1 − φn||E0

))≤ ψ(||φn−1 − φn||E0

).Since ψ is strictly monotonically increasing we have||φn − φn+1||E0 ≤ ||φn−1 − φn||E0 .Therefore the sequence ||φn − φn+1||E0

is a decreasing sequence in R+ andhence it is convergent.Let lim

n→∞||φn − φn+1||E0

= r. We now show that r = 0.

From (4.2), ψ(||φn − φn+1||E0) ≤ F (ψ(||φn−1 − φn||E0

), φ(||φn−1 − φn||E0)).

On applying limits as n → ∞, we get ψ(r) ≤ F (ψ(r), φ(r)) ≤ ψ(r), whichimplies that either ψ(r) = 0 or φ(r) = 0. Therefore r = 0 and hence

(4.3) limn→∞

||φn − φn+1||E0= 0.

Now we show that φn is a Cauchy sequence in Rc.Suppose that the sequence φn is not a Cauchy sequence. By Lemma 2.30,there exist an ε > 0 and two subsequences φmk

and φnk of φn with

mk > nk > k such that ||φnk− φmk

||E0≥ ε, ||φnk

− φmk−1||E0< ε and

(4.4) limk→∞

||φnk− φmk

||E0= ε.

As in the proof of Theorem 3.1, we get

(4.5)

limk→∞

||φnk+1 − φmk+1||E0= ε and

limk→∞

||φnk− φmk+1||E0 = ε = lim

k→∞||φmk

− φnk+1||E0 .

We now show that limk→∞

||φmk+l1 − φnk+l2 ||E0= ε for any l1, l2 ∈ N.

Let l1, l2 ∈ N. We now consider

||φmk+l1 − φnk+l2 ||E0

≤ ||φmk+l1 − φmk+l1−1||E0+ ||φmk+l1−1 − φmk+l1−2||E0

+ . . .+ ||φmk+1 − φmk||E0 + ||φmk

− φnk+1||E0

+||φnk+1 − φnk+2||E0 + ...+ ||φnk+l2−1 − φnk+l2 ||E0 .

On applying limit superior as k →∞ on both sides, we get

(4.6) lim supk→∞

||φmk+l1 − φnk+l2 ||E0 ≤ ε.

PPF dependent fixed points of generalized contractive type mappings ... 59

Now we consider

||φmk− φnk+1||E0

≤ ||φmk− φmk+1||E0

+ ||φmk+1 − φmk+2||E0+ ...

+||φmk+l1−1 − φmk+l1 ||E0+ ||φmk+l1 − φnk+l2 ||E0

+||φnk+l2 − φnk+l2−1||E0+ ...+ ||φnk+2 − φnk+1||E0

.

Now by applying Proposition 1.1 with

ak = ||φmk+l1 − φnk+l2 ||E0and

bk = (||φmk− φmk+1||E0

+ ||φmk+1 − φmk+2||E0

+...+ ||φmk+l1−1 − φmk+l1 ||E0

+||φnk+l2 − φnk+l2−1||E0+ ...+ ||φnk+2 − φnk+1||E0

)

we have

ε ≤ lim infk→∞

||φmk+l1 − φnk+l2 ||E0+ lim sup

k→∞(||φmk

− φmk+1||E0

+||φmk+1 − φmk+2||E0 + ...+ ||φmk+l1−1 − φmk+l1 ||E0

+||φnk+l2 − φnk+l2−1||E0 + ...+ ||φnk+2 − φnk+1||E0).

Hence

(4.7) ε ≤ lim infk→∞

||φmk+l1 − φnk+l2 ||E0.

From (4.6) and (4.7), we get

(4.8) limk→∞

||φmk+l1 − φnk+l2 ||E0= ε for any l1, l2 ∈ N.

We choose l1, l2 ∈ N such that (nk + l2)− (mk + l1) = 1.From (4.1) we get

ψ(||φnk+l2 − φmk+l1 ||E0)

≤ ψ(HE(Tφnk+l2−1, Tφmk+l1−1))

≤ F (ψ(M(φnk+l2−1, φmk+l1−1)), φ(M(φnk+l2−1, φmk+l1−1))).

On applying limits as k →∞, we get ψ(ε) ≤ F (ψ(ε), φ(ε)) ≤ ψ(ε) and henceF (ψ(ε), φ(ε)) = ψ(ε). Therefore ε = 0, a contradiction.Therefore the sequence φn is a Cauchy sequence in Rc ⊆ E0.Since E0 is complete, we have φn → φ∗ as n→∞.Since Rc is topologically closed, we have φ∗ ∈ Rc.Clearly,d(φ∗(c), Tφ∗) ≤M(φn, φ

∗)= max||φn − φ∗||E0

, d(φn(c), Tφn), d(φ∗(c), Tφ∗),12 [d(φn(c), Tφ∗) + d(φ∗(c), Tφn)]

≤ max||φn − φ∗||E0, ||φn(c)− φn+1(c)||E , d(φ∗(c), Tφ∗),

60 Gutti Venkata Ravindranadh Babu, Madugula Vinod Kumar

12 [d(φn(c), Tφ∗) + ||φ∗ − φn+1||E0 ].

On applying limits as n→∞ we getd(φ∗(c), Tφ∗) ≤ lim

n→∞M(φn, φ

∗) ≤ d(φ∗(c), Tφ∗) and hence

(4.9) limn→∞

M(φn, φ∗) = d(φ∗(c), Tφ∗).

Since α(φn(c), φn+1(c)) ≥ α∗(Tφn−1, Tφn) ≥ 1 and from (v), we haveα(φn(c), φ∗(c)) ≥ 1 for all n ∈ N ∪ 0.Since T is α∗−admissible, we have α∗(Tφn, Tφ

∗) ≥ 1.Clearly,d(φn+1(c), Tφ∗) ≤ HE(Tφn, Tφ

∗), which implies thatψ(d(φn+1(c), Tφ∗)) ≤ ψ(HE(Tφn, Tφ

∗))≤ F (ψ(M(φn, φ

∗)), φ(M(φn, φ∗))).

On applying limits as n→∞ and by using (4.9) we getψ(d(φ∗(c), Tφ∗)) ≤ F (ψ(d(φ∗(c), Tφ∗)), φ(d(φ∗(c), Tφ∗))) ≤ ψ(d(φ∗(c), Tφ∗)).Therefore, either ψ(d(φ∗(c), Tφ∗)) = 0 or φ(d(φ∗(c), Tφ∗)) = 0, and henced(φ∗(c), Tφ∗) = 0. Therefore φ∗(c) ∈ Tφ∗ and hence φ∗ is a PPF dependentfixed point of T .

Theorem 4.2. Let c ∈ I. Let T : E0 → K(E), α : E × E → R+,α∗ : K(E) × K(E) → R+ and f : E0 → E0 be four functions satisfying thefollowing conditions:

i) there exist functions ψ, φ ∈ Ψ, with ψ strictly monotonically increasing,such that

(4.10)

α(fγ(c), fη(c)) ≥ 1 =⇒

ψ(HE(Tγ, Tη)) ≤ F (ψ(||fγ − fη||E0), φ(||fγ − fη||E0

))

for any γ, η ∈ E0,ii) T is an f − α∗−admissible mapping,

iii) Rc is algebraically closed with respect to the difference and f(Rc) ⊆ Rc,iv) Tφ ⊆ f(Rc)(c) = fψ(c)/ψ ∈ Rc for any φ ∈ E0,v) if fφn is a sequence in E0 such that fφn → fφ∗ as n→∞ and

α(fφn(c), fφn+1(c)) ≥ 1 for any n ∈ N ∪ 0, then α(fφn(c), fφ∗(c)) ≥ 1for any n ∈ N ∪ 0,

vi) f(Rc) is complete, andvii) there exist φ0 ∈ Rc and fφ1(c) ∈ Tφ0 such that α(fφ0(c), fφ1(c)) ≥ 1.Then T and f have a PPF dependent coincidence point in Rc.

Proof. Let φ0 ∈ Rc and x1 = fφ1(c) ∈ Tφ0 be such thatα(fφ0(c), fφ1(c)) ≥ 1. If fφ1 = fφ0 then φ0 is a PPF dependent coincidencepoint of T and f .Suppose that fφ1 6= fφ0.From (4.10), we get

ψ(HE(Tφ0, Tφ1)) ≤ F (ψ(||fφ0 − fφ1||E0), φ(||fφ0 − fφ1||E0

)).

Since x1 ∈ Tφ0, by Lemma 2.23, there exists x2 ∈ Tφ1 such that||x1 − x2||E ≤ H(Tφ0, Tφ1). Since x2 ∈ Tφ1 and Tφ1 ⊆ f(Rc)(c),

PPF dependent fixed points of generalized contractive type mappings ... 61

we choose φ2 ∈ Rc such that x2 = fφ2(c) ∈ Tφ1.If fφ2 = fφ1 then φ1 is a PPF dependent coincidence point of T and f .Suppose that fφ2 6= fφ1.Since α(fφ0(c), fφ1(c)) ≥ 1 and T is f − α∗−admissible, we haveα∗(Tφ0, Tφ1) ≥ 1.Clearly α(fφ1(c), fφ2(c)) ≥ α∗(Tφ0, Tφ1) ≥ 1 and henceα(fφ1(c), fφ2(c)) ≥ 1.From (4.10), we get

ψ(HE(Tφ1, Tφ2)) ≤ F (ψ(||fφ1 − fφ2||E0), φ(||fφ1 − fφ2||E0)).

Since x2 ∈ Tφ1, by Lemma 2.23 there exists x3 ∈ Tφ2 such that||x2 − x3||E ≤ HE(Tφ1, Tφ2). Continuing this process, we get a sequencefφn in f(Rc) satisfying the following:for any n ∈ N,

(4.11)

fφn 6= fφn−1,xn = fφn(c) ∈ Tφn−1,||fφn − fφn+1||E0

= ||fφn(c)− fφn+1(c)||E= ||xn − xn+1||E ≤ HE(Tφn−1, Tφn),

α(fφn−1(c), fφn(c)) ≥ 1 and henceψ(HE(Tφn−1, Tφn)) ≤ F (ψ(||fφn−1 − fφn||E0),

φ(||fφn−1 − fφn||E0)).

From (4.11), we have ||fφn−fφn+1||E0≤ HE(Tφn−1, Tφn), which implies that

ψ(||fφn − fφn+1||E0)

≤ ψ(HE(Tφn−1, Tφn))

≤ F (ψ(||fφn−1 − fφn||E0), φ(||fφn−1 − fφn||E0

))(4.12)

≤ ψ(||fφn−1 − fφn||E0).

Since ψ is strictly monotonically increasing, we have

||fφn − fφn+1||E0≤ ||fφn−1 − fφn||E0

.

Therefore the sequence ||fφn−fφn+1||E0 is a decreasing sequence in R+ andhence it is convergent. Let lim

n→∞||fφn − fφn+1||E0

= r.

We now show that r = 0.From (4.12), we have

ψ(||fφn − fφn+1||E0) ≤ F (ψ(||fφn−1 − fφn||E0

), φ(||fφn−1 − fφn||E0))

≤ ψ(||fφn−1 − fφn||E0).

On applying limits as n→∞ we obtain that ψ(r) ≤ F (ψ(r), φ(r)) ≤ ψ(r).This implies that either ψ(r) = 0 or φ(r) = 0 and hence r = 0.Therefore

(4.13) limn→∞

||fφn − fφn+1||E0= 0.

62 Gutti Venkata Ravindranadh Babu, Madugula Vinod Kumar

We now show that fφn is a Cauchy sequence in f(Rc).Suppose that the sequence fφn is not a Cauchy sequence.By Lemma 2.30, there exist an ε > 0 and two subsequences fφmk

and fφnk

of fφn with mk > nk > k such that ||fφnk− fφmk

||E0≥ ε,

||fφnk− fφmk−1||E0 < ε and lim

k→∞||fφnk

− fφmk||E0 = ε.

As in the proof of the Theorem 4.1, we getlimk→∞

||fφmk+l1 − fφnk+l2 ||E0= ε for any l1, l2 ∈ N.

We choose l1, l2 ∈ N such that (nk + l2)− (mk + l1) = 1.From (4.11),

ψ(||fφnk+l2 − fφmk+l1 ||E0) ≤ ψ(HE(Tφnk+l2−1, Tφmk+l1−1))

≤ F (ψ(||fφnk+l2−1 − fφmk+l1−1)||E0 ,

φ(||fφnk+l2−1 − fφmk+l1−1||E0))

≤ ψ(||fφnk+l2−1 − fφmk+l1−1||E0).

On applying limits as k →∞, we get ψ(ε) ≤ F (ψ(ε), φ(ε)) ≤ ψ(ε).This implies that F (ψ(ε), φ(ε)) = ψ(ε) and hence ε = 0, a contradiction.Therefore the sequence fφn is a Cauchy sequence in f(Rc).Since f(Rc) is complete, we have fφn → φ∗ as n→∞ for some φ∗ ∈ f(Rc).Since φ∗ ∈ f(Rc), there exists η ∈ Rc such that φ∗ = fη andhence lim

n→∞fφn = fη. From (4.11), we have α(fφn(c), fφn+1(c)) ≥ 1.

From(v), α(fφn(c), fη(c)) ≥ 1 for any n ∈ N ∪ 0.Clearly d(fφn+1(c), Tη) ≤ HE(Tφn, Tη) and hence

ψ(d(fφn+1(c), Tη)) ≤ ψ(HE(Tφn, Tη))

≤ F (ψ(||fφn − fη||E0), φ(||fφn − fη||E0

))

≤ ψ(||fφn − fη||E0).

On applying limits as n→∞ on both sides, we get ψ(d(fη(c), Tη)) ≤ ψ(0) = 0.Therefore ψ(d(fη(c), Tη)) = 0 and hence fη(c) ∈ Tη.Therefore T and f have a PPF dependent coincidence point in Rc.

Corollary 4.3. Let c ∈ I. Let T : E0 → K(E), α : E × E → R+ andα∗ : K(E)×K(E)→ R+ be three functions satisfying the following conditions:

i) there exist functions ψ, φ ∈ Ψ, with ψ strictly monotonically increasing,such that

α(γ(c), η(c)) ≥ 1 =⇒ ψ(HE(Tγ, Tη)) ≤ F (ψ(||γ−η||E0), φ(||γ−η||E0))for any γ, η ∈ E0,

ii) T is an α∗−admissible mapping,iii) Rc is algebraically closed with respect to the difference,iv) Tφ ⊆ Rc(c) for any φ ∈ E0,v) if φn is a sequence in E0 such that φn → φ∗ as n→∞ and

α(φn(c), φn+1(c)) ≥ 1 for any n ∈ N ∪ 0, then α(φn(c), φ∗(c)) ≥ 1for any n ∈ N ∪ 0, and

vi) there exist φ0 ∈ Rc and φ1(c) ∈ Tφ0 such that α(φ0(c), φ1(c)) ≥ 1.Then T has a PPF dependent fixed point in Rc.

PPF dependent fixed points of generalized contractive type mappings ... 63

Proof. By taking f = identity mapping in Theorem 4.2, we obtain the desiredresult.

The following corollary can be obtained directly from Corollary 4.3 bytaking ψ(t) = t and F (s, t) = s− t for any s, t ∈ R+.

Corollary 4.4. Let c ∈ I. Let T : E0 → K(E), α : E × E → R+ andα∗ : K(E)×K(E)→ R+ be three functions satisfying the following conditions:

i) there exists φ ∈ Ψ such thatα(γ(c), η(c)) ≥ 1 =⇒ HE(Tγ, Tη) ≤ ||γ − η||E0

− φ(||γ − η||E0)

for any γ, η ∈ E0,ii) T is an α∗−admissible mapping,

iii) Rc is algebraically closed with respect to the difference,iv) Tφ ⊆ Rc(c) for any φ ∈ E0,v) if φn is a sequence in E0 such that φn → φ∗ as n→∞ and

α(φn(c), φn+1(c)) ≥ 1 for any n ∈ N ∪ 0, then α(φn(c), φ∗(c)) ≥ 1for any n ∈ N ∪ 0, and

vi) there exist φ0 ∈ Rc and φ1(c) ∈ Tφ0 such that α(φ0(c), φ1(c)) ≥ 1.Then T has a PPF dependent fixed point in Rc.

