Neural crack identification in steady state elastodynamics

__ +__ BB

2s

ELSEWIER

Computer methods in applied mechanics and engineering Comput.

Methods

Appl. Mech. Engrg.

Neural crack identification Car010 Wilhelmina

Received

24 October

129-146

in steady state elastodynamics

G.E. Stavroulakis, Institute ofApplied Mechanics,

165 (1998)

H. Antes*

Technical

1997; revised

University, 15 January

D-38023 Braunschweig,

Germany

1998

Abstract An inverse crack identification problem with harmonic excitation in linear elastodynamics is treated here by means of back-propagation neural network methods and boundary element techniques. The problem concerns the determination of the existence and the characteristics of a hidden crack within an elastic structure by means of measurements of the structural response on the accessible boundary for given external time-periodic loadings. The direct problem is solved by a boundary element formulation in the frequency domain which leads to a system of linear equations with frequency-dependent matrices. Thus, for a given frequency, certain similarities with linear elastostatics exist. Feed-forward multilayer neural networks trained by back-propagation are used to learn the (inverse) input-output relation of the structural 01998 Elsevier system. Then, the inverse problem is solved by a simple application of the neural network recalling (production) ability. Science S.A. All rights reserved.

1. Introduction The subject of the present work is the development and the testing of a solution technique for non-destructive crack identification problems. Time-periodic excitations are considered. Thus, on the assumption of linear elastic (or viscoelastic) behaviour for the structure, the static-dynamic similarity principle permits to use elastostaticlike boundary element solution techniques, with matrices which depend on the excitation frequency, for the numerical treatment of the direct problem. The inverse problem, which concerns the calculation of the characteristics of the crack (position, length etc), is solved by means of a back-propagation neural network system. The data for the training of the neural network identification model and its testing are produced, in this preliminary investigation, by a Boundary Element-based computational mechanics technique. The methodology is generally applicable to damage identification, flaw detection and quality control of structures or structural elements. The engineering problem of the dynamic analysis of a two-dimensional elastic structure with a crack, is treated numerically by a one-region, or, if needed, a multi-region boundary element formulation [5]. The assumption of linear, in particular, traction free crack behaviour is made throughout this paper. Thus, the amplitude of the excitation induced vibrations on the structure is assumed to be lower than the existing crack opening, so that no unilateral contact or frictional effects arise between the crack tips (for further discussion on this topic in connection with inverse, identification problems, the reader is referred to [42] for elastostatic problems). Recent expositions of the BEM for elastostatic problems include [5] and for dynamic problems [4,29]. The computer program used here is based on the code QUADPLEH given in [ 191. Note here that the data for a realistic engineering application can also be measured experimentally. This is especially connected to the sonic nondestructive identification techniques, which are widely adopted in industry for quality control tasks.

* Corresponding

author.

0045.7825/98/$19.00 0 1998 Elsevier Science PII: SOO45-7825(98)00035-8

S.A. All rights reserved

130

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Identification problems belong to the class of the so-called inverse problems, in the sense that a given model of a system is defined, a set of input-output data for this system is available, but the values of the parameters which are involved in the system are unknown. Nevertheless, one should note that the distinction between direct and inverse problems is some how arbitrary. The output error identification problem which is used here is formulated as an optimization problem for the difference between the measured (or computed) and the desired responses within the space of the variables which deline the considered structure. Nevertheless, the classical optimization approach is not always advantageous due to the nature of the arising problem. Namely, for inverse problems, small variations of a certain structural parameter may lead to either large or small variations in the structural response depending on the parameter‘s position or type. Due to this reason, the problem is generally an ill-posed one [35]. Moreover, due to the nonlinearity of the structural response mapping, as a function of the crack variables, the arising optimization problem is usually nonconvex. Thus, there exists the possibility that the mathematical model for the inverse problem admits multiple solutions. In terms of optimization, this corresponds to a problem with several local minima. Due to these inefficiencies classical optimization algorithms sometimes suffer from numerical convergence problems. Thus, soft computing techniques which have the ability to overcome local minima and to smartly scale the ill-posed optimization problem are sometimes advantageous. This is also the case here, where a neural network approach is adopted. In this paper, a two-dimensional specimen is considered which contains an unknown crack. The unknown crack has been parametrized by a certain number of parameters, namely, the length of a linear crack, and the coordinates of the middle point with respect to the used global coordinate system. It is assumed that certain boundary displacements can be measured for various time-periodic external loading. The direct mechanical problem is solved numerically by the BEM method, while the identification (inverse) problem is treated by a neural network based optimization technique. A short bibliographical review in the next section places the work reported here within the area of inverse problems studies in mechanics. The harmonic (direct) structural analysis problem and its BEM numerical treatment is outlined in Section 3. The neural network approach for the inverse problem is described in Section 4. The numerical experiments are presented in the last section.

