Computers and Structures 89 (2011) 556–570

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Computers and Structures

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

Experimental application and enhancement of the XFEM–GA algorithmfor the detection of flaws in structures

Eleni N. Chatzi a,⇑, Badri Hiriyur b, Haim Waisman b, Andrew W. Smyth b

a Institute of Structural Engineering, ETH Zürich, Wolfgang-Pauli-Strasse 15, CH-8093 Zürich, Switzerlandb Department of Civil Engineering and Engineering Mechanics, Columbia University, New York, NY 10027, USA

a r t i c l e i n f o

Article history:Received 16 June 2010Accepted 17 December 2010Available online 26 January 2011

Keywords:Structural health monitoringExtended finite element methodGenetic algorithmsNon-destructive techniquesInverse problemFlaws

0045-7949/$ - see front matter � 2010 Elsevier Ltd. Adoi:10.1016/j.compstruc.2010.12.014

⇑ Corresponding author.E-mail address: chatzi@ibk.baug.ethz.ch (E.N. Chat

a b s t r a c t

The extended finite element formulation (XFEM) combined with genetic algorithms (GAs) have previ-ously been shown to be very effective in the detection of flaws in structures. By this approach, the XFEMis used to model the forward problem and a GA is used as the optimization scheme, converging to thetrue flaw. The convergence is obtained by minimizing the error between sensor measurements and dataobtained by solving the forward problem.

The current study proposes several advances of this XFEM–GA algorithm, more specifically: (i) a novelgenetic algorithm that accelerates the convergence of the scheme and alleviates entrapment in localoptima, (ii) a generic XFEM formulation of an elliptical hole which is utilized to detect any type of flaw(cracks or holes) of any shape, and (iii) experimental verification of the approach for an arbitrary crack ina 2D plate.

Convergence studies on various benchmark problems including the experimental verification clearlyshow the potential of this approach to detection of arbitrary flaws.

� 2010 Elsevier Ltd. All rights reserved.

1. Introduction within some minimization algorithm. Hence by minimizing the

Nondestructive evaluation (NDE) of damage has many importantindustrial applications in structural health monitoring. Undoubt-edly, flaw detection is crucial for practicing engineers to assess thereliability and durability of structures and prescribe appropriatemaintenance and inspection procedures. There are many non-inva-sive techniques available to detect flaws in structures. Most of thesemethods are based on the detection and interpretation of the dy-namic properties of the structure, such as frequencies, mode-shapes,transfer functions and electro-mechanical impedance [1–9] throughthe use of ultrasonic, radiographic, thermographic, impact analysis,electrical impedance tomography and similar methods of measure-ment. These methods have advantages and limitations depending onthe application. For instance, conventional ultrasonic methods areunreliable when used to detect cracks in laminated composite platesdue to the interference of wave reflections from the laminarinterfaces.

Numerical methods for flaw-detection are structured as inverseproblems in which measurements of system variables such asdisplacements and strains may be available, while the physicalproperties of the system itself are unknown. Typically, the solutionof inverse problems consists of an iterative process with multiplesolutions of a changing forward problem that has to be solved

ll rights reserved.

zi).

error between the predicted and measured quantities, it is ex-pected that the iteration will converge to the true flaw. However,these algorithms can routinely converge to a local minimum,which may be far from the true solution, and may also take a largenumber of iterations. When cracks are broadly defined as damagezones in structures, the geometry of the forward problem remainsunchanged and iterations are simply repeated with mesh refine-ment. This process, usually addressed in the literature as parame-ter estimation or identification [10], results in a crude estimation offlaw location and size.

Recently a new NDE technique based on the application of theextended finite element method (XFEM) and genetic algorithms(GAs) was proposed by Rabinovich et al. [11]. The authors haveshown that cracks in flat membranes can efficiently be detectedby this algorithm in both static and dynamic excitations. If conven-tional finite element methods are employed for the solution of theforward problem, the mesh would need to be updated in everyiteration of the optimization scheme. Even though fast meshingalgorithms are becoming increasingly available, this task still re-mains a significant drain on computational resources. Hence, theattractiveness of XFEM–GA approach is directly related to model-ing the forward problem with XFEM [12–18]. More recent reviewson XFEM can be found in [19,20].

The work presented herein employs the XFEM approach for thesolution of the forward problem. To solve the unconstrainedminimization problem, genetic algorithms [21] have proven to be

E.N. Chatzi et al. / Computers and Structures 89 (2011) 556–570 557

the most efficient and reliable choice as they work well even whenthere is no straightforward dependency of the objective functionon the design variables such as in maximum flow problems [22],non-destructive assessment techniques [23], optimal design ofstructures [24,25] and machine learning [26]. Employing a generaloptimization scheme within an XFEM framework to identify modelparameters such as the order of various singularities or thicknessof boundary layers have also been addressed in [27]. Other inter-esting variants of XFEM which employ numerically adaptedenrichment functions include the one proposed by Chahine et al.[28,29] and Menk and Bordas [30].

In a recent paper, Waisman et al. [31] extended the XFEM–GAapproach to the detection of flaws in elastostatics. The XFEMenrichment functions were chosen to model strong and weak dis-continuities arising from straight cracks, circular holes and certainnon-circular shaped holes. Nonetheless, in these applications, ahole was detected with an XFEM-circular hole enrichment and acrack with an XFEM-crack enrichment, i.e. a priori knowledgewas required in order to choose the appropriate XFEM formulationfor the forward problem.

In the present work we introduce several improvements to theXFEM–GA algorithm: (i) the application of a novel genetic algo-rithm that accelerates the convergence of the method and allevi-ates entrapment in local optima, (ii) a generic XFEM formulationof an elliptical hole which is utilized to detect any type of flaw(cracks or holes) of any shape, and (iii) experimental verificationon the detection of a crack in a 2D plate.

The proposed GA method utilizes a newly introduced weightedaverage approach to mutate the design parameters toward areas ofincreased fitness. This particular type of mutation operator is ap-plied in order to alleviate the premature convergence problem,i.e. the entrapment in local optima that occurs for many evolution-ary algorithms. Convergence studies on various benchmark prob-lems including the experimental verification clearly show thepotential of this approach in detection of arbitrary flaws.

