Computers & Fluids 46 (2011) 270–276

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Computers & Fluids

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

Aerodynamic shape optimization using efficient evolutionary algorithmsand unstructured CFD solver

A. Jahangirian a,⇑, A. Shahrokhi b

a Department of Aerospace Engineering, Amirkabir University of Technology, Tehran, Iranb Aerospace Research Institute, Ministry of Science Research and Technology, Tehran, Iran

a r t i c l e i n f o

Article history:Received 9 May 2010Received in revised form 31 January 2011Accepted 14 February 2011Available online 22 February 2011

Keywords:Aerodynamic shape optimizationGenetic AlgorithmNeural NetworksCFD solver

0045-7930/$ - see front matter � 2011 Elsevier Ltd. Adoi:10.1016/j.compfluid.2011.02.010

⇑ Corresponding author.E-mail address: ajahan@aut.ac.ir (A. Jahangirian).

a b s t r a c t

An efficient evolutionary algorithm is presented for shape optimization of transonic airfoils. Several tech-niques have been used to improve the efficiency and convergence rate of the optimization Genetic Algo-rithm (GA). A new airfoil shape parameterization method is used which is capable of producing moreefficient shapes at viscous flow conditions. A Real-Coded Population Dispersion (PD) Genetic Algorithmis developed in order to increase the robustness and convergence rate of the Genetic Algorithm. AMulti-Layer Perceptron Neural Network (NN) is utilized to reduce the huge computational cost of theobjective function evaluation. Further improvement in the performance of NN is obtained by usingdynamic retraining and normal distribution of the training data to determine well trained parts of thedesign space to NN. Using the above techniques, the total computational time of optimization algorithmis reduced up to 60% compared with the conventional GA.

� 2011 Elsevier Ltd. All rights reserved.

1. Introduction

Aircraft wings are the main subject of the optimization effortsin aerodynamic design field. Airfoils are the basic elements of thewing geometry that determine a large share of the wing flow phe-nomena. Consequently, in order to have an optimum wing, first ofall we should have an optimum airfoil. Among shape optimizationalgorithms, gradient-based methods are well-known techniquesthat seek the optimum by calculating the local gradient informa-tion. Since they only search a part of the design space, the finalshape obtained from these methods is suspected to be a local opti-mum, especially for aerodynamic design problems. Alternatively,Genetic Algorithms (GAs) are more likely to find a global optimumand are therefore attractive for the aerodynamic design optimiza-tion [1,2]. However, these methods have drawback of beingcomputationally time-consuming especially in aerodynamicoptimization problems where CFD methods are used for fitnessfunction computation.

One can enhance the convergence rate of the optimizationprocess by precise selection of shape parameterization technique.The method should take into consideration the characteristics ofthe flow phenomena which govern the flow regime. Aparameterization method developed by authors [3] is used in thiswork that increases the convergence rate while provides the bestoptimum shape compared with other techniques.

ll rights reserved.

Another essential problem with GA is the massive calculationsrequired for fitness evaluation by CFD flow solver until the conver-gence is reached. The cumulative information from successivegenerations in GA has encouraged the researchers to developtechniques known as Fitness Function Approximation or surrogatemethods which can be supplied instead of Computational FluidDynamics (CFD). Husken et al. [4] and Giannakoglou et al. [5]applied different NN approaches in several aerodynamic optimiza-tion problems and concluded that NNs are quite beneficial fordecreasing the computational time in the optimization algorithm.However, the reliability of these techniques depends on intellec-tual selection of the training set as well as planned criterion forthe selection of the estimated individuals. The normal distributionof the training set is efficiently applied in this research to deter-mine the boundary of the well trained design space. Consequently,NN is able to provide an appropriate estimation of the individualshence, introducing minimum error in prediction of the objectivefunction.

The final technique that is developed in this work, referred asPopulation Dispersion, provides a method for scattering the indi-viduals in the design space in order to improve the search processof the GA and consequently accelerate the convergence to theoptimum.

2. Computational flow solution

The huge numbers of the airfoil shapes that are generated byGenetic Algorithm are evaluated based on the numerical solution

Fig. 2. Airfoil divisions in PARSEC.

