Soft computing techniques in parameter identiﬁcation and ... · considered, namely parameter...

Available online at www.sciencedirect.com

www.elsevier.com/locate/advengsoft

Advances in Engineering Software 39 (2008) 612–624

Soft computing techniques in parameter identificationand probabilistic seismic analysis of structures

Y. Tsompanakis a,*, N.D. Lagaros b, G.E. Stavroulakis c,d

a Department of Applied Sciences, Technical University of Crete, University Campus, Chania 73100, Greeceb School of Civil Engineering, National Technical University of Athens, Zografou Campus, Athens 15780, Greecec Department of Production Engineering and Management, Technical University of Crete, Chania 73100, Greece

d Department of Civil Engineering, Technical University of Braunschweig, Germany

Received 1 December 2006; accepted 8 March 2007Available online 20 September 2007

Abstract

The objective of this paper is to investigate the efficiency of soft computing methods, in particular methodologies based on neuralnetworks, when incorporated into the solution of computationally intensive engineering problems. Two types of applications have beenconsidered, namely parameter (flaw) identification and probabilistic seismic analysis of structures. Artificial neural networks (ANNs)based metamodels are used in order to replace the time-consuming repeated structural analyses. The back-propagation algorithm isemployed for training the ANN, using data derived from selected analyses. The trained ANN is then used to predict the values ofthe necessary data. The numerical tests demonstrate the computational advantages of the proposed methodologies.� 2007 Elsevier Ltd. All rights reserved.

Keywords: Artificial neural networks; Simulation; Inverse problems; Probabilistic seismic analysis; Structural identification

1. Introduction

The advances in computational hardware and softwareresources since the early 1990s resulted in the developmentof new, non-conventional data processing and simulationmethods. Among these methods soft computing has to bementioned as one of the most eminent approaches to theso-called intelligent methods of information processing.Artificial neural networks (ANNs), expert and fuzzy sys-tems, evolutionary methods are the most popular soft com-puting techniques. Especially ANNs have been widely usedin many fields of science and technology, as well as, in anincreasing number of problems in structural engineering.The present paper is an updated and revised version ofthe conference paper [1], and is focused on the implementa-tion of advanced ANN methodologies in demanding prob-lems in computational mechanics.

0965-9978/$ - see front matter � 2007 Elsevier Ltd. All rights reserved.doi:10.1016/j.advengsoft.2007.06.004

* Corresponding author. Tel./fax: +30 28210 37634.E-mail addresses: [email protected] (Y. Tsompanakis), nlagaros@

central.ntua.gr (N.D. Lagaros), [email protected] (G.E. Stavroulakis).

From among general problems that can be analyzed bymeans of ANNs the simulation and identification problemscan be classified as follows [2]:

• simulation is related to direct methods of structural anal-ysis, i.e., for known inputs (e.g., excitations of mechan-ical systems (MSs)) and characteristics of structures ormaterials outputs (responses of MSs) are searched;

• inverse simulation (partial identification, for example, ofan unknown excitation) takes place if inputs correspondto known responses of MSs and excitations are searchedas outputs of ANNs; and

• identification is associated with the inverse analysis ofstructures and materials, i.e., excitations and responsesare known and MS characteristics or parameters of asuitably parametrized model are searched.

Over the last 10 years artificial intelligence techniqueslike ANNs have emerged as a powerful tool that couldbe used to replace time consuming procedures in manyengineering applications. Some of the fields where ANNs

mailto:[email protected]

mailto:nlagaros@ central.ntua.gr

mailto:nlagaros@ central.ntua.gr

mailto:[email protected]

Y. Tsompanakis et al. / Advances in Engineering Software 39 (2008) 612–624 613

have been successfully applied are: (i) pattern recognition,(ii) regression (function approximation/fitting) and (iii)optimization. Additionally, ANNs are presented in [3] asan alternative approach to nonlinear system modeling. Inthe past the first field of application of ANNs was mostlyused for predicting the behavior of a structural system inthe context of structural optimal design [4–8], structuraldamage assessment [9,10] or structural reliability analysis[11,12]. Function approximation involves approximatingthe underlying relationship from a given finite input–out-put data set. Feedforward ANNs, such as multi-layer per-ceptrons (MLP) and radial basis function networks havebeen widely used as an alternative approach to functionapproximation since they provide a generic functional rep-resentation and have been shown to be capable of approx-imating any continuous function with acceptable accuracy[13].

In this work the application of ANNs is focused on thesimulation, (e.g., probabilistic analysis of structures), andidentification, (e.g., flaw detection), problems. Manysources of uncertainty (material, geometry, loads, etc.)are inherent in structural design and functioning. Probabi-listic analysis of structures leads to safety measures that adesign engineer has to take into account due to the afore-mentioned uncertainties. Probabilistic analysis problems,especially when earthquake loadings are considered, arehighly computationally intensive tasks since in order to cal-culate the structural behavior under seismic loads a largenumber of dynamic analyses (such as multi-modal responsespectrum analyses) are required [14]. Soft computing tech-niques are used in order to reduce the aforementioned com-putational cost. An ANN is trained utilizing availableinformation generated from selected multi-modal responsespectrum analyses of a typical multi-storey building. Thetrained ANN is then used to predict the maximum intersto-rey drift due to different sets of random variables. After themaximum interstorey drift is predicted, the probability ofexceeding the design limit state is calculated by means ofMonte Carlo Simulation (MCS). The results of the pro-posed methodology in the test examples show its efficiencyand its potential for treating large-scale practical problems.

Another very promising field of soft computing applica-tions in computational mechanics is flaw or damage detec-tion, which basically can be considered as an inverseproblem. For example, material, shape or parameter iden-tification problems, which can be formulated as output-error optimization problems, can be solved very efficientlywith the proposed technique. For the classical formulationand solution with classical or less classical (e.g., filter-dri-ven or genetic) optimization algorithms and neural net-works details can be found in Refs. [15–17] and thereview article on inverse analysis [2]. In this work the meth-odology is extended by using a neural network techniquefor the replacement of the mechanical analysis modelingand, consequently, the genetic optimization is applied forthe solution of the inverse crack or defect optimizationproblem. The effectiveness of the proposed hybrid scheme

is compared with the results of the previously used single-method techniques in characteristic test examples.

