ORIGINAL PAPER

Marek Engelhardt Æ Georgios E. Stavroulakis

Heinz Antes

Crack and flaw identification in elastodynamics using Kalmanfilter techniques

Received: 24 February 2005 / Accepted: 25 April 2005 / Published online: 15 September 2005� Springer-Verlag 2005

Abstract Filter-driven optimization based on the ex-tended Kalman filter concept is used here for thenumerical solution of crack and flaw identificationproblems in elastodynamics. The mechanical modelingof the studied two-dimensional problem, which includesthe effect of unilateral contact along the sides of thecrack, is done with the help of the boundary elementmethod. The effect of various dynamical test loads andthe applicability of this method for crack and defectidentification in disks are investigated.

Keywords Inverse Problems Æ Kalman Filters Æ CrackIdentification

1 Introduction

The numerical solution of inverse problems follows,usually, an iterative technique. In most cases this is basedon the solution of an appropriately defined error mini-mization problem, which is actually a data-fitting tech-nique in the least-squares sense for a suitablyparametrized model, cf. [23]. It is well-known that inverseproblems are ill-posed, which is demonstrated by the factthat the previously mentioned optimization problemmayadopt multiple (may be local) solutions, or it isill-conditioned. Therefore, the application of classicalnumerical optimization schemes is not the best choice.The use of filter-assisted optimization avoids severalproblems connected with the local nature of classicalnumerical optimization schemes and leads to more ef-fective algorithms. The feasibility of this approach wastested previously, for static crack identification problems.Here, the extension to dynamical problems with morecomplicated two-dimensional inverse problems is de-monstrated.

The mechanical modeling of the elastodynamicalproblem is done with the help of the boundary elementmethod. Special hypersingular boundary elements areused for the crack boundaries. Furthermore the possi-bility of partial or total closure of cracks, the effectwhich is known as unilateral contact, is included in someof our investigations by a combined boundary elementand linear complementarity problem strategy (LCP-BEM, see [2], [22]).

Filter algorithms have been used in several recentpapers for the solution of defect, material or boundarydata identification problems in mechanics. Defect shapeidentification with filter algorithms and boundary ele-ment techniques has been studied in [26]. Circular holeshave been identified in elastodynamics by Kalman andprojection filters in [27]. These references seems to beamong the first papers in this direction. The effect ofunilateral contact mechanisms in filter-driven crack

Comput Mech (2006) 37: 249–265DOI 10.1007/s00466-005-0709-y

The work has been supported by the German Research Foundation(DFG). Partial support has been provided by a Greek-GermanResearch Cooperation Grant (IKYDA2001). This support isgreatfully acknowledged.

M. EngelhardtInstitute of Applied Mechanics,Department of Civil Engineering,Technical University of Braunschweig,Braunschweig, GermanyE-mail: marek.engelhardt@tu-bs.deTel.: +49-531-3917102Fax: +49-531-3915843

G. E. Stavroulakis (&)Institute of Mechanics, Department of Mathematics,University of Ioannina, Ioannina, GreeceE-mail: gestavr@cc.uoi.grTel.: +30-26510-98268Fax: +30-26510-98201

H. AntesInstitute of Applied Mechanics,Department of Civil Engineering,Technical University of Braunschweig,Braunschweig, GermanyE-mail: h.antes@tu-bs.deTel.: +49-531-3917100Fax: +49-531-3915843

identification using static test loadings has been studiedin [24] and is extended to elastodynamical loadings inthis paper. Other techniques, including neural networksand genetic optimization, have been tested on the sameproblem, as it has been reported in other previouspublications of our group, see [23], [8], [25]. Recent ap-plications of crack identification with filter techniquesare focused on adaptive filter methods and modal ana-lysis data, where several filter models are used in parallelfor the identification of various defects in different po-sitions of a structure and, in a second step, the mostsuitable of them is chosen from the actual measureddata, see [21] and [1]. In a more general setting, both(parts of) the applied loads, the defects and structuralreliability can be assessed by means of filter techniques,as it has been reported in [16]. Difficult material para-meter identification problems concerning cohesivecracks or elastoplastic models have also been studiedwith filter techniques, see for instance [3] and [18],respectively. These works are mentioned here with thepurpose to give the reader a general orientation, and arenot discussed further since they are not directly com-parable with the method adopted in this paper.