The outline of this paper is as follows. In Section 2, we describethe XFEM approach in detail. The formulation of the enrichmentfunction for elliptical flaws is also presented. Section 3 is devotedto the description of the WAM-GA tool that has been proposedand used. A variety of benchmark problems were chosen to evalu-ate the performance of this novel approach and the comparativeresults are presented here. In Section 4, the application of thismethod to detect flaws in two dimensional plates is demonstratedon both a numerical and an experimental sample.

2. Motivation and XFEM approach for solution of the forwardproblem

Consider a generic body domain X with some essential bound-ary conditions specified on Cu and natural boundary conditions onCt. The body is subjected to some traction loads �t. In addition sev-eral strain/displacement measurements are made at certain dis-crete sensor locations. Further, the body contains a flaw, either inthe form of a randomly shaped void (e.g. Fig. 1(a)) or a crack (e.g.Fig. 1(b)).

The objective of the present work is to approximate any flawtype with a generic ellipse formulation (within the extended finiteelement formulation) that results in the least error with the simu-lated data and the sensor measurements, as schematically illus-trated in Fig. 1(a) and (b).

The attractiveness of this approach is directly related to model-ing the forward problem with XFEM. In XFEM, the effect of internalfeatures such as cracks, voids or material interfaces are captured byintroducing special functions, which locally enrich the span of ba-sis functions in the discretized system.

The general form of the discretized weak form approximation inXFEM, for the solution variable u(x), is given as follows:

uhðxÞ ¼XI2N

NIðxÞuI þX

J2NenrI

XI2F

UIðxÞaIJ

!ð1Þ

Here N refers to all the nodes in the mesh and NenrI refers to the sub-

set of N that contains all the nodes enriched by the functions UI,called the support of the basis UI. This support is made up of onlythose elements that contain within their domains, the discontinuityinterface modeled by UI. If an element-wise discretized weak formis considered, both N and N

enr (for an enriched element) refer to thenumber of element nodes Ne. The enrichments are controlled by de-grees of freedom aIJ which are additional unknowns to be solved for,in the global system of equations.

When considering XFEM for linear elastic fracture mechanics,the number of enrichment functions F generally equals 1 for ele-ments completely cut by the crack and the Heaviside step functionHðxÞ is used for U. For elements with a crack-tip, F generallyequals 4 and the span of UI is suitably chosen to model tip singu-larity. In the case of XFEM for modeling weak-discontinuities (suchas those caused by material inclusions), a single enrichment func-tion which is C0 continuous is used. The discretized weak form foran enriched element is given as:

uheðxÞ ¼

XI2Ne

NIðxÞ uI þUðxÞaIð Þ ð2Þ

It may be seen from Eq. (2) that the physical displacement at an en-riched node I is provided in terms of both the standard dof uI andthe enriched dof aI. To ensure that the physical displacement solu-tion is completely defined by the standard dof uI, a shifted-basisform of Eq. (2) may be written as follows:

uheðxÞ ¼

XI2Ne

NIðxÞ uI þ UðxÞ � UðxIÞh i

aI

� �ð3Þ

2.1. Enrichment function for elliptical voids

The enrichment function to model holes commonly takes theform of Eq. (4) as explained in [16]:

UAðxÞ ¼1 if x outside hole0 otherwise

�ð4Þ

Another approach to model elliptical voids is to use the formu-lation for bi-material inclusions with a very low stiffness value forthe region inside the void. The enrichment function is modeledafter the equation of the ellipse and takes the following form:

UBð�xÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�x2

a2 þ�y2

b2

s� 1

���������� ð5Þ

where �x refers to the transformed coordinates with the origin at thecenter of the ellipse and oriented along its major axis. The absolutevalue function enforces a weak discontinuity along the ellipticalinterface. While the enrichment function in Eq. (5) is little moreexpensive to compute than the form in Eq. (4), the convergenceproperties are slightly better as seen in Fig. 2. In this figure, a com-parison of the convergence properties of XFEM using the two for-mulations for the standard Kirsch problem (plate with a circularhole [32]) is provided.

The latter approach using UB(�) for the enrichment function isadopted in this work. The ratio of elastic moduli for the matrixand hole region is set to 1e�7.

Fig. 1. Generic body with void/crack modeled with elliptical enrichment.

Fig. 2. Comparison of error norms in XFEM formulation with different enrichment functions (displacement and energy norms).

558 E.N. Chatzi et al. / Computers and Structures 89 (2011) 556–570

2.2. Convergence of elliptical enrichment to crack

A parametric study is performed to study the effectiveness ofusing XFEM with elliptical hole enrichments to model cracks. Asshown in Fig. 3(a), the idea is to study different elliptical aspect ra-tios and varying major axis lengths which will give the closest dis-placement fields to those obtained when a crack (shown in red1) ismodeled. An illustration of the XFEM mesh and the enriched nodesfor an elliptical hole and the corresponding FEM mesh are shown inFig. 3(b) and (c). In these figures, the crack and ellipse are orientedin 45� where the structure containing the crack within its interior,is pinned along the left edge and uniform traction is applied alongits right edge.

The convergence studies, shown in Fig. 4, are performed onthree different cases of crack orientation h = {0�,45�,90�} with re-spect to the global x axis. In all these cases, FEM analyses of thecracks are performed using ABAQUS and used as the referencesolution. The XFEM parametric study involved an ellipse with

1 For interpretation of color in Fig. 3, the reader is referred to the web version ofthis article.

aspect ratio r = b/a (ratio of major axis b to minor axis a) which var-ies from r = 1 (circle) to r = 0.1 (elongated ellipse). The major axis ofthe ellipse is always oriented along the crack direction. In additionwe define the ratio ra = aellipse/acrack (where acrack is the cracklength) which varies from 2 to 1. The XFEM solutions uh obtainedusing elliptical hole enrichments are compared against the corre-sponding results u obtained from FEM (with double nodes at thecrack line to model opening). The comparisons are made in termsof the energy norm and the displacement norm which are definedas follows:

kru�ruhk2 ¼Z

Xðr� rhÞT : ðe� ehÞdX

� �12

ku� uhk2 ¼Z

Xðu� uhÞTðu� uhÞdX

� �12

The plots in Fig. 4 show of the two error norms versus decreas-ing aspect ratios of the ellipse. The results indicate that in almostall the cases, the ellipse approximation gets monotonically betterwith decreasing aspect ratio. It may also be observed that the con-vergence is best when the ellipse major radius is equal to the

crackcrack

with tighter aspect ratiosapproximating elliptical hole

Fig. 3. Illustration of an elliptical hole used to approximate a straight crack. (a) Varying enclosing elliptical holes; (b) enriched nodes used in the elliptical hole modeled byXFEM; (c) FEM mesh with double nodes along crack.