A. Jahangirian, A. Shahrokhi / Computers & Fluids 46 (2011) 270–276 271

of the governing flow equations. Compressible Navier–Stokesequations are solved using a finite volume cell-centered implicitscheme [6]. Since most of the computational time required foroptimization process is consumed by flow solver, CFD solver whichdrives the optimization process must possess high efficiency andconvergence rate. Consequently, a dual-time implicit method isused in the present work that follows the work of Jahangirianand Hadidoolabi [6] for unstructured grids. This method reducesthe computational time due to utilizing convergence accelerationtechniques such as residual smoothing and local time stepping. Atwo-equation k–e turbulence model is used in conjunction withthe Navier–Stokes equations for high Reynolds number viscousflow simulations. The computational field is discretized using tri-angular unstructured grids generated by a successive refinementmethod [7]. During the design process, the mesh is continuouslyupdated as the shape of the geometry changes. In the present work,the primary mesh generated around the initial airfoil is moved tobe fitted to a new generated airfoil using a modified springanalogy [6].

Fig. 3. Sobieczky parameters for trailing edge.

3. Airfoil shape parameterization

The mathematical representation of the airfoil shapes and crea-tion of wide variety of possible choices are the challenging topicsfor all optimization approaches. Majority of the optimization tech-niques employ smoothing algorithms for shape representationwhich are based on the polynomials and sp-lines [8]. One of thebest proposed techniques for shape parameterization is PARSEC,which has been successfully applied by aerodynamic communityfor transonic airfoil design, see e.g. [9,10]. Fig. 1 illustrates elevenbasic parameters for PARSEC method. A linear combination ofshape functions is utilized to present the airfoil shape in this meth-od so that:

Yk ¼X6

n¼1

an;kXn�1

2k ; ð1Þ

where coefficients an are the shape functions and X and Y are theairfoil coordinates. The subscript k changes from 1 to 2 in order toconsider the length on the upper and lower surfaces. In order to ob-tain the shape functions, the airfoil is divided into five regions onthe upper and lower surfaces as illustrated in Fig. 2. Coefficientsan for these regions are calculated using the parameters in eachregion.

To remove the disadvantages of the PARSEC parametric methodSobieczky proposed a new formulation for trailing edge modeling[11,12]. Fig. 3 shows the Sobieczky parameters for trailing edgemodeling. The parameters Da, L1, L2 that control the increment intrailing edge thickness (DY) are added to make the airfoil surfacea divergent trailing edge. The following function is proposed byauthors for DY [3]:

Fig. 1. PARSEC method for airfoil parameterization.

DY ¼ tan Dal � n ½1þ g � nn � ð1� nnÞl�; ð2Þ

where g is set to 0.8 and n is equal to 6. Parameter l is a constantequal to 6 and is considered fixed during the optimization. Param-eter n represent the distance from trailing edge to the location onthe X axis where DY is obtained (Fig. 3). The entire surface isexposed to the above equation, thus parameter L is not used in thismethod. The parameters DYTE and YTE in Eq. (1) are also consideredzero. Therefore, the total number of design variables is reduced to10. More discussions on different parameterization techniquesand the advantages of the new technique can be found in [3].

3.1. Population dispersion Genetic Algorithm

In the present study Genetic Algorithm is applied to the optimi-zation problem of airfoil shape. A simple one point crossover oper-ator is used with 80% probability of combination, as the use ofsmaller values was observed to deteriorate the GA performance.The mutation probability is set to 10% and then the mutationoperator adds a random disturbance to the parameter about 15%of design space that is defined corresponding to each chromosome.The total population of each generation is set to 20.

To treat an optimization problem with a large design space,Adaptive Range GA (ARGA) was proposed by Arakawa and Hagiw-ara [13] for binary genetic algorithms. The idea behind this methodis to adapt the population toward promising design regions accord-ing to the normal distribution of the design variables. It uses thestatistics of the top half of the population to adapt the genes inthe search space. Hence, the adapted population distributes inthe hopeful search region. The method was further extended toreal coded applications of single and multiple objective functionoptimization problems [14]. However, the crucial point forsuccessful application of this method is maintaining the genediversity, longer than the conventional GA, in the initial phase ofgenerations. While the initial gene diversity contributes to the

X

Y

0 0.2 0.4 0.6 0.8 1

-0.04

-0.02

0

0.02

0.04

0.06

PDNN GA Optimum (Viscous case)PDNN GA Optimum (Inviscid case)

Fig. 6. Optimum airfoils for inviscid and viscous cases using accelerationtechniques.