2 Mechanical modeling and boundary element method

A linearly elastic, two-dimensional medium isconsidered, subjected to dynamical loading. From thewell-known relations of elasticity one formulates thetime-dependent boundary integral equations, which aresolved by appropriate space (boundary) and time dis-cretization, i.e., by using a boundary element method.The existence of unilateral effects (here, closure andopening of cracks) is taken into account by appropriatemethods of contact mechanics. Finally, instead of aclassical system of linear equations within each timestep, one solves a linear complementarity problem(LCP).

2.1 Boundary integral equations

Let us consider the elastodynamic problem in a domainX with boundary C subject to plane load. The dis-placement at a point n and time t can be represented interms of the displacements and tractions on theboundary by

cijðnÞujðn;tÞþ�Z

C

Z tþ

0

�p�ijðx;t�s;nÞujðx;sÞdsdC

¼Z

C

Z tþ

0

ðu�ijðx;t�s;nÞpjðx;sÞdsdC

ð1Þ

where u�ijðx; t � s; nÞ is the displacement field in aninfinite medium due to a unit impulse in the direction xilocated at point n and acting at time s; �p�ijðx; t � s; nÞ isthe corresponding traction field obtained from the dis-placements by Hooke’s law; cij ¼ 1=2dij for smooth

boundary points, dij for interior points, and 0 for pointsoutside the domain X; �

Rstands for the Cauchy Principal

Value (CPV) integral andR tþ

0 ¼ lime!0

R tþe0 . Zero initial

conditions and zero body forces bj have been assumed.As it is known, the traction tensor �p�ij contains

Dirac’s delta functions which preclude a directnumerical integration of Eqs. (1) (see [17], [7] fortechnical details). In order to eliminate the Dirac deltaterms, the spatial derivative of the Heaviside functionis related to its time derivative, and an integrationby parts is performed. This leads to the integralequation

cijuj þZ

C

Z tþ

0

ðp�ijuj þ pv�ij _ujÞdsdC

¼Z

C

Z tþ

0

u�ijpjdsdC

ð2Þ

where p�ij is the stress field of u�ij and p��ij is its first timederivative.From the integral representation of the displacements atan interior point (2), the corresponding integral re-presentation for the stress tensor at this point can beobtained by Hooke’s law:

rimðn; tÞ þZ

C

Z tþ

0

�d�imj � pjdsdC

¼Z

C

Z tþ

0

ð�s�ij � uj þ �s�vij � _ujÞdsdC:

ð3Þ

Here,

�d�imj ¼ q dim c21 � 2c22� �

p�kj; k þ c22 p�ij; m þ p�mj; i

� �h ið4Þ

and similar equations hold for the rest of the kernels.A mixed Boundary Element formulation have been

proposed to solve fracture mechanics problems in static[19], [20] and dynamic problems [9]. A more refinedapproach, taken from [12], which is essential for pro-blems with nonzero traction on the crack faces hasbeen used here. After appropriate manipulation, a hy-persingular boundary integral equation involving thcrack opening displacement Dun can be written for theone face of the crack and can be combined with thestandard displacement boundary integral representationfor the rest of the boundary Cc in order to provide acomplete set of equations. These equations are used forthe computation of the tractions and/or displacementor the crack opening displacement in the boundary andthe crack faces, respectively.

2.2 Boundary element discretization

The boundary element discretization with quadraticelements and a constant time-step discretization are usedfor the numerical solution of the previously outlinedboundary integral equations. Technical details as well as

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discussion of the required conditions for an effectiveimplementation can be found in [12].

