FINITE ELEMENT MODEL UPDATING USING RESPONSE SURFACE METHOD

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1 FINITE ELEMENT MODEL UPDATING USING RESPONSE SURFACE METHOD Tshldz Marwala School of Electrcal and Informaton Engneerng Unversty of the Wtwatersrand P/Bag 3, Wts, 050, South Afrca Ths paper proposes the response surface method for fnte element model updatng. The response surface method s mplemented by approxmatng the fnte element model surface response equaton by a mult-layer perceptron. The updated parameters of the fnte element model were calculated usng genetc algorthm by optmzng the surface response equaton. The proposed method was compared to the exstng methods that use smulated annealng or genetc algorthm together wth a full fnte element model for fnte element model updatng. The proposed method was tested on an unsymmetrcal H-shaped structure. It was observed that the proposed method gave the updated natural frequences and mode shapes that were of the same order of accuracy as those gven by smulated annealng and genetc algorthm. Furthermore, t was observed that the response surface method acheved these results at a computatonal speed that was more than.5 tmes as fast as the genetc algorthm and a full fnte element model and 4 tmes faster than the smulated annealng. Introducton Fnte element (FE) models are wdely used to predct the dynamc characterstcs of aerospace structures. These models often gve results that dffer from the measured results and therefore need to be updated to match the measured data. FE model updatng entals tunng the model so that t can better reflect the measured data from the physcal structure beng modeled 1. One fundamental characterstc of an FE model s that t can never be a true reflecton of the physcal structure but t wll forever be an approxmaton. FE model updatng fundamentally mples that we are dentfyng a better approxmaton model of the physcal structure than the orgnal model. The am of ths paper s to ntroduce updatng of fnte element models usng Response Surface Method (RSM). Thus far, the RSM method has not been used to solve the FE updatng problem 1. Ths new approach to FE model updatng s compared to methods that use smulated annealng (SA) or genetc algorthm (GA) together wth full FE models for FE model updatng. FE model updatng methods have been mplemented usng dfferent types of optmzaton methods such as genetc algorthm and conjugate gradent methods 3-5. Levn and Leven 5 proposed the use of SA and GA for FE updatng. RSM s an approxmate optmzaton method that looks at varous desgn varables and ther responses and dentfy the combnaton of desgn varables that gve the best response. The best response, n ths paper, s defned as the one that gves the mnmum dstance between the measured data and the data predcted by the FE model. RSM attempts to replace mplct functons of the orgnal desgn optmzaton problem wth an ap- * Assocate Professor proxmaton model, whch tradtonally s a polynomal and therefore s less expensve to evaluate. Ths makes RSM very useful to FE model updatng because optmzng the FE model to match measured data to FE model generated data s a computatonally expensve exercse. Furthermore, the calculaton of the gradents that are essental when tradtonal optmzaton methods, such as conjugate gradent methods, are used s computatonally expensve and often encounters numercal problems such as ll-condtonng. RSM tends to be mmune to such problems when used for FE model updatng. Ths s largely because RSM solves a crude approxmaton of the FE model rather than the full FE model whch s of hgh dmensonal order. The mult-layer perceptron (MLP) 6 s used to approxmate the response equaton. The RSM s partcularly useful for optmzng systems that are evolvng as a functon of tme, a stuaton that s prevalent n model-based fault dagnostcs found n the manufacturng sector. To date, RSM has been used extensvely to optmze complex models and processes 7,8. In summary, the RSM s used because of the followng reasons: (1) the ease of mplementaton that ncludes low computatonal tme; () the sutablty of the approach to the manufacturng sector where model-based methods are often used to montor structures that evolve as a functon of tme. FE model updatng has been used wdely to detect damage n structures 9. When mplementng FE updatng methods for damage dentfcaton, t s assumed that the FE model s a true dynamc representaton of the structure and ths s acheved through FE model updatng. Ths means that changng any physcal parameter of an element n the FE model s equvalent to ntroducng damage n that regon. There are two approaches that are used n FE updatng: drect methods and teratve 1

