Model Updating Using Bayesian Estimation

Transcription

1 Model Updatng Usng Bayesan Estmaton C. Mares, B. Dratz 1, J.E. Mottershead, M. I. Frswell 3 Brunel Unversty, School of Engneerng and Desgn, Uxbrdge, Mddlesex, UB8 3PH, UK 1 Ecole Centrale de Llle, Mechancal Engneerng, France Unversty of Lverpool, Dept. of Engneerng, Brownlow Hll, Lverpool L69 3GH, UK 3 Unversty of Brstol, Department of Aerospace Engneerng, Queen s Buldng, Unversty Walk,Brstol BS8 ITR, UK Abstract Varablty n real structures, whch could arse from manufacturng processes, and the modellng assumptons and lmtatons requre the creaton of a statstcal model of the relatonshp between expermental and model predctons and the quantfcaton of the uncertanty of ths estmate. In ths paper Markov-Chan Monte Carlo theory (MCMC) s dscussed and appled to model updatng n the case of multple sets of expermental results by usng frequency responses functons. The MCMC method allows the soluton of complex problems n a unfyng framework, by ntegratng over hgh dmensonal probablty dstrbutons n order to make nferences about the model parameters. A smulated three degree-of-freedom system s used to llustrate some aspects of the method, allowng for practcal assumptons to be tested on a smple example wthn the WINBUGS envronment (Bayesan nference Usng Gbbs Samplng). 1 Introducton Model updatng s usually performed by analysng the degree to whch a fnte element model adequately represents a sngle set of expermental data [1], []. The updated parameters should be justfed physcally and the qualty of the fnal model should be assessed wthn the operatng range. Robustness-touncertanty, fdelty-to-data and confdence-n-predcton are aspects whch gve a measure of credblty for the updated structural model [3], [4]. Varablty n real structures, whch could arse from manufacturng processes, and the modellng assumptons and lmtatons have an mportant effect when complcated jonts and nterfaces, such as welds and adhesvely connected surfaces, are present n the actual structure. For such complex structures, predctons based on a sngle calbraton of the model parameters cannot gve a clear measure of confdence n the capablty of an analytcal model to represent the actual structure. The possblty of correcton of a set of analytcal models wth randomzed parameters based on a set of expermental results from a collecton of nomnally dentcal test peces was presented n [5],[6] where a stochastc Monte-Carlo correlaton and nverse uncertanty propagaton was carred out on a smulated example and a benchmark structure wth spot-welds. The three goals presented above are antagonstc and an estmate of the uncertanty leads to the necessty of creatng a model of the relatonshp between expermental and model predctons. In ths paper the Markov-Chan Monte Carlo theory (MCMC) [7],[8], s dscussed and appled to model updatng n the case of multple sets of expermental results by usng frequency response functons. The MCMC method allows the soluton of complex problems n a unfyng framework, by ntegratng over hgh dmensonal probablty dstrbutons n order to make nference about the model parameters. A smulated three degree-of-freedom system s used to llustrate some aspects of the method, allowng for practcal assumptons to be tested on a smple example wthn the WINBUGS envronment (Bayesan nference Usng Gbbs Samplng) [10]. Ths software uses Gbbs samplng [11],[1],[13] and the Metropols algorthm [14] to generate a Markov chan by samplng from the full condtonal dstrbutons. Dfferent practcal assumptons are analysed and an assessment of the consequences of wrongly chosen 607