2. Multi-layer perceptrons

ANNs metamodels have the ability of learning andaccumulating expertise and have found their way intoapplications in many scientific areas. There is an increasingnumber of publications that cover a wide range of compu-tational structures technology applications, where most ofthem are heavily dependent on extensive computerresources that have been investigated or are under develop-ment. This trend demonstrates the great potential ofANNs.

2.1. The back-propagation learning algorithm

A multi-layer perceptron (MLP) is a feed-forwardANN, consisting of a number of units (neurons) linkedtogether and attempts to create a desired relation in aninput/output set of learning patterns. A neural networkconsists of an input layer, one or more hidden layers andan output layer. Each layer has its corresponding neuronsor nodes and weight connections. A single training patternis an I/O vector of pairs of input–output values in the entirematrix of I/O training set.

The inputs xi, i = 1,2, . . . ,n which are received by theinput layer are analogous to the electrochemical signalsreceived by neurons in human brain. In the simplest modelthese input signals are multiplied by connection weightswp,ij and the effective input netp,j to neurons is the weightedsum of the inputs

netp;j ¼Xn

i¼1

wp;ijnetq;i ð1Þ

where wp,ij is the connecting weight of the layer p from the i

neuron in the q (source) layer to the j neuron in the p (tar-get) layer, netq,i is the output produced at the i neuron ofthe layer q and netp,j is the output produced at the j neuronin the layer p. Inputs xi correspond to netq,i for the inputlayer.

In the biological system, a typical neuron may only pro-duce an output signal if the incoming signal builds up to acertain level. This output is expressed in ANNs by

outp;j ¼ F ðnetp;jÞ ð2Þ

where F is an activation function which produce the outputat the j neuron in the p layer. The type of activation func-tion that has been used in the present study is the sigmoidfunction.

At the output layer the computed output(s), otherwiseknown as the observed output(s), are subtracted from thedesired or target output(s) to give the error signal

614 Y. Tsompanakis et al. / Advances in Engineering Software 39 (2008) 612–624

EðwÞ ¼ 1

2mkEðwÞk2 ð3Þ

EiðwÞ ¼X‘j¼1

½outk;j � tark;j� ð4Þ

where m is the number of training pairs, tark,i and outk,i arethe target and the observed output(s) for the node i in theoutput layer k, respectively. This type of ANN training iscalled supervised learning.

A learning algorithm tries to determine the weights, inorder to achieve the right response for each input vectorapplied to the network. The numerical minimization algo-rithms used for the training generate a sequence of weightmatrices through an iterative procedure. To apply an algo-rithmic operator A, a starting value of the weight matrixw(0) is needed, while the iteration formula can be writtenas follows:

wðtþ1Þ ¼ AðwðtÞÞ ¼ wðtÞ þ DwðtÞ ð5ÞAll numerical methods applied in ANNs are based on

the above formula. The changing part of the algorithmDw(t) is further decomposed into two parts as

DwðtÞ ¼ atdðtÞ ð6Þ

where d(t) is a desired search direction of the move and at

the step size in that direction.The training methods can be divided into two catego-

ries. Algorithms that use global knowledge of the state ofthe entire network, such as the direction of the overallweight update vector, which are referred to as global tech-niques. In contrast local adaptation strategies are based onweight specific information only such as the temporalbehavior of the partial derivative of this weight. The localapproach is more closely related to the ANN concept ofdistributed processing in which computations can be madeindependent to each other. Furthermore, it appears that formany applications local strategies achieve faster and reli-able prediction than global techniques despite the fact thatthey use less information [18].

2.2. Global adaptive techniques

The algorithms most frequently used in the ANN train-ing are the steepest descent, the conjugate gradient and theNewton’s methods with the following direction vectors:Steepest descent method: dðtÞ ¼ �rEðwðtÞÞConjugate gradient method: dðtÞ ¼ �rEðwðtÞÞ þ bt�1dðt�1Þ

where bt is defined:

bt�1 ¼ rE t � rE t=rE t�1 � rE t�1 Fletcher–Reeves

Newton’s method: dðtÞ ¼ �½HðwðtÞÞ��1rEðwðtÞÞThe convergence properties of optimization algorithms

for differentiable functions depend on properties of the firstand/or second derivatives of the function to be optimized.When optimization algorithms converge slowly for ANNproblems, this suggests that the corresponding derivativematrices are numerically ill-conditioned. It has been shown

that these algorithms converge slowly when rank-deficien-cies appear in the Jacobian matrix of an ANN, makingthe problem numerically ill-conditioned [19]. Furthermore,like in all local minimization algorithms, training of ANNwill be interrupted without success at local minima of theerror function if the latter, due to the complexity of theproblem at hand, happens to be non-convex. The introduc-tion of momentum terms in the local step may help thealgorithm to avoid premature stop at local minima.

2.3. Local adaptive techniques

In order to improve the performance of weight updat-ing, two completely different approaches have been pro-posed, namely Quickprop [20] and Rprop [21].

2.3.1. The quickprop methodThis method is based on a heuristic learning algorithm

for a multi-layer perceptron, developed by Fahlman [20],which is partially based on the Newton’s method. Quick-prop is one of most frequently used adaptive learning par-adigms. The weight updates are based on estimates of theposition of the minimum for each weight, obtained by solv-ing the following equation for the two following partialderivatives:

oE t�1

owijand

oE t

owijð7Þ

and the weight update is implemented as follows:

DwðtÞij ¼oEtowij

oEt�1

owij� oEt

owij

Dwðt�1Þij ð8Þ

The learning time can be remarkably improved comparedto the global adaptive techniques.