E.N. Chatzi et al. / Computers and Structures 89 (2011) 556–570 559

length of the crack (ra = 1). The results confirm that an ellipse witha sufficiently small aspect ratio can be used as a good approxima-tion to a crack, in particular for the application considered in thispaper.

3. GA–XFEM based identification

Genetic algorithms have been chosen as the optimization toolfor this non-destructive detection scheme. GAs have long beenused as an efficient tool for search, optimization and machinelearning problems. The main concept associated with this particu-lar method is the mimicking of the biological processes of naturalevolution and survival of the fittest candidates [33,34]. Accordingto Holland’s genetic algorithm each potential optimal solution toa problem can be seen as an individual that can be coded by aset of genes. Usually these genes are chosen to be binary bits andthe binary string or alternatively the ‘‘individual’’ is also knownas a chromosome. A random generation of such chromosomes pro-vides us with the initial population. At each cycle of the evolution-ary process a new set of offsprings is produced from the fittest

individuals of the previous generation. Reproduction takes placethrough the recombination of the bit strings or simple bit flips thatoccur with some probability. The purpose of this evolutionary pro-cedure is to eventually lead to the survival of the fittest individual,as this will be the one to produce the largest number of offspringsand thus has the best chances for survival. All genetic algorithmsare based on the following scheme (Fig. 6 – classic GA).

Representation. Depending on the application the parameters ofthe problem can be either integer or real numbers. Usually, theset of parameters can be appropriately encoded into a finite lengthbinary string. Once a representation is decided a number of differ-ent chromosomes is randomly generated to form the initialpopulation.

Fitness evaluation – selection. The evaluation of the fitness foreach member of the population is carried out through the use ofan objective function associated with each problem. The suitabilityof each chromosome is the criterion based on which this individualwill be selected for reproduction. Selection can be performedthrough various schemes like ‘‘roulette wheel’’ selection or ‘‘tour-nament’’ selection. The roulette wheel selection is solely depen-dent upon the performance (fitness) of each individual. In the

Insertion of Randomly Generated Individuals

Generation t

Population Size n(t)

Mean population Size n

T T T

Fig. 5. Population variation scheme of saw-tooth GA.

Fig. 4. Parametric convergence studies of an elliptical hole modeled by XFEM to approximate a straight crack modeled by FEM. The figures illustrate decreasing major ellipseaxis ratios ra to the crack length. Each figure studies three different ellipse/crack orientations: 0�, 45� and 90� and varying ellipse aspect ratios r = b/a. The energy norm isshown in the plots in the top row while the displacement norm is shown in the bottom row.

560 E.N. Chatzi et al. / Computers and Structures 89 (2011) 556–570

case of the second method, the probability of a string selection isalso dependent upon the fitness of the ‘‘competitive’’ strings cho-sen to participate in a tournament round. The latter contributes to-ward the preservation of the diversity of the population.

Crossover. This operator is applied with a certain probability, tothe pairs of the previously selected individuals (parents). In generalthe crossover procedure randomly selects a position in the binarystring and mutually exchanges the parts of the chromosome beforeand after this location in order to produce two offsprings.

Mutation. The use of the crossover procedure as the only meansfor generating new individuals could result in the loss of the diver-sity of the population. This problem can be overcome using themutation operator which involves the random flip of the binarygenes of the chromosome for the case of jump mutation, or thealteration of the phenotypic (real) representation of the designparameters by a small increment or decrement for the case ofcreep mutation.

More information on the basic (classic) GA scheme can befound in [35–37,33]. Apart from the basic operators there is alsoa number of additional techniques that can be applied duringthe GA procedure such as: elitism which ensures the survivalof the fittest individual into the next generation; and nichingwhich provides the genetic algorithm with the possibility ofexploring different local optima by creating and evolving smallersubgroups within the population. Another more recent techniqueis the micro-GA, suggested by Goldberg [35] and first imple-mented by Krishnakumar [38], which has to do with the evolu-tion of a small sized population where the loss of diversity isprevented by restarting the population as soon as it degeneratesbelow some threshold. Finally, there is also the sawtooth-GAtechnique proposed by Koumousis, and Katsaras [39], whichproved to be most helpful for the particular type of identificationproblem considered here. This method uses a variable populationsize of mean value �n, and amplitude D, and a periodic partial re-initialization of the population of a period T, in the form of asaw-tooth function as shown in Fig. 5.

Heuristic methods such as GAs are particularly advantageousfor the class of problems studied here for the following reasons:

(i) They have been fairly well studied in the literature and haveproven to be very efficient on difficult optimization prob-lems (converging to a global minima).

(ii) In contrast to gradient based approaches they do not requirea straightforward analytical relationship connecting thefunction to be optimized to the design parameters of theproblem. Instead, they can handle even loosely defined prob-lems. In our case the connection between the design vari-ables (crack parameters) and the objective function (error)is done through the forward run of the XFEM code (whichkeeps changing with every step). As a result of the previousremark GAs do not impose restrictions such as continuity/differentiability, proper initial interval estimate, etc.

(iii) They can easily be implemented and adjusted to any usersource code.

(v) Finally, parallelizing GAs is relatively easy and straight for-ward (in some sense similar to Monte Carlo methods), whichis a huge benefit for the case of inverse problems where alarge number of forward runs is necessary.

Fig. 6. The XFEM–GA algorithm flowchart.