If we take φ(t) = (1 − k)t for any t ∈ R+ and k ∈ [0, 1) in Corollary 4.4,we get the following.

Corollary 4.5. Let c ∈ I. Let T : E0 → K(E), α : E × E → R+ andα∗ : K(E)×K(E)→ R+ be three functions satisfying the following conditions:

i) for any γ, η ∈ E0,α(γ(c), η(c)) ≥ 1 =⇒ HE(Tγ, Tη) ≤ k||γ − η||E0

where k ∈ [0, 1),ii) T is an α∗−admissible mapping,

iii) Rc is algebraically closed with respect to the difference,iv) Tφ ⊆ Rc(c) for any φ ∈ E0,v) if φn is a sequence in E0 such that φn → φ∗ as n→∞ andα(φn(c), φn+1(c)) ≥ 1 for all n ∈ N ∪ 0, then α(φn(c), φ∗(c)) ≥ 1for all n ∈ N ∪ 0, and

vi) there exist φ0 ∈ Rc and φ1(c) ∈ Tφ0 such that α(φ0(c), φ1(c)) ≥ 1.Then T has a PPF dependent fixed point in Rc.

5. An application

Jachymski[21] introduced the following notation on a Banach space endowedwith a graph.

Let (E, d) be a metric space where d(x, y) = ||x− y||E for all x, y ∈ E and4 denotes the diagonal of the cartesian product of E×E. Consider a directedgraph G such that the set V (G) of its vertices coincides with E, and the setE(G) of its edges contains all loops; that is 4 ⊆ E(G). We assume that G hasno parallel edges, so we can identify G with the pair (V (G), E(G)). Moreover,we may treat G as a weighted graph by assigning to each edge the distancebetween its vertices. If x and y are vertices in a graph G, then a path in G

64 Gutti Venkata Ravindranadh Babu, Madugula Vinod Kumar

from x to y of length N(N ∈ N) is a sequence (xi)Ni=0 of N + 1 vertices such

that x0 = x, xN = y and (xi−1, xi) ∈ E(G) for i = 1, 2, ...N . A graph G isconnected if there is a path between any two vertices, G is weakly connected ifG is connected (where G is the induced undirected graph) and G is transitive if(x, y) ∈ E(G) and (y, z) ∈ E(G) then (x, z) ∈ E(G), for more details we referto [30].

Definition 5.1. ([21]) Let (X, d) be a metric space endowed with a graph G.We say that a self mapping T : X → X is a Banach G-contraction or simply aG-contraction if T preserves the edges of G; that is, for any x, y ∈ X,

(x, y) ∈ E(G) =⇒ (Tx, Ty) ∈ E(G)and T decreases weights of the edges of G in the following way :there exists α ∈ (0, 1) such that for any x, y ∈ X,

(x, y) ∈ E(G) =⇒ d(Tx, Ty) ≤ αd(x, y).

Theorem 5.2. Let c ∈ I. Let T : E0 → E and E endowed with a graph G.Suppose that the following conditions are true.

i) there exist functions ψ, φ ∈ Ψ, with ψ strictly monotonically increasing,such that

(f(c), g(c)) ∈ E(G) =⇒ ψ(||Tf − Tg||E) ≤ F (ψ(M(f, g)), φ(M(f, g))),where M(f, g) = max||f − g||E0 , ||f(c)− Tf ||E , ||g(c)− Tg||E ,

12 [||f(c)− Tg||E + ||g(c)− Tf ||E ]

for any f, g ∈ E0,ii) if (f(c), g(c)) ∈ E(G) then (Tf, Tg) ∈ E(G),

iii) if (f(c), g(c)) ∈ E(G) and (g(c), h(c)) ∈ E(G) then (f(c), h(c)) ∈ E(G)(i.e. G is transitive),

iv) Rc is algebraically closed with respect to the difference,v) if φn is a sequence in E0 such that φn → φ∗ as n→∞ and

(φn(c), φn+1(c)) ∈ E(G) for any n ∈ N ∪ 0, then (φn(c), φ∗(c)) ∈ E(G)for any n ∈ N ∪ 0, and

vi) there exists φ0 ∈ Rc such that (φ0(c), Tφ0) ∈ E(G).Then T has a PPF dependent fixed point in Rc.

Proof. We define α : E × E → R+ by

α(x, y) =

2 if (x, y) ∈ E(G)15 otherwise.

First we show that T is triangular αc−admissible mapping.Let α(f(c), g(c)) ≥ 1. Then (f(c), g(c)) ∈ E(G). From (ii), we have

(Tf, Tg) ∈ E(G) and hence α(Tf, Tg) = 2 ≥ 1. Let α(f(c), g(c)) ≥ 1 andα(g(c), h(c)) ≥ 1. Then (f(c), g(c)) ∈ E(G) and (g(c), h(c)) ∈ E(G). SinceG is transitive, we have (f(c), h(c)) ∈ E(G). Therefore α(f(c), h(c)) ≥ 1 andhence T is triangular αc−admissible mapping. From (vi), we have that thereexists φ0 ∈ Rc such that α(φ0(c), Tφ0) ≥ 1. Let φn be a sequence in E0

such that φn → φ∗ as n → ∞ and α(φn(c), φn+1(c)) ≥ 1 for any n ∈ N ∪ 0.Then (φn(c), φn+1(c)) ∈ E(G). From (v), we have (φn(c), φ∗(c)) ∈ E(G) forany n ∈ N ∪ 0 and hence α(φn(c), φ∗(c)) ≥ 1 for any n ∈ N ∪ 0. Let

PPF dependent fixed points of generalized contractive type mappings ... 65

f, g ∈ E0 be such that α(f(c), g(c)) ≥ 1. Then (f(c), g(c)) ∈ E(G). From (i),we have T is generalized α− ψ − φ− F−contraction type mapping. Thereforeall conditions of Corollary 3.2 are satisfied and hence T has a PPF dependentfixed point in Rc.

Acknowledgement

The authors would like to thank the honorable editor and referee for theirvaluable suggestions.

References

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[2] Ali, M. U., and Kamran, T. On (α∗, ψ)-contractive multi-valued mappings.Fixed Point Theory Appl. 2013 (2013), Article 137, 7 pages.

[3] Ansari, A. H. Note on φ−ψ-contractive type mappings and related fixed pointtheorems. In The 2nd Regional Conference on Mathematics and Applications,Payame Noor University. 2014, pp. 377–380.

[4] Ansari, A. H., and Kaewcharoen, J. C-class functions and fixed pointtheorems for generalized α-η-ψ-φ-F-contraction type mappings in α-η-completemetric spaces. J. Nonlinear Sci. Appl 9, 6 (2016), 4177–4190.

[5] Asl, J. H., Rezapour, S., and Shahzad, N. On fixed points of α-ψ-contractivemultifunctions. Fixed Point Theory Appl. (2012), 2012:212, 6 pages.

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[7] Babu, G. V. R., Satyanarayana, G., and Vinod Kumar, M. Propertiesof Razumikhin class of functions and PPF dependent fixed points of weaklycontractive type maps. Bull.Int. Math. Virtual Inst. 9, 1 (2019), 65–72.

[8] Bae, J. S. Fixed point theorems for weakly contractive multivalued maps. J.Math. Anal. Appl. 284, 2 (2003), 690–697.

[9] Bapurao, C. D. On some common fixed point theorems with PPF dependencein Banach spaces. J. Nonlinear Sci. Appl. 5, 3 (2012), 220–232.

[10] Bernfeld, S. R., Lakshmikantham, V., and Reddy, Y. M. Fixed pointtheorems of operators with PPF dependence in Banach spaces. Appl. Anal. 6,4 (1977), 271–280.

[11] Bose, R. K., and Roychowdhury, M. K. Fixed point theorems for generalizedweakly contractive mappings. Surv. Math. Appl. 4 (2009), 215–238.

[12] Bose, R. K., and Roychowdhury, M. K. Fixed point theorems for multi-valued mappings and fuzzy mappings. Int. J. Pure Appl. Math. 61, 1 (2010),53–72.

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[14] Choudhury, B. S. Unique fixed point theorem for weakly C-contractive map-pings. Kathmandu University J. Sci., Eng. and Tech. 5, 1 (2009), 6–13.

[15] Ciric, L., Alsulami, S. M., Salimi, P., and Vetro, P. PPF dependent fixedpoint results for triangular αc-admissible mappings. Sci. World J. 2014 (2014),Art. ID 673647, 10 pages.

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[18] Haitham, Q., Noorani, M. S. M., Shatanawi, W., and Alsamir, H. Com-mon fixed points for pairs of triangular α-admissible mappings. J. NonlinearSci. and Appl. 10, 12 (2017), 6192–6204.

[19] Hussain, N., Khaleghizadeh, S., Salimi, P., and Akbar, F. New fixedpoint results with PPF dependence in Banach spaces endowed with a graph.Abstr. and Appl. Anal. 2013 (2013), Art. ID 827205, 9 pages.

[20] Hussain, N., Kutbi, M. A., and Salimi, P. Fixed point theory in α-completemetric spaces with applications. Abstr. Appl. Anal. (2014), Art. ID 280817, 11pages.

[21] Jachymski, J. The contraction principle for mappings on a metric space witha graph. Proc. the Amer. Math. Soc. 136, 4 (2008), 1359–1373.

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[23] Karapinar, E., Kumam, P., and Salimi, P. On a α − ψ−Meir-Keeler con-tractive mappings. Fixed point theory and applications 2013 (2013), ‘Article 94,12 pages.

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Received by the editors July 10, 2018First published online June 29, 2019

Vol.:(0123456789)1 3

Journal of the Brazilian Society of Mechanical Sciences and Engineering (2019) 41:535 https://doi.org/10.1007/s40430-019-2049-4

TECHNICAL PAPER

Experimental investigation to optimize tool performance in high‑speed drilling: a comparative study

Balla Srinivasa Prasad1 · D. S. Sai Ravi Kiran2

Received: 21 March 2019 / Accepted: 23 October 2019 © The Brazilian Society of Mechanical Sciences and Engineering 2019

AbstractIn this study, the influence of drilling parameters on circularity error, tool tip temperature and flank wear is investigated while drilling of Ti–6Al–4V alloy specimens with dissimilar cutting tool materials under dry machining conditions. In addition, optimal control factors for circularity error, tool tip temperature and flank wear have been determined using Taguchi–Grey relational analysis. Rotational speed of the spindle, feed rate and drill bit material are considered as control factors. Numerous drilling experimental runs have been performed employing L27 orthogonal array on a CNC vertical machining centre with 12Ø-mm-diameter holes on 10-mm-thick plates. An infrared thermal camera FLIR E60 is employed to record the temperature at tool chip interface, and Kistler 8793 tri-axial accelerometer is used to get hold of vibration data in real time. Analysis of variance has been carried out to ascertain the most substantial control factors among rotational speed, feed rate and drill bit material and also to establish the effects of the same over circularity error (Cr), temperature (T) and flank wear (VB).

Keywords Multi-objective optimization · Circularity error · Drilling · Infrared thermography · Grey relational analysis (GRA)

1 Introduction

In metal cutting machining, the main issues that affect cut-ting tool life are relative vibrations between the tool and the workpiece, chip temperature or tool tip temperature along with machining parameters like feed rate (f), spindle rota-tional speed (N), tool geometry, depth of cut (d), etc. Thus, it turns out to be important for the manufacturing industry to discover the appropriate intensities of cutting process fac-tors for attaining optimum tool life. In metal cutting, heat generation is always a key topic to be studied [1]. As a part of process enhancement, it is certainly beneficial to proac-tive monitoring of quality directly in the machining pro-cess instead of the product; hence, the chances of parts with defects can be reduced or even eliminated.

Drilling is the most significant step in the whole of machining process. With an objective to enhance metal cut-ting procedures to bring down the cost of parts, the required framework level is used in metal cutting process model [2]. For better performance of a cutter, it is necessary to model the interactions of tool chip interface. Various meth-odologies are proposed for better cutter performance such as mechanistic, empirical, numerical and analytical. It has been accounted that one-third course of action of material subtraction processes performed in manufacturing firms is drilling operation [3]. For this motive, drilling is indispen-sable and all things considered adopted material removal procedure. Nevertheless, tool failure might yield in drilling operations as an outcome of tool wear. Consequently, in-process forecasting of tool wear in drilling operations should be explored [4]. An experimental investigation is presented on how the quality of hole is influenced by the parameters of drilling, i.e. feed rate and spindle rotational speed. The assessment included considering the size of hole, circular-ity error, burrs at the exit and entry and chip developments. Moreover, the quality of hole is investigated by establishing the involvement of parameters of cutting using ANOVA as well as conclusions are derived statistically [5]. Kivak et al. [6] studied the influence of cutting parameters on tool wear

Technical Editor: Lincoln Cardoso Brandao.

* Balla Srinivasa Prasad bsp.prasad@gmail.com

1 Mechanical Engineering, GIT, GITAM, Visakhapatnam 530045, India

2 Mechanical Engineering, ANITS, Visakhapatnam, India

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and the quality of the hole, i.e., circularity of hole diameter while the drilling Inconel 718 with uncoated and coated drill bits. It was concluded that the quality of the hole and the performance of the drill bit are very poor at higher rotational speed and feed rates. A severe wear of tool amplification is noticed at higher cutting speeds. The multiple responses of arduous interrelationships can be solved using GRA [7]. An advanced optimization was carried out in drilling operation performed on metal matrix composites by altering differ-ent cutting parameters with GRA of orthogonal array [8]. The multiple responses obtained by different parameters of drilling operation are optimized in this study such as torque, cutting force and roughness of the surface.

In this work, high-speed steel (HSS) drill, tungsten car-bide drill and titanium nitride (TiN) coated are examined for the duration of dry drilling of Ti–6Al–4V alloy under dry machining condition. The drilling experiments were accomplished on 10-mm-thick aerospace quality workpiece material. The experimental runs are performed to probe the prominent outcome of the ideal operation of drilling frame-work and drilling cutter category on the interface tempera-ture, quality of the hole and vibration factor displacement. By using Taguchi technique in GRA performance charac-teristics, the available experimental results were optimized.

2 Proposed methodology in the present work

In the present work, difficult-to-cut material like Ti–6Al–4V alloy was used as workpiece material. An attempt is made to investigate the effect and performance of different cutting tool materials in dry drilling operation. Analysis of infrared thermal images and regression analysis carried out in this work will validate the cutting tool performance physiogno-mies. The performance analyses have shown that the spindle rotational speed and feed rate are the most prevailing fac-tors for the tool wear and cutting temperature, respectively. The confirmation test has certified the validity of the regres-sion model industrialized in the present work. This kind of work is uncommon in the field of machining operations, particularly in drilling. Figure 1 gives the methodology in the present work. The test set-up used for investigation is a 3-axis CNC drilling machine with high-speed 3.5 KW spindle which is capable of drilling up to 18,000 RPM is presented in Fig. 2.

According to methodology proposed, drilling operation is selected to generate holes on most commonly used materials in aircraft and automobile industry such as titanium alloys (Ti–6Al–4V) of dimensions (150 mm × 150 mm × 10 mm). Steps planned to generate experimental data are listed below:

• The drilling operation is performed using a 2-flute drill bit on 3-axis high drilling machine as per combinations listed in Table 6.

• The workpiece is fixed firmly on the work table as pre-sented in Fig. 2.

• An infrared thermal camera FLIR E60 is employed to collect the temperature at tool chip interface in real time.

• Every test is initiated with a new drill bit, and tempera-ture at tool chip interface is acquired using thermal images, and machining is stopped after drilling the fifth hole in every test condition for circularity error under coordinate measuring machine (CMM) and tool wear measurement with direct method.

• Inspected the drilled holes for roundness measurement under CMM.

