JOURNAL OF THEORETICAL AND APPLIED MECHANICS 42, 3, pp. 667-693, Warsaw 2004 APPLICATION OF NEUROCOMPUTING IN THE PARAMETRIC IDENTIFICATION USING DYNAMIC RESPONSES OF STRUCTURAL ELEMENTS – SELECTED PROBLEMS Leonard Ziemiański Bartosz Miller Grzegorz Piątkowski Department of Structural Mechanics, Rzeszów University of Technology e-mail: ziele@prz.edu.pl Some problems of neurocomputing in the dynamics of structures are presented: 1) damage detection using wave propagation, 2) updating of portal frames finite elementmodels, 3) detection of the void and additio- nalmass in cantileverplates, 4) neuralnetworkmodellingof an”artificial boundary condition”. The analysed problems are related to both data prepared by computational systems and that taken from experimental evidence. Key words: neural networks dynamics, identification 1. Introduction In recent years, interest in Artificial Neural Networks (ANN) has grown rapidly (Haykin, 1999; Waszczyszyn, 1999). The main reason is because of their powerful and adaptive abilities to treat various complex problems. Such a co-disciplinary approach can give solutions which are difficult to achieve by methods specializing in the analysis of inward-disciplinary oriented problems. Development of computer hardware and software causes, beyond doubt, for- mulation of new nonstandard methods of data processing. A wide range of applications were evoked bywell known advantages of ANNswhich offer com- plementary possibilities to computer sequential processing. This corresponds, first of all, to computer simulations of ANNs, sometimes called ”neurocom- puting”. 668 L.Ziemiański et al. In the paper, some results related to research done at the Faculty of Ci- vil and Environmental Engineering of Rzeszów University of Technology are discussed in short. Only selected problems are reported, corresponding to dynamics of structures. In the paper, the main attention is given to Back- Propagation Neural Networks which are mostly used in the analysis of me- chanical problems. The achieved results enable us to point out promising prospects of neurocomputing applications in dynamics of structures. Back- PropagationNeural Network (BPNN) applications in analysis of the following dynamics problems are discussed: (1) damage detection using wave propaga- tion (Ziemiański and Piątkowski, 2000), (2) updating of portal frames finite element models (Miller and Ziemiański, 2001, 2003), (3) detection of the void and additional mass in cantilever plates (Piątkowski, 2003; Piątkowski and Ziemiański, 2003), (4) neural network modelling of an ”artificial boundary condition” for infinite domains (Ziemiański, 2003). The analysed problems are related to both data prepared by computational systems and that taken from experimental evidence. Much attention was focused on preprocessing of input data and selection of the best type of a neural network and correct architec- ture. The influence of random noise inserted into data patterns on the neural network generalization process was analyzed. The results of performed calcu- lations show that adding the noise to learning patterns significantly decreases values of MSE errors and increases resistance to the disturbance. 2. Damage detection using wave propagation Non-destructivemethods of detection of change ofmaterial properties and damage in structural elements present an important and valuable tool. The- se methods allow estimating the state of a structure as well as predicting a period of safety usage. An ultrasonic method is one of the most often used non-destructive methods (Thompson, 1983). For structures such as rods, pla- tes, shells another widely used method is the one based on structural waves propagation. In this method, unlike in the classical method of ultrasonic te- sting, the wavelengths are large compared to the characteristic dimension of the structure, and a wave pulse propagates along the whole structure. More- over, the excitation of structural waves is in one pointwhile themeasurements of structural waves velocity are in a few other points instead of surface scan- ning used in the ultrasonic testing. It simplifies the structural wave tests in rods, beams, plates and some other structures. Application of neurocomputing in the parametric identification... 669 The Backpropagation ANNs with the Rprop learning algorithm were ap- plied. Neural networks with one or two hidden layers were tested. The input vectors consisted of a preprocessed time signal. The outputs provided all pa- rameters describing damage. Some of these parameters were identified by the network with satisfactory precision while other with a significant error. Ba- sed on the previous research done by the author (Ziemiański and Piątkowski, 2000). Cascade neural networks were built to improve the generalisation of networks. The first network was fed with the input vector as for a simple net. The first network’s output is the most precisely determined parameter. The second network has got an input vector with an additional value of the parameter predicted by the first net. The output of this network is the next desirable parameter. The cascade networks with two or three stages were used (Figure 1). Fig. 1. A three stage cascade network 2.1. Preprocessing of signals One of the used methods of preprocessing of input data was compression of a time