2 methods 1. Drect methods, whch use the modal propertes, are computatonally effcent to mplement and reproduce the measured modal data exactly. Furthermore, they do not take nto account the physcal parameters that are updated. Consequently, even though the FE model s able to predct measured quanttes, the updated model s lmted n the followng ways: t may lack the connectvty of nodes - connectvty of nodes s a phenomenon that occurs naturally n fnte element modelng because of the physcal realty that the structure s connected; the updated matrces are populated nstead of banded - the fact that structural elements are only connected to ther neghbors ensures that the mass and stffness matrces are dagonally domnated wth few couplngs between elements that are far apart; and there s a possble loss of symmetry of the systems matrces. Iteratve procedures use changes n physcal parameters to update FE models and produce models that are physcally realstc. Iteratve methods that use modal propertes and the RSM for FE model updatng are mplemented n ths paper. The FE models are updated so that the measured modal propertes match the FE model predcted modal propertes. The proposed RSM updatng method s tested on an unsymmetrcal H-shaped structure. Mathematcal Background In ths study, modal propertes,.e. natural frequences and mode shapes, are used as a bass for FE model updatng. For ths reason these parameters are descrbed n ths secton. Modal propertes are related to the physcal propertes of the structure. All elastc structures may be descrbed n terms of ther dstrbuted mass, dampng and stffness matrces n the tme doman through the followng expresson 10 : [ M ]{ X' ' } + [ C ]{ X' } + [ K ]}{ X } = { F } (1) where [M], [C] and [K] are the mass, dampng and stffness matrces respectvely, and {X}, {X } and {X } are the dsplacement, velocty and acceleraton vectors respectvely whle {F} s the appled force vector. If equaton 1 s transformed nto the modal doman to form an egenvalue equaton for the th mode, then 10 : ( ω [ M ] + jω [ C ] + [ K ]){ φ } = {0 } () where j = 1, ω s the th complex egenvalue, wth ts magnary part correspondng to the natural frequency ω, { 0 } s the null vector and {φ } s the th complex mode shape vector wth the real part correspondng to the normalzed mode shape {φ}. From equaton, t may be deduced that the changes n the mass and stffness matrces cause changes n the modal propertes of the structure. Therefore, the modal propertes can be dentfed through the dentfcaton of the correct mass and stffness matrces. The frequency response functons (FRFs) are defned as the rato of the Fourer transformed response to the Fourer transformed force. The FRFs may be expressed n receptance and nertance form. On the one hand, receptance expresson of the FRF s defned as the rato of the Fourer transformed dsplacement to the Fourer transformed force. On the other hand, nertance expresson of the FRF s defned as the rato of the Fourer transformed acceleraton to the Fourer transformed force. The nertance FRF (H) may be wrtten n terms of the modal propertes by usng the modal summaton equaton as follows 10 : = N ω φ φ k l H ( ω ) (3) kl = 1 ω + ζ ω ωj + ω Equaton 3 s an FRF due to exctaton at poston k and response measurement at poston l, ω s the frequency pont, ω s the th natural frequency, N s the number of modes and ζ s the dampng rato of mode. The exctaton and response of the structure and Fourer transform method 10 can be used to calculate the FRFs. Through equaton 3 and a technque called modal analyss 10, the natural frequences and mode shapes can be ndrectly calculated from the measured FRFs. The modal propertes of a dynamc system depend on the mass and stffness matrces