6

8

10

12

14

16

18

20

(xc,yc)=(5,10)

E.N. Chatzi et al. / Computers and Structures 89 (2011) 556–570 561

The problem described in Section 2, consists of finding the param-eters describing the flaw, i.e., for linear cracks-the tip coordinates,for circular holes-the radius and center coordinates, or for ellipticalholes-the center, minor and major axes and orientation angle. Thisis achieved through minimizing the residual of the XFEM–GA esti-mate with respect to the actual data provided at specific sensorlocations, usually along the boundaries (Eq. (6)):

rðbiÞ ¼k�cðbiÞ � �0

ckk�0

ckð6Þ

where k�k denotes the L2 norm, �0c is the real measured data (in most

cases, this is the measured strain obtained from the damaged mod-el) at some locations along the surface and �c are the computedstrains at the exact same points. These computed strains are a func-tion of the optimization parameters bi which change with the for-ward problem as the outer (optimization) iteration proceeds.

The problem setup involves the definition of the number of un-known parameters as well as the discretized search space (upper,lower bounds and number of discrete possibilities per parameter).The XFEM–GA identification process is initiated with the genera-tion of binary bit individuals (chromosomes), which are randomlyselected from the search space, serving as the initial population.The next step involves the fitness evaluation of each individual inthe population group. The fitness function to be minimized is theone defined in Eq. (6). At the next cycle a new set of offsprings isproduced from the fittest individuals of the previous generationusing the tournament selection scheme. The mechanisms of repro-duction (mutation and crossover) are applied and the evolutionarycycle continues for a specific number of total generations.

3.1. Applied Weighted Average Mutation GA (WAM-GA)

Most population-based, reproductive, optimization algorithmssuch as genetic algorithms, ant colony optimization, and particleswarm optimization (PSO) often deal with the premature conver-gence problem. This problem occurs when highly fit parents in a

Table 1WAM-GA scheme.

Mutation pseudocode

for j = 1,npop{for k = 1,nparam{if (rand1 6 pc) /⁄perform mutation)⁄/{if (rand2 6 pwam) /⁄perform weighted average mutation)⁄/child(k, j) = child(k, j) + rand3 � (wmean(j) � child(k, j))else /⁄(perform standard creep mutation)⁄/child(k, j) = child(k, j) + g1(k) � crsgn}}}

population pool rapidly dominate the breeding process, commonlyleading to confinement in local optima. The crossover operator, isnot enough to circumvent this hindrance since it uses onlyacquired information. The mutation operator on the other hand,can move into broader areas within the search space in contrastto the crossover, but it also can not change many bits in individualsbecause the mutation rate is too low. If the mutation rate is set to ahigh value, then genetic algorithms approach the global optimumvery slowly. As a result, it is very difficult to escape this prematureconvergence problem.

This method suggests the use of a guided mutation scheme ap-plied at the phenotype, meaning the real representation of the

0 5 100

2

4

Sensor Locations

Fig. 7. Mesh generation, loading configuration and sensor placement for a hole ofradius 0.3 (units) located at the center of the plate.

100 101 10210−7

10−6

10−5

10−4

10−3

generation

Fitn

ess

L2 error of ε

sawtooth GAWGA

Fig. 8. XFEM–GA algorithm fitness evaluation for alternative GA schemes.

562 E.N. Chatzi et al. / Computers and Structures 89 (2011) 556–570

individual (creep mutations). The standard implementation for thecreep mutation scheme is for a child to differ from its parent byonly a small increment or decrement, usually toward a randomdirection. The novel Weighted Average Mutation (WAM-GA)implemented here, takes advantage of the large amount of infor-mation obtained throughout a course of GA iterations. Each set ofcandidate parameters (individual) is linked to a fitness value, thusproviding a pseudo probability density function (pdf). The

0 10 20 30 40 50.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

gene

r Par

amet

er V

alue

0 10 20 30 40 53.5

4

4.5

5

5.5

6

6.5

gene

x c Par

amet

er V

alue

0 10 20 30 40 50

2

4

6

8

10

12

14

gene

y c Par

amet

er V

alue

Fig. 9. XFEM–GA algorithm parameter es

WAM-GA method utilizes a weighted average approach to mutatethe design parameters of the current population toward areas ofincreased fitness.

There are two controlling parameters involved. The standardcreep mutation probability pc which in common GA schemes isset to a rather low value, usually pc = 0.04. However, higher valuesare proposed in this case, as shown later through the results of arelevant sensitivity analysis. The second characteristic parameterinvolved is the probability of employing the WAM-GA scheme inplace of the standard creep mutation, pwam. This value lies withinthe range of 0.65–0.9. The reason for not setting it equal to 1 isto ensure a potential of motion toward all feasible areas withinthe search space by allowing some probability for the standardcreep mutation to take place. The reasoning of improving themutation scheme by moving toward areas of increased fitnesshas been proposed by Liu and Chen in [40], where a SimulatedAnnealing driven jump mutation scheme, has been employed.The amount of modification for each individual is a random dis-tance between the initial parameter value and the weighted valuefor that parameter. This can be related to the random accelerationoperator employed by the PSO method [41,42]. More specifically,the Weighted Average Mutation GA (WAM-GA) algorithm replacesthe standard creep mutation operator with the pattern outlined inTable 1 for all members of the population:where,

npop is the number of individuals per populationnparam is the number of parameters per candidate solution (indi-

vidual)rand1,2,3 are random numbers uniformly distributed 2 [01]pc is the creep mutation probability

0 60 70 80 90 100ration

True value r=0.5

GA r estimate

WGA r estimate

0 60 70 80 90 100ration

True value xc=5

GA xc estimate

WGA xc estimate

0 60 70 80 90 100ration

True value yc=10

GA yc estimate

WGA yc estimate

timation for alternative GA schemes.

0 2 4 6 8 100

1

2

3

4

5

6

7

8

9

10

X

Y

Fig. 10. Mesh generation, loading configuration, sensor placement and assumed linear crack locations.

0 2 4 6 8 100

2

4

6

8

10

0 2 4 6 8 100

2

4

6

8

10

0 2 4 6 8 100

2

4

6

8

10

0 2 4 6 8 100

2

4

6

8

10

0 2 4 6 8 100

2

4

6

8

10

0 2 4 6 8 100

2

4

6

8

10

0 2 4 6 8 100

2

4

6

8

10

0 2 4 6 8 100

2

4

6

8

10

0 2 4 6 8 100

2

4

6

8

10

actualellips e 81x41 mes hellips e 41x41 mes h

Fig. 11. Estimated elliptical approximations for the assumed linear cracks of size l1 ¼ 1ffiffiffi2p

.