2 608 PROCEEDINGS OF ISMA006 updatng parameters, model structure errors, nose effects, observablty of the model errors through measurements locatons and optmal frequency measurement ponts for the mnmsaton of model errors s carred out. Theory.1 Bayesan Analyss Bayesan analyss provdes a ratonal approach for the soluton of complex problems, n partcular to parameter estmaton problems wth an assumed correct model or n model selecton problems. The learnng process, based on data, s performed usng Bayes theorem n a hypothess space, where probabltes are assgned to each competng hypothess regardng the problem under study. The parameter estmaton problem s then solved by estmatng the dstrbuton of a random parameter wthn an ensemble of data sets and the pror nformaton. The usual form of Bayes theorem s gven by: p( H p( D H, p( H D, = (1) p( D where H proposton assertng the truth of a hypothess of nterest I - proposton representng our pror nformaton D - proposton representng data p( D H, - probablty of obtanng data D, f H and I are true (called the lkelhood functon ) p( H - pror probablty hypothess p( H D, - posteror probablty of H p( D = p( H p( D H, - normalsaton factor ensurng that p( H D, = 1 Equaton (1) shows how the pror probablty of a hypothess H s updated to a posteror probablty p( H D, whch ncludes all the nformaton provded by the data D. The updatng factor s the rato of two terms and only the lkelhood functon (or samplng dstrbuton) p( D H, depends explctly on H, the denomnator p ( D, called the pror predctve probablty or the global lkelhood, beng ndependent of H.. Markov Chan Monte-Carlo Bayesan Inference Quantfcaton of the uncertanty n model parameters when expermental data and model predctons n some output varables of nterest, can be performed by usng the Bayesan nference. The Bayesan soluton requres ntegrals over the model parameter space and Markov Chan Monte Carlo (MCMC) algorthms form the bass of practcal computaton for probablty dstrbutons n several or many unknowns. The MCMC method estmates the requred posteror dstrbutons for hgh dmensonal models by usng sampler algorthms such as the Metropols Hastngs or Gbbs approaches, whch concentrate the samplng n regons wth sgnfcant probablty performng an effcent stochastc search n the model space. The models are constructed to assess the relatonshp between the response varables y and other characterstcs expressed n varables θ usually called covarates. The explanatory varables are lnked j

3 MODEL UPDATING AND CORRELATION 609 wth the response varables va a determnstc functon. For ndependent model responses followng a probablstc rule f ( θ ) the jont dstrbuton contans all the avalable nformaton provded by the sample: y f ( yθ ) = f ( y θ ) A Markov chan s a stochastc process { ( 1) (), ( t, θ θ ) } f ( θ θ () θ L such that (1) ( t+ 1) ( ), L θ ) = f ( θ θ ) the parameter dstrbuton θ at tme t + 1gven all the precedng θ ( t+ 1) ( t) t (t) ( t 1) ( t) depends only on θ and f ( θ + θ ) s ndependent of the tme step t. It can be shown that under (t) (0) certan condtons the dstrbuton θ tends to an equlbrum ndependent of the ntal estmates θ [8] convergng on a posteror dstrbuton called the statonary dstrbuton of the Markov chan. Assumng (0) that the equlbrum dstrbuton s the target dstrbuton f ( yθ ), after ntalzaton θ, the Gbbs sampler algorthm ntroduced by Geman and Geman [11] generates samples of θ wth a probablty densty whch converges on the desred posteror target. The Gbbs sampler algorthm s a specal case of a sngle component Metropols-Hastngs algorthm [1],[13],[14]..3 Stochastc Search Varable Selecton In buldng a model for the estmaton problem, a key aspect s that of the varables to be ncluded. The Bayesan model selecton usng MCMC technques explores the model space, tracng the best models and estmatng ther posteror probablty based on the observed data and the pror probabltes. A possble two-step soluton for a subset of predctors was developed by George and McCulloch [14]. The Stochastc Search Varable Selecton (SSVS) method assgns a probablty dstrbuton on a set of models such that the most approprate models are gven the hghest probablty dstrbuton and the approach s effectve even for a large number of predctors and small number of observatons [14]. The man feature of ths approach s the assgnment of a latent ncluson varable δ to every predctor. Suppose the adopted model * * * * s descrbed by a set of predctors θ1, θ, θ 3 K θ q from a larger set θ 1, θ, θ 3 Kθ n. The components θ are modelled as a normal mxture: π ( θ δ ) = (1 δ ) N(0, τ ) + δ N(0, cτ ) P( δ = 1) = 1 P( δ = 0) wth τ << cτ (3) If δ = 1 then θ s dstrbuted as a normal wth mean zero and varance cτ where c s a large constant pre-multplyng a small default varance τ. If the ncluson of θ s not supported by the data, then the pror wth the default varance N(0, τ ) wll be selected more often. The choce δ = 1 corresponds to retanng the predctor θ whle for δ = 0 the predctor s dstrbuted around zero wth a small varance, meanng that the data does not support the appearance of the predctor among the defnng varables for the model.