After boundary element discretization the previousequation takes the form:

Cu nð Þ þXE

e¼1

XMn¼1

ZCe

t x; nð ÞUkdCx

� �uðeÞn

¼XE

e¼1

XMn¼1

ZCe

u x; nð ÞUndCx

� �tðeÞn

ð5Þ

Finally, for every time step n, the following set ofequations is written in compact matrix notation:

Hnnun ¼ Gnnpn þXn�1m¼1ðHnmum �GnmpmÞ: ð6Þ

Here the vectors um and pm contain the displace-ments and tractions of the nodes on the externalboundary Cc and the crack opening displacements alongthe crack and the tractions for time step m.

This system of equations is solved step by step as inthe standard displacement formulation in time domain.

At this point the boundary conditions of the elasto-mechanical problem are taken into account, known andunknown quantities are separated and a system of linearequations is formulated and solved for all unknownboundary displacements resp. tractions y:

Hu ¼ Gt) Ay ¼ f ð7ÞThe notations used are as follows: n is coordinateintegration point, M is the number of collocation points,x is the coordinate collocation point, E is the number ofelements, y denotes the unknown boundary quantitiesand f denotes the given boundary quantities. Finally UT

denotes the Ansatz (basis) functions and H;G are thefinal influence matrices of the BEM.

2.3 Contact mechanism of cracks

The classical unilateral contact behavior describes, ateach point of a unilateral boundary, a highly nonlinearrelation between a boundary traction normal tothe considered boundary pn and the correspondingdisplacement un: both quantities obey to certaininequalities, expressing the inadmissibility of tensionstresses and interpenetration, respectively, and arecomplementary quantities, which means that at mostone of them may attain a nonzero value. In our caseunilateral contact arises between the adjacent sides ofeach crack, therefore the crack opening (COD) Dun,which is the difference of the displacements at the twosides in the common normal to the crack direction, andthe corresponding normal crack traction are involved. Africtionless contact case is considered, which can bedescribed by the following linear complementarityproblem:

Dun � 0; pn � 0;DuTn tn ¼ 0 ð8Þ

This set of unilateral contact relations is coupled withthe system of equations coming from the boundaryelement method (5), which for the time step (n) is con-sidered in the following obvious partitioned form:

H11 H12

H21 H22

� �u

Dun

� �

¼G11 G12

G21 G22

� �p

pn

� �þ rðn�1Þ

:

ð9Þ

The vector rðn�1Þ indicates the influence of all previoustime steps. The Linear Complementarity Problem (LCP)described by (8) and (9) can be solved, for instance, bythe Lemke algorithm, which is an extension of theSimplex algorithm of linear programming such as totake into account the complementarity relation of (8)3[15]. Friction has not been taken into account in thiswork, although an extension for the treatment of fric-tional contact problems is possible, see [22].

3 Linearized Kalman filter in discrete time

In general a Kalman filter is used for the estimation ofthe unknown parameters of a time-invariant dynamicalsystem. The estimate is done recursively, i.e. the actualestimate results as an update of the values available inthe previous iteration step. The state is assumed to comefrom a dynamical, time-discrete, linear, stochasticmodel. The theory takes into account that only partialmeasurements of the system’s state, corrupted with somenoise, are available. Details of the derivation of theKalman-filter can be found, among others, in [4], [5], [6],[13] or [28]. Representative, relevant applications incrack identification have been mentioned in theintroduction of this paper.

Let us introduce a short notation, in which the de-pendence of every variable on the time step tk is denotedby an index k. Within every time step tk the symbolswhich are used are shown in Table 1. The size of the

Table 1 Symbols used in Kalman filter technique

Symbol Description Dimension

xk vector of state variables ½n� 1�xk vector of estimated values of the

state variables½n� 1�

Uk transition or system matrix ofthe discrete system

½n� n�

zk vector of measurements ½m� 1�Hk measurement matrix ½m� n�wk vector of disturbances (or errors in

the system)½n� 1�

Qk covariance matrix of the disturbances wðkÞ ½n� n�vk vector of measurement error ½m� 1�Rk covariance matrix of measurement error vðkÞ ½m�m�ek vector of estimate error ½n� 1�Pk covariance matrix of estimate error ek ½n� n�Kk Kalman gain matrix ½n�m�

251

vectors or matrices is also shown in the same Table,where n is the number of states and m the number ofmeasurements. Superscript � denotes in the followingpresentation the extrapolated values, which are calcu-lated from the ones of the previous time step tk�1 andwhich have not yet been updated with the help of mea-surements of time step tk.