2.3.2. The Rprop method

Another heuristic learning algorithm with locally adap-tive learning rates based on an adaptive version of theManhattan-learning rule and developed by Riedmillerand Braun [21] is the Resilient backpropagation abbrevi-ated as Rprop. The weight updates can be written

DwðtÞij ¼ �gðtÞij sgnoE t

owij

� �ð9Þ

where

gðtÞij ¼

minða � gðt�1Þij ; gmaxÞ; if oEt

owij� oEt�1

owij> 0

maxðb � gðt�1Þij ; gminÞ; if oEt

owij� oEt�1

owij< 0

gðt�1Þij ; otherwise

8>>><>>>:

ð10Þ

in which a = 1.2, b = 0.5, gmax = 50 and gmin = 0.1 [22].The learning rates are bounded by upper and lower limitsin order to avoid oscillations and arithmetic underflow. Itis interesting to note that, in contrast to other algorithms,


Rprop employs information about the sign and not themagnitude of the gradient components.

3. Structural identification

Inverse analysis concerns the determination of materialor other data (in general, parameter identification) of astructure from measurements of its mechanical responsecaused by a given loading. Dynamic loadings give the mostpromising results and are used in many non-destructiveevaluation procedures in industry (ultrasonic evaluation,ambient vibration or modal testing). Another ‘ambitious’goal of identifying when the status of a structure has beenchanged beyond a critical limit or not, is structural healthmonitoring which has great practical importance.

The inverse crack or damage identification problem hasthe following general formulation as an output-error min-imization problem: (a) the unknown quantities like numberand type of defects, their position and other geometricparameters are expressed with the help of certain variables,(b) a number of mechanical tests are considered, (c) foreach value of the unknown defect parameters the corre-sponding responses of the structure are considered andcompared with the target (measured) responses. Let z bethe vector of unknown parameters involved in the mechan-ical problem, u0 be the measured response of the mechani-cal system and u(z) be the response of the mechanicalsystem for a fixed value of z. The minimization problemreads

min z :1

2ðuðzÞ � u0ÞT �M � ðuðzÞ � u0Þ

� �ð11Þ

where M is an appropriate symmetric and positive semi-definite weight matrix. The implicit nonlinear dependenceof the structural response u on the unknown parametersz, i.e., u(z), makes the above-mentioned least square mini-mization problem complicated. In fact, the error functionmay become non-convex, with local minima, in the caseof difficult and complicated inverse problems where no suf-ficient estimate of the solution is available.

Several methods have been presented for the effectivenumerical solution of this problem (among others, numer-ical optimization, genetic algorithms, soft computing). Acomparison on crack identification problems in elasticityhas been presented in [16]. Summarizing the results onecould state that a method based on genetic optimization,a method of global optimization, which is able to avoidlocal minima, will solve the problem in almost all cases[16]. The usage of more than one loading cases may be nec-essary and certainly makes the solution easier (compare thethree-dimensional defect identification using static datapresented in [23]). On the other hand, the computerresources needed for the use of genetic algorithms arealmost prohibitive for a regular use in an industrial envi-ronment. A cheaper approach, computationally, is basedon the artificial neural network methodology or on filter-

based optimization techniques (for recent results see[24,25]).

ANNs (usually feedforward multi-layer networkstrained with various back-propagation techniques) areused for the direct approximation of the mapping betweenmeasurements and unknown parameters (the so-calledinverse mapping). They have been used with great effi-ciency for the post-processing of impact-echo data intwo-dimensional elastic structures [16]. Other practicalapplications of this method include the crack-depth deter-mination of a vertical crack emanating from the hiddensurface of a plate from ultrasonic back-scattering data[26] and the depth determination of surface-breakingcracks [27].

In the case that the aforementioned optimization prob-lem has local minima the ANN approach alone may beineffective, while the genetic optimization algorithm maystill be effective. From the discussion of training algorithmsin the previous sections, this result could be expected; sincean optimization problem is hidden within the trainingphase of the ANN and this (distributed and parallel opti-mization) technique is able to overcome only moderatenon-convexity and may stop at local minima. In this papera hybrid strategy made of two components is proposed.First a neural network is used for the replacement of thedirect mechanical model. This part has already been usedin the community of structural optimization, it is knownas response surface method and it is especially beneficialfor demanding modeling tasks like dynamical or computa-tional fluid mechanics simulations. The approximation ofthe response mapping is, in general, easier than the approx-imation of the inverse mapping. The resulting model of themechanical system is used in connection with the geneticalgorithm for the solution of the inverse problem. At thispoint the efficiency of the genetic algorithm is enhanced,since the neural network approximator of the structuralresponse is usually far less demanding in terms of compu-tational cost than evaluating the actual mechanical model.

4. Seismic probabilistic analysis

In the design of structural systems, limiting uncertaintiesand increasing safety is an important issue to be consid-ered. Probabilistic analysis of structures, which is definedas the probability that the system meets some specifieddemands for a specified time period under specified envi-ronmental conditions, is used as a probabilistic measureto evaluate the reliability of structural systems. The perfor-mance function of a structural system must be determinedto describe the system’s behavior and to identify the rela-tionship between the basic parameters in the system. Itshould be noted that in the earthquake loading environ-ment the uncertainties related to seismic demand and struc-ture’s capacity are strongly coupled.

The probability of exceedance pexc can be determinedusing a time invariant probabilistic analysis procedure withthe following expression


pexc ¼ p½R < S� ¼Z 1

�1F RðtÞfSðtÞdt

¼ 1�Z 1

�1F SðtÞfRðtÞdt ð12Þ

where R denotes the structure’s bearing capacity and S theexternal loads. The randomness of R and S can be de-scribed by known probability density functions fR(t) andfS(t), with FR(t) = p[R < t], FS(t) = p[S < t] being the cumu-lative probability density functions of R and S,respectively.

Most often a limit state function is defined asG(R,S) = S � R and the probability of exceedance is givenby

pexc ¼ p½GðR; SÞP 0� ¼Z

GP0

fRðRÞfSðSÞdR dS ð13Þ

It is practically impossible to evaluate pexc analyticallyfor complex and/or large-scale structures. In such casesthe integral of Eq. (13) can be calculated only approxi-mately using either simulation methods, such as the MonteCarlo Simulation (MCS), or approximation methods likethe first order reliability method (FORM) and the secondorder reliability method (SORM), or response surfacemethods (RSM). Despite its high computational cost,MCS is considered as one of the most efficient methods,either for comparison with other methods or as a stand-alone probabilistic analysis tool.