4 610 PROCEEDINGS OF ISMA006 3 Example If the model s descrbed by usng a fnte element model, the global matrces are expressed as lnear combnatons of constant element or substructure matrces multpled by the varable parameters and affectng all the terms n the substructure stffness or mass matrx. For example a general parametersaton for the stffness matrx may be wrtten as, e K ( x) = x K (4) j j wth smlar decompostons for the mass or dampng matrces. The choce of updatng parameters and ther type of varaton (bounds and probablty dstrbutons), must be justfed physcally and requres expermental data that are not always accessble. Dfferent measures for goodness-of-ft used n test-analyss correlaton can be used for Bayesan estmaton [15],[16]. In ths paper the lkelhood functon s bult usng a root mean square error (RMS) between expermental and predcted frequency response functons: { measurement pont set} { exctaton pont set} exp ( αβ ( ) ( )) Nfreq Nfreq Ly ( θ ) = ε k = h ωk h αβ ωk; θ k= 1 k= 1 α β (5) where the sum the measured FRFs for a chosen set Nfreq of the frequency lnes, at the exctaton and measurement locatons used n test. For varablty studes, the sum n equaton (5) s augmented to { measurement pont set} { exctaton pont set} exp ( αβ, j ( ), ( )) Nsamples Nfreq Nsamples Nfreq Ly ( θ ) = ε k = h ωk h αβ j ωk; θ j= 1 k= 1 j= 1 k= 1 α β (6) When the nformaton about the system s uncertan, the model predcton ncludes error components due to the measurement and statstcal uncertanty n the model, generally these beng uncorrelated. In the example dscussed n ths paper, only the errors due to the analytcal model are dscussed n order to assess dfferent possble aspects n model updatng. Each updatng varable n each sample, vares accordng to a Gaussan law, expressed by: k ~ N( k 1, τ ); τ = (7) σ The pror nformaton for each mean and varance parameter k, σ s encoded n a probablty dstrbuton to be used wth the lkelhood term, equaton (6) for Bayesan estmaton usng MCMC: k = Normal(0.0,1.0e 6); τ = Gamma(0.001,0.001) (8) The example consdered s the 3 degree-of-freedom system, shown n Fgure 1.

5 MODEL UPDATING AND CORRELATION 611 Fgure 1. Three Degree-of-Freedom Mass-Sprng Example. The nomnal values of the parameters for the expermental system are: m = 1. 0 kg ( = 1,, 3) k = 1. 0 N/m ( = 3,4, ) and k 6 = 3. 0 N/m. The analyss random parameters have Gaussan dstrbutons wth mean values, k 1 = k = k5 = 1. 0 and standard devatons, σ 1 = σ = σ 5 = 0. 0 N/m. For the analytcal model, the erroneous parameters have the ntal values k1 = k = k5 = 1. 5N/m. A set of 10 systems are used for statstcal nference. 10 Expermental Model Set A Set B Set C H 11 (m/n) Frequency(rad/s) Fgure. Frequency pont sets n Case 1. The followng cases are smulated: Case 1: Only the erroneous parameters are updated as n the deal case of perfectly localzed errors. Case : In the expermental model a sprng connecton: k 6 s not present, the analytcal model presentng a connectvty error. The updatng parameters reman the same as n Case 1. Case 3: The same as Case 1 except that all the sprngs are Gaussan random varables wth standard devatons σ = 0. 0 N/m for =1, K5 and σ 6 = 0. 6 N/m.

6 61 PROCEEDINGS OF ISMA006 Case 1: Ths s an deal case wth all the errors contaned n the analytcal model located correctly and t wll be used to evaluate the method s performance. In order to obtan the most accurate predcton, an optmum subset of frequency lnes should be determned wthn the acqured frequency range. The computatonal costs, and many factors nfluencng the relatve accuracy wth whch the response can be measured, precludes the use of all the frequency range n the updatng algorthm. In the case of real data, the nose mpacts upon the results and dfferent frequences can dramatcally alter the results, and each algorthm s affected by a combnaton of measurement and processng errors. Fgure 3. Mean parameter varaton and pdf n Case 1. Expermental Updated Model Parameter Mean Std Mean Std k (-0.36) 0.194(-7.77) k (-.17) 0.181(-.84) k (1.39) 0.57(-7.98) Table 1. Parameter Estmate n Case 1: h11 and h measured, frequency set A.