2. Inverse crack identification

approaches.

A short review

Inverse, flaw, damage or, in our case, crack identification problems can be formulated for static or dynamic processes in mechanics. In the static case the direct problem has a more tractable form and its solution requires, in general, less computational resources. Dynamic problems in the time domain, i.e. by direct integration of the dynamic analysis relations, constitute the more general case. Both cases allow for the consideration of various nonlinearities. The harmonic analysis problem, which is considered here, lies between the two previous cases and allows for the solution of linear elasticity problems with linear viscosity effects, as it will be discussed later on. From another point of view, dynamic signals have more possibilities to activate existing defects, thus they potentially make the identification problem easier. Moreover, they are more suitable from the experimental point of view (cf. the well established modal analysis techniques for the dynamic inspection of large-scale structures and the ultrasonic techniques used for quality control tasks). In addition, one should add here that the identification of nonlinear processes is far more complicated than the one of linear ones. The appropriate choice of the involved nonlinearities, the excitation signal and the larger number of variables which must be identified in the dynamic case are responsible for these complications. The short bibliographical review given in this section gives some information about recent approaches to inverse analysis problems in mechanics, with an emphasis on the study of dynamical problems and on the consideration of neural network techniques for the inverse problem. It serves as an introduction into the problematic of the studied problem and, due to the abundance of the relevant sources, it is by no means complete. For more details the reader is referred to specialized monographs, among others, [ 14,181. Inverse problems for classical structures with static data have been considered for the damage detection [22]. More details on structural identification problems can be found in [35]. Boundary element method techniques and classical minimization algorithms have been used for the identification of flaws in [33] where all defects are assumed to be idealized elliptic ones and steady heat conduction problems are examined. In [45], the inverse elastostatic analysis problem is considered where the shape of the unknown crack is identified by boundary measurements. Analogous techniques have been used in [ 11,28,50]. In principle, an idealized crack is considered