E.N. Chatzi et al. / Computers and Structures 89 (2011) 556–570 563

g1(k) is the incremental value between the upper and lowerbound for each parameter k

crsgn is equal to ±1 with a uniform probabilitypwam is the weighted average mutation probability

child (k,j) is the real value corresponding to parameter k of the jthindividual, and

xm(k) is the weighted average for that parameter value by theend of the previous generation obtained from:

Fig. 12. Estimated elliptical approximations for the assumed linear cracks of size l1 ¼ 0:75ffiffiffi2p

(41 � 41 mesh case).

564 E.N. Chatzi et al. / Computers and Structures 89 (2011) 556–570

P

xmðkÞ ¼ iwichildðk; iÞP

iwi; wi ¼

1kfitnessðiÞk þ � ð7Þ

where, i is an index corresponding to each of the formed individualsup to that point of the evolutionary process, wi is the associatedweight and � is a sufficiently ‘‘small’’ numerical value added forthe case where the fitness of the individual approaches 0. This par-ticular expression has been chosen for the type of problems we aredealing with herein aiming at minimizing a residual (error) functiontoward zero. The formula yielding the weights can be properlymodified for different classes of problems. The authors have testedthe performance of the newly proposed algorithm on a variety oftest functions [43–45], and the performance was generally im-proved against other commonly used GA variants. This generalmutation scheme can be easily fitted to any GA code however wechose to use it in conjunction with the sawtooth GA since accordingto a previous study conducted by the authors [31], that techniqueproved to be the most efficient for the type of identification prob-lems discussed herein.

4. Applications

4.1. Application of the WAM-GA on a circular hole detection problemusing XFEM

We study the performance of the novel GA scheme on the iden-tification of a circular hole which lies within the interior of a rect-angular plate of dimension 10 by 20 (units). For this particularproblem, the number of parameters to be identified is three;

namely, the radius (r) of the hole and the Cartesian coordinates(x,y) of its center. Fixed boundary conditions are applied on thebottom edge of the plate which is undergoing surface tractionalong the vertical direction. In a previous study [31] parametricruns have been conducted, demonstrating the XFEM–GA algorithmperformance on circular holes of different sizes and placementswithin the interior of the plate, for varying sensor placement con-figurations. In this case, the objective is the comparative perfor-mance of the previously used GA scheme (sawtooth-GA) and thenewly proposed one (WAM-GA). The target circular hole dimen-sions are set equal to (xc,yc) = (5,10), while the radius size is chosenequal to 0.5 units. Ten strain gauges are assumed to be distributedon the three free boundaries. The overall geometry of the problemas well as the 41 � 81 node mesh used for the identification proce-dure are presented in Fig. 7.

Two important GA parameters to be set are the chromosomelength and the mean population size. The chromosome size is cho-sen such that the discretizetion of the search space is enough toyield the desired accuracy in the design variable estimate. For allof the applications presented in this chapter, the chromosome sizewas set equal to 45, corresponding to 29 = 512 possibilities per eachof the five design parameters. Hence, the number of possible sam-ples for each parameter is finite and equal to w/512, where w isthe ‘‘width’’ of the relevant search space. The search space for thecircular hole detection problem is defined follows: r 2 [02],x 2 [0.59.5], y 2 [0.519.5]. This level of discretizetion was judgedas sufficient given the employed mesh density. A very fine discret-ization can negatively affect the convergence of the GA, so a balanceshould be maintained between accuracy and computational cost.Another important parameter to be settled is the population size.

E.N. Chatzi et al. / Computers and Structures 89 (2011) 556–570 565

Crude population scaling laws, such as the one suggested in [35] forthe standard GA, relate the optimal population size to the numberof chromosomes used and the number of unknown parameters tobe optimized, ex. Npopsize = O[(nchrome/nparam)2nparam]. Giventhat the size of chromosome and the number of parameters utilizedherein were of the same order as those used for the applications of[31], we maintained a similar population size as the one used in ourprevious study for all of the applications presented in this chapter.Furthermore, elitism and tournament crossover selection wereimplemented for all GA runs of this chapter.

More specifically, a mean population size of seven individualsand a jump mutation probability pm = 0.02 was implemented forboth GA schemes. A creep mutation probability pc = 0.05 was usedfor the sawtooth-GA, while the tuning parameters for the WAM-GA were chosen to be pc = 0.30, pwam = 0.80. The algorithm was setto run for a total of 100 generations. The objective function to beminimized is given by Eq. (6). Figs. 8 and 9 illustrate the performanceof the proposed modified scheme compared to the sawtooth-GAmethod that had proved most effective in the study presented in[31]. It can clearly be seen that the sawtooth-GA stagnates for about10 generations before making any progress while the proposedWAM-GA reduces the error below 10�5 in just a few iterations,which makes this algorithm extremely attractive on such problems.

4.2. Identification of a crack using the elliptical inclusion XFEM–GAscheme

In order to study the validity of the proposed algorithm in thepresence of model error, we apply the following scheme. The

Fig. 13. Estimated elliptical approximations for the assume

numerical results obtained from the XFEM analysis of a 45� ori-ented crack are utilized as the strain measurements in order toidentify the elliptical shaped hole that would best approximate thisstraight crack. Thus, the GA process makes use of an ellipticalenrichment XFEM code for the solution of the forward problem,while the reference measurements are obtained from the strainof an actual embedded crack, which is modeled with a crack for-mulation with XFEM. For this particular problem, the number ofparameters to be identified is five; namely, the Cartesian coordi-nates (x,y) of the center, the major radius (a), the minor radius(b) and the major axis orientation (w). The search space for theelliptical hole fitting problem is defined follows: {x,y} 2 [0.5 9.5],a 2 [0 2], a 2 [0 2], b 2 [0 2], w 2 [0 2]. The flaw in each case lieswithin the interior of a rectangular plate of dimension 10 by 10(units). Fixed boundary conditions are applied on the left verticalboundary of the plate which is in tension along the horizontaldirection. Displacement sensors are this time assumed to be uni-formly distributed along the three free boundaries. Parametric runshave been performed for different crack sizes and locations withinthe interior of the plate, as shown in Fig. 10. Four different cracksizes have been assumed: l1 ¼ 1

