Crack identification by ‘arrival time’ using XFEM and a genetic algorithm

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERINGInt. J. Numer. Meth. Engng 2009; 77:337–359Published online 10 July 2008 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/nme.2416

Crack identification by ‘arrival time’ using XFEMand a genetic algorithm

Daniel Rabinovich1, Dan Givoli1,∗,† and Shmuel Vigdergauz2

1Department of Aerospace Engineering, Technion, Israel Institute of Technology, Haifa 32000, Israel2Israel Electric Corporation Ltd., Research and Development Division, P.O. Box 10, Haifa 31000, Israel

SUMMARY

A computational framework is developed in which cracks in two-dimensional structures are identified, inconjunction with non-destructive testing of specimens. As opposed to a previous study by the authors,which was based on time-harmonic excitation with a single frequency, here the transient response of thestructure to a short-duration signal is measured along part of the external boundary. Crack detection isperformed using the solution of an inverse time-dependent problem. It is shown that the arrival time of theinput signal to the points of measurement is a good criterion for crack identification in the time domain.The inverse problem of identification is solved using a genetic algorithm, while each forward problem issolved by the time-dependent extended finite element method (XFEM). The XFEM scheme is efficientin that it allows the use of a single regular mesh for a large number of forward time response problemswith different crack geometries. Numerical examples involving a crack in a flat membrane are presented.Identification based on ‘arrival time’ is shown to perform better than that based on time-harmonic response.Copyright q 2008 John Wiley & Sons, Ltd.

Received 25 November 2007; Revised 28 May 2008; Accepted 28 May 2008

KEY WORDS: extended finite element method (XFEM); arrival time; genetic algorithm; crack detection;inverse problem; identification; wave scattering; non-destructive testing; health monitoring

1. INTRODUCTION

Identification of flaws in structures is a critical element in the management of maintenance andquality assurance processes in engineering. Non-destructive testing (NDT) techniques based ona wide range of physical principles have been developed and are used in common practice forstructural health monitoring [1, 2]. However, basic NDT techniques are usually limited in their

∗Correspondence to: Dan Givoli, Department of Aerospace Engineering, Technion, Israel Institute of Technology,Haifa 32000, Israel.

†E-mail: [email protected]

Contract/grant sponsor: TechnionContract/grant sponsor: Lawrence and Marie Feldman Chair in Engineering

Copyright q 2008 John Wiley & Sons, Ltd.

338 D. RABINOVICH, D. GIVOLI AND S. VIGDERGAUZ

ability to provide accurate information on the location, dimensions and shapes of flaws that may bepresent in the material. In some applications such information is not needed, namely the structuralhealth monitor is interested only in the question of existence of significant flaws and not in theirdetailed parametrization. In some other applications, such as health monitoring of aircraft structureparts, one may be interested in more specific information on the flaw. One way to extract suchadditional information from the results of the NDT process is to append it with a computationalmodel that provides detailed analysis of the physical process involved and enables the accurateidentification of the flaw parameters [3, 4].

The mathematical problem of identifying a flaw from measurement data is an inverse problemand is often ill-posed [3, 5]. Except in very special cases, the inverse problem associated with flawidentification must be solved numerically. In doing so, some limiting assumption must usually beapplied. For example, when considering crack identification it is usually assumed that a singlecrack is present; otherwise the search space may become too large for practical computation, and,more importantly, this may render the inverse problem severely ill-posed (e.g. experience loss ofuniqueness).

A direct approach to the numerical solution of inverse problems may be described as follows.Scattering data are obtained by simulating many candidate flaws. Each such simulation involves thesolution of a forward problem. The signals such obtained are compared with the actual measuredsignal. Each simulated signal is assigned a score, which indicates its ‘closeness’ to the measuredsignal. An optimization process is performed to obtain the flaw configuration, which producesscattering data most resembling the measured data. The goal of the optimization is to find theglobal minimum. Gradient-based optimization is not effective in such situations (see, e.g. [6]), andone must resort to heavier ‘soft computing’ tools, which are more suitable to this end.

We remark that while in an actual NDT system the measurement data are obtained via anexperimental system, e.g. by using piezoelectric sensors [7] or a scanning laser vibrometer [8–10],in the present paper we generate the measurement data artificially via simulation. It would certainlybe interesting to replace this simulation by actual measurement; however, this is beyond the scopeof the paper, which is completely computational.

Mathematically, one may distinguish among three types of approaches related to NDT viawave excitation. The first approach, which is the most classical one, is the use of the structure’sspectra, i.e. eigenfrequencies and mode shapes, as the primary information for flaw detection.In this case the associated forward problem is an eigenvalue problem. The second approach isbased on the response to time-harmonic excitation, namely to sinusoidal vibration with a singlefrequency (which is different from any of the eigenfrequencies). The forward problem in thiscase is an elliptic boundary-value problem. The third approach, adopted in this paper, is based ontransient response, and the quantity commonly measured is the ‘arrival time,’ namely the time ittakes for the signal to reach the measurement site after leaving the excitation site. In this case,the forward problem is a hyperbolic initial-boundary-value problem. All three types of inverseproblems have been considered in the literature in the context of flaw detection. See [3, 4, 11, 12].See also [7], which describes an experimental system for flaw detection based on time-dependentresponse.