4.1. Monte Carlo simulation

In probabilistic analysis the MCS method is oftenemployed when the analytical solution is not attainable.This is mainly the case in problems of complex nature witha large number of basic variables where all other probabi-listic analysis methods are not applicable. Expressing thelimit state function as G(x) < 0, where x = (x1,x2, . . . ,xM)is the vector of the random variables, Eq. (13) can be writ-ten as

pexc ¼Z

GðxÞP0

fxðxÞdx ð14Þ

where fx(x) denotes the joint probability of exceedance forall random variables. Since MCS is based on the theory oflarge numbers (N1) an unbiased estimator of the probabil-ity of exceedance is given by

pexc ¼1

N1

XN1j¼1

IðxjÞ ð15Þ

in which I(xj) is a binary indicator for exceeding or not sim-ulations defined as

IðxjÞ ¼1 if GðxjÞP 0

0 if GðxjÞ < 0

�ð16Þ

In probabilistic analysis using simulation methods it isimportant to evaluate the probability of exceedance for agiven performance function efficiently and accurately. Inorder to estimate pexc an adequate number of Nsim indepen-dent random samples is produced using a specific, usuallyuniform, probability density function of the vector x. TheMonte Carlo estimation of pexc is given in terms of samplemean by

pexc ffiNH

N sim

ð17Þ

where NH is the number of exceeding simulations. In orderto reduce the number of simulations and the computationalcost of the standard MCS many efficient sampling reductiontechniques have been used [12,28]. In the present study thereis no need of implementing such techniques, since ANNs re-place the computationally expensive structural responseevaluations and the sample size has trivial importance.

4.2. Structural design under seismic loading

The equations of equilibrium for a finite element systemin motion can be written in the usual form

M€uðtÞ þ C _uðtÞ þ KuðtÞ ¼ RðtÞ ð18Þwhere M, C, and K are the mass, damping and stiffnessmatrices; R(t) is the external load vector, whileuðtÞ; _uðtÞ and €uðtÞ are the displacement, velocity, and accel-eration vectors of the finite element assemblage, respec-tively. A design approach based on the Multi-modalResponse Spectrum MmRS) analysis, which, in turn, isbased on the mode superposition approach, has been usedin the present study [28].

The MmRS method is based on a simplification of themode superposition approach with the aim to avoid timehistory analyses which are required by both, the direct inte-gration and mode superposition approaches. In the case ofthe multi-modal response spectrum analysis Eq. (18) ismodified according to the modal superposition approachto a system of independent equations

Mi€yiðtÞ þ Ci _yiðtÞ þ KiyiðtÞ ¼ RiðtÞ ð19Þ

where

Mi ¼ UTi MUi; Ci ¼ UT

i CUi;

Ki ¼ UTi KUi and RðtÞ ¼ UT

i RðtÞ ð20Þ

are the generalized values of the corresponding matricesand the loading vector, while Ui is the ith eigenmode shapematrix. According to the modal superposition approachthe system of N differential equations, which are coupledwith the off-diagonal terms in the mass, damping and stiff-ness matrices, is transformed to a set of N independent nor-mal-coordinate equations. The dynamic response cantherefore be obtained by solving separately for the responseof each normal (modal) coordinate and by superposing theresponse in the original coordinates.

Table 1Input and output variables for the two levels of approximation

Test case First level of approximation (NN1) Second level of approximation (NN2)

Inputs Outputs Inputs Outputs

(a) E Ti, i = 1, . . . , 8 Rdx(Ti), Rdy(Ti), i = 1, . . . , 8 Max drift hmax

(b) bi, hi, i = 1, . . . , 5 Ti, i = 1, . . . , 8 Rdx(Ti), Rdy(Ti), i = 1, . . . , 8 Max drift hmax

(c) E, bi, hi, i = 1, . . . , 5 Ti, i = 1, . . . , 8 Rdx(Ti), Rdy(Ti), i = 1, . . . , 8 Max drift hmax

Fig. 1. Configuration of a composite plate with a defect.


In the MmRS analysis a number of different formulashave been proposed to obtain reasonable estimates of themaximum response based on the spectral values withoutperforming time history analyses for a considerable num-ber of transformed dynamic equations. The simplest andmost popular one is the Square Root of Sum of Squares(SRSS) of the modal responses. According to this estimatethe maximum total displacement is approximated by

umax ¼XN

i¼1

u2i;max

!1=2

ui;max ¼ Uiyi;max

ð21Þ

where ui,max corresponds to the maximum displacementvector corresponding to the ith eigenmode.

4.3. The ANN training for seismic probabilistic analysis

In the present implementation the main objective is toinvestigate the ability of the ANNs to predict the structuralperformance in terms of maximum interstorey drift. Theselection of appropriate I/O training data is an importantpart of the ANN training process. Although the numberof training patterns may not be the only concern, the distri-bution of samples is of greater importance. In the presentstudy the sample space for each random variable is dividedinto equally spaced distances in order to select suitabletraining pairs. Having chosen the ANN architecture andtested the performance of the trained network, predictionsof the probability of exceedance of the design limit-statecan be made quickly. The results are then processed bymeans of MCS to calculate the probability of exceedancepexc of the design limit-state using Eq. (15).

The modulus of elasticity, the dimensions b and h of theI-shape cross section and the seismic loading have beenconsidered as random variables. Three alternative testcases are considered depending on the random variablesconsidered: (a) the modulus of elasticity and the earth-quake loading are considered to be random variables, (b)the dimensions b and h of the I-shape cross section andearthquake loading are taken as random variables and (c)all three groups of random variables are consideredtogether. For the implementation of the ANN-based meta-model in all three test cases a two level approximation isemployed using two different ANNs. The first ANN pre-dicts the eigenperiod values of the significant modes. Theinputs of the ANN are the random variables while the out-puts are the eigenperiod values. The second ANN is used to

predict the maximum interstorey drift, which is used todetermine whether a limit-state has been violated. There-fore, the spectral acceleration values of both X and Y direc-tions are the input values of the ANN while the maximuminterstorey drift is the output. The input and output vari-ables for the two levels of approximation are shown inTable 1.