ffiffiffi2p

, l2 ¼ 0:75ffiffiffi2p

, l3 ¼ 0:5ffiffiffi2p

andl4 ¼ 0:25

ffiffiffi2p

. The XFEM–WAM-GA scheme was set to run for 200generations for a mean population size of 7 individuals, a creepmutation probability pc = 0.30 and a weighted average mutationprobability pwam = 0.80. The performance of the XFEM–GA identifi-cation scheme is satisfactory even in this case of artificial noisepresence. Fig. 11 provides the finally estimated elliptical shapesthat best approximate a crack of size l1 ¼ 1

ffiffiffi2p

, for two mesh cases;a coarser mesh of 41 � 41 nodes and a finer mesh of 81 � 41 nodes.

d linear cracks of size l1 ¼ 0:5ffiffiffi2p

(41 � 41 mesh case).

566 E.N. Chatzi et al. / Computers and Structures 89 (2011) 556–570

It is obvious that the convergence results are improved for a finermesh since this leads to higher accuracy and thus closer finite ele-ment solutions for the linear and elliptical code.

However, a finer mesh corresponds to a higher computationalcost both for the XFEM forward problem and the GA optimization,

2 323

45

67

80

0.05

0.1

0.15

0.2

y dimension

Rel

ativ

e Er

ror

Fig. 15. Parameter vector relative e

Fig. 14. Estimated elliptical approximations for the assumed

hence it is sometimes preferable to utilize a coarser mesh and ob-tain cruder estimates. Figs. 12–14 illustrate the estimated optimalapproximations for the 41 � 41 mesh case and decreasing assumedcrack length. Fig. 15 reflects the estimation accuracy for the ninedifferent flaw locations. The parameter vector relative error is used

4 5 6 7 8

x dimension

r=1r=0.75r=0.5r=0.25

rror with decreasing flaw size.

linear cracks of size l1 ¼ 0:25ffiffiffi2p

(41 � 41 mesh case).

E.N. Chatzi et al. / Computers and Structures 89 (2011) 556–570 567

in order to represent the deviation of the estimate from the actualparameters. The parameter vector relative error is defined as:

Dr ¼ kr � rk2

krk2ð8Þ

where, r is the estimated parameters vector and r is the true param-eter vector which is known for every parametric case. As can be ob-served in the figure, the estimation error is larger near the boundaryarea where the flaw is located far from the sensing units and decaysfor flaw locations closer to the right vertical boundary wherechanges in geometry of the plate have a greater effect on the read-ings. Also, as expected the decrease in the flaw size generally causeslower estimation accuracy, more so for the areas located near thevertical boundary. For a crack length of l ¼ 0:25

ffiffiffi2p

, which is quiteclose to the actual mesh size, the estimation resolution decays sig-nificantly, for all flaw locations. At this point it should be noted thatthe identification of a flaw with a size smaller than the mesh sizewould indeed be problematic. A possible remedy could be the inte-gration of the XFEM–GA approach with some type of adaptive meshrefinement, such as the methods developed by Duflot and Bordas[46], Fries et al. [47] and Bugeda et al. [48].

4.3. Experimental application of the XFEM–GA scheme on the detectionof a linear crack

Finally, we verify the XFEM–GA algorithm on an actual exper-imental test. Showing that this algorithm works on real prob-lems is of significant importance to the structural healthmonitoring community. For this purpose, the setup displayedin Fig. 16 was utilized. As shown in Fig. 17(a), it involves a12 in. by 24.6 in. rectangular steel plate of 0.254 in. width andASTM A322 quality steel, loaded in tension on an MTS universaltesting machine. The plate is instrumented with 12 strain gaugesmeasuring strain in both the horizontal and the vertical direc-tion. Firstly, the Elasticity Modulus and Poisson ratio of the spec-imen’s material where determined via an ASTM A-370 coupontension test with a full stress strain curve. The test was

Fig. 16. Experimental setup.

performed for 3 coupons and the resulting mean values where:E = 29,681,000 lbf/in.2 and m = 0.2826.

The XFEM model shown in Fig. 17(b) was used in order to sim-ulate the experiment and solve the forward problem using a linearflaw XFEM algorithm. The model was calibrated after performing abaseline strain reading test, so as to achieve a satisfactory accor-dance between the predicted and measured strains. Fig. 18 showsthe plot of the measured against the simulated sensor readings forone of the instances during the loading process. Readings fromSensor #7 were not utilized due to the significant divergence thatis observed in the horizontal strain readings. The fitness function tobe minimized in this case can be written as:

r ¼ k�h � �0hk

k�0hk

þ k�v � �0vk

k�0vk

ð9Þ

where k�k denotes the L2 norm, �0h;v are the real measured strains in

the sensor locations for the horizontal and vertical direction respec-tively and �h,v are the computed strains at the exact same pointsthrough the implementation of the forward problem.

Next, a crack of size 0.6 in., oriented in a 45� angle, was cutwithin the interior of the plate as shown in Fig. 17(a). The tensionexperiment was performed once again and the XFEM–WAM-GAscheme, with 61 � 83 nodes, was implemented in order to deter-mine the location of the flaw based on the output of the strain sen-sors. For this particular problem, the number of parameters to beidentified is four; namely, the Cartesian coordinates (xtip1,ytip1),(xtip2,ytip2) of each crack tip. The search space is defined follows:{xtip1,xtip2} 2 [1 11], {ytip1,ytip2} 2 [1 23]. The XFEM–WAM-GAscheme was set to run for 250 generations for a mean populationsize of �n ¼ 7 individuals, a creep mutation probability pc = 0.30and a weighted average mutation probability pwam = 0.85.

The convergence results are shown in Figs. 19 and 20. Fig. 19presents snapshots corresponding to specific iterations throughoutthe optimization process. It is evident from the figure that the algo-rithm converges to quite a satisfactory result. Future work will ex-plore the sensitivity of the proposed scheme and convergence ratefor flaws of different sizes and shapes (elliptical, circular, etc.). Interms of the computational time, an average capacity of an Intelprocessor running for a total of 100 generations required about3 h for the mesh size used here. The computational cost can be sig-nificantly decreased if the scheme is set to run in parallel on a mul-tiple processor unit.