• Experimental runs are derived through Taguchi L27 orthogonal array, and the same procedure has been adopted to acquire experimental data.

• The above listed steps have been repeated to carry out experiments in real time.

Later, ANOVA is employed to determine the most influ-ential process input parameter and the fraction of participa-tion each contributing process parameter on the attributes of quality has been derived. Then, the optimum machin-ing parameters are identified utilizing Taguchi’s S/N ratio charts. The GRA in Taguchi method is adopted to perform multi-objective optimization for process parameters. After that, effect of machining parameters on cutting temperature, circularity error and tool wear is analysed by varying the tool material. Finally, regression analysis is carried out to determine the correlation of cutting temperature, circularity error and tool wear with drilling process parameters.

The multiple responses are assessed by a grey relational grade (GRG). Proportionately, the multiple response opti-mization will be transformed into an individual relational optimization grade. This proposed methodology gives ample scope for applying the combined Taguchi’s with GRA method for optimization of drilling parameters as a part of drill bit tool performance evaluation. Hence, the cur-rent study attempts to present the application of GRA in choosing optimum order for drilling on multi-performance features, specifically the circularity error of the drilled hole, temperature and tool wear. Furthermore, the most substantial factors and the order of prominence of the manageable fac-tors for drilling process have been generated.

2.1 Drill tool and workpiece materials employed in the experiment

The experiments are carried out on Ti–6Al–4V alloy mate-rial, and the chemical composition is tabulated in Table 1.

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Drill bit tools made up of three different types namely high-speed steel, tungsten carbide (WC) and tungsten car-bide with physical vapour deposition (PVD) TiN coating are cast-off in the experimental investigation. Specifica-tions of the drill bit tools are shown in Table 2, and their properties are presented in Table 3. Chemical composition of the cutting tool materials is given in Table 4.

2.2 Equipment details and specifications

As part of the investigation, experiments have been carried out on a three axis high-speed CNC drilling machine and its specifications are mentioned as follows: CNC machine is programmed with Win CUT software package which runs on standard G/M codes.

Spindle motor of this machine is rated at 3.5 KW with a maximum spindle rotational speed up to 18,000 rpm. Kistler model 8793 accelerometer was mounted on spindle housing

Fig. 1 Methodology of the present work Vibration assisted drilling

Experiment design (Taguchi L27 Orthogonal array)

Workpiece and drill bits(Ti-6Al-4V & HSS/WC-C/WC-UC)

Temperature measurement (T)

(FLIR 60)

Circularity error (CR)measurement

(DEA Global CMM)

Drill bit flank wear (VB) measurement

(Direct method)

ANOVA (To find significant input parameters)

Taguchi coupled Grey Relational Analysis (For multi objective optimization)

S/N ratioNormalization, etc.

Grey relational coefficients (GRC)

Grey Relational Grade (GRG)

ANOVA for GRG

Optimum condition (prediction)

Correlation of circularity error, temperature and flank wear (regression analysis)

Mathematical modeling (2nd order equations for)

WC-C

Cr = 0.00297 + 0.000001 N + 0.000057 f - 0.000000 N2 -0.000000 f2 - 0.000000 N* f

T = 94.02 - 0.01104 N - 0.0604 f + 0.000001 N2

+ 0.000122 f2 + 0.000002 N* fVB = 0.01517 - 0.000000 N + 0.000012 f + 0.000000 N2

+ 0.000000 f2 - 0.000000 N* f

93.98

99.51

99.73

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to enable vibration data acquisition for process monitoring in real time. Discussions related to vibration signal acquisi-tion and signal features are intentionally not included in the present paper. Temperature rise during drilling is measured using FLIR E60 thermal camera [9]. Thermal camera is kept at a constant distance, i.e. 1 m away from the machining zone all through the investigation. Emissivity of Ti–6Al–4V is 0.35. The investigational test set-up is as shown in Fig. 2.

2.3 Experiment design

The design of experiments (DOE) facilitates the designers to resolve both the interactive and individual effects of many aspects simultaneously that can disturb the produc-tivity in any design. In addition, DOE also offers a com-plete understanding of interaction among design elements. The DOE is a well-ordered course of action for scheduling experimentations so that attained data can be explored to

Fig. 2 Investigational test set-up

Table 1 Mechanical properties and chemical composition of workpiece materials Ti–6Al–4V alloy

Workpiece (Young’s modu-lus)

Density (kg/m3) Hardness, HV Yield strength (MPa) Tensile strength (MPa) Thermal conductivity (W/m K)

Mechanical properties Ti–6Al–4V (120 GPa) 4420 349 880 950 6.7

Al Fe O Ti V

Chemical composition of workpiece materials Ti–6Al–4V 6 Max.25 Max.25 90 4

Table 2 Specifications of drill bits used for the investigation

Properties HSS drill WC drill PVD (TiN)-coated WC drill

Standard DIN 338 DIN 338 DIN 338Diameter (mm) 10 10 10Material HSS Tungsten carbide Tungsten carbideType 2-flute twist drill 2-flute twist drill 2-flute twist drillLength (mm) 133 133 133Flute length (mm) 87 87 87Number cutting edges 2 2 2Lip length (mm) 4.5 4.5 4.5Web thickness (mm) 2 2 2Margin (mm) 1.25 1.25 1.25Point angle (°) 118 118 118Helix angle (°) 30 30 30Relief angle (°) 8–12 8–12 8–12Lip angle (°) 55 55 55Coating Uncoated Uncoated coatedCoating thickness (μm) 0 0 2–3Thermal conductivity

(W/m K) 200 °C23.5 88 110

Cutting direction Right Right Right

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yield compelling, nearly exact and impartial conclusions. In the present study, three control factors with three lev-els are taken into consideration for investigation. Table 5 presents the selected levels of the process parameters for the investigation.

Identifying the objectives of the experiment is the initial points of DOE. In order to acquire maximum information possible, a well-defined investigational design should be used. Full factorial method is used for the design of experi-ments using Minitab-17 software. In full factorial design, if the number of levels is identical for individual factors, then the possible number of trials, N, can be expressed as:

where L = number of levels for each factor and m = number of factors. The experimentations have been done as per the orthogonal array L27 (33) as presented in Table 6, to estimate the relationship among the input and the output parameters response variables [10].

3 Modelling of output parameters

A statistical tool called Minitab version 17 is used for exper-imental data statistical analysis. The optimum machining parameters are identified utilizing Taguchi’s S/N ratio charts. ANOVA is employed to determine the most influential pro-cess input parameter, and the fraction of participation each contributing process parameter on the attributes of quality has been derived. The GRA in Taguchi method is adopted to

(1)N = Lm

perform multi-objective optimization for process parameters. The primary intension of the present work is to minimize the circularity error, temperature and flank wear. Therefore, in view of the quality features, the “smaller the better” has been preferred for the output response characteristics.

The method of least squares is primarily cast-off for the regression analysis to compute the regression coefficients. In addition, factual functional relationship can be established between Yxt and the set of process parameters in a multi-ple linear regression model. The process variable of linear function will demonstrate the output response variables. The first-order approximating function is presented as follows:

The second-order mode must employ the higher degree polynomial:

(2)y = 0 + 1X1 + 1X1 +⋯ + kXk

(3)y = 𝛽0 +

k∑

i=1

𝛽iXi +

k∑

i=1

𝛽iiXi2 +∑

i

∑

<j

𝛽ijXiXj

Table 3 Properties of the cutting tool materials

Property HSS Uncoated WC PVD-coated (TiN) WC

Coefficient of thermal expansion (W/m K) 41.5 5.5 7.1 × 10−6/KDensity (g/cm3) 2.79 15.6 5.4 g/cm3

Poisson’s ratio 0.27–0.30 0.31 0.25Specific heat (J/kg K) 418 292 184Young’s modulus (GPa) 210 700 250Melting point (°C) 64 2870 2950Thermal conductivity (W/m K) 200 °C 23.5 88 110Hardness (HV) 870 1460 2200Coating thickness (μm) – – 2–3

Table 4 Composition of drill bit tools

HSS Si V Cr Mn Ni Nb Mo Co Fe

3.709% 1.95% 3.97% 0.046% 0.688% 0.792% 6.469% 4.382% 77.993%

WC W Cu Zn Ni Co C

54 20 16 4 3 3

Table 5 Parameters of the experimental design and their levels

Cutting parameters Level 1 Level 2 Level 3

Rotational speed, N (rpm) 4200 8400 10,600Feed rate, f (mm/min) 159 318 400Drill bit material, DM (WC-

UC)HSS Coated tungsten

carbide (WC-C)

Uncoated tungsten carbide

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One of these models is used in engineering problems for their implementation. Although it is uncertain that the polynomial model will provide exact outcomes for the true functional association over the complete space of the input drilling parameters, nevertheless for a comparatively small zone, they regularly work fairly well.

4 Results and discussion

With an increase in machining time, flank wear, temperature at the tip of the tool and circularity error are measured. The circularity error index (Cr) of the drilled hole represents the quality of the hole and can be determined through experi-mentation only. Experimental findings obtained concerning circularity error (Cr), temperature (T) and flank swear (VB) are shown in Table 6.

From the observations presented in results in Table 6, it is established that circularity error drops with rise in rotational

speed of the spindle from 4200 to 10,600 rpm, but it aug-ments with raise in feed rate from 159 to 400 mm/min.

4.1 Effect of cutting parameters on circularity error (Cr)

A GLOBAL performance DEA CMM that is the most suit-able for performing metrology-related measurements has been used. The new PC-DMIS adaptive scanning function permits the users of any level to grasp superlative scan-ning performance with accuracy from 1.5 + L/333 μm. High exactness is ensured in temperature range 16–26 °C. A global performance coordinate measuring machine DEA (07075) is employed to measure the circularity error of drilled holes [11].

From Table 7, it is identified that the spindle rotational speed (N) is the principal effecting drilling parameter which influences the circularity error succeeded by drilling cutter feed rate and material type [12]. Spindle rotational speed affects the circularity error by 39.57%, feed rate by 19.06%

Table 6 Experimental observations

S. no. Spindle rotational speed, N (rpm)

Feed rate, f (mm/min)

Drill bit material (DM)

Circularity error Cr (µm)

Cutting tem-perature, T (°C)

Flank wear, VB (mm)

1 4200 159 HSS 0.022 351 0.182 4200 159 WC-C 0.012 59.9 0.0223 4200 159 WC-UC 0.014 146.2 0.164 4200 318 HSS 0.024 389 0.215 4200 318 WC-C 0.015 61.3 0.0276 4200 318 WC-UC 0.017 151.5 0.187 4200 400 HSS 0.028 408 0.238 4200 400 WC-C 0.018 63.1 0.0339 4200 400 WC-UC 0.02 158.4 0.2010 8400 159 HSS 0.016 462 0.2611 8400 159 WC-C 0.009 64.6 0.03612 8400 159 WC-UC 0.015 169.2 0.2313 8400 318 HSS 0.019 481 0.2714 8400 318 WC-C 0.012 68.7 0.04015 8400 318 WC-UC 0.017 173.5 0.2516 8400 400 HSS 0.02 498 0.3017 8400 400 WC-C 0.014 73.4 0.04218 8400 400 WC-UC 0.019 192.8 0.2919 10,600 159 HSS 0.006 503 0.3420 10,600 159 WC-C 0.005 83.7 0.04721 10,600 159 WC-UC 0.011 249 0.3222 10,600 318 HSS 0.008 557 0.3823 10,600 318 WC-C 0.011 85.5 0.05024 10,600 318 WC-UC 0.014 265.0 0.3425 10,600 400 HSS 0.017 568 0.4026 10,600 400 WC-C 0.009 90 0.05327 10,600 400 WC-UC 0.016 289 0.36

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and drill bit material by 23.22%. In the present model, the p value < 0.005 illustrates that the drill tool material, feed rate and rotational speed are the key process parameters. Where, DF degree of freedom, Seq.SS sequence sum of square, Adj.SS adjacent sum of squares, Adj.MS adjacent mean square, F Fisher’s value P probability of significance.

Figure 3 shows that a minutest quantity of circularity error transpired at the highest rotational speed considered in the study, i.e. 10,600 rpm and a squat feed rate of 159 mm/min with coated tungsten carbide (WC-C) drill bit.

4.2 Effect of machining parameters on cutting temperature

Figure 4 gives the thermal images which indicate the tem-perature variations with respect to different drill bit materials investigated in the present study. Figure 4a shows the drill-ing zone with work piece and drill bit set-up, and Fig. 4b gives the usage of FLIR E60 thermal camera for tempera-ture measurement in real time. Temperature variations with HSS drill while drilling holes on Ti–6Al–4V specimens are presented in Fig. 4c, d. Figure 4c indicates the initial stage, and Fig. 4d represents the final stage with HSS drill tool. Similarly, Fig. 4e (uncoated WC drill bit tool) and Fig. 4g

(PVD-TiN-coated WC drill bit tool) shows the temperature rise at initial stages. Whereas Fig. 4f (uncoated WC drill bit tool) and Fig. 4h (PVD-TiN-coated WC drill bit tool) pre-sents the temperature rise levels at ending stages of respec-tive machining condition programmed in the investigation.

From Table 8, it is identified that the drill bit material type (DM) is the key cutting parameter which influences the temperature followed by the spindle rotational speed (N) and feed rate (f). The drill bit material type affects the tempera-ture by 91.54%, the rotational speed by 5.64% and feed rate by 0.44%.

From Table 8, the parameter p value < 0.005 specifies that the spindle rotational speed and drill tool material remain the major influential parameters of process in the model, whereas for temperature, the feed rate was observed to remain a less significant parameter. The outcome of variable rotational speed of the spindle, feed rate and drill tool mate-rial on temperature is displayed in Fig. 5. It can be observed from Fig. 6 that the temperature increases at augmented rota-tional speed from 4200 to 10,600 rpm and also ascends with amplified feed rate from 159 to 400 mm/min.

From Fig. 5, it is clear that the temperature rise is the highest while drilling the work piece with high-speed steel drill bit tool among three different drill bits, irrespective of

Table 7 Analysis of variance for circularity error

S = 0.0026226; R2 = 95.87%; R2(adj.) = 94.43%

Source DF Seq. SS Adj. SS Adj. MS F p Contribution (%)

N 2 0.000300 0.000300 0.00015 21.83 0.000 39.57f 2 0.000145 0.000145 0.000072 10.52 0.001 19.06DM 2 0.000176 0.000176 0.000088 12.81 0.000 23.22Error 20 0.000138 0.000138 0.000007Total 26 0.000759

Fig. 3 Main effect plots for circularity error

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machining combination. As expected [13], cutting temperature increased with the elevation of rotational speed in titanium drilling with PVD-coated drill as well, but this tendency is

more predominant with HSS drill bit. The reason may be that material removal rate increased significantly with increase in the feed rate which resulted in increased heat generation [14].

Fig. 4 Temperature measure-ments with FLIR 60 thermal camera with different drill bits

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4.3 Effect of cutting parameters on flank wear

Drill bit tool’s flank wear can be measured by direct method as shown in Fig. 1. For a drill bit, flank wear is represented with (VB) in mm, which can be calculated according to the following relation:

(4)VB =

(

1

tan − tan

)

∗

(

ΔD

2

)

mm

where is the rake angle of the drill tool (in degrees), is the helix angle of the drill bit (in degrees) and ΔD is the change in the diameter of the drill bit (mm) due to tool wear. From Table 9, it is found that the drill bit material type is the chief impelling cutting process parameter which influences the drill bit flank wear (VB) followed by the rotational speed of the spindle (N) and feed rate (f) [15]. The drill bit material type affects the flank wear by 78.79%, the rotational speed by 14.61% and feed rate by 1.3%. The statistical parameter p value < 0.05 implies that the drill bit material type and rotational speed are the two foremost process parameters in

Table 8 Analysis of variance for ‘temperature’

S = 30.8906; R2 = 97.63%; R2(adj.) = 96.92%

Source DF Seq. SS Adj. SS Adj. MS F p Contribution (%)

N 2 45,413 45,413 22,707 23.80 0.000 5.64f 2 3554 3554 1777 1.86 0.181 0.44DM 2 737,027 737,027 368,514 386.19 0.000 91.54Error 20 19,085 19,085 954Total 26 805,080

Fig. 5 Main effect plots for temperature

Fig. 6 Main effect plots for flank wear

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the model, and it is also identified that the influence of feed rate is very minimal in the process.