of the system as ndcated by equaton. Therefore, the measured modal propertes can be reproduced by the FE model f the correct mass and stffness matrces are dentfed. FE model updatng s acheved by dentfyng the correct mass and stffness matrces. The correct mass and stffness matrces, n the lght of the measured data, can be obtaned by dentfyng the correct modul of elastcty for varous sectons of the structure under consderaton 1. In ths paper, to correctly dentfy the modul of elastcty of the structure, the followng cost functon that measures the dstance between measured data and FE model calculated data, s mnmzed: E = + β = m ω ω γ m ω N calc m ( 1 dag( MAC({ φ },{ φ } ))) N calc... Here m s for measured, calc s for calculated, N s the number of modes; γ s the weghtng factor that measures the relatve dstance between the ntal estmated natural frequences for mode and the target frequency of the same mode; the parameter β s the weghtng functon on the mode shapes; the MAC s the modal assurance crteron 11 ; and the dag(mac) stands for the th dagonal element of the MAC matrx. The MAC s a measure of the correlaton between two sets of mode shapes of the same dmenson. In equaton 4 the frst (4)

3 part has a functon of ensurng that the natural frequences predcted by the FE model are as close to the measured ones as possble whle the second term ensures that the mode shapes between measurements and those predcted by the FE model are correlated. When two sets of mode shapes are perfectly correlated then the MAC matrx s an dentty matrx. The updated model s evaluated by comparng the natural frequences and mode shapes from the FE models before and after updatng to the measured ones. Updatng Parameters Intal Condtons Updatng Objectve Generaton of surface response data usng the FE model Updatng Space Response Surface Method RSM method s a procedure that operates by generatng a response for a gven nput. The nputs are the parameters to be updated and the response s the error between the measured data and the FE model generated data. Then an approxmaton model of the nput parameters and the response, called a response surface equaton, s constructed. As a consequence of ths, the optmzaton method operates on the surface response. Ths equaton s usually smple and not computatonally ntensve as opposed to a full FE model. RSM has other advantages such as the ease of mplementaton through parallel computaton and the ease at whch parameter senstvty can be calculated. The proposed RSM conssts of these essental components: (1) the response surface approxmaton equaton; and () the optmzaton procedure. There are many technques that have been used for response surface approxmaton such as polynomal approxmaton 1 and neural networks 13. A mult-layer perceptron s used as a response surface approxmaton equaton 6. Further understandng of dfferent approaches to response surface approxmaton may be found n the lterature In ths paper, MLP s used because t has been successfully used to solve complcated regresson problems. The detals of the MLP are descrbed n the next secton. The second component of the RSM s the optmzaton of the response surface. There are many types of optmzaton methods that can be used to optmze the response surface equaton and these nclude the gradent based methods 0 and evolutonary computaton methods 1. The gradent based methods have a shortcomng of dentfyng local optmum solutons whle evolutonary computng methods are better able to dentfy global optmum soluton. As a result of the global optmum advantage of evolutonary methods, n ths study the GA s used to optmze the response surface equaton. The manner n whch the RSM s mplemented s shown n Fgure 1. In ths fgure t shown that the RSM s mplemented by followng these steps: 1) Settng ntal condtons whch are: updatng parameters, updatng objectve, whch s n equaton 4, Functonal approxmaton Global optmzaton Functonal evaluaton at the optmum soluton on the FE model Updatng crtera satsfed? Stop and updatng space. ) The FE model s then used to generate sample response surface data 3) MLP s used to approxmate the response surface approxmaton equaton from the data generated n Step. 4) GA s used to fnd a global optmum soluton. 