5. Conclusions

The XFEM–GA detection algorithm recently proposed in the lit-erature [11,23,31] is further improved by the addition of a novelgenetic algorithm that accelerates the convergence of the schemeand a generic XFEM formulation of an elliptical hole. The ellipticalhole formulation in XFEM makes the algorithm more versatile andachieves a generic simulation of flaw shapes thus facilitating theidentification process. This is an effort to capture most basic flawshapes of elliptical, circular, oval and even linear type using a sim-plified approach with an elliptical formulation which requires fivedesign parameters. Hence, we seek to identify a more generalizedflaw geometry by maintaining simulation simplicity and computa-tional efficiency. However for a complex flaw shape (far from anelliptical shape) or for multiple cracks scattered away from eachother, this approach may not provide reliable results, and addi-tional research is warranted. One possible solution involves thedefinition of identification problems with larger design parametersets.

The Weighted Average Mutation GA (WAM-GA) is imple-mented herein which makes use of a weighted average approachfor the mutation of the design parameters towards areas of in-

0 2 4 6 8 10 12110

100

90

80

70

60

50

40Horizontal Strain Readings

Actual StrainSimulated Strain

0 2 4 6 8 10 12150

200

250

300

350

400Vertical Strain Readings

Actual StrainSimulated Strain

Fig. 18. Measured against simulated baseline strain gauge readings for both horizontal and vertical directions.

Fig. 17. Plate dimensions, sensor placement, crack location and XFEM model for the simulation of the experimental setup.

568 E.N. Chatzi et al. / Computers and Structures 89 (2011) 556–570

0 5 100

5

10

15

20

Generation 1

0 5 100

5

10

15

20

Generation 25

0 5 100

5

10

15

20

Generation 50

0 5 100

5

10

15

20

Generation 75

0 5 100

5

10

15

20

Generation 100

0 5 100

5

10

15

20

Generation 150

0 5 100

5

10

15

20

Generation 180

0 5 100

5

10

15

20

Generation 200

0 5 100

5

10

15

20

Generation 225

0 5 100

5

10

15

20

Generation 250

Fig. 19. Snapshots of the XFEM–GA crack detection scheme.

0 50 100 150 200 2503.5

4

4.5

5

5.5

6

6.5

7

7.5

generation

x tip1 P

aram

eter

Val

ue

True value xtip1=6.3976

GA estimate

0 50 100 150 200 2505

5.5

6

6.5

7

7.5

8

generation

x tip2 P

aram

eter

Val

ue

True value xtip2=6.9882

GA estimate

0 50 100 150 200 25014

15

16

17

18

19

generation

y tip1 P

aram

eter

Val

ue

True value ytip1=14.6457GA estimate

0 50 100 150 200 25014

15

16

17

18

19

20

generation

y tip2 P

aram

eter

Val

ue

True value ytip2=14.6457GA estimate

Fig. 20. Actual against estimated flaw parameters.

E.N. Chatzi et al. / Computers and Structures 89 (2011) 556–570 569

creased fitness. The effectiveness of XFEM–GA based identifica-tion is established through an experimental application involvinga crack in the interior of a steel rectangular plate instrumentedwith strain sensors. The accuracy ultimately achieved is veryencouraging for potential implementation of this algorithm inthe field.

References

[1] Cawley P, Adams R. Location of defects in structures from measurements ofnatural frequencies. J Strain Anal Eng Des 1979;14(2):49–57. cited By (since1996): 377.

[2] Chang P, Liu S. Recent research in nondestructive evaluation of civilinfrastructures. J Mater Civil Eng 2003;15(3):298–304.

570 E.N. Chatzi et al. / Computers and Structures 89 (2011) 556–570

[3] Giurgiutiu V, Reynolds A, Rogers C. Experimental investigation of e/mimpedance health monitoring for spot-welded structural joints. J Intell MaterSyst Struct 1999;10(10):802–12.

[4] Ostachowicz WM. Damage detection of structures using spectral finiteelement method. Comput Struct 2008;86(3–5):454–62. Available from:<http://dx.doi.org/10.1016/j.compstruc.2007.02.004>.

[5] Giurgiutiu V, Zagrai A. Damage detection in thin plates and aerospacestructures with the electro-mechanical impedance method. Struct HealthMonitor 2005;4(2):99–118.

[6] Park S, Yun C, Roh Y, Lee J. Health monitoring of steel structures using impedanceof thickness modes at pzt patches. Smart Struct Syst 2005;1(4):339–53.

[7] Borcea L. Electrical impedance tomography. Inverse Problems 2002;18:r99–r136.[8] Chaudhry Z, Lalande F, Ganino A, Rogers C. Monitoring the integrity of

composite patch structural repair via piezoelectric actuators/sensors. In:Proceedings of the AIAA/ASME/AHS/ASC 36th SDM conference, vol. 4; 1995.p. 2243–48.

[9] Park G, Sohn H, Farrar C, Inman D. Overview of piezoelectric impedance-basedhealth monitoring and path forward. Shock Vib Dig 2003;35(6):451–63.

[10] Gadala M, McCullough A. On the finite element analysis of inverse problems infracture mechanics. Eng Comput 1999;16(4):481–502.

[11] Rabinovich D, Givoli D, Vigdergauz S. XFEM-based crack detection schemeusing a genetic algorithm. Int J Numer Methods Eng 2007;71(9):1051–80.

[12] Moës N, Dolbow J, Belytschko T. A finite element method for crack growthwithout remeshing. Int J Numer Methods Eng 1999;46:131–50.

[13] Belytschko T, Black T. Elastic crack growth in finite elements with minimalremeshing. Int J Numer Methods Eng 1999;45:601–20.

[14] Sukumar N, Chopp DL, Moës N, Belytschko T. Modeling holes and inclusions bylevel sets in the extended finite element method. Comput Methods Appl MechEng 2000;190(46–47):6183–200.