Figure 6 shows the outcome of variable feed rate, spindle rotational speed and drill tool material on flank wear values. From Fig. 6, it can be observed that the wear of the flank is amassed with rise in rotational speed of the spindle from 4200 to 10,600 rpm. Flank wear values exhibit similar trend with increase in feed rate from 159 to 400 mm/min.

From Fig. 6, it is clear that the flank wear values are found to be small as long as machining is carried out with coated tungsten carbide drill.

4.4 Phases aimed at optimization by grey relational analysis (GRA)

In this work, GRA along with the Taguchi technique is used to optimize the process parameters of drilling. Com-puting the resultant responses is the foremost step, i.e. S/N ratio values either smaller is better or larger is better type problem based on the output variables [16, 17]. The normal-ized original order will attain the required objective value. Otherwise, original order can simply be normalized using straightforward methodology, i.e. allow the values of the inventive order be divided by the starting value of the order:

where Xi*(K) is the order after the data pre-processing and Xi

0 is the desired value of Xi0 (K), where i = 1,2,3…m;

k = 1,2,3,…n. Where m is the experimental data item num-ber, n is the parameter number and Xi

0 (K) denotes the origi-nal order [18]. Normalized S/N ratio value of the response variables and their results are tabulated in Table 10.

The next step in the grey relational analysis is to quantify the grey relation grade (GRG). The GRG is defined as the measure of relevancy among two orders or two systems. In the measurement of local grey relation, all the orders are compared with the reference order Xi

0 (K). After processing the data once, the Kth performance characteristics of the GRC in the ith experiment can be operated by:

(5)X∗

i(K) =

X0i(K)

X0i(1)

where ∆oi (K) is the reference order Xo*(K) deviation and the comparability order Xi*(K). ξ is the coefficient of identifica-tion or distinctive well-defined in the range of 0 ≤ ξ ≤ 1 (this value can be in tune with the real-world requirements of the

(6)(K) =Δmin + Δmax

Δ0i(K) + Δmax

Table 9 Analysis of variance for ‘flank wear (VB)’

S = 0.0332778; R2 = 94.71%; R2(adj.) = 93.12%

Source DF Seq. SS Adj. SS Adj. MS F p Contribution (%)

N 2 0.061188 0.061188 0.030594 27.63 0.000 14.61f 2 0.005444 0.005444 0.002722 2.46 0.111 1.30DM 2 0.329867 0.329867 0.164933 148.94 0.000 78.79Error 20 0.022148 0.022148 0.001107Total 26 0.418647

Table 10 Grey relational coefficients and grey relational grade (GRG) values

Exp. no. Grey relational coefficient (GRC)

Grey relational grade (GRG)

Rank

Cr T VB

1 0.657 0.539 0.462 0.512 112 0.418 0.333 0.333 0.361 273 0.451 0.376 0.441 0.400 214 0.742 0.587 0.499 0.560 85 0.469 0.334 0.336 0.379 246 0.511 0.379 0.462 0.425 197 1.000 0.614 0.526 0.650 38 0.535 0.335 0.340 0.401 209 0.590 0.383 0.486 0.453 1510 0.489 0.706 0.574 0.552 911 0.377 0.335 0.342 0.366 2612 0.469 0.389 0.526 0.434 1713 0.561 0.745 0.592 0.595 614 0.418 0.337 0.344 0.375 2515 0.511 0.392 0.558 0.457 1416 0.590 0.784 0.654 0.619 517 0.451 0.339 0.346 0.395 2218 0.561 0.404 0.632 0.494 1319 0.343 0.796 0.759 0.638 420 0.333 0.344 0.349 0.389 2321 0.404 0.443 0.703 0.506 1222 0.365 0.958 0.904 0.771 223 0.404 0.345 0.351 0.430 1824 0.451 0.456 0.759 0.526 1025 0.511 1.000 1.000 0.878 126 0.377 0.347 0.353 0.446 1627 0.489 0.477 0.825 0.568 7

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system). If the value of ξ is the small and the distinguished ability is the large, once grey relational coefficient (GRC) is determined, take the average of the GRC as the GRG. The following relationship is used to calculate GRG:

The GRA is carried out by evaluating the GRC for nor-malized S/N ratio values, and the resulting GRG values are shown in Table 11 using Eqs. (6) and (7).

(7)i =1

n

n∑

k=1

i(K)

4.5 Determination of optimum process parameter combination

From the analysis of drilling process, it is clearly evident that the lower circularity error and lesser cutting tempera-ture value as well as lower value of tool flank wear pro-vide drilled holes with better quality. Consequently, the data orders circularity error, cutting temperature and tool flank wear pinpoint the “smaller-the-better” characteristics. Higher GRG indicates better machinability combination hence the quality of the product is better. Consequently, on the basis of higher GRG or the rank, the optimal process parameters level can be achieved.

4.6 Identification of significant factor

In order to realize the remarkable factors on measurement, analysis of variance ANOVA for GRG is used as an analyti-cal tool by observing the association among the proposed factor and response variable.

Taguchi method cannot choose the outcome of distinct drilling process parameters on the entire process; there-fore, the fraction of involvement is premeditated by means of ANOVA on its behalf. Empirical Eq. (7) is utilized to compute average GRG values and is presented in Table 11. The specific effects of input factors on GRG are presented in Fig. 7. The utmost influencing factor on the GRG is

Table 11 Grey relational grade (GRG) average effects

Level Rotational speed, N

Feed rate, f Drill bit material, DM

1 0.46 0.476 0.5722 0.462 0.501 0.5443 0.641 0.393 0.473Delta 0.181 0.108 0.098Rank 1 2 3

Fig. 7 Variation in factors on the grey relational grade (GRG)

Table 12 ANOVA for grey relational grade

Source DF Seq. SS Adj. SS Adj. MS F Contribu-tion (%)

N 2 0.06622 0.06622 0.033110 16.87 16f 2 0.03085 0.03085 0.015423 7.86 7DM 2 0.28917 0.28917 0.144586 73.67 68Error 20 0.03925 0.03925 0.001963 9Total 26 0.42549

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identified as drill bit material type (68%), followed by the rotational speed (16%) and feed rate (7%), and these results are tabulated in Table 12.

4.7 Prediction of optimum condition

At the optimum condition, an anticipated mean is given by:

where V3 , DB2 and F1 are the average values of GRG con-cerning to the best operational condition input process parameters. TAGG is the average GRG of the complete exper-imentations. Consequently, the optimum condition predicted mean from Eq. (9) is

5 Mathematical modelling

The regression analysis [19] is used to determine the cor-relation of cutting temperature; circularity error and tool wear with drilling process parameters. In the present work, to forecast the minimum cutting temperature, tool wear and circularity error under every condition with an objective to evaluate the tool performance level, regression analysis is used. The second-order equations of the circularity error, cutting temperature and tool wear are presented in Table 13.

6 Confirmation of the experimentation

The confirmation experimentation is carried out at the optimal conditions to confirm the quality features while drilling of Ti–6Al–4V alloy. The response values by the confirmation experimentation [20] succession at the optimal condition are Cr = 0.009 μm, T = 90 °C and VB = 0.053 mm. The GRG value as per the above

(8)pred = V3 + F3 + DB1 − 2xTAGG

(9)pred = 0.9333.

discussion is 0.9512. This outcome is inside the 95% confi-dence level of the anticipated optimal condition, and more-over, GRA value of confirmation experiment enhanced by 2% from the forecasted mean value. Therefore, the GRA built on Taguchi method for the optimization of the multi-response problems is a very beneficial tool for forecasting the circularity error, cutting temperature and drill bit tool wear in the drilling of Ti–6Al–4V alloy specimens.

7 Conclusions

Drilling experimentations have been carried out with high-speed steel, tungsten carbide and TiN-coated tungsten car-bide twist drill and Ti–6Al–4V alloy as work material. Circularity error, cutting temperature and drill bit tool wear values have been collected under different machining conditions. The subsequent conclusions are drawn:

• Grey relational analysis together with Taguchi method for the optimization of the multi-response problems is a very beneficial means for forecasting the circularity error, cutting temperature and tool wear in the drilling.

• From the GRA analysis, it is discovered that cutting tool material, rotational speed of the spindle and feed rate are protuberant issues which disturb the drilling of difficult to cut material like Ti–6Al–4V alloy.

• Drill bit material influences (DM = 73.67%) more, fol-lowed by rotational speed (N = 16.87%) and feed rate (f = 7.86%). The best performance features are attained with TiN-coated tungsten carbide twist drill bit when drilling with higher feed of 400 mm/min and higher rota-tional speed of 10,600 rpm at constant cutting point angle.

• Confirmation test outcomes demonstrated that the deter-mined optimal condition of drilling process parameters to fulfil the real necessities of drilling operation.

Table 13 Modelling for response-based experimental data

Eqn. no. Drill bit material Responses R2 (%)

10 HSS Cr = 0.0277 + 0.000002 N − 0.000086 f − 0.000000 N2 + 0.000000 f 2 + 0.000000 N* f 95.9011 T = 243.8 + 0.0124 N + 0.293 f + 0.000001 N2 –0.000173 f 2 + 0.000003 N* f 99.1812 VB = 0.2691 − 0.000038 N − 0.000097 f + 0.000000 N2 + 0.000000 f 2 + 0.000000 N* f 99.4913 WC-C Cr = 0.00297 + 0.000001 N + 0.000057 f − 0.000000 N2 − 0.000000 f 2 − 0.000000 N* f 93.9814 T = 94.02 − 0.01104 N − 0.0604 f + 0.000001 N2 + 0.000122 f 2 + 0.000002 N* f 99.5115 VB = 0.01517 − 0.000000 N + 0.000012 f + 0.000000 N2 + 0.000000 f 2 − 0.000000 N* f 99.7316 WC-UC Cr = 0.00421 + 0.000003 N − 0.000001 f − 0.000000 N2 + 0.000000 f 2 − 0.000000 N* f 98.7417 T = 362.2 − 0.06577 N − 0.368 f + 0.000005 N2 + 0.000624 f 2 + 0.000016 N* f 99.8018 VB = 0.2162 − 0.000021 N − 0.000286 f + 0.000000 N2 + 0.000001 f 2 + 0.000000 N* f 99.58

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Acknowledgements This work is supported by the Science and Engi-neering Research Board, Department of Science Technology, Govt. of India under Empowerment and Equity Opportunities for Excellence in Science Scheme with a Grant No: SB/EMEQ-265/2014. Authors would like to thank mentor Sri. P.V.S. Ganesh Kumar, Scientist-H, Associ-ate Director, Naval Science Technological Laboratory Visakhapatnam, India for providing the experimentation test facility and necessary equipment.

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Vol.:(0123456789)1 3

Journal of Bio- and Tribo-Corrosion (2019) 5:61 https://doi.org/10.1007/s40735-019-0253-5

Tribological Behavior of Multi‑walled Carbon Nanotubes Grafted Carbon Fiber‑reinforced Friction Material

Naresh Kumar Konada1 · K. N. S. Suman2

Received: 14 June 2018 / Revised: 8 May 2019 / Accepted: 20 May 2019 / Published online: 3 June 2019 © Springer Nature Switzerland AG 2019

AbstractFriction characteristics of multi-walled carbon nanotubes grafted carbon fiber-reinforced friction material (MWCRFM), asbestos fiber-reinforced friction material (AFRFM), and oxidized carbon fiber-reinforced friction materials (OCRFM) are investigated in this work using a wear test apparatus. The surface of carbon fiber (CF) is chemically inert and hydrophobic in nature and possesses poor bonding performance with polymer matrix. Hence, in this work, an attempt is made to improve the bonding behavior between CF and remaining ingredients. CF surface is modified by two surface treatment methods. First, oxidation and second grafting multi-walled carbon nanotubes functionalized (MWCNT-F) on CF surface. CF content after surface modification is taken as (5 wt%) and mixed with remaining ingredients of friction material. Composite sheets are prepared by using hand layup method. Three types of friction materials, (MWCRFM), (AFRFM), and (OCRFM) are devel-oped and analyzed for coefficient of friction and wear rate using wear test apparatus. These materials are also characterized for SEM. MWCNTs-F on CF surface is observed. Sample specimens are cut from the composite sheets and the influence of performance properties like speed, load, and time on friction and wear is studied. The back ground for this research work is to identify the best configuration of materials and surface treatment method on CF for the improvement of tribological properties of friction material. The behavior of the samples is also analyzed using regression analysis L9 (3 × 3) experimen-tal design method for three different loads, time periods, and speeds. The results reveal that braking load and time plays an important role to control the friction and wear characteristics of MWCRFM. It was also observed that MWCNT-F grafted on CF possess less wear rate and high coefficient of friction compared to other formulations of materials.

Keywords Multi-walled carbon nanotubes · Carbon fibers · Chemical grafting · Tribological properties

1 Introduction

Brake friction material present in an automobile converts the kinetic energy of automobile to thermal energy by means of friction generated at the contact surface. Friction material selection primarily depends on the ability of the material to dissipate heat to the surroundings, wear rate, coefficient of friction, fade resistance, squealing, and atmospheric condi-tions. The other parameters that affect the friction material

selection are engine size, drive train gearing system design, braking system design, and tire and road conditions [1].The friction materials selected for any application in an engi-neering field have to provide stable coefficient of friction and low wear rate at various operating speeds, pressures, temperatures, and environmental conditions [2–6]. In the past, asbestos-reinforced friction material is widely used to produce brake pads, due its strength, resistance to heat, and fire proof. Since 1980s, asbestos was known as harmful content and was banned from being used as an ingredient to produce brake pads, because it can cause lung cancer and other health problems [7]. Therefore, non-asbestos contents like Kevlar (aramid fiber), steel fiber, glass fiber, and carbon fiber were used to replace asbestos. Brake friction material present in an automobile consists of more than nine different ingredients. These ingredients often contain fiber, binder, lubricant, abrasive, fillers, and other friction modifiers.

* Naresh Kumar Konada nareshkonada@gmail.com

K. N. S. Suman sumankoka@yahoo.com

1 Department of Mechanical Engineering, Anits Engineering College, Visakhapatnam, India

2 Department of Mechanical Engineering, Andhra University College of Engineering, Visakhapatnam, India

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The reinforcing fibers and binder resin used in friction materials have substantial influence in determining the friction characteristics. The frictional heat generated at the interface between brake disk and pad can easily raise the temperature beyond the transition temperature zone, which results in abrupt change in friction force during brak-ing. This occurs mainly due to degradation mechanism of binder and other ingredients. Fade resistance is a phenom-enon that describes the decreasing of frictional force at high temperature. Therefore, binder resin is usually preferred to be heat resistant. The type and amount of these ingredients are determined mostly based on experience, observation, or a trail and error method to make a new formulation. At present, Carbon fiber-reinforced polymer composites have been gaining their importance in all the fields of engineer-ing sectors, namely mechanical, civil, aerospace, etc. CF is primarily preferred for composite materials due to its excel-lent properties, such as high specific strength and stiffness, thermal stability, corrosion resistance, high tensile modulus, self lubrication, lower density, good electrical conductiv-ity, lower linear coefficient of expansion, and outstanding fatigue characteristics. They also find applications where high damping and chemical inertness are important. Hence, carbon fiber reinforcements can be used in friction materi-als to have high wear resistance compared to other fibers [8, 9]. The performance and mechanical properties of carbon fiber-reinforced composite primarily depend on the inter-facial adhesion between fiber and matrix and ingredients selected for the composite. However, CF surface is having poor damage resistance, early fiber matrix de bonding, trans-verse cracks and delamination in several applications and it is compensated by overdimensioning of composite parts. The best promising method to improve damage resistance is by growing carbon nanotubes (CNTs) on CF. The inter-facial properties of the CF and polymer can be improved by modifying the surface of CF and introducing hydroxyl or carboxyl groups on the surface. Therefore, many efforts have been carried out from the past to improve the surface properties of CF by using different treatment techniques like sizing, plasma, chemical oxidation, γ-ray irradiation, electrochemical, dip coating, MWCNT grafting on carbon fiber surface by chemical vapor deposition (CVD), etc. [10]. Among different techniques available to improve the inter-facial adhesion between fiber and matrix, MWCNT grafting on carbon fiber surface under CVD achieved good bonding strength between polymer and CF surface. Song et al. [11], observed that grafting straight carbon nanotubes radially on carbon fiber surface has improved its mechanical proper-ties. The mechanical properties such as compressive strength and interlaminar shear strength are increased by 275% and 138% compared to pure C/C Composite. Sharma et al. [12] identified that coating carbon fibers by using CVD on CF surface improves its tensile strength to 69% for CF/epoxy/

amine polymer matrix composites. Qian et al. [13] modified the carbon fiber surface by grafting carbon nanotubes on carbon fibers by CVD technique and observed that there is 26% increase in interlaminar shear strength for carbon fiber and polymethyl methacrylate (CF/PMMA) composite. Based on the work carried by many researchers to graft CNTs on CF surface, the use of CVD technique improved the perfor-mance of composite in terms of its interfacial shear strength, mechanical and tribological properties to a greater extent. But, grafting CNTs on CF surface by using CVD is a cost-lier process and involves more attention towards control of operating temperatures involved in the process. Therefore, an alternative method is followed in this work by modifying CF surface by two surface treatment methods and, the best surface treatment method is selected with optimum ingre-dients for improving the tribological properties of a friction material [14–17]. Konada and Suman also conducted experi-mental studies on usage of carbon fiber more effectively in friction materials [18–21].