The global optimization approach used in the present paper is that of a genetic algorithm (GA).GAs have been applied in the past to the identification of flaws in structures, mainly with modalinformation employed and simple structures such as beams, trusses and frames (see References[13–16]). Time-harmonic response data have been used with GA optimization for flaws in sandwichplates by Liu and Chen [17] and for a certain class of cracks in plates by Wu et al. [18].

Copyright q 2008 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2009; 77:337–359DOI: 10.1002/nme

CRACK IDENTIFICATION BY ‘ARRIVAL TIME’ 339

The flaw type we concentrate on here is a crack. Solution of the inverse crack identificationproblem by GA requires that the forward problem of wave scattering be solved many times fordifferent crack configurations. Modeling this problem by the standard finite element (FE) methodinvolves remeshing of the computational domain for each new crack configuration and is thereforehighly inefficient. The present paper, following a previous work by the authors [19], uses theextended finite element method (XFEM), which allows modeling arbitrary discontinuities indepen-dent of the FE mesh and thus allows us to avoid remeshing altogether. In fact, it is this capability ofthe XFEM that makes the identification problem tractable. XFEM was developed by Belytschko’sgroup in [20–24], based on the principles of the partition of unity method (PUM) [25, 26]. In thepast few years a great deal of work has been done to improve and analyze the original XFEM forvarious configurations; we mention [27–31] as important examples.

Recently [19] we have proposed a crack detection scheme based on XFEM and GA. The NDTmethod underlying this scheme was of the time-harmonic type, namely it uses the measuredamplitudes of a scattered single-frequency signal to identify the crack. It has been shown in [19]that the detection method is quite successful in predicting the parameters of the crack based onboundary measurements. However, in recent years, ultrasonic NDT practice relies more and moreon time-dependent pulse methods for detection. It is the goal of the present paper to extend ourcrack detection scheme of [19] to the time domain.

To this end, we shall use XFEM for time-dependent wave scattering. XFEM, which was orig-inally developed for static problems, was extended to the time domain in the context of phasetransformation by Dolbow and Merle [32], Ji et al. [33] and Chessa et al. [34], and for crackgrowth simulations by Belytschko et al. [35] and Rethore et al. [36, 37]. We will use a time-dependent formulation of XFEM in combination with the unconditionally stable implicit Newmarktime-stepping method [38].

We consider a two-dimensional model problem of crack detection in a flat membrane basedon measured boundary information. Among other things we show that the arrival time is a goodcriterion for crack identification in the time domain. Moreover, by comparing our results here withthose obtained in [19] it turns out that identification based on the arrival time performs betterthan that based on time-harmonic response. This is consistent with known industrial practice [39].(This is not necessarily true in the case of multiple cracks, which is not dealt with here.) Asmentioned above, we shall use a GA scheme to perform the optimization and will solve eachforward problem using XFEM and implicit time-stepping. The resulting detection scheme providescomplete characterization of the suspected crack. We will demonstrate its performance using somenumerical examples.

We mention that the basic computational approach taken by Ishak et al. [8–10] for crackidentification is quite similar to the one taken here, although the details in the two cases aresignificantly different. Ishak et al. consider longitudinal two-dimensional cracks in long three-dimensional beams, whereas here one-dimensional cracks in two-dimensional membranes areconsidered. Ishak et al. employ a beam model, which is semi-analytical, and an optimizationmethod based on multilayer perceptron networks (MLPs); this is in contrast to the XFEM and GAused here. In addition, the cost function used by Ishak et al. is the L2 norm of the signal difference(which we call C1 in the sequel; see Section 2), whereas here the travel time is found to be moreeffective for the cases under study.

The outline of the remainder of this paper is as follows. In Section 2 the inverse identificationproblem under study is stated. Section 3 describes the GA optimization scheme in general terms. InSection 4 the solution of the forward problem via XFEM is discussed along with the time-stepping



procedure. In Section 5 numerical examples are presented, and the capabilities of the schemeare tested, with and without added noise. The notions of detectability and forward sensitivityare discussed as well in the present context. The paper ends with some concluding remarks inSection 6.

2. THE IDENTIFICATION PROBLEM

The lateral dynamics of a two-dimensional flat linear membrane are considered. The membraneoccupies the domain � (an open set) bounded by �. The lateral displacement u(x, t) of themembrane is governed by the two-dimensional wave equation

c2∇2u− �2u�t2

=0 in � (1)

Here c is a given constant wave speed, which is a property of the membrane material. Theboundary � comprises four disjoint parts, �=�S∪�D∪�N ∪�c, on which different boundaryconditions are imposed. The boundary parts �S , �D and �N together constitute the externalboundary, which we denote by �, and correspond, respectively, to the ‘source boundary,’ the ‘fixedboundary’ (with homogeneous Dirichlet condition) and the ‘free boundary’ (with homogeneousNeumann condition). The part of the boundary denoted by �c is the boundary of a crack, whichmay be contained in � entirely or may intersect the external boundary �. On �c we assume a free(homogeneous Neumann) boundary condition. Thus, the boundary conditions are