signal (Ghaboussi and Lin, 1998; Haykin, 1999; Waszczyszyn and Ziemiański, 2004). The compression was performed by the dedicated neural network called replicator shown in Fig.2. Fig. 2. The use of a replicator in a damage identification problem 670 L.Ziemiański et al. Networks of architecture n-h-n were learned to replicate at the output the time signal data points given at the input – the input variables we- re mapped into the same output variables. The compression networks of 200-8-200 and 200-12-200 architectures were created. After training, the out- puts from hidden neurons (see Fig.2) were used as preprocessed data for the damage identification net. 2.2. Identification of cross-section changes The next analysed task was a rodwith a defect simulated as a notch (Zie- miański and Piątkowski, 2000). Removal of certain finite elements simulated the failure of the cross-section. Various widths b=0.1, . . . ,1.1m, various he- ights h=0.1, . . . ,0.8m and various locations l=3.5, . . . ,5.5m of notch were considered (Figure 3a). Fig. 3. A test on propagation of structural waves Table 1.Comparison of errors of identification of cross-section parameters St. par. Height of defect Location of defect Width of defect Learn Test Learn Test Learn Test Standard net R2 0.996 0.994 0.994 0.994 0.990 0.988 Stε 0.0149 0.0177 0.0193 0.0192 0.0215 0.0232 Cascade net R2 1.000 0.999 0.999 0.997 0.994 0.986 Stε 0.0043 0.0071 0.0094 0.0121 0.0165 0.0253 The input vector was described with values of first three peaks of the time signal (Ziemiański and Piątkowski, 2000) and by use of the replicator. Both standard and cascade nets were used at this stage. Better results were obtained fromneural networkswith two hidden layers. Thewidth of the defect Application of neurocomputing in the parametric identification... 671 has no influence on the quality of results.Damage identification is significantly better in the case of use of cascade networks. In Table 1 the results of neural simulation are put together. 2.3. Identification of the yielding zone A group of finite elements with an elasto-plastic material occurred inside the rod. A plastic-bilinear material model was used. Young’s modulus and strain hardeningmoduluswere fixed.A failurewas simulated by the change of yield stress σ0.Thewidth band location lof thedefectwere changed (Fig.3d). A concentrated force with a sine characteristic and duration of 400µs was applied to the free end of the rod. The measured points were located behind the zone of yielding. For the identification networks the input vectors consisted of compressed time signals. A compression network with 8 neurons in the hidden layerwasused.Theoutputvectors consistedof three failureparameters (σ0,b, l). When simple nets were used, the yield strength parameter and the location of the yield zone were identified correctly. The width of the defect was identified significantly worse. The use of cascade networks improved the identification of width of the defect (see Table 2). Table 2.Learning errors in the identification of parameters of the yielding zone St. par. Yield stress Location Width Standard net R2 0.922 0.995 0.769 Stε 0.0222 0.0117 0.0970 Cascade net R2 1.000 1.000 0.886 Stε 0.0039 0.0012 0.0680 3. Updating of finite element models of portal frames 3.1. Models and the experiment This section presents an application of ANNs in updating models of two frames: one-storey portal frame and two-storey frame (Miller, 2002;Miller and Ziemiański, 2001). The height of both frameswas 40cm, the spanwas 46.9cm. 672 L.Ziemiański et al. The beam and columns had a rectangular cross-section of 2.6cm by 0.6cm. The beam-to-column connections and the footings were treated as rigid or flexible, depending on the used dynamic model. Several FEmodels of each frame were built, the differences between them were as follows: • the column footing and the beam-to-column connection were treated either as rigid or as flexible, • the effective length of both columns was in the range of 40cm (overall length of the columns) and 30cm (the length of the columns above the steel connection plate); in dynamicmodels the columns lengthwas 35cm (flexible column footing) or 40cm (clamped columns), • the models were considered to be symmetrical or not. Each model consisted of 12 (one-storey frame) or 16 (two-storey frame) finite beam elements and it had 39 or 48 DOFs. The laboratory models of both considered frames are shown in Fig.4, while the FEmodels are shown in Fig.5. Fig. 4. Laboratorymodels of both considered frames Vibrations of the laboratory models were excited by an impact. The re- sponse of the structure was measured in the range of 0 to 1024Hz with the step of 0.25Hz. The measurements were done using eight accelerometers at- tached in points corresponding to the nodes of the computational model. The FEmodels presented in this paper took into account concentratedmasses cor- responding to the masses of the accelerometers (10.2g). The laboratory set- up involved PCB accelerometers, Brüel&Kjær modal hammer and CADA-X acquiring systemwithmultichannel analyser Scadas III. Application of neurocomputing in the parametric identification... 