The current work aims to evaluate and investigate the tribological properties of three brake friction materials AFRFM, OCRFM, and MWCRFM. Tribo tests were car-ried using a pin-on-disk machine. All the friction materials were tested against EN-32 case-hardened steel rotor disk. These tests were conducted at a nominal contact load of (19.62 N, 39.24 N, and 58.86 N), time (5 min, 10 min, and 15 min), and speed of the rotor is taken as (300 rpm, 600 rpm, and 900 rpm). Results are tabulated at all the conditions of the test and the best possible conditions are considered for achieving high coefficient of friction and low wear rate of the friction material.

2 Experimental Work

2.1 Materials and Methods

The material selection for the brake pad depends mainly on the ability of the material to withstand the given pressure and to satisfy all the important characteristics of a fric-tion material. The wear rate of friction material depends on the type of friction material used, pressure applied on the pads, friction material temperature, friction material contact area, friction material finish, heat removal rate, etc. The selection of ingredients for the friction material has to be carried based on the characteristics of a friction material.

The main characteristics that the friction materials should possess are [22]

(a) Maintain a sufficiently high coefficient of friction with the brake disk.

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(b) Not to decompose or break down in such a way that the friction coefficient with the brake disk is compromised, at high temperatures.

c) Exhibit stable and consistent coefficient of friction with the brake disk.

(d) Wear resistant.(e) Able to dissipate heat to the surroundings.(f) Having sufficient fade resistance.(g) Induce less squealing action and should be operated

over different atmospheric conditions.

2.2 Carbon Fiber

Commercially available, polyacrylonitrile (PAN)-based chopped carbon fibers were used for this study. Chopped carbon fibers having a carbon content of 95% are used in the present work. The properties of chopped carbon fibers given by the supplier are given in Table 1.

2.3 Multi‑walled Carbon Nanotubes (MWCNT)

Multi-walled carbon nanotubes (MWCNT) used in the pre-sent study are produced by using chemical vapor deposi-tion method (CVD). MWCNTs are used widely, as filler materials in polymer matrix composites in applications like structural, industrial, and aerospace sectors. The damp-ing characteristics and toughness behavior of the compos-ites reinforced with MWCNTs are observed to be greatly improved. Hence, in this work, MWCNTs are selected for use in organic friction material composites. The properties of MWCNT supplied by the supplier for the given study are given in Table 2. Chemicals such as NaOH, HNO3, H2SO4, and acetone solutions used in the current study are purchased from chemical laboratories, Visakhapatnam, India.

2.4 Carbon Fiber Surface Treatment Method

2.4.1 MWCNT Surface Oxidation Treatment

The multi-walled carbon nanotubes (MWCNTs) are treated to attach carboxylic acid groups on the surface to form func-tionalized MWCNTs called MWCNT-F. In this study, 2 g of MWCNT is added to 100 ml of concentrated sulphuric acid and 35 ml of nitric acid (Purity of 98.08% H2SO4 and 70% HNO3) (3:1 by volume ratio). The mixture is sonicated in a bath for 3 h at a temperature of 70 °C. The reaction mixture is then diluted with 200 ml of deionised water fol-lowed by vacuum filtration process using filter paper of 2 µm porosity. This washing operation is repeated three times to remove metal particles adsorbed by MWCNTs. The sample is allowed to dry in an oven at 100 °C. After drying opera-tion, the collected powder particles are immersed in 40 ml of acetone solution. Finally, MWCNT powder after vacuum filtration process is added with acetone solution and placed on ultra-sound bath sonicator to allow the acetone solution to completely mix with MWCNTs. During sonication process, acetone gases will leave the sample because the low boiling point of acetone is at 30 °C. The sample is later dried in oven at 100 °C for 4 h followed by drying in the open air to form MWCNT-F powder. This process will completely remove metal particles present in MWCNT powder in order to use MWCNT-F more effectively in polymer matrix composites. The process of using MWCNT-F on CF surface increases the damping characteristics and reduces squeal generation at the contact region between brake disk and pad [20].

2.4.2 Grafting MWCNT‑F on Oxidized CF Surface

Multi-walled carbon nanotubes functionalized (MWCNT-F) of qty 1.5 g obtained after filtration and drying operation is dispersed in a 20 ml of acetone solvent using an ultra-sound bath at 70 °C for 1 h. Complete mixing operation is car-ried out using a ultra-sound bath sonicator. Now, chopped carbon fibers are placed on a glass substrate and suspension

Table 1 Properties of carbon fiber [20]

Material Diameter (µm) Length (mm) Tensile strength (MPa)

Tensile modulus (GPa)

Sizing (%) Resistivity Ω/cm Carbon content (%)

Carbon fiber 6.9 6 4810 225 1–1.2 1.54 × 10−3 95

Table 2 Properties of MWCNT [20]

Material Diameter (nm) Length (microns) Metal particles Amorphous carbon Specific surface area (m2/g)

Bulk density (g/cm3)

Nanotubes purity

MWCNT 10 to 30 10 < 4% < 1% 330 0.04– 0.06 > 95%

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containing MWCNT-F is deposited drop by drop using a doppler over the entire surface of carbon fiber. The deposi-tion operation is repeated several times with evaporation of the solvent between each deposition. Heat treatment process is carried out on the resulting CFs grafted MWCNT-F at 100 °C for 2 h [20].

2.4.3 Characterization of MWCRFM by Fourier Transform Infrared Spectroscopy (FTIR)

In this analysis, the bonding behavior of all the three com-posite sheets, OCRFM, AFRFM, and MWCRFM fabricated by using hand lay-up method were analyzed for interlaminar shear force attraction between all the ingredients present in the composite. FTIR of make shimadzu, IR tracer 100, and high signal-to-noise ratio of around 6000 is used to deter-mine the extent of absorption or emission of solid, liquid on the three composite materials. This analysis uses attenuated total reflectance module (ATR) for detecting the functional groups and characterization of covalent bonding information present in the composite. In this ATR technique, composite specimen was placed on a diamond crystal, and exposed to an infrared beam across a range of wavelengths, normally from 2.5 to 25 µm and wave number rages from 4000 to 500 cm−1. Infrared light will be absorbed or emitted depend-ing upon the chemical bonding within the material and the subsequent spectral fingerprints generated can be interro-gated to determine the material composition of the sample. Three composite sheets were analyzed for bonding behavior with increase of wavelength. The bonding strength obtained by this method improves the strength of the composite used for many applications.

It was observed from Fig. 1 that AFRFM exhibits lower value of fluctuations in the value of absorption spectrum at

the beginning stage of wavelength value of 4000 cm−1. A sudden drop in absorption spectrum was noticed at a wave-length of 2400 cm−1. It was also observed from the graph that AFRFM possess more fluctuations in the absorption spectrum value in the wavelength range between 1700 and 1200 cm−1. AFRFM bonding behavior is good in the range of wavelength from 4000 to 2400 cm−1. But, poor bonding performance was observed at the intermediate value of wavelength from 1700 to 1200 cm−1. Figures 2 and 3, 4 represent the FTIR analysis results performed on OCRFM and MWCRFM. Based on the observations of graphs, both these materials OCRFM and MWCRFM pos-sess good bonding behavior with increase of wavelength value from 4000 to 500 cm−1. Comparing the results of both these materials, it was observed that OCRFM pos-sess more fluctuations in the absorption spectrum value in the wavelength range of 1750 cm−1 to 750 cm−1 and sudden drop in absorption spectrum was noticed at the wavelength of 2350 cm−1. This concludes that the bond-ing behavior was not good in the specified range of wavelength values. MWCRFM exhibits superior bond-ing behavior with increase of wavelength. It was also noticed that, MWCRFM possess fewer fluctuation with increase of wavelength and absorption spectrum is domi-nating the other two materials. Hence, MWCRFM exhibits good bonding behavior with increase of wavelength and the interlaminar shear force and covalent bond attraction between all the ingredients are superior compared to other formulations. This method of multi-walled carbon nano-tubes grafting on CF is a cost effective process and yields good results in terms of all mechanical properties and can further extend its application in all the fields of engineer-ing, mechanical, civil, automobile, aerospace, medical, sports, etc.

Fig. 1 FTIR analysis on AFRFM

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3 Preparation of Composite Sheets

Hand lay-up method is used to fabricate friction composite sheets for the ease of fabrication and control of all operat-ing variables. CF content (5 wt%) oxidized and grafted with MWCNTs-F is mixed with the remaining ingredients and phenolic polymer matrix. Similar procedure is car-ried for fabrication of AFRFM by considering asbestos content as (5 wt%) along with the remaining ingredients and polymer matrix. The details of ingredients selected for fabrication of composite sheets are given in Table 3.

Fabrication of composite sheet starts with an initial step of preparing a die with dimensions of 42 cm × 22 cm × 1.2 cm made of mild steel material. The die is finished to remove the unwanted material of thickness 2 cm on all the sides by using gas welding. The mixing operation of all the ingredients is carried using a foculator. Initially, resin is taken in a foculator and all the ingredients are added slowly, one after the other with an interval time

Fig. 2 FTIR analysis on OCRFM

Fig. 3 FTIR analysis on MWCRFM

132 139 133 145 146

204187

152

394

5 10 15 10 155

155 10

300 300 300

600 600 600

900 900 900

19.62 39.24 58.86 19.62 39.24 58.86 19.62 39.24 58.860

50

100

150

200

250

300

350

400

450

Wear(micromts)Time (min)Speed(rpm)

Load(N)

Wea

r(micr

omts

)

0

200

400

600

800

1000

Spee

d(rp

m)

Fig. 4 Wear performance of AFRFM

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gap period of 10 min. Total mixing operation for all the ingredients was carried for 30 min for each sample sheet. Two composite sheets with variation of surface treatment performed on CF and one composite sheet with AF content were fabricated. Sample specimens are cut according to ASTM G-99 standard to evaluate tribological properties of the composite specimens.

4 Test Set‑Up and Procedure

Pin-on-disk technique was used for performing the test with inbuilt software data acquisition system as (WIN DUCOM; TL-20). The surface of the rotor is made of hardened ground steel disk (EN-32) having hardness 65 HRC and surface rough-ness value (Ra) of 0.5 μm. Wear samples are prepared with the dimensions of square pin (30 mm × 5 mm × 5 mm) as per ASTM G99 standards. Before starting of the test, initially, the contact surfaces were cleaned with 600-grit silicon carbide paper and finally with acetone solution. The disk rotates with the help of a D.C. motor having a speed range of 0–1000 rpm, track diameter 70 mm, and a load of 0 to 100 N. Load is to be applied on pin (specimen) by dead weight through pulley string arrangement. In this experiment, the load applied on the friction material is varied from 19.62 N, 39.24 N, and 58.86 N for a time interval period of 5 min, 10 min, and 15 min. The speed of the rotor disk was varied with the values of 300 rpm, 600 rpm, and 900 rpm. Counter disk temperature was meas-ured during the test by using non-contacting infrared (IR) thermometer. The thermometer is placed at approximately 15 cm away from the trailing edge of the friction material. Frictional force at the sliding interface was measured using strain gauge mounted on the level arm that holds the speci-men, while the wear rate was determined by weighing the specimen before and after each test. The values of coefficient

of friction, wear rate, frictional force, and temperature gener-ated at the contact region during the application of load were measured and graphs are drawn. The morphologies of worn surfaces are observed by using Scanning Electron Microscope. Table 4 represents the chemical composition of grade EN-32 case-hardened steel rotor disk.

In the full factorial design approach, the total number of experiments required to analyze the behavior of friction material was (3 × 3) = 27. But, using L9 orthogonal array, it requires only 9 experiments to analyze the behavior of the friction composite against the application of load. The plan of experiments for the present study starts with L9 Orthogo-nal Array. These experiments were carried to analyze the influence of all the derating factors on dry sliding wear behavior of fiber-reinforced friction material. The control factors and their levels for L9 orthogonal array of experi-ments are shown in Tables 5 and 6.

4.1 Speed and Load Sensitivity

The behavior of three samples AFRFM, OCRFM, and MWCRFM is evaluated and coefficient of friction, wear rate, and frictional force obtained after conducting the

Table 3 Composition of friction material studied in this study

Ingredient Wt%

Phenolic resin 70CF (oxidized and MWCNT-F grafted) and AF 5Zirconium silicate 15Graphite powder 1.5Barium sulfate 2.5Rubber powder 3Molybdenum disulphide Balance

Table 4 Chemical composition (Vol%) of grade EN-32 case-hardened steel rotor disk

Material Element (Vol%) Tensile strength (N/mm2) Elongation (%) Impact strength Izod (J)

C Mn Si S

En-32 Case hardening steel 0.10–0.18 0.60–1.00 0.005 0.005 430 18 39

Table 5 Control factors and their levels

Factors Level 1 Level 2 Level 3

Speed (rpm) 300 600 900Load (kg) 2 4 6Time (min) 5 10 15

Table 6 Orthogonal array of experiments conducted for wear test for all samples (MWCRFM, AFRFM, and OCRFM)

Ex no. Speed (rpm) Load (Kg) Time (min)

1 300 2 52 300 4 103 300 6 154 600 2 105 600 4 156 600 6 57 900 2 158 900 4 59 900 6 10

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experiments are shown in Tables 7, 8, and 9. It was observed that, at lowest speed i.e., 300 rpm and under a load of (19.62 N, 39.24 N, and 58.86 N), and time period of (5 min, 10 min, and 15 min), the µ is observed to be in the range of 0.449 to 0.443 for (AFRFM). It also possesses less wear rate of 132 to 139 μm for the same loading and time conditions. At moderate speed conditions of the rotor i.e., 600 rpm and the same loading and time conditions, the µ is in the range of 0.343 to 0.35 and wear rate is observed to be 145 to 204 μm. At heavy speed conditions of the rotor i.e., 900 rpm and the same loading and time conditions, the µ is in the range

of 0.387 to 0.258 and wear rate is observed to be 187 to 394 μm as shown in Figs. 4, 5 and 6. Based on the results, it was observed that AFRFM possess more wear rate and less coefficient of friction under moderate speed conditions of the automobile. It was also observed that the stability of coefficient of friction is maintained constant between 0.25 and 0.4. The fade resistance is greatly improved for AFRFM with increase in temperature.