�u�n

(x, t) = S(x, t) on �S (2a)

u(x, t) = 0 on �D (2b)

�u�n

(x, t) = 0 on �N (2c)

�u�n

(x, t) = 0 on �c (2d)

The differential equation (1) and boundary conditions (2) hold for t>0. The solution u is allowedto be discontinuous across the crack surface �c at all times. The given source function S drivesthe response of the membrane and in the present case will be a transient function over a short timeperiod. Appended to (1) and (2) are the initial conditions

u(x,0) = 0 in �

�u�t

(x,0) = 0 in �(3)

The initial values are zero since the membrane is assumed to be at rest before the NDT procedurestarts. In the absence of body forces, we conclude from the wave equation that the initial accelerationalso vanishes.



For given �,�S,�D,�N ,c, S and the crack �c, the forward problem consists in finding thevalues u� of u on some given manifold � (typically part of the external boundary) at times 0�t�T ,where T is the measurement time, subject to (1), (2) and (3).

The associated inverse problem (identification problem) is: Given �,�S,�D,�N ,c, S as wellas the manifold � and u�, find the crack boundary �c such that u satisfies (1), (2) and (3) as wellas u=u� on � for 0�t�T .

Owing to the basic ill-posedness of the inverse problem, the common practice of relaxing thestatement of the above inverse problem is adopted; instead of attempting to find �c which satisfiesall the equations above (which may be non-existent due to noise), a crack �c is sought such thatthe function u satisfies (1), (2) and (3), and some cost functional C[u] is minimized.

The performance of the optimization scheme in identifying the crack varies for different choicesof the cost functional. The performance of three different cost functionals is checked and comparedhere. The first cost functional is defined as

C1=‖u−u�‖�,T (4)

where ‖·‖�,T is the space-time L2 norm over � and over measurement time. The second costfunctional will be defined momentarily.

The third cost functional is based on the arrival time � of a signal to measurement points.This criterion is inspired by applications in geophysics (e.g. oil exploration) where arrival time isused routinely for identifying the material properties of the earth layers, although modern NDTultrasonic techniques are also based on this concept [39]. To describe the arrival-time cost, wedistinguish among four signals:

• u(x�, t) is the sought displacement, calculated on the boundary �. Each given crack candidateis associated with a certain signal u computed for this crack.

• u�(x�, t) is the given measured displacement on � in the presence of the true crack.• u0(x�, t) is the calculated displacement for a membrane without any crack. The same type ofnumerical model is used for finding u and u0. The calculation of u0 is performed only once,as a preprocess.

• u�0(x�, t) is the measured displacement for a membrane without any crack. This measurementis done once and for all.

Now, the arrival time � is defined with respect to the perturbation signal u−u0 rather thanwith respect to u itself. We have found that this definition provides a much better sensitivityfor identification. The arrival time at a certain point x� ∈� is defined as the time at which thisperturbation signal first exceeds a predefined small threshold �:

�(x�)=max

{t ′

∣∣∣∣ max0�t�t ′

{u(x�, t)−u0(x�, t)}<�

}(5)

Similarly, we define the measurement arrival time as

��(x�)=max

{t ′

∣∣∣∣ max0�t�t ′

{u�(x�, t)−u�0(x�, t)}<�

}(6)

The associated cost functional is then defined as

C3=‖�−��‖� (7)



where ‖·‖� is the L2 norm over �. The condition ‘<�’ cannot be replaced by ‘=0’ due to thepresence of numerical and measurement noise; see more details in Section 5.3. The interpretationof the arrival-time cost is as follows: this cost is small if the time it takes for the calculatedperturbation signal to reach the measurement region is close to the time it takes for the measuredperturbation signal to reach this region.

Coming back to the cost C2, it is based, similar to C1, on the L2 norm of the difference betweenthe wave signals. However, rather than taking the measured and computed signals themselves, weconsider the difference between their perturbations with respect to the signals obtained without acrack, as in C3. Thus,

C2=‖(u−u0)−(u�−u�0)‖�,T (8)

In the implementation of the GA it is convenient to consider the score of a crack candidaterather than its cost. The score is a value between 0 and 1; the value 1 is attained when the signalgenerated by the candidate crack is indistinguishable from the measurement signal, while thevalue 0 is approached when the two signals are arbitrarily different from each other. For each ofthe three costs (n=1,2,3) the score is defined as

Sn = 1

1+Cn(9)

Some assumption on the regularity of the curve describing the crack is required. On the levelof the forward problem, we shall assume that the membrane contains at most one non-branchingcrack and that the crack curve may (exactly or approximately) be represented by a connectedsequence of straight segments. In the context of XFEM we shall assume that in each element cutor slit by the crack, the crack is straight.

3. THE GENETIC OPTIMIZATION SCHEME

As described above, the solution of the inverse problem involves minimization of a cost functionalcalculated for all candidate cracks. Any of the cost functionals defined in Section 2 typicallyhas many local minima, and this fact precludes the use of a gradient-based descent optimizationmethod. GA was chosen as the global method of optimization in [19] and in the present paper.