7 MODEL UPDATING AND CORRELATION 613 The fnte element model s updated usng the lkelhood functon, equaton (1): h1,1, h, evaluated for set A consstng of 5 equally spaced frequency lnes n the nterval rad [ 0.0,4.0] ( ) ; s h for set A; 1, 1 h1,1, h, evaluated for set B consstng of 14 frequency lnes avodng the areas close to the resonances or ant-resonances; h1,1, h, evaluated for set C wth 16 frequency lnes n the areas close to the resonances. The locaton for each frequency set s shown n Fg. for a nomnal expermental model. The results for the mean and standard devaton obtaned after updatng n each case after 6000 teratons, are presented n Tables 1-4. The convergence hstory for each updated parameter and the fnal probablty densty functons when usng set A and h 1,1, h, are presented n Fg. 3.In ths case, the estmated mean and standard devaton of the expermental sample are not exactly the same as the dstrbuton that was used to produce them. The updated parameters converge upon the statstcs of the expermental sample rather than the underlyng dstrbuton. The results are mproved when usng two frequency response functons due to ncreased nformaton contaned n the response functons. The errors for the mean values are reduced when usng frequency lnes avodng the resonances (set B) as opposed to a equally spaced selected frequences (set A) but both sets present standard devatons errors (ncreased for set B). Set C presents bg errors for all the estmates showng that n general t s better to avod frequences located n the resonance or ant-resonance areas. The expermental sample s small, as encountered usually n ndustral applcatons and t can be expected that f more expermental samples were avalable, the MCMC process would estmate better the model parameters. Expermental Updated Model Parameter ean Std Mean Std k (3.5) 0.07(-15.0) k (-5.54) 0.158(10.) k (.8) 0.59(-8.8) Table. Parameter Estmate n Case 1: h11 measured, frequency set A. Expermental Updated Model Parameter Mean Std Mean Std k (0.3) 0.197(-9.44) k (0.19) 0.187(-6.5) k (-0.10) 0.63(-10.50) Table 3. Parameter Estmate n Case 1: h11 and h measured, frequency set B. Expermental Updated Model Parameter Mean Std Mean Std k (-38.40) 0.398(-11.11) k (-1.55) 0.0(-14.77) k (1.63) 0.94(-3.5) Table 4. Parameter Estmate n Case 1: h11 and h measured, frequency set C.

8 614 PROCEEDINGS OF ISMA006 Case The model correcton s carred out n two phases, frst by analyzng the structure of the analytcal model and then by an estmaton phase smlar to that dscussed for Case 1. The possblty of havng a dfferent structure of the analytcal model s nvestgated by usng the model selecton procedure proposed by George and McCulloch [14]. In the Stochastc Search Varable Selecton (SSVS) method, the relatonshp between the output varables and a set of predctor varables s descrbed by a latent ncluson varable. Three dfferent models are consdered for comparson, each havng a dfferent structure by addng a sprng connecton k 5, k 6 or both, to a baselne system as shown n Table 5. The ntal sprng stffnesses are k = k = k 1. 5N/m. 1 5 = Model Elements 1 k1,k,k3,k4,k5 k1,k,k3,k4,k6 3 k1,k,k3,k4,k5, k6 Table 5. Models used n SSVS. As prevously, the frequency responses h11 and h are used n the lkelhood functon, equaton (4). The probablty of appearance for each model, named p, s defned as a categorcal dstrbuton, p = (1,...,3) 3 and P ( p) = 1. A pror dstrbuton for p s gven by the nverse of the number of models contanng k = 1 each k. A lnk s created between the models and varables as descrbed by (9): model model 0 1 model k5 k6 (9) The varable parameters are defned by equaton (3) as a normal mxture and the stochastc search wll determne a posteror dstrbuton for the ncluson varable whch wll defne the model wth the best structure. The results for a smulaton wth 5000 updates are presented n Fgure 4, where the probablty of occurrence for the frst model, ranks t as havng the most approprate for the analysed data. In ths case the connecton represented by the sprng k 6 should not exst n the fnal analytcal model ( model 1 s the most probable). Case 3 Table 6 shows the results n ths case. All of the stffnesses are randomsed and the estmated parameters ncorporate all the varablty presented by the expermental set. The two models come from dfferent famles wth dfferent statstcal propertes and both the mean and standard devatons are n error. The same type of result can be obtaned when a localsed error s not located pror to the updatng process and consequently not ncluded n the updatng loop.