G.E. Stavroulakis,

H. Antes

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131

and it is parametrized by means of a certain number of parameters (for instance, through parameter-defined approximation of the crack sides). The determination of these parameters constitutes the inverse problem. The latter reference, [50], includes elements of regularization techniques for the effective numerical treatment of the ill-posed inverse problem. A number of several inverse problems studied by boundary integral techniques and, almost exclusively, iterative solution techniques based on numerical optimization, can be found in [24]. The work of Tosaka [48] deals with the identification of elliptical defects in 2-D or spherical defects in 3-D problems, where the classical numerical minimization scheme is replaced by a Kalman filtering based iterative algorithm which, according to the cited reference, substantially reduces the computational effort. Some comparison on the performance of neural network methods on damage identification using static loading data can be found in [8]. Other approaches include genetic algorithms [16], Kalman filtering techniques in [48] and neural networks in [37]. All these techniques belong to the so-called soft computing techniques, whose use in inverse analysis problems steadily gains on interest. This is due to the fact that these methods are in most cases model free, thus they avoid complicated analytic studies. Moreover, they have the potential to overcome the inherent ill-posedness of the inverse analysis problem through their self-adapting scaling procedures. Another advantage is the robustness against (measurement) data inaccuracies. The generality, for instance, of neural network methods to deal with very complicated fault detection and general pattern correlation problems can be appreciated from the applications on chemical processes reported in [59]. Examples of dynamic analysis problems treated with neural networks will be cited in the sequel. In this respect, for a certain type of unilateral contact nonlinearity and for crack identification problems by means of static test loadings, the authors have proposed a neural network technique in [42,43]. The extension of this technique to cope with time-periodic dynamic problems is reported here. Note that due to the formal similarity of the harmonic dynamical problem with the static one the techniques used for static identification, including multiple loadings, can be used in some extend here as well. Dynamic identification problems have both been studied by means of classical, optimization-based techniques and by means of more contemporary, soft computing tools. Eigenvalue and eigenvector data have been used for the correlation of a given design or defect configuration with a measured modal signature (see, among others [55,56]). In fact, crack detection has traditionally been studied by considering the changes of the system’s eigenvectors and eigenvalues due to the appearance of the crack (see, among others [20,30,6]). Beyond the classical, general purpose model updating techniques (see e.g. [35]), more specialized algorithms have been proposed for the crack identification problem. Among them, let us mention the multi-hypothesis reliability based crack diagnosis algorithms of Ben-Haim [9, p. 1271, the reciprocity gap approach of Andrieux and Ben Abda [2] and the cluster analysis technique of Meltzer and E&hold [32]. In particular, neural network identification techniques with modal data have been proposed and tested in [22,21,49,38,56]. Especially, the evaluation of ultrasonic data has been performed in several cases by means of neural network models (see, among others [13,40,25,26,44,53,54,37]).Usually, some kind of preprocessing is involved in order to reduce the size of the data which are used in the neural network model. For instance, in [55], after solving the wave propagation problem in a medium with a defect, characteristic values of the response signal which are influenced from the existence and the data of the defect are extracted. In that case, these values are the first peak in the measured waveform, its height and the arrival time. A detailed investigation of the application of back propagation neural networks in this area with both binary and analogue response units can be found in [44]. For a recent review of neural network applications in computational mechanics with some applications on inverse problems, see [52]. In this paper, the magnitude of a steady state, periodic excitation is used for the formulation and the solution of the inverse problem. There are several reasons for doing so. First, in order to reduce the amount of data involved without losing the advantages of having a dynamical signal, a time-periodic dynamical problem is used. Thus, the only remaining possibility would be to use the modal data of the specimen. But, eigenvalues and eigenvectors are not always significantly influenced by small changes of the geometrical and stiffness data of a structure. This may lead to accuracy problems for crack identification tasks. Moreover, for large changes of the crack quantities, a change of order in the systems eigenmodal quantities may arise, i.e. the first mode may become the second for a certain change of the position of the crack. This would require additional effort to trace these changes. Finally, for small-scale specimens it is not always efficient to perform an accurate modal analysis test due to the fact that, in this case, the mass and stiffness of the required apparati (shakers, sensors) are of the same order with the ones of the structure itself and, thus, influence the measured quantities. In view of all these

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facts, the choice of the boundary element method, in the form outlined in the next section, has been natural. This would not be the case, had we preferred to use modal quantities (they are extracted more easily by finite element techniques). Harmonic elastodynamic response for structural identification has been used in [47] and [15] in connection with iterative techniques based on optimization algorithms and computational mechanics modelling.

3. BEM harmonic

dynamic

analysis

A short outline of the formulation of the reduced elastodynamic equations for homogeneous, elastic and isotropic bodies subjected to time-harmonic excitation and the corresponding boundary element solution method are given in this section. More details can be found in the specialized literature (see, among others [3,5,29,19]). Let us consider the equations of motion, written in a Cartesian coordinate system, for each point x of an elastic body which occupies the area R: ~,.,(.& t) - p(-W,(x, t) + P,(X>t) = 0 ,

x E fI .

(1)

Here q, is the stress tensor, p is the mass density of the body, ui is the displacement vector and p, is the loading vector. Moreover, i, j run over the values 1, . . . , 3 for three-dimensional problems (resp. the values 1, . . . ,2 for two-dimensional problems) and the usual Einstein’s summation assumption for repeated indices is adopted. On the assumptions of linear elastic material behaviour and of small amplitude vibrations, i.e. a small displacements and deformations theory, Eq. (1) takes the form: (c;(x)

- &,u,,,;(x,

p (X> I) t) - ii,@> f) + ‘--=o, &)

t) + c;(x)u,,j,(x,

XER.