The GA operates by constructing sets of candidate cracks and solving the forward problemsassociated with each of them. These sets of candidates are arranged in generations and are repre-sented by binary strings. In the present paper, binary encoding with ‘gray code’ is used [40]. Thefirst generation comprises randomly chosen cracks. Certain ‘genetic operations’ are applied to therepresentative strings of the current generation of cracks to form a new generation. The generationsevolve, and if the optimization process is successful, the cracks of each generation are better inthe broad sense than those of the previous one. The genetic operations include selection of thosecandidate cracks that will pass to the next generation, based on the cost function used, whichis followed by the operations of crossover and mutation—the former interchanges bits betweentwo sequences of crack representations, while the latter changes a group of bits in a single crackrepresentation. In the GA scheme adopted here, the best string from the current generation isalways kept unchanged and transferred to the next generation; this is a practice called elitism.

The stopping criterion is a key issue in GAs, as there is no practical way of assessing theactual error during the run in real applications. Progress is measured with respect to the results



obtained in previous generations; when from a certain generation the scheme does not yield furtherimprovement, the process is stopped. Despite the stochastic nature of GAs, which practically impliesthat different GA runs may result in different solutions, it is hoped that the process converges toa solution, which is ‘close’ to the exact global minimum. Sometimes the optimized solution issignificantly different from the true minimum, which means either that the optimization processfailed due to some inherent difficulty or that in that specific run the (random) circumstances weresuch that the process was stopped prematurely. To distinguish the latter case from truly pathologicalcases, multiple GA runs were performed in this study for each given problem.

The GA computational parameters used here are the same as those used in the frequency-domainscheme described in [19] and proved to lead to good performance in the time-dependent case aswell. This includes a representation of 7 bits per crack coordinate, a population of 80 strings pergeneration, a 2-out-of-3 tournament selection, standard 1-point crossover with a probability of 1,standard mutation with a bitwise probability of 0.0035 and creep mutation with a probability of0.07. For additional details on the way cracks are generated, stored and represented see [19].

4. SOLUTION OF THE FORWARD PROBLEM

The standard technique for solving time-dependent problems is by discretization in space first,followed by discretization in time [38]. However, in the present case there is an advantage indiscretizing the problem in time first, since in combination with implicit time integration this resultsin a sequence of elliptic time-independent problems, which bear similarity to the elliptic forwardproblems associated with frequency-response crack detection [19]. The difference between the twoelliptic problems is that the frequency-response one is associated with the Helmholtz equation,whereas the semi-discrete time-dependent one is associated with the modified Helmholtz equation.See [41] for a similar formulation in another context. This allows us to use the same code for thetwo types of forward problems and compare their results with ease. However, it must be stressedthat this approach is completely equivalent to the standard practice and is used here merely forthe sake of convenience.

We apply the predictor–corrector form of the implicit Newmark time discretization scheme [38]to the strong form of the problem (1)–(3). We use the standard Newmark parameters �=0.25 and�=0.5, for which the method is unconditionally stable and second-order accurate. The result is asequence of elliptic boundary-value problems that have to be solved in each time step.

To solve this problem at time step n+1, we employ an XFEM procedure similar to the onedescribed in [19]. We consider single continuous cracks, which are straight lines in each element.The crack intersects the regular mesh lines, as shown in Figure 1. The effect of the crack isintroduced into the FE formulation via two enrichment functions: �J , the enrichment function atglobal node J , which allows for the discontinuity across the crack, and �M,P , the Pth enrichmentfunction at the M th crack tip, which incorporates the asymptotic behavior near the crack tip. Thus,the global-level XFEM approximation for un+1(x) is

uhn+1(x)=∑J∈�

NJ (x)(dJ,n+1+d∗J,n+1�J (x))+

Ntip∑M=1

Nsing∑P=1

d∗∗M,P,n+1�M,P(x) (10)

Here � is the set of all nodes not on �D , NJ is the standard FE shape function correspondingto node J , Ntip is the number of crack tips in the membrane, Nsing is the number of singularity



Discontinuity enriched nodes

1e1

e2

4

3

2

Figure 1. XFEM crack and mesh configuration.

functions used to represent the solution field near the crack tip, and dJ , d∗J and d∗∗

MP(with the

additional subscript n+1) are the unknown values of the standard, discontinuity enrichment andsingularity enrichment degrees of freedom, respectively. The nodes for which the d∗

J enrichmentdegrees of freedom are not zero are those belonging to the set {J ∈�|supp(NJ )∩�c =∅} (seeFigure 1). For more details see the XFEM-related papers referenced in the Introduction.

The discontinuity enrichment functions � and the singularity enrichment functions � areconstructed as in [19]. Thus, we use ‘localized’ discontinuity enrichment as proposed in [28], andwe set Nsing=1; since we are not interested in the precise behavior of the forward solution nearthe crack tip, but rather in the far field behavior of the solution, only the crack-tip element itself issingularly enriched. The discontinuity enrichment function � depends only on the geometry, andhence it is identical in the time-dependent and time-harmonic cases. The singularity enrichmentis potentially different, because it has to be chosen so that it represents correctly the asymptoticfield near the crack tip when the modified Helmholtz equation (and not the Helmholtz equation asin [19]) governs. Despite this, it can be shown that it is possible to use the same singularity enrich-ment (with a square-root singularity) in the present case as in the frequency-response case [42].