5) The new optmum soluton s used to evaluate the response from the full FE model. 6) If the optmum soluton does not satsfy the objectve, then the new optmum and the correspondng FE model calculated response replaces the canddate wth the worst response n data set generated n Step and then steps 3 to 5 are repeated. If the objectve s satsfed then stop and the optmum soluton becomes the ultmate soluton. Step 6 ensures that the smulaton always operates n the regon of the optmum soluton. The next secton Y N Replace the worst surface response data wth the optmum pont and ts FE response Fgure 1. The flowchart of the RSM. Here N stands for no and Y stands for yes. 3

4 descrbes an MLP, whch s used for functonal approxmaton. Mult-layer Perceptron Mult-layer perceptron s a type of neural networks whch used n the present study. Ths secton gves the over-vew of the MLP n the context of functonal approxmaton. The MLP s vewed n ths paper as parameterzed graphs that make probablstc assumptons about data. Learnng algorthms are vewed as methods for fndng parameter values that look probable n the lght of the data. Supervsed learnng s used to dentfy the mappng functon between the updatng parameters (x) and the response (y). The response s calculated usng equaton 4. The reason why the MLP s used s because t provdes a dstrbuted representaton wth respect to the nput space due to cross-couplng between nput, hdden and output layers. The MLP archtecture contans a hyperbolc tangent bass functon n the hdden unts and lnear bass functons n the output unts 6. A schematc llustraton of the MLP s shown n Fgure. Ths network archtecture contans hdden unts and output unts and has one hdden layer. The bas parameters n the frst layer are shown as weghts from an extra z 1 x d y 1 Input Unts Output Unts z M Genetc Algorthms GA was nspred by Darwn s theory of natural evoluton. Genetc algorthm s a smulaton of natural evolux 1 y c x 0 z 0 Hdden Unts bas Fgure. Feed-forward network havng two layers of adaptve weghts nput havng a fxed value of x 0 =1. The bas parameters n the second layer are shown as weghts from an extra hdden unt, wth the actvaton fxed at z 0 =1. The model n Fgure s able to take nto account the ntrnsc dsc dmensonalty of the data. Models of ths form can approxmate any contnuous functon to arbtrary accuracy f the number of hdden unts M s suffcently large. The relatonshp between the output y, representng error between the model and measured data, and nput, x, representng updatng parameters may be wrtten as follows 6 : M d ( ) ( 1 ) ( 1 ) ( ) y = w tanh w x + w + w j j0 k 0 (5) k kj j = 1 = 1 ( 1 ) ( ) Here, w and w ndcate weghts n the frst and second layers, respectvely, gong from nput to hdden j j unt j, M s the number of nput unts, d s the number of ( 1 ) output unts whle w ndcates the bas for the hdden j0 unt j. Tranng the neural network dentfes the weghts n equatons 5 and a cost functon must be chosen to dentfy these weghts. A cost functon s a mathematcal representaton of the overall objectve of the problem. The man objectve, ths s used to construct a cost functon, s to dentfy a set of neural network weghts gven updatng parameters and the error between the FE model and the measured data. If the tranng set N D = { x,t } s used and assumng that the targets t k k k = 1 are sampled ndependently gven the nputs x k and the weght parameters, w kj, the cost functon, E, may be wrtten as follows usng the sum-of-square error functon 6 : K E { t y } (6) nk nk = N n= 1 k = 1 The sum-of-square error functon s chosen because t has been found to be suted to regresson problems 6. In equaton 6, N s the number of tranng examples and K s the number of output unts. In ths paper, N s equal to 150, whle K s equal to 1. Before the MLP s traned, the network archtecture needs to be constructed by choosng the number of hdden unts, M. If M s too small, the MLP wll be nsuffcently flexble and wll gve poor generalzaton of the data because of hgh bas. However, f M s too large, the neural network wll be unnecessarly flexble and wll gve poor generalzaton due to a phenomenon known as over-fttng caused by hgh varance. In ths study, we choose M such that the number of weghts s at most fewer than the number of response data. Ths s n lne wth the basc mathematcal prncple whch states that n order to solve a set of equatons wth n varables you need at least n ndependent data ponts. The next secton descrbes the GA, whch s a method that s used to solve for the optmum soluton of the response surface approxmaton equaton. 4