PDNN GA Optimum (Viscous case)PDNN GA Optimum (Inviscid case)

272 A. Jahangirian, A. Shahrokhi / Computers & Fluids 46 (2011) 270–276

robustness of the method, the adaptive feature improves its localsearch capability.

The new technique developed in this paper referred as Popula-tion Dispersion (PD) is actually used in the initial phase of genera-tions to provide a better diversity in the design space while usingGenetic Algorithm. To produce the new generations in the initialphase, PD alters the limits of the design space to a new specifiedrange such that it allows the optimization algorithm to explorethe parts of the design space which has not been investigatedyet. The adapted range is defined using normal distribution ofthe individuals in the previous generations. The new populationis then obtained from preceding generations without interferingGA selection operator. Considering N sample of each design vari-able Xk, the standard deviation for design variable Xk is obtainedfrom the following equation:

rk ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1N

XN

i¼1

ðxi;k � lkÞ2

vuut ; ð3Þ

where xi,k is the ith sample of design variable Xk. In the new ap-proach the average li and the standard deviation ri are calculatedusing the information of the previously generated individuals. Nor-mal distribution is determined using the values of the average andthe standard deviation. The selected range for generation of the new

Data which are in thisregion are excludedfrom design space

Nor

mal

Fun

ctio

n

Design Parameter

New

Des

ign

Spac

e

New

Des

ign

Spac

e

Fig. 4. New range for Population Dispersion.

Exc

lude

d R

egio

n

New design space

Nor

mal

Fun

ctio

n

Design Parameter

Exc

lude

d R

egio

n

Fig. 5. New range for Population Concentration.

X

CP

0 0.2 0.4 0.6 0.8 1

-1

-0.5

0

0.5

1

Fig. 7. Pressure distribution of the optimum airfoils for inviscid and viscous casesusing acceleration techniques.

X

Y

0 0.5 1-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

Fig. 8. Unstructured grid around optimum airfoil.

Fig. 9. Normal distribution of the 20th generation for simple GA and PDGA.

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274 A. Jahangirian, A. Shahrokhi / Computers & Fluids 46 (2011) 270–276

individuals is adapted so that it excludes those parts of the designspace that consist the majority of the previous individuals. Fig. 4illustrates the normal distribution of a sample data set and the cor-responding adapted range. New data are generated using conven-tional GA; however they are checked to be located within thespecified range. Consequently, it increases the robustness of GA infinding the global optimum.

Once the generated data spread out enough in the design space,the new GA members are concentrated in the promising region ofthe design space, i.e. the region between standard deviationand the average. This region is shown in Fig. 5. The important sub-ject here is the selection of the particular generation to switch frompopulation dispersion to the population concentration (AR) mode.In the present study a probability factor P(c) is considered whichpermits switching from dispersion to concentration as:

piðcÞ ¼ sinh4ðg stepiÞ;if piðcÞ > 1 then piðcÞ ¼ 1:0;

ð4Þ

g step1 ¼ 0:42;g stepi ¼ g stepi�1 þ 0:038;

ð5Þ

Subscript i is the generation number. The parameter g_stepi

incrementally increases while the generations evolve. Eq. (5) forcalculation of g_stepi is set in a way that pi(c) becomes one in thegeneration number about 25 when the new populations are thenconcentrated in the design space for next generations. pi(c) is

0.0

1.0

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35

Prob

abilt

y of

Con

cent

ratio

n

Generation Number

Fig. 10. Probability of the Adaptive Range utilization in the optimization process.

Fig. 11. Convergence history for viscous case.

calculated for each generation and the optimization algorithmdecides to scatter or concentrate the new individuals using concen-tration probability. The concentration probability is obtainedthrough a random process in the optimization algorithm. If theconcentration probability becomes equal to zero the new dataare spread in the design space and vice versa. The higher valuesof pi(c) increase the chance of data concentration. Therefore, asGA proceeds, the probability of dispersion occurrence decreasesand the probability of the concentration increases. Concentrationfactor and pi(c) are quantitatively discussed in the results section.