[15] Daux C, Moës N, Dolbow J, Sukumar N, Belytschko T. Arbitrary branched andintersecting cracks with the extended finite element method. Int J NumerMethods Eng 2000;48(12):1741–60.

[16] Sukumar N, Moës N, Moran B, Belytschko T. Extended finite element method forthree-dimensional crack modelling. Int J Numer Methods Eng 2000;48:1549–70.

[17] Dolbow J, Moës N, Belytschko T. Modeling fracture in Mindlin–Reissner plateswith the extended finite element method. Int J Solids Struct 2000;37:7161–83.

[18] Duarte CA, Reno LG, Simone A. A high-order generalized FEM for through-the-thickness branched cracks. Int J Numer Methods Eng 2007;72(3):325–51.

[19] Belytschko T, Gracie R, Ventura G. A review of extended/generalized finiteelement methods for material modeling. Model Simul Mater Sci Eng;17(4).doi:10.1088/0965-0393/17/4/043001. <http://dx.doi.org/10.1088/0965-0393/17/4/043001>.

[20] Fries TP, Belytschko T. The extended/generalized finite element method: anoverview of the method and its applications. Int J Numer Methods Eng2010;84(3):253–304.

[21] Holland JH. Adaptation in natural and artificial systems. Cambridge, MA,USA: MIT Press; 1992.

[22] Munakata T, Hashier D. A genetic algorithm applied to the maximum flowproblem. In: Proceedings of the fifth international conference on geneticalgorithms; 1993. p. 488–93.

[23] Rabinovich D, Givoli D, Vigdergauz S. Crack identification by arrival time usingXFEM and a genetic algorithm. Int J Numer Methods Eng 2009;77(3):337–59.

[24] Chatzi EN, Koumousis VK. Optimal inelastic design of multistorey reinforcedconcrete buildings towards uniform distribution of minimal damage. In:International conference on structural engineering dynamics, Ericeira,Portugal; June 2009.

[25] Triantafillou SP, Koumousis VK. Form finding and structural optimization oftension structures using multi-objective genetic algorithms. In: Topping BHV,

Papadrakakis M, editors. Proceedings of the ninth international conference oncomputational structures technology. Stirlingshire, UK: Civil-Comp Press;2008. doi:10.4203/ccp.88.271. Paper 271.

[26] Booker L, Goldberg D, Holland J. Classifier systems and genetic algorithms.Tech. rep., Technical report, No. 8, University of Michigan; 1987.

[27] Waisman H, Belytschko T. Parametric enrichment adaptivity by the extendedfinite element method. Int J Numer Methods Eng 2008;73:1671–92.

[28] Chahine E, Laborde P, Renard YA. Spider-XFEM, an extended finite elementvariant for partially unknown crack-tip displacement. Eur J Comput Mech2008;15(5–7):625–36.

[29] Chahine E, Laborde P, Renard Y. A reduced basis enrichment for the extendedfinite element method. Math Model Nat Phenom 2009;4(1):88–105.

[30] Menk A, Bordas S. Numerically determined enrichment functions for theextended finite element method and applications to bi-material anisotropicfracture and polycrystals. Int J Numer Methods Eng 2010;83(7):805–28.

[31] Waisman H, Chatzi E, Smyth AW. Detection and quantification of flaws instructures by the extended finite element method and genetic algorithms. Int JNumer Methods Eng 2010;82:303–28.

[32] Grinchenko VT, Ulitko AF. An exact solution of the Kirsch problem. Int ApplMech 1970;6:455–61. doi:10.1007/BF00901835.

[33] Whitley D. A genetic algorithm tutorial. Stat Comput 1994;4:6585.[34] Miller BL, Miller BL, Goldberg DE. Genetic algorithms, tournament selection,

and the effects of noise. Complex Syst 1995;9:193–212.[35] Goldberg DE. Sizing populations for serial and parallel genetic algorithms, In:

Proceedings of the 3rd International conference on genetic algorithms, MorganKaufman; 1989. p. 70–9.

[36] Mitchell M. An introduction to genetic algorithms. Cambridge, MA: MIT Press;1998.

[37] Back T, Fogel DB, Michalewicz Z. Handbook of evolutionarycomputation. Bristol, UK: IOP Publishing Ltd; 1997.

[38] Krishnakumar K. Micro-genetic algorithms for stationary and non-stationaryfunction optimization. In: SPIE proceedings: intelligent control and adaptivesystems.

[39] Koumousis VK, Katsaras CP. A saw-tooth genetic algorithm combining theeffects of variable population size and reinitialization to enhance performance.IEEE Trans Evol Comput 2006;10(1):19–28.

[40] Liu J-L, Chen C-M. Improved intelligent genetic algorithm applied to long-endurance airfoil optimization design. Eng Optim 2009;41:137–54.

[41] Kennedy J, Eberhart RC. Particle swarm optimization. In: Proceedings of IEEEinternational conference on neural networks, Piscataway, NJ; 1995. p. 1942–8.

[42] Clerc M. Discrete particle swarm optimization. New optimization techniquesin engineering. Springer-Verlag; 2004.

[43] Eiben AE, Bäck T, Empirical investigation of multi-parent recombinationoperators in evolution strategies. J Evol Comput, 1997;5(3):347–65.

[44] Ortiz-boyer D, Hervs-martfnez C, Garcfa-pedrajas N. Cixl2 – a crossoveroperator for evolutionary algorithms based on population features. J ArtifIntell Res 2005;24:2005.

[45] Goldberg DE, Richardson J. Genetic algorithms with sharing for multimodalfunction optimization. In: Proceedings of the second international conferenceon genetic algorithms on genetic algorithms and their application.

[46] Duflot M, Bordas S. A posteriori error estimation for extended finite elementsby an extended global recovery. Int J Numer Methods Eng 2008(76):1123–38.

[47] Fries T-P, Byfut A, Alizada A, Cheng KW, Schröder A. Hanging nodes and XFEM.Int J Numer Methods Eng. in press. doi:10.1002/nme.3024.

[48] Bugeda G, Ródenas JJ, Oñate E. An integration of a low cost adaptive remeshingstrategy in the solution of structural shape optimization problems usingevolutionary methods. Comput Struct 2008;86(13–14):1563–78.