Bayesian evidence for Finite element model updating

Transcription

1 Bayesan evdence for Fnte element model updatng Lnda Mthembu, PhD student, Electrcal and Informaton Engneerng, Unversty of the Wtwatersrand, Johannesburg Prvate Bag 3, Johannesburg, 050, South Afrca Tshldz Marwala, Professor of Electrcal and Informaton Engneerng, Unversty of the Wtwatersrand, Prvate Bag 3, Johannesburg, 050, South Afrca Mchael I. Frswell, Sr George Whte Professor of Aerospace Engneerng, Department of Aerospace Engneerng, Queens Buldng, Unversty of Brstol, Brstol BS8 1TR, UK Sondpon Adhkar, Char of Aerospace Engneerng, School of Engneerng, Swansea Unversty, Sngleton Park, Swansea SA 8PP, Unted Kngdom Abstract Ths paper consders the problem of model selecton wthn the context of fnte element model updatng. Gven that a number of FEM updatng models, wth dfferent updatng parameters, can be desgned, ths paper proposes usng the Bayesan evdence statstc to assess the probablty of each updatng model. Ths makes t possble then to evaluate the need for alternatve updatng parameters n the updatng of the ntal FE model. The model evdences are compared usng the Bayes factor, whch s the rato of evdences. The Jeffrey s scale s used to determne the dfferences n the models. The Bayesan evdence s calculated by ntegratng the lkelhood of the data gven the model and ts parameters over the a pror model parameter space usng the new nested samplng algorthm. The nested algorthm samples ths lkelhood dstrbuton by usng a hard lkelhood-value constrant on the samplng regon whle provdng the posteror samples of the updatng model parameters as a by-product. Ths method s used to calculate the evdence of a number of plausble fnte element models. Nomenclature θ D H Z N Max L π c k m & x& ( t) x& ( t) ( t) w φ Model Parameters Real measured system data Fnte element model (Mathematcal model) Evdence Number of samples Maxmum number of teratons Lkelhood probablty Pror Probablty Dampng matrx Stffness matrx Mass matrx Node acceleraton Node velocty x Node dsplacement Natural frequency vector Mode shape vector FEM Fnte element model FEMUP Fnte element model updatng problem PDF Probablty Dstrbuton Functon MCMC Markov Chan Monte Carlo

2 1. Introducton System modelng forms an mportant stage of many engneerng desgn problems. The results from the model ether confrm or hghlght lmtatons of the desgn. An analyst s usually nterested n the accuracy, confdence range and more crtcally the correctness of the assumed mathematcal model. In ths paper the model doman s structural fnte element models (FEMs). These models are used to approxmate the dynamcs of structural systems, e.g. tran chasss, arcraft fuselages, bcycle frames, cvl structures etc. The fnte element model updatng problem (FEMUP) arses when a real system s dynamc behavor s measured (e.g. the natural frequences at whch partcular system deformatons occur) and the results of the mathematcal model of that system do not correspond to the measured data [6, 8]. Ths problem s compounded by the fact that a multtude of mathematcal models of the structure, wth varyng levels of complexty, can be developed, leadng to non-unque solutons for a partcular system. Fnte element models are lmted by defnton; they are an approxmaton of a real system and wll thus never produce dynamc results that are equal to the measured system s data. The challenge s then what can be done to the ntal model for t to better reflect the real system s dynamc results? Ths leads to the need for ntellgently mproved or updated models. Specfcally an automatc methodology of determnng salent model parameters that requre updatng needs to be developed. Ths has to be attaned whlst usng realstc characterstc parameters of the system n queston. Two man drectons of research have been establshed n the area of fnte element model updatng (FEMU); drect and ndrect (teratve) methods [8]. In the drect model updatng paradgm [3, 5, 8] the model modal parameters are drectly equated to the measured modal data. Model updatng s then characterzed by the drect updatng of mass or stffness matrx elements. Ths effectvely constrans the modal propertes and frees the system matrces for updatng. Ths approach often results n unrealstc elements n the system matrces e.g. large and physcally mpossble mass elements. In the ndrect or teratve model updatng approach the updatng problem s formulated as a relaxed optmzaton problem, often approached by the use of maxmum lkelhood methods [8, 9, 1 and 19]. A select few parameters are vared n model elements and the resultng model s optmzed to mnmze ts dfference from the measured data. Gven the non-unqueness of updatng models, ths paper proposes gong back to the ntal stage of modelng and questons the choce of the ntal fnte element model(s). We propose the FEMUP should ntally be approached from a model selecton [, 10, 17] perspectve. Before model updatng, a number of fundamental questons need to be addressed, these are; () Is the model we want to update the correct one? () Whch aspects of the model do we need to update? () Havng determned the updatng parameters, wth what certanty can we guarantee that our updated model s correct? The frst pont can be recast as; what evdence do we have that our model s the correct one gven that a number of models can be generated? Ths we propose ought to be an essental and necessary statstc to establsh before any model updatng scheme can proceed. Havng establshed that the most probable model s found from a set of possble models we can then focus on the updatng process. In ths paper we focus on the frst problem whch lends tself well to Bayesan nference. Bayesan analyss allows one to deal wth ntal and subsequent fnte element model uncertantes n an ntutve and systematc way [, 10]. Ths paper s concerned wth calculatng the evdence of fnte element models. In the next secton we formally present the fnte element model formulaton and very compactly revew the current approaches to the FEMUP. In secton 3 we revew Bayes theorem and ntroduce the evdence calculaton n the Bayesan framework. Secton 4 presents the evdence calculaton algorthm. Secton 5 presents proof of concept smulatons of evdence calculaton usng a smple beam structure. We then conclude the paper.. Fnte element background.1 Formal defnton In engneerng, dynamc structures are often analyzed from bottom to systems (component) level. At the bottom most level the structure s dscretzed nto consttuent elements, for mathematcal analyss and computatonal feasblty, to a system of a second-order matrx of the form [6, 8]: M & x ( t) + Cx& ( t) + Kx( t) = 0 (1)