673 Fig. 5. The FEmodels of considered frames: (a) one-storey framemodel FRAME1a, (b) one-storey framemodel FRAME1b, (c) two-storey framemodel FRAME2 3.2. The updating procedure Themodel updating procedure consists of the following steps: 1. generation of a set of training data vectors based on the dynamicmodel, 2. training of the neural network with the training data, 3. exposition of expected data to obtain a set of changes, 4. application of the changes to the original model in order to generate a newmodel, 5. repetition of the previous steps if necessary. This procedure leads to an updated model that can be updated again using the samemethod. Multi layer feed-forward networks and networkswith the radial basis func- tion were used. The input information was the values of the first four eigenva- lues.On the basis of eigenfrequencies, the networks updated one, two, three or four parameters of themodels (for example rotational or translational stiffnes- ses of the supports and beam-to-column connection). The learning data were obtained from numerical simulations and then contaminated by an artificial 674 L.Ziemiański et al. noise. The trained networks were fed on the data obtained from the experi- ment, and they predicted the updated values of selected model parameters. The updated models were then used to calculate the eigenfrequencies of the considered frames. FRAME1a, FRAME1b and FRAME2 models were updated on the basis of measurements of the laboratory model with attached eight accelerometers. 3.3. Model FRAME1a The updated parameters were the stiffnesses of the joints in column fo- otings. The updating ANNs were of architecture 4-h-1, the learning and te- sting patterns were disturbed by the artificial noise (Gaussian) of variation 0.006. The number of learning patternswas equal to the number of the testing patterns, both sets consisted of 1005 patterns. The results of updating are shown in Table 3. Table 3. The results of model FRAME1a updating on the basis of eigen- frequencies – symmetrical model Eigenfrequency Measured Calculated Relative [Hz] value value error f1 27.0 26 3.7% f2 81.5 85 −4.3% f3 169.8 174 −2.5% f4 176.5 182 −3.1% RMSE=3.5×10−2 The RMSE is Root Mean Square Errors of the eigenfrequencies obtained from the updatedmodel. RMSE is defined by the formula RMSE= √ √ √ √ 1 n−k+1 n ∑ i=k (f0i−fi f0i )2 (3.1) where k – first of considered eigenfrequencies n – last of considered eigenfrequencies i – number of eigenfrequency, i= k,k+1, . . . ,n f0i – ith eigenfrequency obtained from ”measurements” fi – ith eigenfrequency obtained from numerical simulations. Application of neurocomputing in the parametric identification... 675 The comparison of eigenforms obtained from numerical calculations and frommeasurements is shown in Fig.6. The calculated eigenforms were, before the comparison, reduced using the static method (see Friswell andMottershe- ad, 1996). Fig. 6. The comparison of eigenforms obtained from calculations (symmetrical model FRAME1a) and frommeasurements: (a) first, (b) second, (c) third Fig. 7. The comparison of eigenforms obtained from calculations (non-symmetrical model FRAME1a) and frommeasurements: (a) first, (b) second, (c) third The matrix MAC (see Friswell and Mottershead, 1996) for the first four eigenvectors is as follows MACij = ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ 0.9918 0.0008 0.2719 0.0288 0.0004 0.9845 0.0003 0.0044 0.1576 0.0000 0.9421 0.1060 0.0001 0.0084 0.0507 0.8873 ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ The values on themain diagonal of theMACmatrix should be, in the case of proper updating, higher than 0.8, all other values should not exceed 0.1 (Friswell andMottershead, 1996). In the presentedmatrix two elements have 676 L.Ziemiański et al. values outside the expected range: MAC3,1 and MAC1,3 have values consi- derably higher than 0.1. It is caused by the linear dependence of adequate measured eigenforms. TheMACmatrix calculated formeasured eigenforms is as follows MACij = ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ 1.0000 0.0001 0.2099 0.0200 0.0001 1.0000 0.0000 0.0019 0.2099 0.0000 1.0000 0.0110 0.0200 0.0019 0.0110 1.0000 ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ The elements MAC3,1 andMAC1,3 are twice as high as they should be. The third and fourth eigenforms shown in Fig.6c andFig.6d indicate that the considered frame in non-symmetrical. In the next step the values of the stiffnesses of the joints in the columns footings were updated independently. The ANNs were of architecture 4-h-2, the learning and testing patterns were disturbedby the artificial noise of variation 0.006. Thenumber of learning and testing patterns was 5780 each. The results of updating are shown in Table 4, the comparison of eigenforms is shown in Fig.7. Table 4. The results of model FRAME1a updating on the basis of eigen- frequencies – non-symmetrical model435 Eigenfrequency Measured Calculated Relative [Hz] value value error f1 27.0 26 3.7% f2 81.5 85 −4.3% f3 169.8 174 −2.5% f4 176.5 183 −3.7% RMSE=3.6×10−2 Thematrix MAC for the first four eigenvectors is as follows MACij = ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ 0.9917 0.0010 0.2721 0.0275 0.0004 0.9847 0.0004 0.0041 0.1558 0.0001 0.9839 0.0390 0.0018 0.0081 0.0112 0.9499 ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ The independent updating of rotational stiffnesses of both joints made it possible to obtain amodel reproducing the eigenformswith a higher accuracy. The improvement is evident also in theMACmatrix, all elements on themain diagonal are equal or higher than 0.95. Application of neurocomputing in the parametric identification... 