Regardless to the way that the next generation is produced, thenew population may not locate within the specified range. In thiscase, the generation of the new offspring is repeated using thesame parents until the range requirements are satisfied. However,if this way does not work, a wider range is provided for that spe-cific couple. The range enlargement continues until the new off-spring is obtained. Numerical experiments showed that it doesnot influence the overall efficiency of the proposed technique.

3.2. Fitness function approximation

In order to reduce the computational time required for the fit-ness function evaluation, several surrogate methods have beenproposed in the literature [15,16]. Among fitness function approx-imation methods, Neural Network (NN) is widely applied for theestimation of the expensive objective functions. Feed-forwardMulti-Layer Perceptron (MLP), one of the most common NeuralNetwork models, is implemented in this research with two hiddenlayers. Layers are fully connected and each layer contains tenneurons.

Once NN has been trained with the data provided from succes-sive generations, it is used to evaluate the fitness values in the sub-sequent generations of the optimization algorithm. In the presentwork, the total number of training data within different genera-tions is kept fixed by excluding the worse individuals from thetraining list as the populations evolve.

Table 1Initial and optimum Airfoil parameters (viscous case).

Simple GA, PARSEC PDNN, new parameterization

Initial value Final value Initial value Final value

rle 0.0095 0.0082 0.0095 0.0083xup 0.44 0.5334 0.44 0.5200zup 0.06286 0.0592 0.06286 0.0616zxxup �0.4469 �0.4425 �0.4469 �0.4542a 7.6 6.54 7.6 7.6465b 8.8 11.16 8.8 10.89xl 0.3549 0.3838 0.3549 0.3688zl �0.0592 �0.0567 �0.0592 �0.0512zxxl 0.825 0.677 0.825 0.830at – – 0.0 5.991

Table 2The comparison of conventional GA and accelerated optimization methods (Viscouscase).

Aerodynamiccoefficient

Simple GA, PARSEC PDNN, newparameterization

Initialvalue

Finalvalue

Initialvalue

Finalvalue

Cl 0.822 0.799 0.822 0.799Cd 0.02784 0.0160 0.02784 0.0140Cm �0.1109 �0.0827 �0.1109 �0.102Cl/Cd 29.52 49.93 29.52 57.07

Number of CFD calls

Max

imum

Obj

ectiv

e Va

lue

500 1000

50

55

60

65

70

75

80

85

90

PARSEC (case 1)PARSEC (case 2)NN (case 1)NN (case 2)PD (case 1)PD (case 2)PD+NN (case 1)PD+NN (case 2)

Fig. 12. Convergence history for inviscid case.

A. Jahangirian, A. Shahrokhi / Computers & Fluids 46 (2011) 270–276 275

In order to control the discrepancy between the fitness obtainedfrom the accurate solution (CFD) and NN estimation and also toavoid the incorrect optimum convergence during the optimizationprocess, the normal function distribution is proposed in this work.Normal distribution of the training test data illustrates thescattering of the data in the design space. In the case that thenew member is confirmed to be surrounded by the most populatedpart of the training set, NN would be a good candidate forevaluating the objective function. Otherwise, it is evaluated bythe CFD flow solver. Numerical experiments show that themaximum error incorporated in the function estimation is reducedby about 30% when the present approach is used as a measure forthe fitness function evaluation in NN [17].

3.3. Results

The proposed method is applied to the aerodynamic design oftransonic airfoil shape. To demonstrate the capabilities of themethod a comparison is made between the results obtained byconventional GA using PARSEC parameterization method and thosefrom GA with proposed acceleration techniques. The flight condi-tions are Mach number 0.75 and incidence angle 2.8 degrees. Bothinviscid and viscous flow conditions are studied in order to showthe applicability of the method with different evaluation functions.For the viscous case, Reynolds number is 6.5 millions and fully tur-bulent flow is considered. The RAE-2822 airfoil is selected as initialairfoil and the wing section efficiency factor (Cl/Cd)max is consid-ered as the objective function.