In (2), c, is the dilatational (or pressure) and c2 is the distortional (or shear) wave propagation velocity. For plane stress two dimensional isotropic elasticity applications, c,, c2 are related with the elasticity modulus E and the Poisson’s ratio v as follows: (-;

zz

c; =

E

2

p(l+V)(l-V)’

E (3)

c2=cf=2p(l+2,).

A linear elastic, homogeneous and isotropic material law is assumed hereafter. Let us further assume that all elastodynamic quantities of the studied problem are time harmonic. Thus, for a given frequency w the excitation takes the form p,(+ t) = $,(x) elw’ with i = \‘. Accordingly, the response of the system is harmonic as well, thus we use the Ansatz u;(x, t) = i&(x) elw’. Under this transformation the time-dependent equations of motion ( 1)-( 3) take the following, frequency-dependent form:

(c:(x)- c:(x,,c;,.,,(x) + c;(4u^,,,j(x) + Aqx) +

m p(x)

=

0,

x E fi

By using the reciprocal theorem of Green and adequate fundamental solutions and by assuming that only boundary excitations are applied on the structure, one may obtain the boundary integral equation of the system at each point 5 E 0: d( 6% 5) = I, [6(x@ *(x> 5;

K, ,

$1 - G(d$ *(X. 5;

K, ,

K2)1dT .

(5)

Here, K, = w/c,, ~2 = w/c,, are the wave numbers, 5 and x are the points on the boundary r or in the body .1& &*(x, 6; K,, ~2) (resp. @*(x, 5; K,, K~)) denotes the fundamental solution (resp. its normal derivative fi z on the boundary) and the jump factor d( 5) is calculated as usual in the BEM (for instance, d( 5) = 0.5 for a smooth part of the boundary). Moreover, in the derivation of (5) all loading has assumed to be applied on the boundary of the structure. After appropriate point collocation and boundary element discretization one gets the discretized form of (5): H(K, >K,b = ‘%, , K2)t >

(6)

G.E. Stavroulakis,

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where u (resp. t) denotes the boundary nodal displacement (resp. boundary traction) vector and the influence matrices H, G depend on the assumed excitation frequency o. Further processing of (6) i.e. taking into account the boundary conditions of the structure, separating known and unknown elements of vectors u and t according to the boundary conditions of the structure, forming the system of equations, etc., follows the classical techniques in the BEM and will not be discussed in detail here. For further reference we write here the form of the system of equations which arises: A(w)x(w) = b(w) ,

(7)

with solution denoted by x(w, b) for a given frequency w and a given ‘loading’ vector b. One should mention here that, by using the correspondence principle of linear viscoelasticity (attributed to Hashin [23]) certain types of material (viscous or hysteretic) damping can be included in the previous formulation. In this case the elasticity constants E and v become complex numbers and, accordingly, one has to work with complex numbers in Eqs. (4)-(6). For instance, a viscous material damping would require an elasticity modulus of the form: E*=E+iwC. A hysteretic,

frequency

independent

damping

can be represented

by an elasticity

modulus

of the form

E*=E(l+il), where 6 is the loss (or damping) factor and El is sometimes referred to as the loss elasticity modulus. From the real (res. the imaginary) parts of the boundary quantities u and t of (6) one may extract information about the oscillatory and the decaying behaviour of the solution (see [19, Chap. 21 for more details). Although the BEM program used in this investigation has this ability, no attempt has been made in this paper to use this information for identification tasks. Thus, only the real parts of the corresponding quantities have been utilized in the sequel, i.e. only the oscillatory behaviour of the system is studied. Nevertheless, as it will be discussed later on, a small damping is added in order to avoid excessive numerical problems with resonance frequencies.