OCRFM also possess good coefficient of friction values in the range of 0.46 to 0.317. At lower speed conditions of the rotor i.e., 300 rpm OCRFM exhibits coefficient of

Table 7 Results of AFRFM Speed (RPM) Load (N) Time (min) Wear (μm) Frictional force (N)

Coefficient of friction

300 19.62 5 132 9.1 0.449300 39.24 10 139 17.4 0.447300 58.86 15 133 20.8 0.343600 19.62 10 145 8.1 0.373600 39.24 15 146 16 0.35600 58.86 5 204 21.9 0.35900 19.62 15 187 7.5 0.387900 39.24 5 152 12.9 0.304900 58.86 10 394 15.1 0.258

Table 8 Results of OCRFM Speed (RPM) Load (N) Time (min) Wear (μm) Frictional force (N)

Coefficient of friction

300 19.62 5 104 10.8 0.46300 39.24 10 75 18.2 0.43300 58.86 15 138 28.6 0.476600 19.62 10 115 10.4 0.478600 39.24 15 96 15.9 0.433600 58.86 5 198 22.1 0.351900 19.62 15 112 8.9 0.417900 39.24 5 128 15.9 0.365900 58.86 10 217 23.1 0.317

Table 9 Results of MWCRFM Speed (RPM) Load (N) Time (min) Wear (μm) Frictional force (N)

Coefficient of friction

300 19.62 5 119 12.3 0.512300 39.24 10 212 16.8 0.414300 58.86 15 166 29.2 0.483600 19.62 10 140 8.2 0.395600 39.24 15 149 17 0.404600 58.86 5 89 26.3 0.419900 19.62 15 62 10.7 0.49900 39.24 5 111 15.2 0.336900 58.86 10 268 22.3 0.392

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friction value of 0.46 and low wear rate of 104 μm. With the increase of speed of the rotor to 600 rpm, it was observed that coefficient of friction value is maintained stable but the wear rate increases at the starting condition of the rotor and there after decreases to a value of 96 μm. This is shown in Table 8, and Figs. 7 and 8. At higher speed conditions of the rotor i, 900 rpm µ value is observed to be decreased slowly from 0.417 to 0.317 and the wear rate is observed to be less at the starting i.e., 119 μm and increases to 212 μm with same speed and different operating parameters shown in Fig. 9. Based on all the observations, OCRFM material possesses low wear rate and good stable value of coeffi-cient of friction. The main reason to possess low wear rate is oxidizing treatment performed on CF which makes all the ingredients and polymer matrix to bond together (Figs. 7, 8, 10)

The third friction material MWCRFM subjected to same loading and operating conditions is observed to possess µ in the range of 0.512 to 0.392 and wear rate of 119 to 268 μm for a speed range of 300 rpm to 900 rpm shown in Figs. 9, 11

9.1

17.4

20.8

8.1

16

21.9

7.5

12.9

15.1

5

10

15

10

15

5

15

5

10

300 300 300

600 600 600

900 900 900

19.62 39.24 58.86 19.62 39.24 58.86 19.62 39.24 58.860

5

10

15

20

25

Frictional Force(N)Time (min)Speed(rpm)

Load(N)

Frict

ional

Forc

e(N)

0

200

400

600

800

1000

Spee

d(rp

m)

Fig. 5 Frictional force variation of AFRFM

0.449 0.447 0.343 0.373 0.35 0.35 0.387 0.304 0.258

5

10

15

10

15

5

15

5

10

300 300 300

600 600 600

900 900 900

19.62 39.24 58.86 19.62 39.24 58.86 19.62 39.24 58.860

2

4

6

8

10

12

14

16

Coefficient of frictionTime (min)Speed(rpm)

Load(N)

Coeff

icien

t of fr

iction

0

200

400

600

800

1000Sp

eed(

rpm)

Fig. 6 Coefficient of friction variation of AFRFM

104

75

138

11596

198

112128

217

510

1510

155

155

10

300 300 300

600 600 600

900 900 900

19.62 39.24 58.86 19.62 39.24 58.86 19.62 39.24 58.860

50

100

150

200

250

300

Wear(micromts)Time (min)Speed(rpm)

Load(N)

Wea

r(micr

omts

)

0

200

400

600

800

1000

Spee

d(rp

m)

Fig. 7 Wear performance of OCRFM

10.8

18.2

28.6

10.4

15.9

22.1

8.9

15.9

23.1

5

10

15

10

15

5

15

5

10300 300 300

600 600 600

900 900 900

19.62 39.24 58.86 19.62 39.24 58.86 19.62 39.24 58.860

5

10

15

20

25

30

Frictional force(N)Time (min)Speed(rpm)

Load(N)

Frict

ional

forc

e(N)

0

200

400

600

800

1000

Spee

d(rp

m)

Fig. 8 Frictional force variation of OCRFM

119

212

166

140149

89

62

111

268

510

1510

155

155

10

300 300 300

600 600 600

900 900 900

19.62 39.24 58.86 19.62 39.24 58.86 19.62 39.24 58.860

50

100

150

200

250

300

Wear(micromts)Time (min)Speed(rpm)

Load(N)

Wea

r(micr

omts

)

0

200

400

600

800

1000Sp

eed(

rpm

)

Fig. 9 Wear performance of MWCRFM

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and 12. It was observed that, with increase in speed, the wear rate decreases from beginning to ending speed of the rotor. MWCRFM possesses high value of stable coefficient of fric-tion and less wear values at moderate speed conditions of the

rotor and exhibits best values compared to remaining formu-lations of the materials shown in Table 9. The main reason to have consistent stable coefficient of friction and wear rate is grafting MWCNT-F on CF surface which increases the bonding and tribological behavior of MWCRFM.

4.2 S/N Ratio Analysis

Signal-to-noise ratio (S/N) is a measure of signal strength relative to background noise. The ratio is usually measured in decibels (dB) using a signal-to-noise ratio formula. The influence of control parameters sliding speed, load, and time on wear rate, frictional force, and coefficient of friction is evaluated using S/N ratio response analysis. The wear rate, frictional force, and coefficient of friction are considered as the quality characteristic by considering smaller is better for wear rate and higher is better for coefficient of friction. This was calculated by using the following equation.

S∕N ratio = −10 Log10

1

n

∑

y2

. Here n is 1 & y is response value.

Process parameter settings with the highest S/N ratio always yield the optimum quality with minimum variance. The control parameter with the strongest influence was determined by the difference between the maximum and minimum value of the mean of S/N ratios. Higher the dif-ference between the mean of S/N ratios, the more influential will be the control parameter.

4.2.1 Analysis of Variance (ANOVA)

ANOVA was used to determine the design parameters sig-nificant influence on response values. This analysis was evaluated for a confidence level of 95%, and the parameter response indicates the degree of influence on the result.

Total sum of squares = Sum of squares groups + Sum of squares with in group

F-test is a statistical test in which the test static has an F-distribution under null hypothesis. Exact F-test is mainly a rise when the models have been squares. F statistic-like regression tries to find the connection between the two P values of equal or smaller than 0.05.

P is probability of obtaining a result at least as extreme as the one that was actually observed, by giving null hypothesis as true. Delta is the difference between the maximum and minimum average S/N ratio for factors. Rank is the rank of each delta and the largest delta.

Mean squares (MS) =Sum of squares

Degrees of freedom.

F =Variance of group means

Mean in group variance

0.46 0.43 0.476 0.478 0.433 0.351 0.417 0.365 0.317

5

10

15

10

15

5

15

5

10300 300 300

600 600 600

900 900 900

19.62 39.24 58.86 19.62 39.24 58.86 19.62 39.24 58.860

5

10

15

20

25

30

Coefficient of frictionTime (min)Speed(rpm)

Load(N)

Coef

ficien

t of f

rictio

n

0

200

400

600

800

1000

Spee

d(rp

m)

Fig. 10 Coefficient of friction variation of OCRFM

12.3

16.8

29.2

8.2

17

26.3

10.7

15.2

22.3

5

10

15

10

15

5

15

5

10300 300 300

600 600 600

900 900 900

19.62 39.24 58.86 19.62 39.24 58.86 19.62 39.24 58.860

5

10

15

20

25

30

Frictional force(N)Time (min)Speed(rpm)

Load(N)

Frict

ional

force

(N)

0

200

400

600

800

1000Sp

eed(

rpm)

Fig. 11 Frictional force variation of MWCRFM

0.512 0.414 0.483 0.395 0.404 0.419 0.49 0.336 0.392

5

10

15

10

15

5

15

5

10

300 300 300

600 600 600

900 900 900

19.62 39.24 58.86 19.62 39.24 58.86 19.62 39.24 58.860

2

4

6

8

10

12

14

16

Coefficient of frictionTime (min)Speed(rpm)

Load(N)

Coef

ficien

t of f

rictio

n

0

200

400

600

800

1000

Spee

d(rp

m)

Fig. 12 Coefficient of friction variation of MWCRFM

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Table 10 indicates L9 (3 × 3) orthogonal array analysis of variance for MWCRFM for wear rate. It can be observed from Table 11 that time was considered as the most sig-nificant parameter having the highest statistical influence (45.81%) on the dry sliding wear of composites followed by load (23.59%) and speed (9.49%). When the P value for the model is less than 0.05, then the parameter or interaction can be considered as statistically significant. From the results, it was observed that the interaction affect of load and time is having significant influence on wear rate of MWCRFM.

Table 12 gives rankings to influence of parameters. Based on the values it was observed that time is considered as the most influencing parameter for wear rate for MWCRFM.

Figure 13 shows the wear main effective plots of S/N ratio for MWCRFM. It was observed that if the speed of the rotor increases then, S/N ratio of wear increases. Similarly, it was observed that if load increases then S/N ratio of wear decreases. Finally, with increase of time, S/N ratio of wear decreases initially up to 10 min and thereafter it increases.

Figure 14 shows the wear main effective plots of con-trol factors for MWCRFM material. It is observed that if

speed increases then, the mean of wear decreases. Simi-larly, it was observed that if load increases then, the mean of wear increases. It was also observed that with increase in time, the mean of wear increases initially up to 10 min and thereafter it decreases. Figure 15 gives the informa-tion related to coefficient of friction main effective plots of S/N ratio for MWCRFM. It is observed that if speed increases then S/N ratio of coefficient of friction decreases. Similarly, it was observed that if load increases, then S/N ratio of coefficient of friction decreases up to 39.24 N load and thereafter S/N ratio increases. It is also observed that

Table 10 L9 (3 × 3) orthogonal array results for MWCRFM

Speed (RPM) Load (N) Time (min) Wear (μm) SNRA1 FITS_SN1 RESI_SN1 %Error

300 19.62 5 119 41.5109 − 39.5686 − 1.94235 4.908817300 39.24 10 212 − 46.5267 − 48.6427 2.11598 − 4.35005300 58.86 15 166 − 44.4022 − 44.2285 − 0.17363 0.392575600 19.62 10 140 − 42.9226 − 42.7489 − 0.17363 0.406162600 39.24 15 149 − 43.4637 − 41.5214 − 1.94235 4.677949600 58.86 5 89 − 38.9878 − 41.1038 2.11598 − 5.14789900 19.62 15 62 − 35.8478 − 37.9638 2.11598 − 5.57368900 39.24 5 111 − 40.9065 − 40.7328 − 0.17363 0.426266900 58.86 10 268 − 48.5627 − 46.6203 − 1.94235 4.166318

Table 11 Analysis of variance for S/N ratios of MWCRFM

Source DF Seq SS ADJ MS F P Contribution

Speed (RPM) 2 11.18 5.592 0.45 0.69 9.493886Load (N) 2 27.78 13.89 1.12 0.472 23.59035Time (min) 2 53.95 26.975 2.17 0.315 45.81352Residual error 2 24.84 12.42 21.09375Total 8 117.76 100

Table 12 Response table for S/N ratio smaller is better for MWCRFM

Level Speed (RPM) Load (N) Time (min)

1 − 44.15 − 40.09 − 40.472 − 41.79 − 43.63 − 463 − 41.77 − 43.98 − 41.24Delta 2.37 3.89 5.54Rank 3 2 1

MeanofSNra

tios

900600300

-40

-42

-44

-46

642

15105

-40

-42

-44

-46

SPEED(RPM) LOAD(KG)

TIME(MIN)

Signal-to-noise: Smaller is better

Fig. 13 Wear plots of S/N ratio for MWCRFM

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with increase in time S/N ratio of coefficient of friction decreases initially up to 10 min and there after it increases slowly.

Figure 16 gives the information related to coeffi-cient of friction main effective plots of control factors for MWCRFM. It is observed from the graph that if speed increases then mean of coefficient of friction decreases. Similarly, it is observed that with increase in load mean of coefficient of friction decreases initially up to 39.24 N and thereafter it increases. Finally, with increase of time, mean of coefficient of friction decreases initially up to 10 min and thereafter increases.

Table 13 shows L9 (3 × 3) orthogonal array and ana-lyzed taguchi design for coefficient of friction MWCRFM. It can be observed from Table 14 that load was the most significant parameter having the highest statistical influence of (38.39%) on the coefficient of friction of the composite material followed by Speed (31.005%) and Time (19.86%). When the P value for the model is less than 0.05, then the parameter or interaction can be considered as statistically significant. From the results, it was observed that the inter-action effect of load and time is having significant influence on coefficient of friction for MWCRFM. Table 15 gives rankings to the most influencing parameter for coefficient of friction for MWCRFM. It was observed that application of load on rotor disk plays an important role for coefficient of friction induced in MWCRFM (Table 16).

5 Scanning Electron Microscope Results

SEM images are observed using S 3700 N Hitachi make given in Fig. 17a–e. Figure 17a, b gives the SEM image taken on carbon fiber surface treated with oxidation treat-ment and MWCNT-F grafted on its surface. It can be clearly observed from Fig. 17b that the fiber surface becomes rougher after this treatment to increase hydroxyl groups on the surface. Figure 17c–e gives SEM images for AFRFM, OCRFM, and MWCRFM. The fiber surface after oxida-tion treatment shown in Fig. 17d is observed to be soft and smooth and only little improvement in the hydrophobicity of the surface occurs after this treatment.

MeanofWEAR(M

ICROMETERS

900600300

200

175

150

125

100642

15105

200

175

150

125

100

SPEED(RPM) LOAD(KG)

TIME(MIN)

Fig. 14 Main effect plots of wear for MWCRFM

900600300

-6.5

-7.0

-7.5

-8.0

-8.5642 15105

SPPED(RPM)

MeanofSN

ratios

LOAD(KG) TIME(MIN)

Main Effects Plot for SN ratiosData Means

Signal-to-noise: Larger is better

Fig. 15 Coefficient of friction S/N ratio plots MWCRFM

MeanofCOF

900600300

0.46

0.44

0.42

0.40

0.38642

15105

0.46

0.44

0.42

0.40

0.38

SPPED(RPM) LOAD(KG)

TIME(MIN)

Fig. 16 Main effect plots of friction for MWCRFM

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61 Page 12 of 13

From Fig. 17d after oxidation treatment performed on CF, the surface of CF is improved with moderate rough-ness. It is clearly observed from Fig. 17e that the com-posite specimen containing CF grafted with MWCNT-F exhibits high covalent bonding strength and good inter-laminar shear force attraction between all the ingredients

compared to remaining formulations of materials. There is a greater possibility of enhancement of strength of the composite after CF grafted with MWCNT-F because of uniform distribution of all the grains in the structure along with good absorptive characteristics shown with all the ingredients and polymer matrix.

6 Conclusion

Three types of friction materials AFRFM, OCRFM, and MWCRFM were investigated in this work using a wear test apparatus for three different loading conditions, speeds, and time under dry conditions. It was observed from the results that, MWCRFM possess low wear rate and high coef-ficient of friction compared to remaining formulations of the materials. It was also observed from analysis of variance results that the most influencing parameter for MWCRFM is observed to be time for wear rate and load for coefficient of friction. The surface treatments performed on CF by graft-ing MWCNT-F on its surface increase the bonding strength between CF, Polymer matrix, and remaining ingredients. MWCNT-F grafting on CF is the main reason to have higher wear resistance and high stable coefficient of friction com-pared to other formulations of materials. Hence, MWCRFM can be effectively used to replace current existing friction materials for automobile applications.