3.1 Alternative form of the Kalman filter

The equations of the regular Kalman filter can bemanipulated algebraically, and lead to a more efficient,for numerical algorithms, form. An outline of thisreformulation is presented in [5]. Subsequently, theadvantages of this form for the inverse problem arestudied in this paper. The equations of the alternativeKalman filter and the flow chart are given in Fig. 1.

3.2 Application of the alternative Kalman filter

1. The alternative form of the Kalman filter is mean-ingful, if no information about the values of thestate variables xk exists. From the stochastic pointof view, this means that the scattering r2

e�0i

of theestimated error e�0 and, therefore, the inputs in theprincipal diagonal of the corresponding covariancematrix P�0 are infinite. Since these quantities areused in the regular Kalman filter, this leads tonumerical difficulties, a problem which does notarise in the alternative form of the filter algorithmwhere the inverse of the error covariance matrixP�0� ��1

is applied.

P�0 ¼ 1 ! P�0� ��1¼ 0 ð10Þ

2. In the regular Kalman filter, the matrix HkP�k

�HT

k þ RkÞ must be inverted for the calculation of theKalman-gain matrix. The size of this matrix is m� m,where m is the number of measurements zj.

In the alternative Kalman filter the matrix which mustbe inverted, Pk, has the size n� n, where n is the numberof state variables to be estimated xi.In the here studied case the following relation holdsn� m, therefore the cost required for the inversion ofthe involved matrix is considerably reduced in thealternative Kalman filter.

Finally one should mention that the calculation ofsensitivity quantities for a small number of unknownscan be based on finite differences. The algorithm re-mained stable in all numerical examples reported here.Larger applications may require more referred techni-ques.

4 Flaw and crack identification in time domain

4.1 Formulation of the problem

As a starting point for the identification we use thedeformation of the studied, dynamically excited disk.Here, the mechanical response of the system is modeledwith the previously outlined boundary element method,while in a future extension of this investigation experi-mental measurements could be considered. Thereforethis work is based on pseudo experimental data. Havingmeasured the deformations of the defected disk, onefollows with the formulation of the inverse problem. Thestudied elastic body is modeled numerically. The defectis first embedded in the structure with the help of certainparameters. For example, for a rectilinear crack in atwo-dimensional domain one uses the coordinates of it’s

Fig. 1 Alternative Kalman filteralgorithm

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center, it’s length and it’s orientation as suitableparameters. Analogously an ellipsoidal hole can beparametrized by means of it’s center, the orientation andthe two half-axes. For every estimate of the parametersdefining the defect and for the same loading andboundary conditions as the one used in the experiment,one solves the mechanical problem. In our case theboundary element method is used, with classical orunilateral behavior along cracks. Therefore the existenceof partial or totally closed cracks is modeled, as it is thecase with real-life. The resulting values of the displace-ments are compared, through the Kalman filter, with theexperimentally obtained measurements. Within the filterthe parameters of the defect are updated. This is done bymeans of minimizing the difference between mechanicalresponses of the parametrized system with the onescoming from measurements (or pseudo-experiments).Therefore the error F is used as indicator of the esti-mation quality, and is defined as the sum of squareddifferences between measured and calculated displace-ments at each degree of freedom (i.e., point and direc-tion) of the disk:

F ðuðxÞÞ ¼Xm�Dt

j¼1

X2n

i¼1ðu0ijðx0Þ � uijðxÞÞ2: ð11Þ

In this paper we present results for a modeltwo-dimensional quadrilateral disk, with dimensionsequal to 100� 100 and one defect which has the form ofeither an elliptical hole or of a unilaterally workingcrack. For the constitutive modeling of the elastic ma-terial, the elastic constant E, the Poisson’s ratio m, andthe density of the disk . are needed. They are assumed tohave the values:

E ¼ 1 � 106; m ¼ 0; 3 and . ¼ 2450:

All quantities are considered to be given in compatibleunits. For the used boundary element method, time andspace discretization are not independent. The quality ofthe results depend on the quality of factor:

b ¼ c1Dtre

ð12Þ

where Dt denotes the equidistant time step, c1 is thevelocity of compression waves and re is the size of theboundary elements. The maximum value b ¼ 1:0, whichmeans that the compression wave passes through onecomplete element within one time step, should not beviolated. Moreover, values lower than b < 0:7 lead tonumerical instabilities and must be avoided. Accordingto our numerical experience, a value b ¼ 0:9 gives thebest results. For the examined problem, each side ofthe quadrilateral disk is discretized with 20 nodes. Forthe approximation of both displacements and stressesquadratic elements have been used. Therefore, each sideof the disk is discretized by 10 elements, each one havinga length equal to 10, as it is shown in Fig. 2. From Eq.(12) and the required optimal value of this factor, onecalculates a wave velocity c1 ¼ 23:44 and a time stepequal to Dt ¼ 0:384.

The internal boundaries are discretized for ellipticdefects with five and for cracks with two quadraticelements. Finer discretizations of the external bound-aries and of the defects have been tested withoutpractical enhancement of the identification results. Atime interval of 23 time steps has been used. This is theminimal time required by a compression wave startingfrom the upper side of the disk to run within a diskwithout defects and, after reflection from the oppositeside, to come again to the initial point, and to create aresponse of the system.

4.2 Modification of the Kalman filter

Taking into account the previous sections on Kalmanfilters, the proposed estimation algorithm can bedescribed, briefly, as follows:

The variables to be estimated xk correspond to theparameters of the unknown defect. Since the identifica-tion of an existing defect without any changes isstudied, the value of these parameters does not changethroughout the entire duration of the non-destructiveevaluation test. In otherwords, one has a stationary, time-invariant process. Therefore, the transition matrix Ukwhich describes the transition of the system from time steptk to time step tkþ1, is equal to the unity matrix I. Ac-cordingly, we set the error in the system wk equal to zero.Finally the equation of the considered process reads:

xkþ1 ¼ xk: ð13ÞThe measurement vector zk corresponds to the dis-placements at accessible parts of the external boundaryof the disk which are measured during the experiment.They depend nonlinearly on the defect parametersthrough the mechanical model, i.e., the boundaryelement model

zk ¼ hðxkÞ þ vk ð14ÞThe application of the Kalman filter technique for thesolution of the defect identification problem requireswithin every iteration step the minimization of a linearFig. 2 Discretization of boundary and defects

253

approximation of relation (14). The theory of extendedKalman filter is used where the linearized measurementmatrix Hk is defined within each iteration step by meansof a sensitivity analysis. The vector zk is covered with theexperimentally obtained (or calculated) displacements ofeach discretization point of the external boundary foreach considered time step. For the previously describeddiscretization a vector with 3496 components results. Inthe simplest case the defect of the disk is described by fourparameters, therefore vector xk as well has four para-meters. Furthermore at the beginning of the algorithmwesuppose that no initial knowledge concerning form andposition of the defect is available. Therefore, according tothe discussion of Sect. 3.1, the application of the alter-native formulation of Kalman filter seems appropriate.

5 Numerical results and discussion

5.1 Deformation of the disk

The existence of defects and their parameters (position,shape, etc.) changes the mechanical behavior of thestructure. The difference between the response of anintact disk and the same disk with a defect (see Fig. 3)can be calculated by the boundary element method. Theloading has the form of the Heaviside function, which isdefined later in Sect. 5.4. Figure 7(a) shows the verticaldisplacement of the point positioned at the middle of theupper side for the disk without and with the crack. Atotal of 100 time steps has been considered. A differencecan be observed after an initial time of approximatelyt ¼ 4. Indeed for the material data we used and thecorresponding wave velocity c1, the reflection from thecrack is expected at time

t ¼ sc1¼ 100

23:44¼ 4:26

where s denotes the path of the wave. In fact, the com-pression wave is reflected from the crack and leads to therequired difference in the response of the system, whichwill be used for identification purposes.