5. Numerical examples

5.1. Defect identification in a composite beam using static

measurements

This test example consists of a clamped beam subjectedto a distributed loading. The beam is equipped with piezo-electric actuators and sensors, which can be used for defectidentification (see schematic configuration in Fig. 1). Mea-surements of the deflections from the piezoelectric sensorsare used for the identification of the defect. The finite ele-ment model has been developed and tested for static anddynamic optimal control problems in [29,30], respectively,where further technical details can be found. The elastichost beam has a length of 0.30 m, width equal to 0.04 mand depth equal to 0.0096 m with elasticity constantsE = 150 · 109 Pa and m = 0.30. On the top and the bottomof the beam piezoelectric layers with thickness equal to0.0002 m are perfectly bonded and used as sensors or actu-ators. The elastic constants of the piezoelectric layers areE = 63 · 109 Pa and m = 0.30 and the piezoelectric constantd31 = 254 · 10�12 m/V. The beam has been discretized with30 finite elements. A defect is approximately modelled byreduction of the stiffness in the corresponding finite elementwith a smeared-crack-like approach. The first assumptionis that the complete deformation of the beam can be mea-sured by using the distributed piezoelectric sensors or othersuitable measurements. The second assumption is that onlyone defect will arise, therefore the two unknown variablesof the defect identification problem are: (a) the position(number of damaged element) and (b) the extent of thedefect. For the position it is assumed that this continuousvariable takes values in the interval [1,30], which indicates

Table 2Identification of the position and size of one defect in the composite beam using static measurements

Example no. Exact position of defect Calculated position of defect Exact extent of damage (%) Calculated extent of damage Fitness value

1 5 4.9860 10 9.9915 42.93952 15 15.1133 10 10.078 41.00513 25 25.2182 10 10.463 42.9711


the damaged element (after appropriate trimming togive the discrete value corresponding to the element num-ber). The extent of damage is approximated by a continu-ous variable, taking values in the interval [1,10], which islinearly dependent on the extent of damage (between 0%and 50% of the nominal stiffness, respectively).

The direct approximation of the inverse mapping (i.e.,the relation between measurements and damage variable)with neural networks does not lead to useful results. In par-ticular the procedure developed in [16,17], for static prob-lems, which has been extended for the treatment ofdynamic problems in [24,25] has been used. The problemis that the resulting inverse mapping is too complicated.Therefore every attempt to train a suitable neural networkresulted to problems with overtraining, where the trainednetwork is able to reproduce the reply of the known databut is unable to generalize and produce reasonableresponses for the unknown inputs. The genetic algorithmsolves the inverse problem. Although this problem is rathersmall, the mechanical problem can be replaced by aresponse function approach, which is based either on inter-

Fig. 2. Documentation of one genetic optimization solution with bestindividual and mean value for each generation.

Table 3Damage identification of a defect at element no. 15 with extent of damage eq

Example no. Number of measurements Calculated positio

1 30 15.00092 15 14.97833 7 15.0058

polation functions or on neural networks. The accuracy ofthe inverse problem is not influenced by either of the twoapproaches while the required computational time isreduced. For a constant population size equal to 15 thesolution of the inverse problem is documented in the fol-lowing Table 2. The fitness variable transforms the errorminimization problem to a maximization problem andincludes a logarithmic scaling. The fitness maximizationfor this example is demonstrated in Fig. 2.

In this example all vertical displacements of the beam,i.e., 30 measurements, have been taken into account forthe solution of the inverse problem. By reducing the num-ber of measurements (again, equally distributed along thelength of the cantilever) the solution of the inverse problemis not affected, as it can be shown in Table 3.

5.2. Crack identification problem

In this example an orthogonal plane stress domain rig-idly supported at the lower boundary and loaded at theupper boundary is considered. From measurements at thefree boundaries of the plate one can identity cracks andholes in the interior of the plate [24]. A schematic descrip-tion of the problem is shown in Fig. 3. First one example ofthe neural network based solution of the inverse problem ispresented. Using a set of 206 measurements of displace-ments at the boundary of the plate in several representativetime instances a suitably trained 206-7-2 neural networkcan be trained to reproduce the position of a given crack(horizontal, of given length) with acceptable accuracy.Here a boundary element model is used for the productionof the training and test examples. A set of 81 training and64 test examples has been used. In the following figures(Figs. 4–10) the training procedure, the accuracy of predict-ing the training data and of the test data are demonstrated.

In order to enhance the accuracy of the results the sameproblem is solved with the genetic algorithm optimizationprocedure. In fact, in more complicated forms of the plate,only the genetic optimization can solve the inverse problem(see [23] for relevant investigations). For the technical real-ization of this concept the genetic optimization tool can be

ual to 25% loss of stiffness: effect of reduced number of measurements

n of defect Calculated extent of defect Fitness value

25.0035 45.471824.9285 41.846625.0155 46.3440

Fig. 3. Configuration of a plate with a defect. Measurements at theexternal boundary can be used for defect identification.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 510-4

10-3

10-2

10-1

100

101

5 Epochs

Trai

ning

-Blu

e G

oal-B

lack

Performance is 0.000455743, Goal is 001

Fig. 4. Training of the neural network for the defect identificationproblem.

0 10 20 30 40 50 60 70 80 90 1000

10

20

30

40

50

60

70

80

90

100

Fig. 5. Training data: Predicted (+) and real (o) position of the defect fordifferent places of the defect.