Substituting the approximation (10) into the weak form of the problem yields the linear algebraicFE system

Kdn+1=Fn+1 (11)

where the stiffness matrix K and the force vector Fn+1 are obtained, as usual, by the assembly ofelement-level arrays ke and fen+1, respectively. See [42] for the expressions of the entries of ke andfen+1. The numerical integration used to evaluate these expressions is carried out for quadrilateralelements in the manner described in [19].

5. NUMERICAL EXAMPLES

5.1. Basic model and arrival time

The numerical examples presented here involve cation of a single straight crack. Forsimplicity, this fact was used a priori in the identification process by restricting the cracks



Figure 2. The cracked square membrane problem.

generated in the genetic optimization process to be globally straight. Such a simplification reducesthe search space of cracks in the domain and thus the computational load incurred by the scheme,but it does not affect the generality of the method and the conclusions drawn from the numericalresults below.

We consider crack detection in a square membrane. The membrane domain � is of size L×L=20×20. We set c=1, and we apply the boundary conditions shown in Figure 2. Thus, three of themembrane sides are free, while an incident pulse �u/�n= S(t) is applied on the fourth (bottom)side. The source function S is defined by

S(t)={0.125sin(t/16), 0�t�16

0, t>16(12)

At the single corner point A (see Figure 2) we fix the displacement by imposing u=0 at thispoint. This is not essential from the well-posedness perspective in the present hyperbolic case sincethe time-dependent problem is well posed even when a Neumann condition is imposed along theentire boundary; however, this reduces the rate in which the error increases in time, and it breaksthe symmetry around the mid vertical axis and thus yields a more complicated wave pattern. Thecoordinate s runs along �, starting from the point A (see Figure 2).

The ‘measured signals’ u� and u�0 are generated on all of �, namely �=�. These data aresimulated by solving the forward problem with the true crack (for u�) and without any crack(for u�0). These solutions are performed with a mesh, which is significantly finer than the meshused in the optimization process itself, so that an ‘inverse crime’ is avoided; see the discussionin [19]. Uniform structured meshes of 10×10 and 20×20 elements for the coarse and fine mesh,respectively, were used; see Figure 3. The crack-less solution u0 is also found by using the coarsemesh. Generating u0, u� and u�0 is done one time as a pre-process for a given problem.

For the membrane under study, the time required for an unobstructed input signal to reach theopposite side of the rectangular domain and return is 40 time units. The time span for the measuredsignal should therefore be at least that long. To keep the error in the forward calculation small, itis desirable to keep the wavelength of waves generated by the input signal at 10 element lengths atleast, which translates into a wave with a period of T>10h/c=20, where h=2 is the coarse meshparameter. For this reason the input signal was chosen to be the half-period sine wave specified



(A) (B)

Figure 3. Meshes used in the identification tests. Mesh A (10×10) is used for the opti-mization itself (and also to find the crack-less solution u0) whereas mesh B (20×20) is

used to produce the measurement input.

above, lasting 16 time units. The time step used in the computation was chosen as �t=1, so that16 time steps fit into the half sine wave of the source excitation in the present test case. Eachforward time-dependent solution was run for 250 time steps.

Figure 4 shows the evolution of the wave field obtained from the solution of a forward problemat various times. The left column shows the response field u in � when a horizontal crack oflength 5 is present in the middle of the domain. The center column shows the response u0 in �without the presence of the crack. The difference between the two signals, u−u0, which illustratesthe net effect of the crack on the solution, is shown in the right column. Note that this differenceis the perturbation used in the definition of the cost functionals C2 and C3 as discussed inSection 2.

At t=10 the wave front does not yet reach the crack, and hence the perturbation u−u0 vanishes.For later times, the perturbation is non-zero and itself propagates in a wave-like manner. It firstreaches the boundary �=� at about t=25. As Figure 4 shows, the boundary points most influencedby this perturbation wave are those in the middle of the upper and lower sides of the membrane,corresponding to s=30 and 70. As time proceeds, the point with the largest amplitude shiftssideways. This has direct relevance on the cost C2, which involves the amplitude of u−u0.

We recall that C3 is based on the arrival time and depends on the given small threshold �.Figure 5 shows the arrival times obtained for different points on � and for two values of �. Theboundary points to which the perturbation wave reaches last are the points in the middle of theright and left sides of the membrane, corresponding to s=10 and 50. This is also apparent fromthe perturbation field in Figure 4.