Table 13 L9 (3 × 3) orthogonal array and analyzed Taguchi design for coefficient of Friction in MWCRFM

Speed (RPM) Load (N) Time (min) COF SNRA1 FITS_SN1 RESI_SN1 %ERROR

300 19.62 5 0.512 5.8146 5.99589 − 0.18129 − 3.02359300 39.24 10 0.414 7.65999 7.97777 − 0.31778 − 3.98329300 58.86 15 0.483 6.32106 5.82199 0.499069 8.572138600 19.62 10 0.395 8.06806 7.56899 0.499069 6.593601600 39.24 15 0.404 7.87237 8.05366 − 0.18129 − 2.25104600 58.86 5 0.419 7.55572 7.8735 − 0.31778 − 4.03604900 19.62 15 0.49 6.19608 6.51386 − 0.31778 − 4.87849900 39.24 5 0.336 9.47321 8.97415 0.499069 5.561184900 58.86 10 0.392 8.13428 8.31557 − 0.18129 − 2.18014

Table 14 Analysis of variance for S/N ratios for MWCRFM

Source DF Seq ss M F P Contribution

Speed (RPM) 2 3.317 1.6584 2.89 0.257 31.0058Load (N) 2 4.108 2.0542 3.58 0.219 38.3997Time (min) 2 2.125 1.0623 1.85 0.351 19.86353Residual 2 1.149 0.5744 10.74033Total 8 10.698 100

Table 15 Response table for S/N ratio larger is better for MWCRFM

Level Speed (RPM) Load (KG) Time (min)

1 6.599 6.693 7.6152 7.832 8.335 7.9543 7.935 7.337 6.797Delta 1.336 1.642 1.158Rank 2 1 3

Table 16 Most influencing parameters for three composite materials

Material Wear Frictional force Coefficient of friction

AFRFM Speed Load SpeedMWCRFM Time Load LoadOCRFM Load Load Speed

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Acknowledgements The authors would like to thank Anits Engineer-ing College for providing the laboratory facilities for performing this experiment.

References

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10. Tang LG, Kardos JL (1997) A review of methods for improving the interfacial adhesion between carbon fiber and polymer matrix. Polym Compos 18:100–113

11. Song Q, Li KZ et al (2012) Grafting straight carbon nano tubes radially on to carbon fibers and their effect on the mechanical prop-erties of Carbon/Carbon composites. Carbon 50(10):3949–3952

12. Sharma SP, Lakkad SC (2011) Effect of carbon fibers on the tensile strength of CNTs grown carbon fiber reinforced polymer matrix composites. Composites Part A 42:8

13. Hui Q, Bismarck A et al (2010) Carbon nano tubes grafted carbon fibers: a study of wetting and fibre fragmentation. Composites Part A 41:1107–1114

14. De Greef N et al (2015) Direct growth of carbon nano tubes on carbon fibers: effect of the CVD parameters on the degradation of mechanical properties of carbon fibers. Diam Relat Mater 51:39–48

15. Rahaman A, Kar KK (2014) Carbon nano materials grown on E-glass fibers and their application in composite. Compos Sci Technol 101:1–10

16. Severini F, Formaro L, Pegoraro M et al (2002) Chemical modi-fication of carbon fiber surfaces. Carbon 40:735–741

17. Sharma M, Gao S, Mader E et al (2014) Carbon fiber surfaces and composite interface. Compos Sci Technol 102:35–50

18. Kumar KN, Suman KNS (2017) Effect of pad and disc materials on the behaviour of disc brake against fluctuating loading condi-tions. J Automob Eng Appl 4(2):22–34

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20. Konada NK, Suman KNS (2018) Effect of surface treatments on tensile and flexural properties of carbon fiber reinforced friction material. Int J Eng Manuf 3(8):23–39

21. Konada NK, Suman KNS (2018) Damping behaviour of multi walled carbon nano tubes grafting on carbon fiber reinforced fric-tion material. J Soc Automot Eng 2(2):127–140

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Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Fig. 17 a–e SEM images of CF composite sheets

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Performance Analysis of Various Differential Privacy Preserving

Data Distortion Techniques using Privacy Class Utility Metric

K. Sandhya Rani Kundra1, Dr.J Hyma2, Prof.P.V.G.D.Prasad Reddy3,Prof.K.Venkata Rao4

1 Asst.Professor,Dept of I.T, G.V.P.Collage of Engg(A),Visakhapatnam

2 Associate Professor, Dept of CSE, ANITS Engg.college(A), Visakhapatnam 3 Sr.Professor, Dept of CS&SE, Andhra University, Visakhapatnam.

4 Professor, Dept of CS&SE, Andhra University, Visakhapatnam.

1Email:sandhyaranikks38@gmail.com, 2Email:jhyma.cse@anits.edu.in,

3Email:prasadreddy.vizag@gmail.com, 4Email:professorvenkat@yahoo.com

Abstract

Statistical database security focuses on the protection of confidential individual data, stored in

databases for statistical purposes. One of the techniques used for preserving statistical database

privacy is noise addition. In this technique, in response to the queries, the statistical data provided as

answers are only approximate rather than exact. In this background analysis of various techniques

with heterogeneous data distortion is presented in this paper. An attempt is made, to study the effect

of application of various statistical measures on the distorted data, and their impact on ensuring the

privacy of the original data. Experimental results show that the proposed solution outperforms

traditional differential privacy in terms of Statistical Metrics on a group of queries. The performance

of heterogeneous data distortion is evaluated with three types of techniques namely homogeneous

with differential privacy, heterogeneous with differential privacy and also sigmoid technique

(Learning model) with differential privacy. It is observed that the sigmoid technique can successfully

retain the utility of published data while preserving privacy.

Keywords: Differential Privacy, Heterogeneous, Sigmoid technique

1. Introduction

Statistical database security is concerned with protecting privacy of individuals whose conditional

data is collected through surveys or other means. In this context individuals can refer to person’s

households, companies or other entities. With the digitization being encompassing all walks of life,

there is accumulation of enormous data on a daily basis. This massive collection of data and its likely

sharing has added to the already growing public concern about its misuse and breach of privacy.

Privacy Preserving Data Mining (PPDM) is a prominent area concerned about the data

disclosure. Its primary task in data mining is to develop models about aggregate data without letting

access to original data records [1] [2]. Several privacy preserving techniques have been proposed and

used in various applications. Mostly these methods followed a homogeneous data distortion technique

in privacy preservation in data utility. In real time, it is not adequate because, the level of compromise

in privacy is an individual choice of data disclosure and changes among datasets.

Recently Dinur and Nissim[3] and Dwork and Nissim [4] tried to provide a rigorous

mathematical treatment for protecting the privacy of individuals while attempting any statistical

analysis. Based on the work of Dwork et. al.[5] our research has yielded a robust privacy guarantee

of differential privacy, which guarantees that the outcome of analysis adjacent databases that differ

only in one participants information is very similar. In particular, differential privacy guarantees that

participation in the analysis does not incur significant additional risk for individuals.

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A central question in this line of research regards the tradeoff between utility and privacy. In an

interactive setting the queries are specified in a sharing and adaptive manner. Here the challenge lies

in answering large number of queries accurately without compromising on privacy. In a recent work,

Roth and Roughgarder [6] presented a new mechanism for answering queries. This efficient

implementation guarantees privacy for all input databases and also gives accurate results.

To address this issue, a study is carried out on various heterogeneous data distortion techniques

in data preservation and utility and reported in the present communication.

2. Related work

The problem of protecting individual privacy in the process of data collection, querying, mining

and release has been researched extensively. Mainly there are two scenarios in the data privacy

protection. One is the privacy preserving data publishing scenario, in which a trusted server releases

datasets of individual information or answer queries on such data sets. The second one is the data

collection scenario, in which an untrusted server collects personal information from different sources.

A large number of privacy preserving publishing models based on anonymity techniques such

as k-anonymity [7][8], L-diversity[9] and t-closeness[10] have been proposed. Some other reports

showed the implementation of privacy preserving data clustering by data transformation [11] [12]. In

the perturbation approach the data is modified by the inclusion of noise component [13] [14]. Random

data perturbation technique along with the necessary theoretical foundation is proposed by Kargupta

et.al. [14]. they applied perturbation technique to many experimental results and observed that in most

of the cases random data distortion technique failed to preserve data privacy.

The Exiting perturbation techniques follow one–size-fits-all approach which is relatively

inflexible. To enhance the scope of exiting perturbation methods, Lu et.al.[15] have performed the

perturbation at two different levels with different intervals. Our group attempted to study on methods

to achieve maximum utility while protecting privacy in the data publishing scenario by noise-addition

technique. In our analysis, we adopt the rigorous differential privacy introduced by Dwork et.al [16]

that has been widely studied in the data publishing or statistical query answering scenario. The work

of Lu et.al.[15] Have motivated us to perturb the data values in a heterogeneous manner. In this

approach the quality of data distortion is measured in terms of various utility and privacy measures

[17][18].

An empirical evaluation on amazon dataset is conducted and the performance of the different

proposed techniques using differential privacy (Inverse Laplace mechanism) on heterogeneous data

are compared and their statistical measures are reported. One advantage of the use of the randomized

response in the data collection scenario is that the collected data can be released freely for analysis

without worrying too much about privacy disclosure. This is different from the output perturbation

where each additional analysis consumes further privacy budget. Moreover, the use of the randomized

response for collecting data incurs less utility loss, than the output perturbation when the sensitivity of

functions is high. This was demonstrated in the present study during the application of different

techniques on heterogeneous data while trying to preserve differential privacy.

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3. Proposed technique

As single level privacy approach is not advisable for a better privacy protection and data utility, a

new heterogeneous data distortion technique is reported in the previous work [19]. In this data has

been classified into three different classes namely High, Medium and Low. In order to perform this

classification a different privacy analysis approach is proposed. Here the privacy preference of the

owner, privacy decision of the data collector and exiting correlations are taken into consideration.

Using these validations, the data could be mapped to any of the convenient classes. Then accordingly

perturbation with various threshold levels is introduced for different privacy classes [19] using ε-

Differential privacy technique.

The stepwise details of the proposed work are presented in the following flow chart.

Figure 1: Flow Chart of the Proposed Work

In this approach, first data mapping is to be done into different privacy classes and then specific

data distortion is performed to each of these classes. In the earlier work the performance of

heterogeneous data distortion with differential privacy - three query model [19] and sigmoid learning

model [20] is discussed. In sigmoid model it is demonstrated how learning models can be applied to

analyze the data sensitivity and classify them into various privacy classes. Once the privacy class

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distribution is done the model applies Inverse Laplacian query model to check the data utility without

compromising on privacy. Basing on this background the given experimental study succeeded in

training the network to perform privacy analysis under a modest privacy budget, complexity training

efficiency and data utility. Here it has to be ensured that the distorted data preserves its essential

properties to prove its effectiveness in data utility while ensuring privacy protection with an

acceptable deviation. In order to do this, the outcome of various statistical measures performed on

transformed data against original data is evaluated.

3.1 Differential Privacy

This technique works by adding aptly chosen random noise to the original data to generate an

answer to a query while taking care that the added noise does not deviate the answer too much from

the original. In the work published earlier [21] on Differential Privacy it has been stated to enact ε-

Differential Privacy by adding a random noise whose magnitude is chosen on the basis of query

posed. The amount of noise added depends on the optimum change a single entity can withstand to

give a meaningful and useful result.

Definition :- f : D → Rd,

The L1-sensitivity of f is-

Δf = max D1,D2 || f(D1) - f(D2) ||

For all D1, D2 differing in at most one element.

Here, Δf is the sensitivity of the function f.

There are divergent noise adding mechanisms such as Laplacian, exponential, and posterior

sampling used to achieve differential privacy. Laplacian and Inverse-Laplacian methods are proven to

be useful in adding controlled noise to the dataset [5][22]. Here, the proposed model deals with the

use of Inverse-Laplacian Mechanism to achieve differential privacy.

3.2 Inverse Laplacian Noise

The Inverse-Laplace mechanism adds a noise from the Inverse-Laplace distribution [23], which

can be expressed as in Eq. 1.

noise(y) ∝ exp(−|y| λ⁄ )….. Eq. 1

which has a mean of zero and standard deviationλ. Now in this case the output function of A [23] is

defined as a real valued function called as the transcript output, TA by A and is given in Eq. 2.

TA(x) = f(x) + Y…. Eq.2

where Y~Lap−1(λ) and f is the original real valued query or function that is planned to execute on

the database. Now clearly TA(x) can be considered to be a continuous random variable [23] given in

Eqs. 3 and 4 [22].

pdf(TA,D1(x)=t)

pdf(TA,D2(x)=t)=

noise(t−f(D1))

noise(t−f(D2)) ..............Eq. 3 [23]

Lap−1(u,m, bx) = m −bxS ∗ sgn(u) ∗ ln(|1 − 2 ∗ (u)|………Eq. 4

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∆(f)

λ being the privacy factorϵ which is at the most e

|f(D1)−f(D2)|

λ ≤ e∆(f)

λ [24]. Thus T follows a

differentially private mechanism (as can be seen from the definition above). It is a derived fact that in

order to have A as the ϵ - differential private algorithm [23] we need to have λ =1

ϵ .

Final Value

TA(x) = Originalvalue(f(x)) +Lap−1(u,m, bx)

Where

Lap−1= Inverse Laplacian Distribution,

u = Uniform (0,1),

m = Mean,

bx = Scaling Parameter ∆(f)

ϵ ,

∆f is global sensitivity and ϵ is the privacy budget.

3.3 Heterogeneous Differential privacy

One of the major drawbacks of Homogeneous Noise addition is that it adds a fixed noise to each

and every data set. Here, even if one of the entries is known, the noise can be calculated easily by the

adversary during data extraction. This leads to violation of privacy which defeats our prime interest.

Another method to add noise is using random noise addition. But sometimes an unacceptable level of

noise generation results during random noise addition.

The work on Heterogeneous Differential Privacy discussed in [24], acknowledges the fact that

Privacy requirements are not homogeneous across users and among the attributes from the same user.

This concept of people having varied preferences can help us create a basis for adding heterogeneous

noise for the posed query.

Thus, this work proposes a model where one can divide Data-Set into groups based on their

privacy requirements and adding a different chunk of noise to each sub-group. This makes it difficult

to find the amount of noise added, as it isn’t uniform throughout.

3.4 Sigmoid-Learning based Technique

Sigmoid functions give a better deal, while dealing with the non-linear data, by providing a

continuous output between 0 and 1, as a probability range. Whereas, other neural network functions

like perceptron, gives a step function as output which has a disadvantage while dealing with the non-

linear data. Sigmoid function output is an S shaped curve which is smoother than the step functions in

the perceptron neural network. In perceptron, for every small change the result might be a complete

flip, whereas in sigmoid with its S shaped output, the transaction is smooth and for every small

change the result might not change drastically.

Input:

The input to sigmoid are real numbers and the output will be in the range of 0 to 1, whereby allowing

to choose options for threshold values to be classified into binary.

Step-1: Initialize the parameters w, b

Step-2: Iterate until satisfied

Compute L (w, b)

w(t + 1) = wt − ηΔwt

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b(t + 1) = bt − ηΔbt

Here, w and b are initialized randomly and iterated through the data. After each iteration, the squared

error is computed and depending on its value the parameters are updated in such a way that the

squared error is minimized.

L (w,b) > L(w +ηΔw, b+ηΔb)

The loss function is defined as follows

Loss = ∑i (Zi- Ẑi)2

Where Ẑ = 1

1+e−(wx+b)

In sigmoid the main purpose is to update the parameters w and b so that the overall loss function of

the model is reduced.

3.5 Statistical Measures

Every data modification process has to be evaluated carefully. Any drastic change may negatively

affect the data utility and moderate change will not add anything to preserve privacy. Hence judicious

balance of these properties needs to be ensured. The following properties shown in Table -1 are used

to perform this evaluation.