677 3.4. Model FRAME1b The frame with eight accelerometers was modelled also with clamped co- lumns.The local stiffnessmatrices of the clamped elements were calculated as a product of the stiffness matrix of another column element (all others were the same) and the computational coefficient α, whichwas being updated.Mo- del FRAME1bwas updatedwith different values of coefficients α1 and α2, so the ANNwas of architecture 4-h-2. The results of model FRAME1b updating are shown in Table 5. Table 5.The results of model FRAME1b updating on the basis of eigen- frequencies Eigenfrequency Measured Calculated Relative [Hz] value value error f1 27.0 27 0.0% f2 81.5 85 -4.3% f3 169.6 169 0.5% f4 176.5 177 -0.3% RMSE=2.2×10−2 The results are significantly better than those obtained frommodel FRA- ME1a. The error of eigenfrequency prediction exceeds 4% only for the second eigenfrequency f2, in other cases the error is smaller than 1%. The matrix MAC for the first four eigenvectors is as follows MACij = ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ 0.9918 0.0008 0.2719 0.0288 0.0004 0.9845 0.0003 0.0044 0.1576 0.0000 0.9421 0.1060 0.0001 0.0084 0.0507 0.8873 ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ 3.5. Model FRAME2 The comparison of exemplary results obtained from the updated model (Fig.5c) and the datameasured on the laboratorymodel are shown inTable 6. The accuracy of the updated model was determined by the comparison of the eigenforms and eigenfrequencies obtained from the measurements and from the updatedmodel, the maximum difference between the measured and calculated eigenfrequencies did not exceed 5%, whichmust be considered as a very good result. The results obtained from the updatedmodels are precise in the range of the eigenfrequencies used as input information of the ANNs. 678 L.Ziemiański et al. Table 6.The results of updating of the FEmodel of the two-storey frame Eigenfrequency Measured Calculated Relative [Hz] value value error f1 33.0 312 3.0% f2 91.1 95 −4.3% f3 106.1 105 2.9% f4 123.0 121 1.6% RMSE=3.1×10−2 4. Detection of the void and additional mass in a cantilever plate The assessment of the structure state is a task which requires continu- ous researches and their development. There are a lot of methods based on a non-destructive testing of structural elements. Some of these methods utilize dynamical parameters of the monitored structure, such as modal characteri- stics (resonance frequencies), response of the structure to an impulse forcing and structuralwaves analysis (Yagawa andOkuda, 1996). One of themost im- portant objectives of these methods are detection and localisation of changes, failures and defects in structural elements. This section presents the possi- bility of application of Artificial Neural Networks (ANN) for non-destructive detection of a circular void and additionalmass in cantilever plates. Detection methodsarebasedon the analysis of eigenfrequencies andanalysis of the struc- ture response to a harmonic excitation. The backpropagation neural networks with the Levenberg-Marquardt training function were applied. Standard ne- tworks and cascade sets ofMLFF backpropagation neural networks were used (Waszczyszyn and Ziemiański, 2001; Piątkowski and Ziemiański, 2002). 4.1. Identification of a circular hole location Numerical tests were carried out for rectangular plates with an internal defect in the form of a circular hole (Piątkowski and Ziemiański, 2003). The geometry of a plate is presented in Fig.8. The plate has a unit thickness. The position of the void centre is defined by two parameters which are OY and OZ coordinates. These coordinates and the diameter are unknownparameters of the inverse problem. The extreme locations of centre of the hole are located 0.05m from the plate edges. The location of the void is changed with a step of 0.01m. It has Application of neurocomputing in the parametric identification... 679 given 341 different positions. In eigenproblemanalysis, due to symmetry of the plate, only the 186differentpositionsof thehole are takenunderconsideration. The centres are located under or on the axis of symmetry of the plate. Fig. 8. The numerical model of the plate-scheme andmain dimensions For each considered location of the void, a FEM model of the plate was found by means of the Adina finite element system. Numerical models were madeof 2D solidplane stress elements [1].Thus, only the in-planevibrations of the platewere analysed.Themeshes of elementswere generated automatically on the basis of sets of solid geometry faces. An isotropic linear elastic material model was used. The plate had the following material properties: Young’s modulus E=215GPa, Poisson’s ratio ν =0.3, density ρ=7850kg/m3. The artificial neural network analysis of the calculated eigenfrequencies was performed to find the hole centre coordinates and its diameter. The first neural experiment consisted in identification of two coordinates of the centre of the void with diameter D = 0.002m on the basis of eigen- frequencies. The computations showed that the acceptable accuracy could be achieved by the input vector, which composed of only five elements. These elements