4. Neural network solution of the inverse problem Let a given structure be considered which contains an unknown crack. The crack is characterized by a set of parameters z = [z,, . . . , zJT. Here, the coordinates of the crack center and the length of the crack are used as identification parameters. Moreover, let the response of the structural system for a given loading b’, 1= 1, . . ,I, and a given frequency mm, m = 1, . . , m, and for a given crack z be given by the vector J?(w~, z, b’) as solution of Eq. (7) for b’(w”). Here, I, is the total number of different loading cases and m, is the total number of frequencies used. Obviously the response of (7) is parametrized by the unknown crack parameters z. Moreover, let the response of the examined structure with a known crack subjected to the same loading b’ and for the same frequency w”’ be denoted by TO(m”‘, z, b’). Note that, while in this investigation the elements of f,(w”, Z, b’) are produced by computational mechanics techniques, the same procedure for the solution of the inverse analysis problem can be used if these data are obtained from experiments. Instead of formulating and solving the inverse problem as, e.g. an output least square error minimization problem (cf. minimization of the error function (9), see below), a direct treatment of the inverse relation by means of back propagation neural networks is chosen. Recall that, in view of the nonlinearity in the response vector as a function of the crack parameters, the classical error minimization approach leads to nonconvex optimization problems (see relevant discussion in the next section and in [42]). Here, a multilayer back propagation error driven neural network is used to learn the relation

*(mm,z, b’)

+z

for a given value of loading vector b’ and for a set of excitations w”‘. The couples of data composed of the vectors i(w”, z, b’) and the corresponding parameter vectors z are used as training examples. In the production mode, the nonlinear network reproduces the relation x +z, i.e. for a given set of measurements x” (different from the ones used in training) it gives a prediction for the variables characterising the internal crack.

134

5. Numerical

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examples

5. I. Direct problem

A plane stress plate with a crack at its lower boundary is considered, as it is shown in Fig. 1. The material constants are: elasticity modulus E = 100000.0 + i10000.0, mass density p = 100.0 and Poisson’s ratio v = 0.43. An artificial damping i = 0.10 is used throughout. All quantities used here are in compatible units. Thus, the wave velocities are equal to c = 59.2345 + i 2.954359, c = 3 1.662 18 + i 1.579 17 1. The external dimensions of the plate are 10.00 X 10.06. We assume a uniform vertical loading on the upper boundary of the plate with amplitude equal to pJ = 1000.0 and varying frequency w, as given later. For the BEM discretization the external boundary of the plate (abed in Fig. 1) is discretized by means of 67 quadratic boundary elements, i.e. a total of 134 nodes are used. For further reference the nodal numbering is as follows: node 1 at corner a, node 5 1 at corner b, node 69 at corner c‘ and node 117 at corner Lz.The boundary element discretization is uniform at all boundaries of the plate. Moreover, by varying the position of the assumed crack at the lower boundary ab, a uniform continuous change of the nodes is assumed so that the effect of the influence of the discretization on the response of the structure is reduced. Finally, only the response of the structure at selected nodes of the upper boundary cd is used here for identification purposes. The influence of the existence of a crack on the response of the system is shown in Figs. 2-5 for various rack configurations and for excitation frequencies equal to 5, 10, 15. 20, 25, 30, 35 and 40. The crack variables z = [x,-, I,] (cf. Fig. 1) are as follows: z = L5.20, I.401 for Figs. 2 and 5, z = 14.20, 1.601 for Fig. 3 and no crack for Fig. 4. Moreover, vertical loading is assumed for Figs. 2-4 and horizontal loading at the upper boundary cd

Fig. I. Configuration

Fig. 2. Structural

of the plate with a crack.

response for a vertical loading, a crack of length equal to

I .40 and center at (5.10.0.0)

and various excitation

frequencies.

G.E. Stavroulakis, H. Antes I Comput. Methods Appl. Mech. Engrg. 165 (1998) 129-146

Fig. 3. Structural

response for a vertical loading, a crack of length equal to 1.60 and center at (4.20.0.0)

Fig. 4. Structural

Fig. 5. Structural frequencies.

response

response

for a vertical loading,

no crack and various excitation

and various excitation

loading,

a crack

of length

equal to 1.40 and center

frequencies.