5.2 Crack identification

The Kalman filter uses the difference of displacements ateachmeasurement point and each considered direction. Inorder to facilitate the comparison with more classicaloptimization methods, we calculated the difference in thesense of the square error method. This sum of squareddifferences, which is actually not directly required by thefilter-driven algorithm and is only used at the stoppingcriterion, is called in the sequel error value. Localization ofa defect in a disk requires an improvement of the errorvalue. If the point of the defect is found, the error valueachieves a global minimum. The negative values of theerror are given in Fig. 7(b) as a function of the coordinatesof the crack’s center, for the whole area of the disk. Forthis investigation the length and the orientation of a rec-tilinear crack are kept constant. The actual defect lies inthe middle of the disk, as it is shown in Fig. 3. The Hea-viside loading is used in this case as well. It is obvious thatthe existence as well as the localization of a defect is pos-sible, with the use of the boundary element simulation.

For the crack identification the Kalman filter requiressuitable starting estimates. Starting from the crackparameters x

�0 ¼ ½70 20 20 1� the identification of a

crack with parameters x ¼ ½50 50 10 0�, as shown in Fig.3, has been solved. The performance of the algorithm isdemonstrated in Fig. 4. The bold lines denote the wantedsolution of the optimization. From Fig. 4 one recognizesthat the position, length, and orientation of the crack areapproximated satisfactorily after 23 iterations. TheKalman filter is a suitable method for the defect iden-tification. The error at the end of the algorithm is equalto 1� 10�6. From our numerical experiments we havefound that this limit is suitable and corresponds to asatisfactory defect identification. Therefore this valuehas been used as stopping criterion.

5.3 Influence of the form of the defect

Two forms of material defects are considered in thispaper: cracks and elliptical holes. Furthermore, the ef-fect of opening and closing (unilateral contact) is con-sidered for the cracks. The influence of the unilateralcontact at the measured displacements of the externalboundaries is enhanced in static loadings. In fact, crackclosure remains, if activated, under static loading, whilein dynamics this phenomenon is instantaneous and doesnot prevent wave reflection from the crack, cf. [23].From the technological point of view, a crack-typedefect is more dangerous than an elliptic hole, due to thehigher stress intensities. Concerning the solution of theinverse problem one should mention that elliptic defectsare more extended material defects than cracks. There-fore, the resulting boundary displacements of the loadeddeformable body are, in general, higher. Consequently,if all other parameters (material, dimensions, loading,etc) are comparable, the solution of the inverse problemis expected to be easier.Fig. 3 Loading of a disk with and without defect

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Fig. 4 Iterations and problemsetting for starting point [70 2020 1]

Fig. 5 Iterations and problemsetting for starting value [30 208 2]

255

The solution steps for a crack identification withstarting point x

�0 ¼ ½30 20 8 2� and real crack char-

acterized by x ¼ ½50 50 10 0� are shown in Fig. 5. Thesolution required 42 iterations. For an elliptic defectidentification problem with comparable parameters withx ¼ ½50 50 10 10� and x

�0 ¼ ½30 20 8 12� the iterations of

the algorithm are documented in Fig. 6. For the iden-tification of the comparable elliptical defect only 8iterations are needed.

This example demonstrates the general trend of thenumerical experiments: elliptical holes can easier beidentified than cracks with comparable dimensions. Theremaining numerical experiments are restricted to crack-type defects, which lead to more difficult identificationproblems.

5.4 Influence of the loading

The previous investigations have been restricted to aHeaviside loading. For identification purposes otherloadings are more practical and usually more effective.Three different loadings are applied on the same iden-tification problem concerning the disk of Fig. 3 in thissection: a Heaviside-, a Ricker-impulse-, and a sinusoi-dal loading.

Heaviside: The Heaviside function over the 23 con-sidered time steps, with a constant intensity equal to one:

HðtÞ ¼ 1; for t > 00; for t < 0.