included relative changes of first five eigenfrequencies. The relative changes were calculated as follows ∆fi = fi−f o i foi (4.1) where foi are the eigenfrequencies of the homogeneous plate (without a void) and fi respective values corresponding to the plate with the void. 680 L.Ziemiański et al. Better results were obtained from a cascade network (Piątkowski and Zie- miański, 2002). The maximum error of prediction of centres locations for the training data set was 1.4mm and for the testing dataset was 1.5mm. The identified locations are shown in Fig.9, where the centers from the training and testing sets aremarked by x in gray and black colours. This figure shows the lower half of the plate due to symmetry of the solution to the eigenproblem for the plate under consideration. Fig. 9. Results for the OY .OZ cascade network of architecture 5−7−1.6−7−1 Next, the identification of three parameters of the void {OY,OZ,D} was performed. In this task, the number of void positions was decreased to 39. The voids were located in the lower half of the plate. Nine diameters of each void were simulated in the range of 0.004-0.02m. Thus, the total number of patternswas 351.Relied on experiences fromthe previous expirement, a three- stage cascade neural network was applied. It was experimentally determined that the best results were obtained by the cascade network with the following prediction order of parameters: D.OY .OZ. The results presented in Fig.10multiplemarks xwhich relate to different diameters of voids at a given position.One can see that the vertical coordinate of voids was predicted with a higher error than the horizontal one. The histograms presented inFig.11 show that themaximum relative error of the OY coordinate identificationwas ±2%,both for the trainingandtesting sets of patterns. The OZ coordinates were predicted with a relative error of ±10%. The third parameter of the void, its diameter, was identified with a relative error of ±4% in the diameter range 0.004-0.020m. Application of neurocomputing in the parametric identification... 681 Fig. 10. Results for the D.OY .OZ cascade network of architecture 5-7-1.6-7-1.7-7-1 Fig. 11. Histograms for (a) diameter of the hole D, (b) coordinate OY , (c) coordinate OZ for D.OY .OZ cascade network of architecture 5-7-1.6-7-1.7-7-1 4.2. Detection of an additional mass Two laboratory models were built as it is shown in Fig.12 (Waszczyszyn and Ziemiański, 2004). The models were made of steel and aluminium alloy plates, fixed to a massive stand by high-tensile bolts. The experimental set consisted of eight PCB acceleration sensors. Sca- das III analyzer with LMS software was applied to measure the response of the structure. The out-plane vibrations of the plate were forced with a mo- dal hammer. The locations of sensors and the point of impact are shown in Fig.12b. 682 L.Ziemiański et al. Fig. 12. (a) The experimental model of the steel plate, (b) scheme of the plate, main dimensions The additional mass, whose position was detected, had a mass of 3% or 8%of the platemass, respectively for the steel or the aluminiumalloymodels. The mass was located in nodes of the measurement grid. The grids for both models are shown in Fig.12b. The 27 locations of the additional mass in the steel model were considered. For the aluminium alloy model the grid was changed. Thus, the vibrations of the plate for 40 locations of the additional mass were measured. The acceleration signals in the time domain were transformed by means of Fast Fourier Transformation into the frequency domain. The obtained fre- quency characteristics had resolution of 0.25Hz. These characteristics were analysed in selected bands. Consequently, within these bands the harmonics with the highest amplitudes were searched out. Finally, the sets of frequencies of vibration of the plates with the additional mass were obtained. The total number of frequency characteristics was equal to 8 ·27=216 as the result of conducting ofmeasurementswith eight sensors for 27 locations of the mass. The set of frequencies for the steel plate consisted of [216×5] fre- quencies, because a searc formodal frequencies within five selected bandswas performed. The inputs vectors for ANN training were prepared by a process Application of neurocomputing in the parametric identification... 683 of elimination of outstandingmeasurements. After that, the average frequency in each band for each location of the mass was calculated. Consequently, the [27×5] input data set was prepared for the training of the networks used for mass identification. In the case of the aluminium alloy plate the number of frequency cha- racteristics was much higher and was equal to 1528. The number of 1528 characteristics was a result of considering 40 locations of the mass, conduc- ting measurements with eight sensors and recording plate vibrations for at least three hammer impacts. These characteristics were analysed in six selec- ted bands. Finally the [1528×6] set of characteristics was obtained. In order to prepare input data, on averaging of the whole set of frequencies was perfor- med using neural networks. Six networks of 2-10-1 architecture were built – separately for six bands. These networks were learnt to re-map the two coor- dinates of the additional mass intomodal frequency within the selected band. Consequently, the [40×6] input data set was prepared. Due to a small number of patterns (27 for the steel plate and 40 for the aluminiumalloy plate) itwas decided tofindadditional patterns by adding the artificial Gaussian noise with the variance σ=0.001. The number of patterns was increased to four hundreds that way. The two coordinates of location of the additional mass were determined separately by means of neural networks of architecture 5-5-1. 