5 ton where the law of the survval of the fttest s appled to a populaton of ndvduals. Ths natural optmzaton method s used to optmze ether the response surface approxmaton equaton or the error between the FE model and the measured data. GA s mplemented by generatng a populaton and creatng a new populaton by performng the followng procedures: (1) crossover; () mutaton; (3) and reproducton. The detals of these procedures can be found n Holland 1 and Goldberg. The crossover operator mxes genetc nformaton n the populaton by cuttng pars of chromosomes at random ponts along ther length and exchangng over the cut sectons. Ths has a potental of jonng successful operators together. Arthmetc crossover technque s used n ths paper. Arthmetc crossover takes two parents and performs an nterpolaton along the lne formed by the two parents. For example f two parents p1 and p undergo crossover, then a random number a whch les n the nterval [0,1] s generated and the new offsprngs formed are p1(a-1) and pa. Mutaton s a process that ntroduces to a populaton, new nformaton. Non-unform mutaton was used and t changes one of the parameters of the parent based on a non-unform probablty dstrbuton. The Gaussan dstrbuton starts wth a hgh varance and narrows to a pont dstrbuton as the current generaton approaches the maxmum generaton. Reproducton takes successful chromosomes and reproduces them n accordance to ther ftness functons. In ths study normalzed geometrc selecton method was used. Ths method s a rankng selecton functon whch s based on the normalzed geometrc dstrbuton. Usng ths method the least ft members of the populaton are gradually drven out of the populaton. The basc genetc algorthm was mplemented n ths paper as follows: 1) Randomly create an ntal populaton of a certan sze. ) Evaluate all of the ndvduals n the populaton usng the objectve functon n equaton 4. 3) Use the normalzed geometrc selecton method to select a new populaton from the old populaton based on the ftness of the ndvduals as gven by the objectve functon. 4) Apply some genetc operators, non-unform mutaton and arthmetc crossover, to members of the populaton to create new solutons. 5) Repeat steps -6, whch s termed one generaton, untl a certan fxed number of generatons has been acheved The next secton descrbes smulated annealng whch s used to update an FE model usng a FE model. Smulated Annealng Smulated Annealng s a Monte Carlo method that s used to nvestgate the equatons of state and frozen states of n degrees of freedom system 3. SA was nspred by the process of annealng where objects, such as metals, re-crystallze or lquds freeze. In the annealng process the object s heated untl t s molten, then t s slowly cooled down such that the metal at any gven tme s approxmately n thermodynamc equlbrum. As the temperature of the object s lowered, the system becomes more ordered and approaches a frozen state at T=0. If the coolng process s conducted nsuffcently or the ntal temperature of the object s not suffcently hgh, the system may become quenched formng defects or freezng out n metastable states. Ths ndcates that the system s trapped n a local mnmum energy state. The process that s followed to smulate the annealng process was proposed by Metropols et al. 4 and t nvolves choosng the ntal state wth energy E old (see equaton 4) and temperature T and holdng T constant and perturbng the ntal confguraton and computng E new at the new state. If E new s lower than E old, then accept the new state, otherwse f the opposte s the case then accept ths state wth a probablty of exp -(de/t) where de s the change n energy. Ths process can be mathematcally represented as follows: f E new < E old accept state E E E (7) new old else accept E wth probablty exp new T Ths processes s repeated such that the samplng statstcs for the