�ð15Þ

Ricker-Impulse: The Ricker-impulse is an impulsivecontinuous loading which is usually applied for themodeling of ultrasonic waves. The continuity avoidsnumerical instabilities in numerical simulations. Thisloading is defined by

RðtÞ ¼ a2e�a2 t22 ða2t2 � 1Þ: ð16Þ

The coefficient a is chosen to be equal to one. In order toavoid compressive loading, which could activate crackclosure and make the inverse problem considerablymore difficult, we take a time-shifted form of Eq. (16)with tension-only contributions:

R�ðtÞ ¼ e�ðtþ1Þ2

2 ðt þ 1Þ2 � 1� �

: ð17Þ

Sinus: A sinusoidal loading is also considered, where sixtime steps include one complete circle of the loading:

SðtÞ ¼ sin pDt þ t6 � Dt

� �ð18Þ

Fig. 6 Iterations and problemsetting for starting point [30 208 12]

256

All previous loadings are multiplied with a factor equalto 1000. The differences of the displacement at a pointlying in the middle of the upper side of the disk aredepicted in Fig. 7. The existence of a crack and thereflection of the waves from this defect can be identifiedwith all three aforementioned loadings. Therefore, inprinciple, they can be used as test loadings for crackidentification purposes. On the right-hand-side of Fig. 7the negative error values are depicted as a function ofthe crack position. From this figure a comparison be-tween the different loadings is firstly done. The intervalsof the results coming from a Heaviside- and a Sinus-

loadings are similar, while the same results from aRicker-impulse lead to narrower expressions.

The evolution of the optimization for a Ricker-im-pulse and a sinusoidal function is depicted in Figs. 8 and9. The starting values are x

�0 ¼ ½30 20 8 2� and the actual

defect parameters are x ¼ ½50 50 10 0�. In both cases theidentification algorithm stopped at a local minimum ofthe error, near the lower boundary of the disk, as it isclearly recognized from Fig. 7. The stopping criterion ofthe algorithm is never reached. Therefore, no useful re-sults are provided. On the contrary, a Heaviside loadingwith the same starting point is able to solve the problem

Fig. 7 Displacements at theupper boundary of the disk anderror values as a function of thecrack center coordinates fordifferent loadings, a crackdescribed by [50 50 10 0]

257

Fig. 8 Ricker-Impulse: Iterationsand geometry for a startingpoint [30 20 8 2]

Fig. 9 Sinusoidal function:iterations and geometry for astarting point [30 20 8 2]

258

Fig. 10 Ricker-Impulse: problemsetting and iterations fromstarting point [70 75 7 1,1]

Fig. 11 Sinus-Function: problemsetting and iterations fromstarting point [70 75 7 1,1]

259

Fig. 12 Heaviside-Function:problem setting and iterationsfrom starting point [70 75 7 1,1]

Fig. 13 Compressive loading:problem setting and iterationsfor a starting value [40 60 10 0]

260

and leads to useful results for all unknown crack para-meters, as it is demonstrated in Fig. 5. The iterations ofthe algorithm for all three loadings and for a differentstarting point, x

�0 ¼ ½70 75 7 1; 1�, are shown in figures

10, 11 and 12. In this case, for all three test loadings theidentification was successful. Nevertheless the algorithmneeds 25 iterations, for a Ricker-impulsive loading andapproximately 15 iterations for a sinusoidal or a Hea-viside loading. This is in accordance with the observa-tion, supported from Fig. 7, that the optimizationalgorithm is slower, due to the more even error dis-tribution.

As a conclusion, one can observe that the best defectidentification results have been obtained with a Heavi-side-loading. The other two loadings are useful, pro-

vided that suitable starting points are used. The Ricker-Impulse needs, as a rule, more iterations than the sinu-soidal loading.