Table 1: List of Measures

STATISTICAL

MEASURE EQUATION

Mean 𝜇 =Σx

𝑁

Standard deviation σ=√Σ(X−μ)2

N

Signal to Noise

Ratio SNR =

μ

σ

Mean Square Error 1

𝑁∑(µ − 𝑋)2𝑁

𝐼=1

Mean Absolute

Error

1

𝑁∑|𝑋 − µ|

𝑁

𝐼=1

Utility Measure Equation

Information Loss (N-O) / (U-L)

Data Utility Metric (Information Loss-IL)

A new metric is introduced to measure the Utility. The basic idea is drawn from the metric

proposed earlier [25]. In the proposed work the data distortion is performed at various classes, hence

a variant of existing metric is imposed to measure the information loss in each of the privacy

classes.

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ILClass = (Nh—Xh ) / (Uh — Lh) ..….. Eq. - 5

Nh — New Distorted Data Xh — Original Data

Uh — Max Value in Class h , Lh- — Min Value in class h

The extent of data distortion can be assessed on the basis of information loss metric value. If the

ILattribute measure returns a ‘0’ value, then it means that there is no distortion and if it is ‘1’ it implies

out of range distortion. The results are shown in Fig.2. So the administrator has to take a decision to

fix this parameter to optimize data utility and minimize privacy loss. This measure hopefully helps

us to provide a balancing factor between the data utility and privacy.

Fig. 2: Data Utility Vs Data Privacy

4. Experimental Analysis

An experiment is performed on three different data sets given in Table 2 and an illustration with

sample data is shown in Table 3.

Dataset Attributes Instances Classes

Amazon Data Set 5 2518 2

Adult Data Set 15 32561 2

Income Data set

9 18000 -

Table 2: Data Set Description

00.10.20.30.40.50.60.70.80.9

1

1 2 3 4 5 6 7 8 9 1011

Attributes

Data Utility

Data Privacy

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Table 3: Sample Data on Amazon Data Set

Result analysis is carried out on three different transformations and finally checked with

various statistical parameters. In this work we proposed a Utility metric for data modification. The

administrator can check the value and accordingly can fix the threshold parameter that is privacy

budget ε - value. Comparison plots for different proposed techniques are given in Figure 3 and the

statistical Metric evaluation applied on Amazon data set is given in Table 4.

The Graphical representation sown in Figure 4 on three proposed transformations proved that the

data pre-ordering technique showed desirable performance with respect to all statistical metrics with

different deviations. The deviation rate is high in Sigmoid-Differential Privacy followed by

Heterogeneous-differential Privacy followed by Homogeneous Differential Privacy. Practically, thus

statically analysis is clearly shown in Figures 4 to 6.

Original Data

Distorted Data using Homo-D differential Privacy

Distorted Data using Hetero- differential Privacy

Distorted Data using Sigmoid- hetero differential Privacy

15.62 16.62 17.12 18.28

15.59 17.09 17.34 18.06

15.37 15.87 16.12 16.37

15.3 16.05 16.3 16.62

15.43 15.68 16.29 16.51

15.44 15.55 15.77 16.76

15.53 17.03 17.14 18.01

15.44 15.66 16.1 16.99

15.44 15.57 15.9 16.64

15.29 15.51 15.65 16.51

15.37 15.38 17.38 18.35

15.48 15.92 18.92 19.76

15.42 16.42 16.52 16.64

15.51 16.51 16.61 16.95

ISSN: 2005-4297 IJCA

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International Journal of Control and Automation

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821

Fig. 3: Comparison of Proposed Techniques

Table 4: Metric evaluation on Amazon data set

Metrics Original Homo Hetero Sigmoid

Mean 26.51342 27.5137 28.11009 28.09897

Std

Deviation 8.364222 8.371208 8.294888 8.271612

SNR 3.169861 3.286706 3.388845 3.397037

MSE 0 1.005127 2.625724 2.588186

MAE 0 1.000278 1.596668 1.585551

Figure 4: Comparison Plot on Statistical Measures

0

5

10

15

20

25

1 3 5 7 9 11 13 15

Original Vs Distorted Values

Original Data

Distorted Datausing Homo-D

Distorted Datausing Hetero-differential Privacy

Distorted Datausing Sigmoid-hetero differentialPrivacy

05

1015202530

Comparison on Statistical Measures

Original

Homo

Hetero

Sigmoid

ISSN: 2005-4297 IJCA

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Figure 5: Comparison Plot on Mean

Figure 6: Comparison Plot on Signal to Noise Ratio

One more experimental statistical evaluation is conducted on different attributes by using Amazon

Data Set. That is given in Table 4. Graphical representation plots for different proposed techniques are

given in Figures 7 to 9.

Table 4: Metric evaluation on Attribute

Metrics Original Homo Hetero Sigmoid

Mean 112.6851 113.6878 114.3967 114.2891

Std

Deviation 80.31156 80.42424 80.39715 80.57795

SNR 1.403099 1.413601 1.422896 1.418367

MSE 0 5.617072 5.61518 5.627808

MAE 0 113.6878 114.3967 114.2891

2526272829

Mean

Mean

3

3.2

3.4

3.6

Signal to Noise Ratio

Signal toNoise Ratio

ISSN: 2005-4297 IJCA

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International Journal of Control and Automation

Vol. 13, No. 2, (2020), pp. 813 - 826

823

Figure 7: Comparison Plot on Statistical Measures

Figure 8: Comparison Plot on MSE

Figure 9: Comparison Plot on MAE

The information metric proposed in section 4 is applied on Amazon Data set and is evaluated

for different classes of data. The closing price attribute with the Heterogeneous differential privacy

has produced an IL value 0.3132 for class-1 and 0.1243 for class-2. In the attribute Daily Percent, the

author has noticed an IL value of 0.5432 on class-1 and 0.3456 on class-2. The Information Loss on

0

20

40

60

80

100

120

140

Comparison on Statistical Measures

Original

Homo-DifferentialPrivacyHetero-DifferentialPrivacySigmoid-DifferentialPrivacy

0

2

4

6

MSE

MSE

0

50

100

150

MAE

MAE

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824

Amazon Data set with the maximum limitation as ‘1’ representing full distortion is shown in Table 5.

Graphical representation on Utility Metric is given in Figure 10.

Table 5: Data Utility on Amazon Data set

Attribute Privacy

Class

Utility

Metric

Closing Price

Class1 0.3132

Class2 0.1243

High 1

Daily Percent

Return

Class 1 0.5432

Class 2 0.3456

Figure 10: Graphical Representation on Utility Metric

5. Conclusion

In this paper a comparative analysis of various data transformation techniques with

homogeneous and heterogeneous data distortion methods is proposed. The techniques namely

homogeneous differential privacy, heterogeneous differential privacy, and sigmoid heterogeneous

differential privacy have been applied to transform the data for privacy protection. These

normalizations are applied at various privacy classes. The distorted data is evaluated against various

distortion measures and privacy measures. A new privacy measure is implemented to measure the

level of data distortion in each of the privacy class. The data analysist can take the decision on

amount of noise to be added depending upon the Utility metric (IL).All the three transformation

techniques are performed in accordance to the data perturbation with different data deviation rates.

The present approach of data categorization into various privacy classes is adoptable to any distortion

and enhances the privacy protection. The results obtained by the application of the proposed

sigmoid heterogeneous differential privacy data perturbation method showed that better utility and

privacy are ensured.

00.20.40.60.8

1

Cla

ss1

Cla

ss2

Hig

h

Cla

ss 1

Cla

ss 2

Closing PriceDaily Percent Return

Dis

tort

ion

Me

asu

re

Attribute

Utility Metric

Utility Metric

ISSN: 2005-4297 IJCA

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6. References

[1] R. Agrawal and R. Srikant. "Privacy-preserving data mining," In Proc. SIGMOD, pp. 439-450, 2000.

[2] D. Agrawal and C. Aggarwal. "On the design and quantification of privacy pre-serving data mining

algorithms", In Proc. of the Twentieth ACM SIGACT-SIGMOD-SIGART Symposium on Principles of

Database Systems, Santa Barbara, California,USA, May 2001.

[3] I. Dinur and K. Nissim. “Revealing information while preserving privacy”. In Proc. 22nd PODS, pages 202–

210. ACM, 2003.

[4] C. Dwork and K. Nissim. “Privacy-preserving data mining on vertically partitioned databases” , In Proc.

24thCRYPTO,pages 528–544. Springer, 2004.

[5] C. Dwork, F. McSherry, K. Nissim, and A. Smith. “Calibrating noise to sensitivity in private data analysis”. In

Proc. 3rd TCC, pages 265–284. Springer, 2006.

[6] A. Roth and T. Roughgarden.,” Interactive privacy via the median mechanism”, In STOC, pages 765–774,

2010.

[7] P. Samarati and L. Sweeney, “Protecting privacy when disclosing information: k-anonymity and its

enforcement through generalization and suppression”, In Technical Report SRI-CSL-9804, SRI Computer

Science Laboratory, 1998.

[8] L. Sweeney, “k-anonymity: a model for protecting privacy”, International Journal on Uncertainty, Fuzziness

and Knowledgebased Systems, pp. 557-570, 2002.

[9] A.Machanavajjhala, J.Gehrke, and D.Kifer, “ℓ-diversity: Privacy beyond k-anonymity”, In Proc. of ICDE,

Apr.2006.

[10] N. Li, T. Li, and S. Venkatasubramanian, “t-Closeness: Privacy Beyond k-anonymity and ℓ-Diversity”, In

Proc. of ICDE, pp. 106-115, 2007.

[11] Stanley R. M. Oliveira1, Osmar R. Zaane, “Privacy Preserving Clustering by Data Transformation”, Journal

of Information and Data Management”, Vol. 1, No. 1, Pages 37, February 2010.

[12] Md Zahidul Islam, Ljiljana Brankovic “Privacy preserving data mining: A noise addition framework using a

novel clustering technique”, Knowledge-Based Systems, Volume 24, Issue 8, Pages 1214–1223, December

2011.

[13] Olivera, S.R.M. and Zaiane, O.R., “Privacy Preserving Clustering by Data Transformation”, Proceedings of

the 18t Brazillian Symposium on Data bases, Manaus Brazil, pp.304 -318 ,2003.

[14] H. Kargupta, S.Datta, Q. Wang, K. Sivakumar, “The privacy preserving properties of random data

perturbation techniques”, ICDM, IEEE Computer Society, pp. 99-106, 2003.

[15] ] Li Lu, Murat Kantarcioglu, Bhavani Thuraisingham “The applicability of the perturbation based privacy

preserving data mining for real-world data”, ELSEVIER, 2007.

[16] C. Dwork, F. McSherry, K. Nissim, and A. Smith. “Calibrating noise to sensitivity in private data analysis”,

Theory of Cryptography, pages 265–284, 2006.

[17] Yang Xu, Tinghuai Ma, Meili Tang and Wei Tian ,” A surery of Privacy Preserving Data Publishing using

Generalization and Suppression”, International Journal of Applied Mathematics& Information Sciences, 8, No.

3, 1103-1116. 2014.

[18]. Santosh Kumar Bhandare,”Data Distortion Based Privacy Preserving Method for Data Mining System”,

International Journal of Emerging Trends & Technology in Computer Science, Volume 2, Issue 3, May – June

2013.

[19]. K Sandhya Rani Kundra, Dr.J.Hyma, Prof P.V.G.D Reddy, Prof.K.Venkata Rao “Privacy preserving query

model using inverse laplacian differential technique”, IOP Conf. Series: Journal of Physics, 2019.

[20]. K Sandhya Rani Kundra, Dr.J.Hyma, Prof P.V.G.D Reddy, Prof.K.Venkata Rao “A Sigmoid based Learning

in Heterogeneous Distortion for Data Privacy”, IJITEE, Volume-8 Issue-11, September 2019.

[21]. Cynthia Dwork and Adam Smith,”Differential Privacy for Statistics: What we Know and What we Want to

Learn”, Journal of Privacy and Confidentiality. V1, pp: 135- 154, 2009.

[22] Rathindra Sarathy, Krish Muralidhar,” Differential privacy for Numeric Data “ in proceedings Joint

UNECE/Eurostat work session on statistical data confidentiality ,Bilbao, Spain, 2-4 December 2009.

[23]Differential Privacy (From Wikipedia,the free encyclopedia)

https://en.wikipedia.org/wiki/Differential_privacy.

[24] Mohammad Alaggan, ebastien Gambs and Anne-Marie Kermarrec “ Heterogeneous Di erential Privacy”,

Journal of Privacy and Con dentiality, V7, Number 2, 127-158, 2016.

[25] Yang Xu, Tinghuai Ma, Meili Tang, Wei Tiam “A Survey of Privacy Preserving Data Publishing using

Generalization and Suppression” ,Applied Mathematics & Information Sciences, 8(3):1103-1116 , May-

2014.

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Author-2

P hoto

or-3

Photo

AUTHORS PROFILE

Mrs K.SandhyaRani Kundra ,Asst.Professor in the department of Information

Technology,Gayatri Vidhya Parishad College Of Engineering(A),currently perusing

Ph.D. in Andhra University, Visakhapatnam. Interested areas are Privacy and security

and information security.

Dr.J.Hyma,Associate Professor in the department of computer science engineering,

ANITS. Her area of interest in Data Science, Internet of Things, Privacy and Security.

Prof P.V.G.D Reddy is presently Vice Chancellor, ANDHRA UNIVERSITY and

Sr.Professor of Computer Science & Systems Engineering department which is the

largest department in entire South India, and also serving as MEMBER, Executive

Council of VIGNAN Deemed University, Guntur. He has been awarded the Best

Teacher Award for the year 2011 by the Govt. of Andhra Pradesh in the combined state. Prof.

Reddy’s Research areas include Soft Computing, Software Architectures, knowledge Discovery from

Databases, Image Processing, Number theory & Cryptosystems. He has 3 Patents granted.

Prof.K.Venkata Rao, presently Academic Dean, ANDHRA UNIVERSITY, and

Professor of Computer Science & Systems Engineering department. He is currently

director & chairman; teachers mutually aided cooperative society ltd. and. Honorary

director and web master, Andhra University.

ISBN 978-81-265-3956-7 | Pages: 948 ` 599 |

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Soil Foundation Eng neering

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Table of Contents

Soil Foundation Engineering

Mechanics &

CD ROM and of wa e com i a on co y i ht © 2014 W l y nd a Pvt td A l ig ts e e ved

I d v dua p og ams a e c py gh ed by he r e pe t ve wne s and m y r qu re ep r te

l c n i g T is D OM may ot be e i t bu ed w th ut p or w i t n pe mi s on r m

t e pub s er T e r gh to ed s i ute he nd v d al ro r m on t e CD OM

d pen s on e ch p og am s l en e on ul ea h pr g am or e a ls W i ey

t e W ey ogo re r dema k and eg s e ed r dema ks f Wi ey

and o i s a f i t s l ot er r dem r s a e pr pe ty f t e r

r sp c i e own rs

F EE i h he b ok No to e s l se a a e yo l M c an s & ou da on ng n e i g

Soil Foundation Engineering

Mechanics & Soil Foundation Engineering

Mechanics &

B N D Na a i ga R o

I BN 9 8 81 2 5 3 56 7

1 Introduction

2 Origin and Formation of Soils

3 Soil Mineralogy and Structure

4 Physical Properties of Soils

5 Plasticity Characteristics of Soils

6 Soil Classification

7 Stresses Due to Self-Weight

8 Vertical Stress Due to Applied Loads

9 Permeability of Soils

10 Seepage Analysis

11 Consolidation

12 Compaction

13 Shear Strength

14 Soil Exploration

15 Lateral Earth Pressure

16 Retaining Structures

17 Stability of Earth Slopes

18 Bearing Capacity of Shallow Foundations

19 Settlement of Shallow Foundations

20 Pile Foundations

21 Well Foundations

In CD

22 Soil Dynamics and Machine Foundations

23 Ground Improvement Techniques

24 Foundations in Expansive Soils

Bibliography

List of IS Codes