current temperature s adequate, and then the temperature s decreased and the process s repeated untl a frozen state where T=0 s acheved. SA was frst appled to optmzaton problems by Krkpatrck, et al. 3. The current state s the current updatng soluton, the energy equaton s the objectve functon n equaton 4, and the ground state s the global optmum soluton. Example: Asymmetrcal H-structure An unsymmetrcal H-shaped alumnum structure shown n Fgure 3 was used to valdate the proposed method. Ths structure was also used by Marwala and Heyns 4 as well as Marwala 5. Ths structure had three thn cuts of 1mm that went half-way through the crosssecton of the beam. These cuts were ntroduced to elements 3, 4 and 5. The structure wth these cuts was used so that the ntal FE model gves data that are far from the measured data and, thereby test the proposed proce- new 5

6 dure on a dffcult FE model updatng problem. The structure was suspended usng elastc rubber bands. The structure was excted usng an electromagnetc shaker and the response was measured usng an accelerometer. The structure was dvded nto 1 elements. It was excted at a poston ndcated by double-arrows, n Fgure 3, and acceleraton was measured at 15 postons ndcated by sngle-arrows n Fgure 3. The structure was tested freely-suspended, and a set of 15 frequency response functons were calculated. A rovng accelerometer was used for the testng. The mass of the accelerometer was found to be neglgble compared to the mass was run 150 tmes to generate the data for functonal approxmaton. The MLP mplemented had 1 nput varables correspondng to the 1 elements n the FE model, 8 hdden unts and one output unt correspondng to the error n equaton 4. As descrbed before, the MLP had a hyperbolc tangent actvaton functon n the hdden layer and lnear actvaton functon n the output layer. The RSM functonal approxmaton va the MLP was evaluated 10 tmes (teratons) each tme usng the GA to calculate the optmum pont and evaluatng ths optmum pont on the FE model and then storng the prevous optmum pont n the data set for the current A 400mm 00mm B 600mm y Cross-secton ndcated by lne AB x 9.8mm of the structure. The number of measured coordnates was 15. Thereafter, the fnte element model was constructed usng the Structural Dynamcs Toolbox 6. The FE model used Euler-Bernoull beam elements. The FE model contaned 1 elements. The modul of elastcty of these elements were used as updatng parameters. When the FE updatng was mplemented the modul of elastcty was restrcted to vary from 6.00x10 10 to 8.00x10 10 N.m -. The weghtng factors, n the frst term n equaton 4, were calculated for each mode as the square of the error between the measured natural frequency and the natural frequency calculated from the ntal model and the weghtng functon for the second term n equaton 4 was set to When the RSM, SA and GA were mplemented for model updatng the results shown n Table 1 were obtaned. On mplementng the proposed RSM, the FE model Fgure 3. Irregular H-shaped structure functonal approxmaton. The scaled conjugate gradent method was used to tran the MLP, prmarly because of ts computatonal effcency 7. The ntal functonal approxmaton was obtaned by tranng the MLP for 150 tranng cycles and on a subsequent functonal Table 1. Results showng measured frequences, the ntal frequences and the frequences obtaned when the FE model s updated usng the RSM, SA and GA Measured Freq (Hz) Intal Freq (Hz) 3.mm Frequences from RSM Updated Model (Hz) Frequences from SA Updated Model (Hz) Frequences from GA Updated Model (Hz)

7 approxmaton, where the data set had the prevous optmum soluton added to t, used 5 tranng cycles. On usng the RSM, the MLP was only ntalzed once. The GA was mplemented on a populaton sze of 50 and 00 generatons. The normalzed geometrc dstrbuton was mplemented wth a probablty of selectng the best canddate set to 8%, mutaton rate of 0.3% and crossover rate of 60%. When SA and a full FE model was mplemented for FE updatng, the scale of the coolng schedule was set to 4 and the number of ndvdual annealng runs was set to 3. When the smulaton was run, the frst run nvolved 7008 FE model calculatons, n the second run 6546 FE model