10 20 30 40 50 60 70 80 900

10

20

30

40

50

60

70

80

90

100

T

A

Best Linear Fit: A = (1) T + (-0.0919)

R = 0.999

Data PointsBest Linear FitA = T

Fig. 6. Training data: Accuracy of x-coordinate prediction.


combined with the mechanical modeling software, if this ispossible (for example if the analysis code is open, it is writ-ten in a compatible computer language, etc.). The require-ments on computer time and storage may become verylarge. Alternatively one may use the proposed hybrid strat-egy. First, a number of small neural networks approximatethe response of each boundary point of interest as a func-tion of the crack position (for all other parameters of themechanical problem, like loading or boundary conditions,fixed). A similar performance can be obtained by usingclassical interpolation functions. The genetic optimization

uses then the approximators to construct the fitness func-tion and proceeds with the solution of the output-errorminimization problem and, eventually, of the inverse prob-lem as in the previous example.

5.3. Seismic probabilistic analysis test example

The six storey space frame, shown in Fig. 11, has beenconsidered for the purpose of the current study in orderto assess the proposed metamodel assisted structural prob-abilistic analysis methodology. The space frame consists of63 structural elements which are divided into five groupshaving the following cross sections: (1) HEB650, (2)HEB650, (3) IPE450, (4) IPE400 and (5) HEB450. Thestructure is loaded with a permanent action of G = 3 kPa

10 20 30 40 50 60 70 80 900

10

20

30

40

50

60

70

80

90

T

ABest Linear Fit: A = (1.04) T + (-2.38)

R = 0.998


Fig. 10. Test data: Accuracy of y-coordinate prediction.

10 20 30 40 50 60 70 80 900

10

20

30

40

50

60

70

80

90

100

T

A

Best Linear Fit: A = (0.996) T + (0.169)

R = 0.998


Fig. 7. Training data: Accuracy of y-coordinate prediction.

0 10 20 30 40 50 60 70 80 90 1000

10

20

30

40

50

60

70

80

90

100

Fig. 8. Test data: Predicted (+) and real (o) position of the defect fordifferent places of the defect.

10 20 30 40 50 60 70 80 9010

20

30

40

50

60

70

80

90

T

A

Best Linear Fit: A = (1.01) T + (-0.503)

R = 0.999

Data PointsBest Linear Fi A = T

Fig. 9. Test data: Accuracy of x-coordinate prediction.


and a variable action of Q = 5 kPa. In addition, in order totake into account the nonlinear behavior of the structurewhen the MmRS method is used the seismic loads arereduced by the behavior factor q = 4.0 as Eurocode 8(EC8) suggests for steel frames [31]. The three test casescorresponding to different combinations of random vari-ables are considered here.

The most common approach for the definition of theseismic input is the use of design code response spectrum.This is a general approach, which is easy to implement.However if higher precision is required the use of spectraderived from natural earthquake records is more appropri-ate. Since significant dispersion on the structural responsedue to the use of different natural records has beenobserved, these spectra must be scaled to the same desired

earthquake intensity. The most commonly applied scalingprocedure is based on the peak ground acceleration (PGA).

In this study a set of nineteen natural accelerograms,shown in Table 4, is used. Each record corresponds to dif-ferent earthquake magnitudes and soil properties. Thesetime histories are from different earthquakes. Two are fromthe 1992 Cape Mendocino earthquake, two are from the1978 Tabas, Iran earthquake and fifteen are from the1999 Chi–Chi, Taiwan earthquake. The records are scaled,to the same peak ground acceleration of 0.32 g in order toensure compatibility between the records. The responsespectra for each scaled record, in x and y directions, areshown in Figs. 12 and 13, respectively. The type of proba-bility density functions, mean values and standard devia-tions for all variables are presented in Table 5.

Fig. 11. The six-storey space frame.

Table 4List of the natural records

Earthquake Station Distance Site

Tabas 16 September 1978 Dayhook 14 rockTabas 1.1 rock

Cape Mendocino 25 April 1992 Cape Mendocino 6.9 rockPetrolia 8.1 soil

Chi–Chi 20 September 1999 TCU052 1.4 soilTCU065 5.0 soilTCU067 2.4 soilTCU068 0.2 soilTCU071 2.9 soilTCU072 5.9 soilTCU074 12.2 soilTCU075 5.6 soilTCU076 5.1 soilTCU078 6.9 soilTCU079 9.3 soilTCU089 7.0 rockTCU101 4.9 soilTCU102 3.8 soilTCU129 3.9 soil


It has been observed that the response spectra follow thelognormal distribution [32]. Therefore, the median spec-trum x̂, also shown in Figs. 12 and 13, and the standarddeviation d are calculated from the above set of spectrausing the following expressions:

x̂ ¼ exp

Pni¼1 lnðRd;iðT ÞÞ

n

� �ð22Þ

d ¼Pn

i¼1ðlnðRd;iðT ÞÞ � lnðx̂ÞÞ2

n� 1

" #1=2

ð23Þ

where Rd,i(T) is the response spectrum value for periodequal to T of the ith record (i = 1, . . . ,n, where n = 19 in

this study). For a given period value, the acceleration Rd

is obtained as a random variable following the log-normaldistribution whose mean value is equal to x̂ and standarddeviation is equal to d.

In the first test case where the modulus of elasticity andthe earthquake loading are considered as random variablesone hundred values of the modulus of elasticity are selectedin order to train the first ANN and one hundred combina-tions of the spectral values are selected for the trainingof the second ANN. Ten out of them are selected fortesting the generalization capabilities of the trained ANNs.In the second test case the training–testing set was com-posed by one hundred fifty pairs while in the case of thethird test case the set was composed by two hundred pairs.

As mentioned previously, a two level ANN-basedapproximation was used in all three test cases examined.As shown in Fig. 14, both ANNs consisted of three layers:one input, one hidden, and one output layer with varyingnumber of nodes per layer. After an initial investigationon the optimum number of hidden layers and their nodes,one hidden layer was used for both ANN configurationshaving 10 and 20 nodes, respectively. The input and theoutput data of the first ANN are the random variablesand the eigenperiod values of the significant modes, respec-tively. It has been observed that for the six storey testexample considered, the number of significant eigenperiodsis eight. For the second ANN the input data are the corre-sponding sixteen spectral acceleration values of both X andY directions, while the output is one, i.e., the maximuminterstorey drift value, which defines the limit-state viola-tion. Thus, the two level ANN configurations that wereused are the following: (i) For the first test case: NN1: 1-10-8 and NN2: 16-20-1, (ii) For the second test case:

0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

16.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0

T (sec)

SA (m

/sec

2 )

Dayhook TabasCape Mendocino PetroliaTCU052 TCU065TCU067 TCU068TCU071 TCU072TCU074 TCU075TCU076 TCU078TCU079 TCU089TCU101 TCU102TCU129 Median X

Fig. 12. Natural record response spectra and their median (x-axis).