Figure 6 illustrates the progress of a typical genetic optimization run, in which identification ofthe crack shown in Figure 4 was carried out by minimizing the arrival-time cost C3. The plot onthe left shows the cost obtained for the best candidate crack in each generation. The plot on theright shows the best candidate cracks themselves in the membrane domain at certain generations.The number by each crack represents the generation at which it was found. The true crack isplotted with a thick line. We see from the left plot that ‘convergence’ is achieved after about 70generations, but the right plot shows that the correct orientation of the true crack is captured muchearlier. We note that the cost reduces in ‘leaps’; for several generations there is no noticeable change



Figure 4. Evolution of a signal radiated from the source at the bottomedge and reflected from the crack.

and then the results are improved in a sharp leap. This behavior is typical to GA performance andwas also observed in [19].

5.2. Detectability and choice of cost function

We study the property of detectability for this problem, namely the capability of the schemeto identify various cracks as a function of the crack parameters and the cost function used inthe optimization. To this end, a ‘distance parameter’ is defined to measure the ‘closeness’ of a



Figure 4. Continued.

candidate crack to the true crack. The definition chosen is

D =min{√

(x1− x1)2+(y1− y1)2+(x2− x2)2+(y2− y2)2,√(x2− x1)2+(y2− y1)2+(x1− x2)2+(y1− y2)2} (13)

where (x1, y1) and (x2, y2) are the coordinates of the two tips of the candidate crack, and (x1, y1)and (x2, y2) are the coordinates of the tips of the true crack. The minimum in (13) is taken overtwo possibilities due to the arbitrariness in numbering the two tips of each crack. According to



0 10 20 30 40 50 60 70 800

10

20

30

40

50

60

70

80

90

100

s

arriv

al ti

me

threshold=0.01threshold=0.1

Figure 5. Arrival time of a signal with different thresholds �, for thehorizontal crack problem of Figure 4.

0 10 20 30 40 50 60 70 80 90 1000

5

10

15

20

Generation0 5 10 15 20

–10

–8

–6

–4

–2

0

2

4

6

8

10

0

52450

Figure 6. The crack identification problem: progress of a typical genetic optimization run.

definition (13), D is zero when the candidate crack coincides with the true crack and becomeslarge when the cracks are further apart.

The detectability performance of the three cost functionals C1 (see (4)), C2 (see (8)) and C3(see (7)) is now compared by applying the GA identification scheme to the case of a straightcrack with length 5, parallel to the source boundary and located approximately in the middle ofthe domain. We use 40 measurement points on �. The results of the identification runs are shownin Figure 7. Here and in subsequent figures, we show representative results obtained in five runsfor each case; they differ due to the stochastic character of the GA scheme.

As Figure 7 shows, the values of D obtained by the cost C1 are in all cases higher than 14,which are not much better than the values of D corresponding to randomly generated cracks in the



0

2

4

6

8

10

12

14

16

18

20

Dis

tanc

e pa

ram

eter

D

Figure 7. Detectability of horizontal crack with length 5 and center at(10.5,0.5) for different cost functionals.

10–2 10–1 1000

2

4

6

8

10

12

14

16

18

20

Dis

tanc

e pa

ram

eter

D

Arrival time threshold, displacement units

Figure 8. Identification of horizontal crack with length 5 and center at (10.5,0.5) using arrival time ascost, with different threshold values �.

membrane. This suggests that a simple comparison of the measured signal at all the measurementtimes in the L2 norm, as was done in the time-harmonic case, does not produce satisfactoryidentification results in the transient case. Better results are obtained by the cost C2, which isbased on the perturbation with respect to a no-crack configuration. In this case the values of Dare between 5 and 12.



With the arrival-time cost functional, C3, we use the value 0.01 for the threshold �, which isapproximately 1

200 of the maximal value of the signal obtained in the cracked membrane withinthe entire measurement time window. It is clear from Figure 7 that the detectability of this cost isthe best of the three, with identified cracks, which are very close to the target crack. The best resultgave D�1. The arrival-time cost functional C3 was therefore used in all the following numericaltests.

The choice of the threshold � defining the arrival time (see (5)) strongly affects the identificationprocess. It is therefore desirable to find the values of �, which are best for identification. Figure 8shows the results of a detectability test with a horizontal crack, where the value of � was varied inthe range [0.01,1]. For large values of this threshold, detectability deteriorates, since � becomes thetime for which there are significant differences between the responses of the cracked domain andthe domain without crack at the measurement points, and it no longer represents the ‘arrival time’in the desired sense. However, as Figure 8 shows, the detectability of the crack is not impairedfor threshold values below �=0.2. The precise value of � for which deterioration starts to occurdepends on the parameters of the configuration. It is generally advised to use values of �, whichare as small as possible, yet still larger than the measurement noise level; see Section 5.3.

5.3. The effect of noise

Up till now the detection procedure was applied to noiseless input signals. In order to test theeffect of measurement noise on the performance of the scheme, a sinusoidal noise function ofthe form

f(s, t)=

2

[sin

(�t+2

s

smax

)+sin

(ks+2

t

tmax

)](14)

is now added to the original measurement signal on �, i.e. to u�. In (14), represents the noiseintensity. No noise is added to the crack-less membrane measurement u�0, since this measurementis assumed to be done once in advance under ideal conditions. (Besides, any noise present in thecrack-less membrane measurement can be attributed in effect to noise in the cracked membranemeasurement, since only the difference between the two signals appear in C2 and C3.) High-frequency noise is taken in both space and time, with �=100, k=100 in (14). Owing to these highvalues and the fact that the arguments of the sines in (14) strongly depend on time t and on thelocation on the measurement boundary s, the noise function f may be regarded as pseudo-random.The smax is the reference length defining the added phase in time, here taken as the length ofthe measurement boundary, which is 80 in the following examples, and tmax is the reference timedefining the added phase in space, taken as the duration of the measurement window, namely 250.