frequencies

q u e”ps cl 0 gl El n

for a horizontal

135

at (5.10,O.O)

and various

excitation

of the plate for Fig. 5. A uniform scaling factor equal to 50.0 is used for plotting the deformed shaper in the previous figures. Roughly speaking, the difference between the deformed shapes, measured at appropriate measurement points, will be used for identification of the crack parameters. To this end, note here that not all excitation loadings and frequencies are able to activate a given crack. For instance, a vertical loading with a frequency equal to 20 does not ‘see’ the crack in Figs. 2 and 3 (last case in the upper row). A horizontal loading does not work effectively either (cf. Fig. 5). These facts show the importance of using several appropriately chosen excitation frequencies and loadings so that the influence of the crack parameters on the measurements is sufficiently strong (see also 1151). One more point is worth mentioning. If an excitation frequency lies near an eigenfrequency of the analysed structure, then resonance reactions occur and the arising results are, for our purpose, useless. Such a case is obvious in the third plot of the upper row in Fig. 5. Results of this kind must be excluded from further consideration in the study of the inverse problem. Here, this resonance phenomenon will be used for an overall check of the quality of the BEM model. In fact, by observing the peak(s) of the structural response at a given point for various excitation frequencies, the eigenfrequency of the system can be specified. This is shown in Fig. 6 where the vertical displacement of node 100 (at the coordinates x = 3.0, y = 10.0, see Fig. 1) is plotted for a plate with a crack (1, = 1.60, xc = 3.80) and

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165 (1998)

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J 7.5

8

8.5

Fig. 6. Eigenfrequency

9.5

10

10.5

11

shift for a plate with a crack

the same plate without crack. The extracted eigenfrequencies, equal to 8.30 and 8.51, respectively, agree well with the results of a separate finite element calculation. The first four computed eigenmodes for the plate without the crack are given in Fig. 7. For instance, the fourth eigenfrequency is estimated from the plot to be equal to 8.5 1while a FEM calculation by using 25 quadrilateral finite elements and the program PCFEAP (see [58]) leads to a value equal to 8.537. An investigation of the inverse problem for the plate of Fig. 1 is performed here. The range of cracks with a length lying in the range of lc = [0.40 - 1.81 and a crack center lying in the range of xc = ]I .5 - 4.01 is investigated. First, let us consider a least-square type measure of the measurements’ difference. By considering all vertical displacements of the upper boundary cd in the Fig. 1 (elements i of the subvector u of x, denoted as XL here, at the nodes i = 69-l 17, for several frequencies m (w E (15, 20, 25, 30, 35, 40)) this measure reads: @(z) = @(I,, x, ) = log ~=h4,,,m~,6{~~(~~,z,b’)-x:(~~.Z_h’)J2. c

(9)

The case of @ = --m has been set equal to -20 for the plots. Moreover, before using the displacement values of the boundary nodes, they have been normalized, for each loading case separately, between values 0.0-1.0. The previous measure for several positions of the crack calculated by means of all six previously given excitation frequencies or only the last value of them is plotted in Figs. 8 and 9. From the results of Figs. 8 and 9 and from similar parametric investigations, which cannot be given here due to lack of space, one may observe some interesting features of an error function like (9). First of all, the considered signal has all the required information for identifying both the position and the magnitude of a given crack. Clearly, a global minimum of Q(Z) always appears at the positions of the assumed crack. The ill-posedness of the problem, i.e. the fact that the effect of the crack variables on the result has a different magnitude for different values of these variables, is obvious from the need of using a logarithmic scaling in (9) (otherwise the plots would have been less informative). Moreover, the nonlinearity of the mapping between the crack variables and the structural response makes the (otherwise convex) quadratic error function nonconvex. Some evidence of this latter fact is shown in Figs 8 and 9, where local minima arise. An analogous investigation by using the position of a crack of given length and static analysis data has been performed in [42]. Note that the case here is much more complicated, since the size of the measured data is

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137

2nd ebenmoda 5.021

(b) 1

Fig. 7. (a)-(d)

First four eigenmodes.

Results of PCFEAP

with quadrilateral

finite elements.