0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0

T (sec)

SA (

m/s

ec2 )

Dayhook TabasCape Mendocino PetroliaTCU052 TCU065

TCU067 TCU068TCU071 TCU072TCU074 TCU075TCU076 TCU078

TCU079 TCU089

TCU101 TCU102TCU129 Median Y

Fig. 13. Natural record response spectra and their median (y-axis).

Table 5Characteristics of the random variables

Randomvariable

Probability densityfunction

Meanvalue

Standarddeviation

E N 210 10 (%)b N b* 2 (%)h N h* 2 (%)Seismic load Log-N x̂, (Eq.

(22))d (Eq. (23))

* Dimensions from the IPE and HEB databases.

output layer hidden layerinput layer

Fig. 14. A typical neural network configuration.


NN1: 10-20-8 and NN2: 16-20-1 and (iii) For the third testcase: NN1: 11-10-8 and NN2: 16-20-1.

The influence of the three groups of random variableswith respect to the number of simulations is shown in

0.00

0.50

1.00

1.50

2.00

2.50

0 50 100

200

500

1000

2000

5000

1000

020

000

5000

0

1000

00

Simulations

p exc

(%)

Mod. Elast., Earth.Cross Sect., Earth.Mod. Elast., Cross Sect., Earth.

Fig. 15. Influence of the number of MC simulations on the value of pexc for the three test cases.

Table 6‘‘Exact’’ and predicted values of pexc and the required CPU time

Number of simulations Test case 1 Test case 2 Test case 3

‘‘exact’’ Pexc (%) NN Pexc (%) ‘‘exact’’ Pexc (%) NN Pexc (%) ‘‘exact’’ Pexc (%) NN Pexc (%)

50 0.00 0.00 0.00 0.00 2.00 0.00100 1.00 2.00 1.00 2.00 2.00 1.00200 2.00 1.60 1.00 1.00 1.00 3.00500 1.60 1.26 1.00 1.70 0.40 0.701000 0.90 0.81 0.90 0.67 1.00 0.862000 1.15 1.06 1.45 1.41 1.10 0.975000 1.12 1.04 1.30 1.24 1.32 1.1810,000 1.09 1.03 1.21 1.14 1.42 1.2920,000 1.21 1.14 1.25 1.31 1.31 1.1950,000 1.26 1.16 1.22 1.31 1.36 1.21100,000 1.28 1.16 1.24 1.31 1.37 1.21

CPU time in seconds

Pattern selection – 2 – 3 – 5Training – 7 – 9 – 12Propagation – 25 – 25 – 25Total 1154 34 1154 37 1154 42


Fig. 15. It can be seen that 20–50 thousand simulations arerequired in order to calculate with an adequate accuracythe probability of exceedance of the design limit state. Itis also observed that considering both modulus of elasticityand the cross section dimensions as random variables leadsto an increase of 7% of the value of the probability ofexceedance (1.37% for the third test case instead of 1.28%of the first and 1.24% of the second test case).

Once an acceptable trained ANN in predicting the max-imum drift is obtained, the probability of exceedance foreach test case is estimated by means of ANN-based MonteCarlo Simulation. The results for various numbers of sim-ulations are depicted in Table 6 for the three test casesexamined. From these results it can be observed that, in

the case of basic MCS, the error of the predicted probabil-ity of failure with respect to the ‘‘exact’’ one is rather mar-ginal. On the other hand, the computational cost isdrastically decreased, approximately 30 times, for all testcases.

6. Conclusions

This paper presented applications of soft computingtechniques, involving ANNs, in computationally demand-ing tasks in mechanics. For inverse and parameter identifi-cation problems there exist, in principle, the possibility touse directly ANNs for the approximation of the inversestructural mapping. Nevertheless, this mapping becomes


complicated for real-life applications and the training ofthe ANNs may be difficult or even impossible. In contrast,the direct mapping relating unknown parameters withstructural response is much easier to be approximated. Thiscan be done very efficiently with ANNs. The resultingtrained ANN can be combined with a powerful numericaloptimization algorithm, like genetic algorithms, for thesolution of the inverse problem. This approach has beendemonstrated in this paper. On the other hand, the compu-tational effort involved in the conventional MCS becomesexcessive in large-scale problems especially when earth-quake loading is considered because of the enormous sam-ple size and the computing time required for each MonteCarlo run. The use of ANNs can practically eliminateany limitation on the scale of the problem and the samplesize used for MCS.

Acknowledgement

Partial support from the Greek-Italian Bilateral Scien-tific Cooperation Project is gratefully acknowledged.

References

[1] Tsompanakis Y, Lagaros ND, Stavroulakis GE. Efficient neuralnetwork models for structural reliability analysis and identificationproblems. In: Proceedings of the eighth international conference onthe application of artificial intelligence to civil, structural andenvironmental engineering, Topping BHV (Ed), Civil-Comp Press,Stirling, United Kingdom, Paper 41, 2005.

[2] Stavroulakis GE, Bolzon G, Waszczyszyn Z, Ziemianski L. Inverseanalysis. In: Karihaloo B, Ritchie RO, Milne I, editors in Chief.Comprehensive structural integrity. In: de Borst R, Mang HA,volume editors. Numerical and computational methods, vol. 3,Elsevier Publishers, 2003, p. 685–718 [Chapter 13].

[3] Gurley KR, Tognarelli M, Kareem A. Analysis and simulation toolsfor wind engineering. Prob Eng Mech 1997;12:9–31.