An arrival-time threshold of �=0.07 is used in the detectability test in the presence of noise withvarying intensity. The results are shown in Figure 9. These results as well as others (not shownhere) demonstrate that whenever the noise amplitude is smaller than the threshold � detectabilityis good, whereas if >� detectability sharply deteriorates. This is, of course, expected and caneasily be explained. In the latter case, noise causes the signals with and without a crack to differsignificantly at the very beginning of wave evolution, long before the input signal even reachesthe crack. This results in incorrect evaluation of the arrival time from the measurement signal,which in turn produces wrong cost values for candidate cracks and thus causes the sharp decreasein detectability. Some high-frequency filtering may therefore be required in the presence of noise



0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.20

2

4

6

8

10

12

14

16

18

20

Dis

tanc

e pa

ram

eter

D

Noise level, displacement units

Figure 9. Identification of a horizontal crack with length 5 and center at (10.5,0.5) in the presence ofsinusoidal noise with varying intensity.

whose intensity exceeds �. Alternatively, if one can estimate the noise level a priori, one is ableto choose �> accordingly.

5.4. The effect of number of measurement points

The tests described above were performed using a large number of measurement points along �,i.e. 40. Now we examine the detectability of the scheme for a varying number of measurementpoints. The number of measurement points on � is varied between 4 (one point in the middle ofeach edge of the membrane) and 40 (ten points on each edge). The results of the identificationprocess are shown in Figure 10.

It is clear that the crack can be identified quite well with much less measurement points thanthe number used in the previous examples. Small distance D values are obtained for as few aseight measurement points along �. However, when the number of measurement points is decreasedfurther to 4, identification is impaired. The reason for this becomes clear when we consider thecost C3 as a function of the number of the measurement points used, as shown in Figure 11.The scores associated with the best found cracks generally increase with decreasing number ofmeasurement points, and for 4 measurement points the score jumps to the value 1, namely to aperfect score. This indicates that for too few measurement points the score (or cost) functional isuseless; the GA finds cracks that are very different from the true crack yet which produce arrivaltimes similar to the true arrival times at those few measurement points. This does not occur whenthe number of measurement points is sufficiently large.

5.5. The effect of crack orientation

Up till now, only true cracks that are parallel to the wave-source boundary �S , and hence alsoparallel to the incident wave front, have been tested. Now we test the detectability for true cracks,which are set at different orientations with respect to the wave front coming from �S .



0 10 20 30 40 500

5

10

15

Dis

tanc

e pa

ram

eter

D

Number of measurement points

Figure 10. Identification of a horizontal crack of length 5 and center at (10.5,0.5) with different numbersof measurement points along �.

0 5 10 15 20 25 30 35 400.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Ave

rage

sco

re

Number of measurement points

Figure 11. Scores of identified cracks with different numbers of measurement points along �.

Figure 12 shows the value of the distance D for true cracks positioned at the center of thedomain at different angles to �S . The plot shows that there is a decrease in detectability whenthe wave incidence becomes more oblique. At 90◦ the crack has little influence on the solutionfield and is therefore very hard to detect. (Note that the influence is not altogether null, due tothe fact that the bottom right point A is held fixed, as in Figure 2, and thus breaks completesymmetry.) This it to be expected, as cracks perpendicular to the input wave front have no effect



0 10 20 30 40 50 60 70 80 900

2

4

6

8

10

12

14

16

18

20

Dis

tanc

e pa

ram

eter

D

angle[degrees]

Figure 12. Identification of crack of length 5 and center at (10.5,0.5) orientedat different angles with respect to �S .

on the measured displacement and are therefore undetectable. Good detectability is obtained forcracks of up to 60◦. This is to be contrasted with the fact that in the time-harmonic case, asreported in [19], only cracks with angles of up to 30◦ yield good detectability. Thus, our resultsseem to indicate that identification based on arrival time performs better than identification basedon frequency response, at least as far as oblique crack detection is concerned.

5.6. Sensitivity of the forward problem

In [19] it was found for the time-harmonic case that for a certain true crack configuration there isa strong correlation between crack detectability and the sensitivity of the solution of the forwardproblem, which can be termed forward sensitivity. A crack configuration is said to be forward-sensitive if a slight change in the parameters of the crack causes significant change in the forwardsolution. Lack of forward sensitivity may therefore be used as a priori indication of the difficultyin identifying a certain type of crack. If one relies on this correlation, one may obtain a completecharacterization of detectability for various classes of cracks without solving any inverse problemat all.

Here we apply an analogous procedure for the time-dependent case. The forward problem issolved repeatedly for a sequence of nominal cracks positioned at the center of the membrane atdifferent angles with respect to �S . Each of these nominal cracks is slightly perturbed by changingthe positions of its crack tips. The resulting change in the L2 norm of the arrival times is recordedand is shown in Figure 13.