5.5 Influence of the unilateral contact

Tensile dynamical loadings have been used in the pre-vious experiments. The problem of identification withcompressive loadings, which enhance the influence of theunilateral contact effects, will be investigated in thissection. Let us consider the previous disk with a defect,loaded with a Heaviside, compressive loading. Theidentification was not successful, as it is shown inFig. 13. Different configurations and especially direction

Fig. 14 Tensile loading: problemsetting, iterations for thestarting point [40 60 10 0]

Fig. 15 comparison of identifi-cation: —– tension, —– com-pression

261

of the crack in relation with the direction of the loadingwave lead to successful identification results. Theinability of the Kalman filter, to identify the crack in theprevious example, could be explained as follows: Thecrack is perpendicular to the loading direction. There-fore, all changes in the boundary displacements are dueto the wave reflected from the crack, which nevertheless,due to the unilateral contact, does not has a significantcrack opening. This leads to relatively small, measurable

displacements. The error value, which is used as anindicator of the estimation quality, has very small va-lues, as one sees in Fig. 13(c). Due to the bad scaling ofthe error function, it seems that the Kalman filter doesnot work effectively. One should also mention that allthese difficulties disappear for a crack without unilateralcontact conditions. This is exactly the simplificationused in the majority of fracture mechanics publications.The defect case shown in Fig. 13 corresponds to acompression applied on a crack which is perpendicularto the loading. Fig. 14 shows that the crack can beidentified with the dynamic tensile loading in only 21iterations.

Fig. 16 Identification results and problem setting for four unknowncracks within the disk

Fig. 17 Three different appliedloadcases as excitation for thedisc

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The iterations of the algorithm for the crack identi-fication of Fig. 13, for the same starting points anddifferently oriented loadings are shown in Fig. 15. Usinga tensile loading the crack parameters are identified in afew iterations. On the contrary for a compressive load-ing the estimated values after 200 iterations are not yetsatisfactory.

As a conclusion one should mention that the crackidentification with compressive loadings is not alwayspossible. The effectiveness depends on the wave propa-gation, the loading directions, the position of the crack,and the shape of the specimen. Provided that all otherquestions of the inverse problem have been addressed,the main additional difficulty comes from the activationof the unilateral contact mechanism. Therefore, a tensileloading is preferable for which the optimization algo-rithm converges, even for the case of Fig. 13. A tensileloading must be preferred, if this is technologicallyfeasible.

5.6 Identification of four cracks within the disk

More than one crack can be identified simultaneously.The results from the identification of four cracks, with16 unknown parameters, with the Kalman filter tech-nique are presented in Fig. 16. This problem requiredthe use of two loading cases. For this purpose weconsidered the loading case of Fig. 3 and a similarloading which resulted by loading the right hand side

Fig. 18 Dimensions and discretization of the disc

Fig. 19 Identification results for applying loadcase 1 and oneembedded crack in the loaded flange of the U-shape

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of the plate with a horizontal loading and supportingthe left hand side plate.

5.7 Crack identification in U-formed shape

In the following numerical examples an U-shaped diskwith an embedded crack is investigated. For the solu-tion of the identification problem several differentloadcases are used as the excitation of the disc. Fig. 17shows all applied loadcases. The dimensions and thediscretization of the U-formed structure is representedin Fig. 18.

In general, in an U-formed structure three differentpossibilities to define one hidden crack exist. The crackcan be defined in one of the two flanges or directly in theweb of the shape. This crack has to be found bychoosing the right loadcase for the discs excitation.

Figure 19 demonstrates the solution procedure forsearching a crack lying in the excited flange of thestructure. The disk is loaded by the loadcase 1. Thestart values for the estimated crack are equal to a crackposition in the unloaded flange. In Fig. 19a the eva-luation of the center coordinates of the crack is givenfor the performed iterations. It is clearly shown thatafter 20 calculated iterations the estimated crack posi-tions move around the flange across the web. At theend of the simulation the identification finished suc-cessfully by finding the final crack position. As opposedto the numerical experiment of Fig. 19, the identifica-

tion failed for the loadcase 2 and the same initialguesses for the crack variables. The identification pro-cess is illustrated in Fig. 20. The estimated coordinatesof the crack’s center do not escape from the neigh-borhood of the starting position. It appears obviousthat a crack identification using the loadcase 3 leads toa good result if the crack lies in one the flanges of theU-profile. The results of this optimization procedurecan be found among others in [8].

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