[4] Adeli H, Park HS. Optimization of space structures by neuralnetworks. J Neural Networks 1995;8:769–82.

[5] McCorkle DS, Bryden KM, Carmichael CG. A new methodology forevolutionary optimization of energy systems. Comp Meth Appl MechEng 2003;192:5021–36.

[6] Papadrakakis M, Lagaros ND. Reliability-based structural optimi-zation using neural networks and Monte Carlo simulation. CompMeth Appl Mech Eng 2002;191:3491–507.

[7] Sakata S, Ashida F, Zako M. Structural optimization using Krigingapproximation. Comp Meth Appl Mech Eng 2003;192:923–39.

[8] Zhang L, Subbarayan G. An evaluation of back-propagation neuralnetworks for the optimal design of structural systems: Part I. Trainingprocedures. Comp Meth Appl Mech Eng 2002;191:2873–86.

[9] Liu GR, Xu YG, Wu ZP. Total solution for structural mechanicsproblems. Comp Meth Appl Mech Eng 2001;191:989–1012.

[10] Zacharias J, Hartmann C, Delgado A. Damage detection on crates ofbeverages by artificial neural networks trained with finite-elementdata. Comp Meth Appl Mech Eng 2004;193:561–74.

[11] Hurtado JE, Alvarez DA. Neural-network-based reliability analysis:a comparative study. Comp Meth Appl Mech Eng 2002;191:113–32.

[12] Papadrakakis M, Papadopoulos V, Lagaros ND. Structural reliabil-ity analysis of elastic–plastic structures using neural networks andMonte Carlo simulation. Comp Meth Appl Mech Eng1996;136:145–63.

[13] Hwang JN. Nonparametric multivariate density estimation: a com-parative study. IEEE Trans Signal Process 1994;42:2795–810.

[14] Tsompanakis Y, Lagaros ND, Papadrakakis M. Reliability analysisof structures under seismic loading. In: Mang HA, RammerstorferFG, Eberhardsteiner J, editors. Proc. 5th world congress on compu-tational mechanics WCCM-V, Vienna, Austria, July 7–12, 2002.

[15] Stavroulakis GE, Antes H. Flaw identification in elastomechanics:BEM simulation with local and genetic optimisation. Struct Optim1998;16(2/3):162–75.

[16] Stavroulakis GE. Inverse and crack identification problems inengineering mechanics. Habilitation Thesis, Technical UniversityBraunschweig. Kluwer Academic Publishers – Springer Verlag,Dordrecht, Boston, London, 2000.

[17] Engelhardt M, Likas A, Stavroulakis GE. Neural crack identification.In: Mang HA, Rammerstorfer FG, Eberhardsteiner J, editors. Proc.5th world congress on computational mechanics WCCM-V, Vienna,Austria, July 7–12, 2002.

[18] Schiffmann W, Joost M, Werner R. Optimization of the back-propagation algorithm for training multi-layer perceptrons. TechnicalReport, Institute of Physics, University of Koblenz, 1993.

[19] Lagaros ND, Papadrakakis M. Learning improvement of neuralnetworks used in structural optimisation. Adv Eng Software2004;35:9–25.

[20] Fahlman S. An empirical study of learning speed in back-propagationnetworks. Technical Report, Carnegie Mellon: CMU-CS-88-162,1988.

[21] Riedmiller M, Braun H. A direct adaptive method for faster back-propagation learning: The RPROP algorithm. In: Ruspini H, editor.Proc. of the IEEE international conference on neural networks(ICNN), San Francisco, 1993, p. 586–91.

[22] Riedmiller M. Advanced supervised learning in multi-layer percep-trons: from back-propagation to adaptive learning algorithms.Technical Report, University of Karlsruhe: W-76128 Karlsruhe, 1994.

[23] Engelhardt M. Numerische Verfahren zur Identifizierung von Fehlst-ellen aus Randdaten. PhD Thesis, Carolo Wilhelmina TechnicalUniversity, Braunschweig, Germany, Braunschweig Series onMechanics No. 56, 2004.

[24] Stavroulakis GE, Engelhardt M, Likas A, Gallego R, Antes H.Neural network assisted crack and flaw identification in transientdynamics. J Theor Appl Mech Polish Acad Sci 2004;42(3):629–49.

[25] Engelhardt M, Stavroulakis GE, Antes H. Crack and flaw identifi-cation in elastodynamics using Kalman filter techniques. ComputMech 2006;37(3):249–65.

[26] Oishi A, Yamada K, Yoshimura A, Yagawa G. Quantitativenondestructive evaluation with ultrasonic method using neuralnetworks and computational mechanics. Comput Mech1995;15:521–33.

[27] Kitahara M, Achenbach JD, Guo QC, Peterson ML, Ogi T, NotakeM. Depth determination of surface-breaking cracks by a neuralnetwork. Rev Progr Qual Nondestructive Eval 1991;10:689–96.

[28] Papadrakakis M, Tsompanakis Y, Lagaros ND, Fragiadakis M.Reliability based optimization of steel frames under seismic loadingconditions using evolutionary computation. J Theor Appl MechPolish Acad Sci 2004;42(3):585–608.

[29] Hadjigeorgiou EP, Foutsitzi G, Stavroulakis GE. Shape control ofbeams with piezoelectric actuators. HERMIS Int J 2004;5:65–78.

[30] Stavroulakis GE, Foutsitzi G, Hadjigeorgiou E, Marinova D,Baniotopoulos CC. Design and robust optimal control of smartbeams with application on vibration suppression. Adv Eng Software2005;36(11–12):806–13.

[31] Eurocode 8: Design provisions for earthquake resistant structures.CEN, ENV, 1998-1-1/2/3, 1994.

[32] Chintanapakdee C, Chopra AK. Evaluation of modal pushoveranalysis using generic frames. Earth Eng Str Dyn 2003;32:417–42.

Soft computing techniques in parameter identiﬁcation and ... · considered, namely parameter...

Documents

Transcript of Soft computing techniques in parameter identiﬁcation and ... · considered, namely parameter...