calculatons and n the thrd run 5931 FE model calculatons were made. On mplementng the GA and a full FE model, the same optons as those that were used n the mplementaton of the RSM were used. The results showng the modul of elastcty of the ntal FE model, RSM updated FE model, SA updated FE model and GA updated FE model are shown n Fgure 4. Table 1 shows the measured natural frequences, ntal natural frequences and natural frequences obtaned by the RSM, SA and GA updated FE models. The error between the frst Modulus of Elastcty (e10) Element Intal RSM GA SA Fgure 4. Graph showng the ntal modul of elastcty and the modul of elastcty obtaned when the FE model s updated usng the RSM, GA and SA. Here e10 ndcates 10 to the power 10 and the unts are Nm - measured natural frequency and that from the ntal FE model, whch was obtaned when the modulus of elastcty of 7.00x10 10 N.m - was assumed, was 4.3%. When the RSM was used for FE updatng, ths error was reduced to 3.1% whle usng SA t was reduced to 0.% and usng the GA approach t was reduced to 0%. The error between the second measured natural frequency and that from the ntal model was 8.4%. When the RSM was used, ths error was reduced to 0.9% whle usng SA t was reduced to 1.3% and usng the GA t was reduced to.4%. The error of the thrd natural frequences between the measured data and the ntal FE model was 9.6%. When the RSM was used, ths error was reduced to 0.5% whle usng SA reduced t to 0.6% and usng the GA and a full FE model reduced t to 1.4%. The error between the fourth measured natural frequency and that from the ntal model was 3.7%. When the RSM was used for FE updatng, ths error was reduced to 1.1% whle usng the SA reduced t to 0.1% and usng the GA and a full FE model reduced t to 0.%. The error between the ffth measured natural frequency and that from the ntal model was 1.6%. When the RSM was used, ths error was ncreased to.8% whle usng SA ncreased t to.1% and usng the GA and a full FE model the error was reduced to 1.5%. Overall, the SA gave the best results wth an average error, calculated over all the fve natural frequences, of 0.9% followed by the GA wth an average error of 1.1% and then RSM wth an average error of 1.7%. All the three methods on average mproved when compared to the average error between the ntal FE model and the measured data, whch was 5.5%. The updated FE models mplemented were also valdated on the mode shapes they predcted. To make ths assessment possble the MAC 11 was used. The mean of the dagonal of the MAC vector was used to compare the mode shapes predcted by the updated and ntal FE models to the measured mode shapes. The average MAC calculated between the mode shapes from an ntal FE model and the measured mode shapes was When the average MAC was calculated between the measured data and data obtaned from the updated FE models, t was observed that the RSM, SA and GA updated FE model gave the mproved average of the dagonal of the MAC matrx of , and , respectvely. Therefore, the SA gave the best MAC followed by the GA whch was followed by the RSM. However, these dfferences n accuraces of the MAC and natural frequences were not sgnfcant. The computatonal tme taken to run the complete RSM method was 46 CPU seconds, whle the SA and a full FE model took 19 CPU mnutes to run and the GA and a full FE model took 117 CPU seconds. The RSM was found to be faster than the GA whch was n turn much faster than the SA whch was faster that the GA. On mplementng the RSM, 160 FE model evaluatons were made, whle on mplementng the SA FE model evaluatons were made and on mplementng 7