The optimum airfoil shapes obtained from present method areshown in Fig. 6 for both inviscid and viscous flow cases. The pres-sure coefficient distributions around optimum airfoils are illus-trated in Fig. 7. The final unstructured viscous grid around theoptimized airfoil is shown in Fig. 8. The normal distribution of eachdesign variable at 20th generation is shown in Fig. 9 for the present

Table 3The comparison of conventional GA and accelerated optimization methods (Inviscid case)

Optimizationmethod

Parameterization Maximum objectivevalue (Cl/Cd)max

Total time (h)

GA PARSEC 88.97 111.85GA + PD New technique 89.33 85.80GA + NN New technique 87.34 77.64GA + NN + PD New technique 89.55 44.24

method and simple GA. As shown in this figure, the PDGA permitsthe search tool to seek the optimum in a wider range of the designspace. The probability of Population Dispersion is decreased as theGA evolves. The changes in probability of concentration and p(c)are shown in Fig. 10. The new individuals are concentrated whenthe concentration probability equals one and they are spread inthe design space when it equals zero. According to Fig. 10, the indi-viduals that are generated in the first twenty generations aremostly dispersed in the design space while after generation 25where the probability of data concentration is increased the indi-viduals are all concentrated in the promising region. Fig. 11 showsthe convergence histories against the number of CFD calls insteadof the computational time since this part of the optimization rou-tine accounts for about 99% of total computational time. It is evi-dent from this figure that the new method is capable of reducingthe number of CFD calls (or computational time) by a factor ofabout two while at the same time improving the quality of theoptimum airfoil. The initial and final values of the airfoil shapeparameters are provided in Table 1 for this case. A comparison ofthe aerodynamic coefficients is also given in Table 2. As shown inthis table the values of the lift coefficient are decreased in optimumairfoils compared with the initial airfoil but the drag coefficient isremarkably decreased as well. Note that the drag reduction is moresignificant for the present technique. Therefore, the aerodynamicefficiency factor has increased from 49.93 for conventional GA to57.07 for the present method.

To provide a better understanding of the influence of eachacceleration technique in reducing the computational time, theinviscid results are shown separately for each technique inFig. 12. The results are repeated for each case using differentrandom seeds. According to this figure, NN and PD provide similarreduction in time. However, applying all acceleration techniquestogether, the total computational time for optimization process isreduced by 60%. The reduction in time obtained through eachtechnique is briefly stated in Table 3. The computations are carriedout on a Pentium IV personal computer with 2.81 GHz speed. It isnoted that each viscous flow solution and inviscid flow solutiontake about 11 and 3 min on the above PC, respectively. Four groupsof results are presented in this table. The first one is for conven-tional GA with PARSEC, the second and third cases use PD andNN both with new parameterization. The last case explores thecombination of all acceleration techniques. According to the table,the total optimization time is reduced by 23% when PD and thenew parameterization techniques are applied. The correspondingreduction for NN is about 31%. No considerable changes are seenin the value of the objective function in four cases. Only in case 3the objective function is decreased a little due to the differencesbetween NN estimations and CFD flow solver. However, someimprovements are obtained when PD is also incorporated in thesolution (case 4). The maximum and average RMS errors are pre-sented in this table for the cases that NN is associated with. Asmentioned the total reduction in computational time is increasedto about 60% when all acceleration techniques are used (case 4).It is evident from this table that using PD technique in combinationwith NN can improve the efficiency of the NN. The reason would bewidening the acceptable region for NN estimations when PD is

.

Total reduction in time (%) Maximum RMS error Average RMS error

– – –23.29 – –30.58 4.99 1.9560.04 4.25 2.7

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applied to distribute the data in the design space. Thus the numberof members that are evaluated approximately is increased andconsequently the total CPU time is decreased in the last case.

4. Conclusions

In this research, the efficiency and convergence rate of evolu-tionary aerodynamic shape optimization was improved employingseveral techniques. The proposed airfoil shape parameterizationand the Population Dispersion Genetic Algorithm (PDGA) pre-vented the premature convergence and ease finding the optimum.The NN was adopted for estimating the expensive objective func-tions. The normal distribution of the training data was employedin order to determine well trained parts of the design space toNN. Using the aforementioned acceleration techniques, the totalcomputational time of optimization algorithm was remarkably re-duced comparing to the conventional GA.

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