10% of the patterns was selected to testing. Table 7. Statistical parameters for networks 5-5-1 and 5-5-2 5-5-1 5-5-2 OY OZ OY OZ R2 0.992 0.991 0.939 0.980 Stε ·102 2.97 2.60 8.21 3.75 Next, two coordinates of location of the additional mass were determined by means of the neural network of architecture 5-5-2. Statistical parameters, which describe the trained networks are collected in Table 7. The first part of this table refers to two 5-5-1 networks, which were used for separate de- termination of two coordinates of location of the additional mass. The second part refers to 5-5-2 network. This network did not manage to determine both coordinates simultaneously. The results of the identification of location of the mass are presented in Fig.13a. The horizontal and vertical bars are proportional to standard devia- tion of determination of the OY and OZ coordinates due to noised input vectors. The best accuracy was obtained from the OY .OZ cascade neural 684 L.Ziemiański et al. Fig. 13. Results of the identification of the additional mass in the steel plate, (a) standard network 5-5-2, (b) cascade network 5-5-1.6-5-1 network of architecture 5-5-1.6-5-1. The results for the training (gray marks) and testing sets (black marks) are shown in Fig.13b. For the aluminiumalloy plate, the resultswere analogous to those obtained for the steelmodel. Again, the best determination of location of the additional mass was obtained from the cascade network, which was fed with the input vector noised with the variance σ=0.001. The results for the training (gray marks) and testing patterns (black marks) are shown in Fig.14. 5. Modelling of an ”artificial boundary condition” by neural network In many fields of engineering an infinite or semi-infinite domain (unboun- ded structure) has to be analyzed. This kind of structure is considered inwave propagation problems such as soil-structure interaction, fluid-structure inte- raction, acoustics, electromagnetism and so on Wolf and Song (1996), Givoli Application of neurocomputing in the parametric identification... 685 Fig. 14. Results of the identification of the additional mass in the aluminium alloy plate – cascade network 6-5-1.7-5-1 (1992). The numerical study of problems originally formulated on unbounded domains requires the implementation of special techniques for the treatment of ”infinity”. A number of methods were formulated for the reduction of the problem to a bounded domain such as direct and indirect boundary element methods, infinite elements, DtNmethod (Wolf and Song, 1996; Givoli, 1992). One of the corresponding techniques is based on an artificial truncation of the original infinite domain at a certain distance away from the region of interest, which implies that one must set special boundary conditions on the exter- nal (artificial) boundary of the newly formed finite computational domain. A surface which encloses the most important part of the structure is often in- troduced. The model can be obtained by the application of an approximate boundary condition to the artificial surface – the so called Artificial Bounda- ry Condition (ABC). In wave propagation problems, ABC is also the called non-reflecting boundary condition or transmitting boundary (Givoli, 1992). The modelling of the transmitting boundary using neural networks was discussed by Ziemiański (2003). The assumption that the simulation of the artificial boundary condition on the artificial surface is carried out by corre- spondingneural networks ismade (Ziemiański, 2003). The task of these neural 686 L.Ziemiański et al. networks is to simulate boundary conditions fulfilling the radiation condition and avoiding appearance of the reflected wave. Back Propagation Neural Ne- tworks are used for this task (Waszczyszyn and Ziemiański, 2001). 5.1. A semi-infinite strip with notch In the presented example thewave propagation in a semi-infinite stripwith notch is discussed (see Fig.15). Fig. 15. A semi-infinite strip: (a) analyzed problem, (b) finite element mesh for the extendedmodel, (c) finite element mesh for the truncated model Thewave originates at the free end of the rodby a displacement excitation in the form of a pulse with various durations and amplitudes. An extended finite elementmesh (Fig.15b) was used for computation of dynamic responses Application of neurocomputing in the parametric identification... 