larger (roughly speaking, the size of the static problem multiplied by the number of excitation frequencies). Moreover, variables of different nature (position and length of the crack) are involved in the investigation of this paper. Unfortunately, from our limited numerical experience we could not extract rules of general validity concerning the number and magnitude of appropriate excitation signals, so that to optimize the performance of the identification scheme. Moreover, the solution of the complete identification problem (both position and magnitude of a crack) as posed previously, cannot be performed effectively by using a unified neural network strategy (analogous to the one used in [42] for a static crack identification problem). The first attempts either reach the upper bound of allowable computing time, or, lead to problems of ‘over-learning’. The last phenomenon is relatively well known in the neural network literature: due to insufficient choice of the size of the network (or of the scaling in the used parameters) the network is able to learn the given examples (data) with sufficient accuracy but the accuracy of the prediction from unknown data is not good. This effect is also referred to as the saturation of learning or as an overfitting condition. For this reason, a multiple network (cascading) splitting strategy is adopted here (see also relevant discussions and applications in [1,57,43]). This way the complexity of the problem is simplified, the arising subproblems can be solved quickly, and the whole problem arises as a combination of several steps. The essence of this splitting is to solve a problem, first for finding the size of the crack, then the length, etc., i.e. all elements of the crack variables vector by using separate neural network systems. As previously, the range of cracks I, = [0.40-1.81 and xC = [IS-4.01 is investigated. These quantities are in

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(b)

2

1.5 0.4

Fig. 8. (a)-(c) Plot of measurement

1

1.2

error function,

1.4

1.8

use of six excitation

0

frequencies.

G.E. Stavroulakis,

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e-9

/

I

Fig. 9. (a)-(c)

Mech.

Plot of measurement

error function,

Engrg.

16.5 (1998)

T”_.

“‘----I

use of one excitation

129-146

I

frequency.