Contrary to the time-harmonic case, no positive correlation can be found between forwardsensitivity in arrival time and crack detectability. In fact, there is a negative correlation. Forexample, the slope of the curve obtained for a 90◦ crack in the vicinity of 0 variation in the crackparameters is much larger than the slope corresponding to a 0◦ crack, which means that the 90◦crack configuration is much more forward-sensitive, despite the fact that such a crack is obviously



–0.5 –0.4 –0.3 –0.2 –0.1 0 0.1 0.2 0.3 0.4 0.50

0.1

0.2

0.3

0.4

0.5

variation in all crack parameters, length units

rela

tive

L2

chan

ge in

arr

ival

tim

e ||t

ar–t

ar(0

)|| 2/

||tar

(0)|

| 2

0 °15°30°45°60°75°90°

Figure 13. Solution sensitivity to crack parameters of the forward problem for a centered crack of length 5.

Figure 14. Domain and FE mesh for the arch problem.

not well detectable whereas the 0◦ crack is highly detectable. Thus, forward sensitivity seems notto be a useful criterion for detectability in the time-dependent case. This significant differencebetween the behaviors of the transient scheme used here and the time-harmonic scheme employedin [19] may be attributed to the major difference in nature between the two forward problems (onebeing elliptic, and the other hyperbolic), and the major difference between the cost functions usedin the two cases (one based on the L2 norm, and the other on the arrival time).

5.7. An additional example

As an additional example we consider crack identification in a membrane in the shape of an arch.The domain, true crack and FE mesh are shown in Figure 14. The inner and outer radii of the arch



0 5 10 150

20

40

60

80

100

Generation

Cos

t, C

Figure 15. Cost evolution for the arch problem.

0 5 10 150

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Generation

Dis

tanc

e

Figure 16. Distance measure evolution for the arch problem.

are 10 and 12, and the crack’s length is 0.8. A uniform impulse is applied on the bottom straightedge of the arch. The response is measured along the inner curved boundary (of radius 10).

Figures 15 and 16 show the evolution of the cost value and distance measure, respectively, forthe best cracks detected in each generation. The good correlation between the cost and the distancemeasure shows that the latter is a good predictor for the former. Identification using the arrival-timecriterion is very successful in this case as well. While the (x, y) coordinates of the end points ofthe true crack are (8.0,8.0) and (8.6,8.6), the coordinates of the detected crack are (7.93,8.00)and (8.59,8.66). Thus, the detection error (in terms of the crack parameters) is less than 1%.

6. CONCLUDING REMARKS

The present study aims at making a step forward in equipping NDT techniques with a model-basedinformation in order to enable accurate identification and full characterization of cracks and otherfaults in structures. We have used arrival-time information for identifying cracks in flat membranes.



The two main computational procedures used here were GA for finding a global minimum for thearrival-time functional and XFEM for solving many forward problems without need for remeshing.

Good identification results were obtained for a wide range of possible cracks, with the exceptionof cracks that are nearly perpendicular to the incident wave front. Repeated measurements withmodified source waves may alleviate this drawback similar to the remedies demonstrated in [19].We also showed here that the number of measurements required for identification of the cracksneed not be large.

One of our conclusions from the present study is that crack detectability is generally betterwith a transient procedure based on arrival time than with a frequency-response procedure asin [19]. This fact is consistent with the experience from industrial practice [39]. However, arelative disadvantage of the time-dependent identification scheme proposed here is the much largercomputational load associated with it due to the time-stepping required in the solution of eachforward problem. Developing methods for making time-stepping more efficient would be veryimportant in this context. One obvious idea is to use explicit time-integration schemes, which areonly conditionally stable but may be much more efficient in the present application.

Having established the usefulness of the method for the simple membrane model problem, itis now necessary to extend it to three-dimensional elasticity problems with realistic geometriesand meshes. In addition, in order to adapt the method to practical identification, realistic outputmeasurements should be tested and the effect of noise should be investigated under these conditions.

Admittedly, the most serious obstacle in turning the proposed methodology into a useful workingtool in the service of NDT is the large computational effort that is still required for realisticproblems. Despite the significant savings that XFEM brings with it compared with standard FEanalysis in the present context, the large amount of forward problems to be solved with differentcandidate cracks poses a considerable challenge. This is the reason why all the cracks detected inthis paper have been relatively long; identifying a much shorter crack would require a much finermesh, which in turn would increase significantly the computational effort. The purpose here wasto introduce the methodology, and we have shown this methodology to be successful in principle.Advanced methods to reduce the computational effort without hampering accuracy are now calledfor. One idea is to use sophisticated model reduction algorithms such as those proposed in [43].

ACKNOWLEDGEMENTS

This research was funded in part by the Fund for the Promotion of Research at the Technion and by thefund provided through the Lawrence and Marie Feldman Chair in Engineering. The authors would liketo thank Dr Hagit Saguy for her assistance.

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Crack identification by ‘arrival time’ using XFEM and a genetic algorithm