687 in the nodes placed on the artificial boundary. The results of computation were used to train the back-propagation neural network (BPNN). The input for the neural network was defined as a vector including horizontal and ver- tical displacements at the time t− 1 in three consecutive nodes on the line perpendicular to the artificial boundary.The output vector determines the dy- namic reactions R(t) (two components Rh and Rv) in nodes on the truncated boundary (ABC) at the time t. The learning patternswere obtained from the extendedFEmesh. The tra- inedneuralnetworkpredicted thedynamic reactions R(t),whichwere checked by the comparison of the displacement, velocities and accelerations of selected nodes obtained from the extendedmesh and those obtained from the trunca- ted model with the neural networks used for the prediction of the dynamic reactions on ABC. After some calculations and multi-fold cross-validations, two-hidden-layer networks of architecture 6-12-8-2 were applied and the re- sponses of the truncated hybrid NN/FE model were computed. In the Fig.16 the response of horizontal and vertical displacements are displayed and traces of the difference between two signals are also shown. Fig. 16. Responses of horizontal displacements at points A and B and errors of displacements 5.2. A semi-infinite wedge In the second example, the wave propagates in a semi-infinite wedge fixed on one edge, along the direction of the wave propagation. The considered FE 688 L.Ziemiański et al. model is shown inFig.17.Thewedgeangle equals 30degrees, thedistance from the center of the angle to the beginning of the wedge and the width of the wedge both equal 80m. The wave originates at the narrower (internal) end of the wedge by the horizontal displacement excitation with various durations t and amplitudes A (see Fig.18a). The excitation is triangle-shaped, with the amplitude A varying with the location on the excitated edge from zero at the bounded end of the edge to the maximum value at the free end of the edge. Fig. 17. A semi-infinite wedge: FEmesh Fig. 18. (a) The triangle-shaped displacement excitation; (b) selected nodes of FE mesh Application of neurocomputing in the parametric identification... 689 The extended model, shown in Fig.15d, is used for computation of dyna- mic responses in the nodes located at the artificial boundary. The values of displacements, velocities and accelerations at the time t−1 calculated at se- lected points using the extendedmodel were used to build up the input vector of neural networks. The output vector determined the dynamic reactions R(t) in nodes on the ABC at the time t. The learned network was used to predict the reactions, whichwere applied to the truncatedmodel in order to avoid the reflection of the propagating wave. The BPNN networks trained using the Levenberg-Marquardt algorithm were applied. The networks had one or two hidden layers, the output vector consisted of the values of two components (horizontal and vertical) of the dynamic reaction in a selected point. The input vector consisted of: • horizontal andvertical components of displacements of three consecutive points on the line perpendicular to the artificial boundary (points 1, 2 and 3 in Fig.18b) • horizontal and vertical components of displacements of three consecuti- ve points on the line perpendicular to the artificial boundary and the dynamic reaction in point 1 (see Fig.18b) at the time t−1 • horizontal and vertical components of displacements of four points (po- ints 1, 2, 4 and 5 in Fig.18b) • horizontal and vertical components of the displacement, velocity and acceleration in point 1. The dynamic reaction was predicted only in point 1, in the middle of the artificial boundary. The accuracy of the neural prediction was determined by the comparison of the data obtained from the extended FE model and from the truncatedmodelwith dynamic reactions applied to the artificial boundary (one of themwas predicted by the neural network). In order to train and test the neural network 61 different excitation signals were generated. The dynamic reaction in each point on ABC was calculated numerically in 326 timemoments with the step ∆t=20×10−6 s. The overall time of observationwas 0.0065s.The excitation signals haddifferentdurations and amplitudes. The duration of the excitation signal varied from 1.2×10−4 s to 3.6×10−4 swith the step of 0.48×10−4 s (6 different values).Theamplitude of the signal varied from 6×10−2 to 14×10−2 with the step of 1.6×10−2 (6 different values), so together 36 excitation signals and 36 sets of dynamic reactions (each consisting of 326 points) were obtained. The second set of excitation signals and dynamic reactions, used to test the neural network, was 690 L.Ziemiański et al. obtained using 5 different durations and 5 different amplitudes (with values between the values used to generate the learning signals). The number of testing signals and the number of sets of dynamic reactions equaled 25. In the first of considered examples, the input vector consists of six values: vertical and horizontal displacements in points 1, 2 and 3 at the time t−1. Theneural networkwas trained to predict the vertical andhorizontal dynamic reaction in point 1 at the time t. The learning set consisted of 11700 patterns (36 excitation signals, 326 dynamic reactions calculated using each excitation signal, one reaction rejected from each group in order to obtain patters consi- sting of inputs at the time t−1 and outputs at the time t). The testing set consisted of 8125 patters (25 excitation signals, 326 minus 1 dynamic reac- tions). Two hidden layer networks of architecture 6-12-8-2 were applied. The network was able to predict the dynamic reaction on the basis of the displa- cements in the input vector. However, the goal of the ANN application was not to predict the proper values of dynamic reactions but to obtain the trun- cated model with the transmitting boundary, without the excitation due to the reflected wave. The results of calculations done using the truncatedmodel were used to test the accuracy of the artificial boundary built using the neural networks.The results obtained from this network are presented graphically in Fig.19. 