139

140

G.E. Stavroulakis,

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Trainin for 125 Epochs

g

-I

lod

~~~:-:‘.:-

0.4 0.6 0.5 aadc len@h (normalized)

(a)

0.7

0.9

0.9

1

Trainiq for 127 Epochs lo0 ,

I

1 40

60

60

120

100

Epoch

‘aiven. + mdiiad

0

0.1

0.2

0.3

0.6 0.4 0.5 crack length (normalized)

0.7

0.9

0.9

1

(b)

~~_, 0

0.1

0.2

Cc)

Fig. IO. (a)-(c)

Neural prediction

0.3

0.6 0.4 0.5 crsk IwJth (mxmalii)

of the

crack

length,

0.7

for

given

0.9

crack

0.9

center

1

position.

G.E. Stavroulakis,

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I Comput. Methods Appl. Mech. Engrg. 165 (1998) 129-146

Trainitqfor741 Epochs lo0 [

I

10-f

1

100

0

300

200

400 Emdl

500

700

900

*given, +pmoiied

~-‘--_:: 0

0.1

02

0.4 0.5 0.6 cmckle~(nonnal~

0.3

0.7

0.9

0.9

1

(4 Tminiqfor29OEpcchs

1

10-t 50

0

100

200

s

250

ids_:: 0

0.1

02

0.3

0.4 0.5 0.6 uacklen@h(mti~

0.7

0.9

0.9

1

0.7

0.9

0.9

1

(b)

Epoa 'tJivm,+pmdiied

alar’ 0

0.2 (c)

Fig. 11

(a)-(c) Neural

0.3

0.6 0.4 0.5 cmklemJth(normaiii

prediction of the crack

length, for

given crack center position.

141

142

C.E. Sta~~roulnkis. H. Antes

I Cornput.

Methods

Appl.

Med.

Iji~~ 0

100

50

150

Engrg.

165 (1998)

200

256

129% 146

Ewh + predicted

*goJan.

ii:l.:-_I-_ 0

0.2

0.3

(a)

0.7 0.6 0.5 0.4 crackp&ion (normalized)

0.6

0.9

1

J

150

100

50 Epoch 'gwen,+pmdi%d

i.----;-Tj 0

0.1

0.2

(b)

0.3

0.4 0.5 0.6 0.7 crackpmition(nommlized)

0.6

0.9

1

Training for134 Epochs

i.c_~

~~1

0 Cc)

Fig. 12. (a)-(c) Neural prediction

0.4 crackpositiin (nommlimd)

of the crack center position,

0.9

1

for a crack of given length.

G.E. Stavroulukis, Table 1. Effectiveness Net/case x= 150 x = 50 x= 10

Table 2 Effectiveness Nodes Iterations Accuracy

H. Antes

of neural learning.

Fig. 10 125/17.21% 127117.49% 156/ 16.96%

I Comput. Methods Appl. Mech. Engrg. 165 (1998) 129-146

Number of learning

Fig. 11 741/1.39% 290/1.91% 360/2.48%

epochs and percentage

accuracy

143

of predictions

Fig. 12 250/ 13.52% 152/14.13% 134/15.97%

of using less measurements 48 125 17.21%

12 106 18.18%

6 102 19.05%

3 95 18.99%

I 87 21.13%

the sequel scaled in the range [O.O-1.01. Using all vertical displacement data of the upper boundary, for two frequency excitations (w = 5.0 and 25.0) the length of an unknown crack (with given crack venter position) can be identified by means of a neural network system. For xc = 1.40 and x,. = 2.0 the learning history and the results are presented in Figs. lO(a-c) and 1 l(a-c). For a given crack length lc = 0.4, the learning and prediction of the crack position x,. is done by an analogous procedure and the results are given in Figs. 12(a-c). A feed-forward neural network, trained by the back propagation learning algorithm is used. The network has 96 input nodes (i.e., all 48 nodal displacements for two frequency cases), two internal layers with equal number of nodes and one output node, which measures the unknown length of the crack. The neural configurations 96 - X -X - 1 have been tested, with X = 150 for Figs. 10(a), 1 l(a), 12(a), X = 50 for Figs. 10(b), 1 l(b), 12(b) and X = 10 for Figs. 10(c), 1 l(c), 12(c). In all these results three sets of data have been used for learning and three, different sets of data are used for testing the efficiency of the prediction. Thus, in the graphical representation of the predictions in Figs. 10-12, the vertical difference between the predicted value, plotted by a + , and the diagonal line shows the accuracy of the obtained prediction. Thus, the first, third, fifth and seventh point (from the left-hand side) denoted by a * in figures, has been used for learning. The remaining points are used for testing. The learning procedure has been performed by a momentum backpropagation algorithm with logarithmic activation functions, a momentum constant equal to 0.95 and a stopping sum-squared error equal to 0.0001. As expected, neural prediction near the boundaries of the given data, or extrapolation from known data, is in general of less accuracy. Nevertheless, for this application, and considering that learning has been based on only three sets of data (i.e. the solution of the problem for only three different crack configurations, for each case, is needed) the obtained accuracy is acceptable. Moreover, use of only a limited number of measurements seems to lead also to acceptable results. These findings are summarized in Tables 1 and 2. The number of required epochs (steps) in the learning procedure and the achieved accuracy as a percent of the known values is given in Table 1, where the net configuration is given by the variable X, as previously, and the three columns correspond to Figs. 10, 11 and 12, respectively. The effect of using less input variables (measurements) has been examined by taking the case of Fig. 10(a), and reducing the number of input variables. Since this investigation is restricted to an academic example we have chosen elements of a small number of nodes, homogeneously distributed on the upper boundary and starting from corner c, i.e. from node 69. The effect of using a reduced number of nodes on the number of iterations and on the accuracy of the prediction is documented in Table 2. Note that the actual number of inputs for the neural network is twice the number of measurement nodes (due to the fact that two excitation frequencies have been used). Note also that for this application even one measurement point is sufficient for an acceptable accuracy. Nevertheless, in any case, one should have in mind that the given numbers are indicative and have a stochastic nature since the neural network application starts from randomly determined internal variables of the network and the result depends on several parameters (including the accuracy of the computer used, in which series the learning examples are considered, etc.). All reported experiments have been done by using the Neural Network Toolbox of MATLAB, installed on an IBM RISC/6000 System at the Institute of Applied Mechanics of the TU Braunschweig.

6. Conclusions The effectiveness of using back-propagation neural networks for a class of inverse crack identification problems in harmonic elastodynamics has been investigated. A BEM computational mechanics technique has been applied for the production of the needed data. A cascadic, multi-network splitting technique has proven to be efficient for this class of problem. The application of other kinds of neural network models and of data-reduction schemes, which would allow for the study of more complicated effects, including multiple crack identification problems, are left open for further investigations. Moreover, the use of other appropriate data of the viscoelastic problem (e.g., phase hysteresis), which can be produced by analogous computational techniques. may be of interest for the solution of related inverse problems in mechanics.

Acknowledgement The work reported

here has been supported

by the European

Union (TMR Grant ERBFMBICT960987).

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and its