8 the GA FE model calculatons were made. In ths paper, a smple FE model wth 39 degrees of freedom s updated. It can, therefore, be concluded that f the FE model had several thousand degrees of freedom, the RSM wll be substantally faster than the other methods. Ths concluson should be understood n the lght of the fact that FE models usually have many degrees of freedom. Concluson In ths study, RSM s proposed for FE model updatng. The proposed RSM was mplemented wthn the framework of the MLP for functonal approxmaton and GA for optmzaton of the MLP response surface functon. Ths procedure was compared to the GA and SA. When these technques were tested on the unsymmetrcal H-shaped structure, t was observed that the RSM was faster than the SA and GA wthout much compromse on the accuracy of the predcted modal propertes. Acknowledgment The author would lke to thank Stefan Heyns, the now Natonal Research Foundaton as well as the AECI, Ltd for ther assstance n ths work. References 1 Frswell, M.I., and Mottershead, J.E., Fnte element model updatng n structural dynamcs, Kluwer Academc Publshers Group, Norwell, MA, 1995, pp Montgomery, D.C., Desgn and analyss of experments, 4th Edton, John Wley and Sons, NY, 1995, Chapter Marwala, T., Fnte element model updatng usng wavelet data and genetc algorthm Journal of Arcraft, 39(4), 00, pp Marwala, T., and Heyns, P.S., A multple crteron method for detectng damage on structures AIAA Journal, 195(), 1998, pp Levn, R.I. and Leven, N.A.J, Dynamc fnte element model updatng usng smulated annealng and genetc algorthms Mechancal Systems and Sgnal Processng, 1(1), pp Bshop, C.M., Neural Networks for Pattern Recognton. Oxford: Clarendon, Lee, S.H., Km, H.Y., and Oh, S.I., Cylndrcal tube optmzaton usng response surface method based on stochastc process Journal of Materals Processng Technology, (0), 00, pp Edwards, I.M., and Jutan, A., Optmzaton and control usng response surface methods Computers & Chemcal Engneerng, 1(4), 1997, pp Doeblng, S.W., Farrar, C.R., Prme, M.B., and Shevtz, D.W., Damage dentfcaton and health montorng of structural and mechancal systems from changes n ther vbraton characterstcs: a lterature revew Los Alamos Natonal Laboratory Report LA MS, Ewns, D.J., Modal testng: theory and practce, Research Studes Press, Letchworth, U.K, Allemang, R.J. and Brown, D.L., A correlaton coeffcent for modal vector analyss Proceedngs of the 1st Internatonal Modal Analyss Conference, 1-18, Sacks, J., Welch, W.J., Mtchell, T.J., and Wynn, H.P., Desgn and analyss of computer experments Statstcal Scence, 4(4), 1989, pp Varaajan, S., Chen, W., and Pelka, C.J., Robust concept exploraton of propulson systems wth enhanced model approxmaton Engneerng Optmzaton, 3(3), 000, pp Gunta, A. A., and Watson, L. T., A Comparson of Approxmaton Modelng Technques: Polynomal Versus Interpolatng Models, AIAA , Amercan Insttute of Aeronautcs and Astronautcs, Inc., 1998, pp Koch, P. N., Smpson, T. W., Allen, J. K., and Mstree, F., 1999, Statstcal Approxmatons for Multdscplnary Desgn Optmzaton: The Problem of Sze, Journal of Arcraft, 36(1), 1999, pp Jn, R., Chen, W., and Smpson, T., 000, Comparatve Studes of Metamodelng Technques under Multple Modelng Crtera, 8th AIAA/NASA/USAF/ISSMO Symposum on Multdscplnary Analyss and Optmzaton, Long Beach, CA, September 6-8, Ln, Y., Krshnapur, K., Allen, J. K., and Mstree, F., 000, Robust Concept Exploraton n Engneerng Desgn: Metamodelng Technques and Goal Formulatons, Proceedngs of the 000 ASME Desgn Engneerng Techncal Conferences, DETC000/DAC- 1483, September 10-14, 000, Baltmore, Maryland. 18 Wang, G.G., 003, Adaptve response surface method usng nherted Latn hypercube desgn ponts, Transactons of the ASME, Journal of Mechancal Desgn,15, pp Smpson, T. W., Peplnsk, J. D., Koch, P. N., and Allen, J. K., Metamodels for Computer-based Engneerng Desgn: Survey and Recommendatons, Engneerng wth Computers, 17, 001, pp Fletcher, R., Practcal Methods of Optmzaton. nd edton, New York: Wley, Holland, J, Adaptaton n Natural and Artfcal Systems, Ann Arbor: Unversty of Mchgan Press, Goldberg, D.E., Genetc algorthms n search, optmzaton and machne learnng, Addson-Wesley, Readng, MA, Metropols, N, Rosenbluth, A.W., Rosenbluth, M.N., Teller, A.H., and Teller, E., Equatons of state 8

9 calculatons by fast computng machnes, Journal of Chemcal Physcs, 1, 1953, pp Krkpatrck, S., Gelatt, C.D., and Vecch, M.P., Optmzaton by smulated annealng, Scence, 0, 1983, pp Marwala, T, A multple crteron updatng method for damage detecton on structures. Unversty of Pretora Masters Thess, Balmès, E., Structural Dynamcs Toolbox User s Manual Verson.1, Scentfc Software Group, Sèvres, France, Møller, M. A scaled conjugate gradent algorthm for fast supervsed learnng Neural Networks, vol. 6, 1993, pp