6. Final remarks Onthebasis of theachieved results, the following conclusions canbedrawn: • ANNs can be efficiently applied in the field of dynamics of structures • ANNs can deal with data prepared by computational systems and those taken from experiments (also the data can bemixed) • ANNs make it possible to use structural wave propagation analysis for detection and assessment of damage in structural elements • the hybrid approach (neural network – finite element) proves to be an effective tool to solve the problem of the model updating • ANNs can be effectively used as a tool for non-destructive detection of a void and additional mass in plates Application of neurocomputing in the parametric identification... 691 Fig. 19. Displacements (a,b) and accelerations (c,d) in point 1 in relation with the time calculated using the truncated model and 8-12-8-2 network in closed loop (the input vector consists of displacements in points 1, 2 and 3) • BPNNs can be efficiently applied to the implementation of an approxi- mate boundary condition on an artificial surface. • Application of cascade networks improves accuracy of neural approxi- mation in all analysed cases. References 1. ADINA System Online Manuals, Theory and modeling guide, 2001, ADINA R&D Inc.,Watertown 2. Friswell M.I., Mottershead J.E., 1996, Finite Element Model Updating in Structural Dynamics, Kluwer Academic Publishers, Dordrecht 3. Ghaboussi J., Lin C-C.J., 1998, Newmethod generating spectrum compati- ble accelerogramsusing neural networks,Earthquake Engineering and Structu- ral Dynamics, 27, 377-396 692 L.Ziemiański et al. 4. Givoli D., 1992,Numerical Methods for Problems in Infinite Domains, Else- vier, Amsterdam 5. Haykin S., 1999,Neural Networks, a Comprehensive Foundation, second ed., Prentice Hall, Upper Saddle River 6. MillerB., 2002,Updating ofMathematicalModels of EngineeringStructures (in polish), PhD dissertation, RzeszówUniversity of Technology 7. Miller B., Ziemiański L., 2001, Updating of mathematical models using neural networks,Proc. 2nd European Conference onComputational Mechanics, Kraków, I, 386-387 8. Miller B., Ziemiański L., 2003, Neural Networks in Updating of Dynamic Models with Experimental Verification, In: L. Rutkowski, J. Kacprzyk, eds., Advances in Soft Computing, Physica-Verlag a Springer-Verlag Company 9. Piątkowski G., 2003, Detection of Damage in Structural ElementsUsingAr- tificial Neural Network (in polish), PhD dissertation, Rzeszów University of Technology 10. Piątkowski G., Ziemiański L., 2002, Identification of Circular Hole in Rec- tangular Plate Using Neural Networks,Proceedings of AI-Meth Symposium on Methods of Artificial Intelligence, 329-332 11. PiątkowskiG.,ZiemiańskiL., 2003,Neuralnetwork identificationof a circu- lar hole in the rectangular plate, In: L. Rutkowski, J. Kacprzyk, eds.,Advances in Soft Computing, Physica-Verlag a Springer-Verlag Company 12. Thompson R.B., 1983,Quantitative ultrasonic nondestructive evaluationme- thods, Journal of Applied Mechanics, 1191-1201 13. Waszczyszyn Z., edit., 1999,Neural Networks in the Analysis and Design of Structures, CISMCourses and Lectures No. 404, Springer,Wien-NewYork 14. WaszczyszynZ.,ZiemiańskiL., 2001,Neuralnetworks inmechanicsof struc- tures andmaterials – new results and prospects of applications,Computers and Structures, 79, 2261-2276 15. Waszczyszyn Z., Ziemiański L., 2004, Neural networks in the identifica- tion analysis of structuralmechanics problems, In: Z.Mroz, G.E. Stavroulakis, eds.,Parameter Identification ofMaterials and Structures,CISMLectureNotes Vol.469, Springer,Wien-NewYork (in print) 16. Wolf J.P., Song C., 1996, Finite-Element Modelling of Unbounded Media, JohnWiley & Sons, Chichester 17. Yagawa G., Okuda H., 1996, Neural networks in computational mechanics, Archives of Computational Methods in Engineering, (3-4), 435-512 18. Ziemiański L., 2003, Hybrid neural network/finite element modelling of wave propagation in infinite domains,Computers and Structures, 81, 1099-1109 Application of neurocomputing in the parametric identification... 693 19. Ziemiański L., Piątkowski G., 2000, Use of neural networks for damage detection in structural elements using wave propagation, In: B.H.V. Topping, editor,Computational Engineering UsingMetaphors fromNature, 25-30,Edin- burgh, Civil-Comp Ltd. Identyfikacja parametrów konstrukcji na podstawie dynamicznych odpowiedzi z wykorzystaniem sieci neuronowych – wybrane zagadnienia Streszczenie W artykule przedstawiono zastosowanie sieci neuronowych w wybranych zagad- nieniach dynamiki konstrukcji: 1) wykrywanie uszkodzeńw elementach prętowych na podstawie propagacji fali, 2) dostrajanie modeli MES ram portalowych, 3) wykry- wanie pustki i dodatkowej masy w drgającej płycie wspornikowej, 4) modelowanie „sztucznej granicy” w zagadnieniu propagacji fali. Rozpatrywane problemy dotyczą zarównomodeli numerycznych, jak i eksperymentalnych. Manuscript received February 2, 2004; accepted for print April 26, 2004