9 MODEL UPDATING AND CORRELATION 615 a) b) c) Fgure 4. Probablty of appearance for each model: a) convergence of the latent varable; b) detal of the convergence process; c) fnal probabltes, model 1 beng the most probable one. Expermental Updated Model Parameter Mean Std Mean Std k (-.58) 0.67(-56.14) k (0.0) 0.166(-9.93) k (.31) 0.11(15.30) Table 6. Parameter Estmate n Case 3. 4 Conclusons In ths paper a non-lnear least squares analyss seen from a Bayesan perspectve s used for model updatng. The uncertanty about some model parameter or about the model structure s encoded based on pror nformaton and used n a Bayesan nference for the determnaton of a posteror probablty densty functon based on measured frequency response data. A smulated example s used to demonstrate the potental of the method for applcatons related to real lfe applcatons. It s accepted nowadays that the structure of the model (the parametersaton used to descrbe the model dynamcs) s the most mportant aspect related to the updatng process. A two-stage methodology for varable selecton and an analyss of measurement locatons and frequency lnes dstrbuton are presented together wth the effects on parameter estmaton.

10 616 PROCEEDINGS OF ISMA006 References [1] J. E. Mottershead and M. I. Frswell, Model updatng n structural dynamcs: a survey, Journal of Sound and Vbraton, 16(), (1993), pp [] M. I. Frswell, J. E. Mottershead, Fnte Element Model Updatng n Structural Dynamcs, Dordrecht, Kluwer Academc Press, (1995). [3] F. M. Hemez, S. W. Doeblng, M. C. Anderson, A bref tutoral on verfcaton and valdaton, Internatonal Modal Analyss Conference, Dearborn, USA, (003). [4] K. F. Alvn, W. L. Oberkampf, K. V. Degert, B. M. Rutherford, Uncertanty quantfcaton n structural dynamcs: a new paradgm for model valdaton, Internatonal Modal Analyss Conference, USA, (1998). [5] C. Mares, J. E. Mottershead and M. I. Frswell, Stochastc Model Updatng - Part 1: Theory and Smulated Examples, Mechancal Systems and Sgnal Processng, 0(7), (006), pp [6] J. E. Mottershead, C. Mares, S. James and M. I. Frswell, Stochastc Model Updatng: Part : Applcaton to a Set of Physcal Structures, Mechancal Systems and Sgnal Processng, n press. [7] D. Gamerman, Markov Chan Monte Carlo Stochastc Smulaton for Bayesan Inference, (1997). [8] W.R. Glks, S. Rchardson and D.J. Spegelhalter, Markov Chan Monte Carlo n Practce, Chapman &Hall, (1996). [9] D.J. Spegelhalter, A. Thomas, N. J. Best, D. Lunn, WnBUGS verson1.4 Users Manual, MRC Bostatcs Unt Cambrdge, URL:// (003). [10] N. Metropols, A. Rosenbluth, M. Rosenbluth, H. Teller, E. Teller, Equaton of State Calculatons by Fast Computng Machnes, Journal of Chemcal Physcs, 1, (1953), pp [11] S. Geman, D. Geman, Stochastc Relaxatons, Gbbs Dstrbutons and the Bayesan Restoraton of Images, IEEE Transactons of Pattern Analyss and machne Intellgence, 6, (1984), pp [1] A. E. Gelfand, A. F. M. Smth, Samplng-Based Approaches to Calculatng Margnal Denstes, Journal of the Amercan Statstcal Assocaton, 85, (1990), pp [13] G. Casella, E. George, Explanng the Gbbs Sampler, Amercan Statstcan, 46, (199), pp [14] E. I. George, R. E. McCulloch, Varable Selecton Va Gbbs Samplng, Journal of Amercan Statstcal Assocaton, 88, (1993), pp [15] G. Kerschen, J-C. Golnval, F. M. Hemez, Bayesan Model Screenng for the dentfcaton of Nonlnear Mechancal Structures, Journal of Vbraton and Acoustcs, 15, (003), pp [16] T. Marwala, S. Sbs, Fnte Element Model Updatng Usng Bayesan Framework and Modal Propertes, Journal of Arcraft, 4(1), (005), pp