3 where M, C and K are of equal sze and are called the mass, dampng and stffness matrces respectvely or system matrces. Assumng each fnte element s dynamc dsplacement response behaves accordng to wt x( t) = φ e equaton 1 above s transformed to: [ w M + wc + K] φ = 0 () In cases where the structure s small, lghtly damped or undamped the dampng matrx C = 0 the above equaton reduces to: where { ε } w and [ w M + K] φ = { ε } φ are the th system natural frequency and mode shapes, together known as modal propertes, s the th error of the error vector. Gven a set of measured real system modal data, the FEMU problem s then for the model to realstcally approxmate the mass and stffness matrces that wll produce modal data that s as close to the real systems as possble. If the model does not result n matchng modal propertes the error vector s non-zero and some model parameters wll need to be updated. Basc physcs relates elastc element stffness wth Young s modulus; the mass s a functon of ts geometry and densty. By consderng these varables as random values that are defned wthn certan ntervals for partcular materals, t s possble to closely approxmate the measured modal values n a formal way. In ths paper we treat the Young s modulus as a varable to be updated. Measurement of structural dynamcs produces one set of, not necessarly repeatable data, e.g. buldng vbratons, earthquakes and vehcle dynamc performances. Tradtonal approaches to prevent data basng models are to repeat the measurements a number of tmes to generate a general trend and averagng the data. Ths s not a smple process and can become very expensve [6]. Ths stuaton s well suted to the Bayesan approach as opposed to a frequentst paradgm. In Bayesan nference we are gven one observed data set whch our model s supposed to approxmate well. In the frequentst approach the data acquston or experment s assumed to be repeatable, such that a pattern could be determned from a large dataset. The Bayesan perspectve s bold n that t nfers a lot from one dataset []. In the next secton we present the tools of ths paradgm, Bayes theorem and then explan Bayesan nference n the context of the fnte element updatng problem. 3. Bayesan Inference In current tmes of massve data, the ablty to quantfy the model s posteror confdence and to compare one model s ablty to approxmate data wth another has become the focus of data analyss. One paradgm n ths drecton that has seen a recent resurgence s Bayesan nference [, 10 and 0]. Bayesan nference allows one to quantfy uncertantes n quanttes of nterest n a formal way. Bayesan nference s often mplemented n two settngs; parameter estmaton and model selecton. Parameter estmaton s concerned wth the plausblty of a gven model s parameters and ths s often carred out usng standard samplng methods e.g. Markov Chan Monte Carlo (MCMC) [, 10, 11 and see 16 for recent advances n these technques]. Model selecton on the other hand deals wth the evdence for each canddate mathematcal model to approxmate a partcular observed dataset. 3.1 Parameter estmaton In parameter estmaton the mathematcal model s assumed to be true, the model s then ftted to the data and the posteror plausblty of the model parameters can be nferred. Ths probablty s calculated va Bayes theorem as follows: D θ, H ) θ H ) θ D, H ) = D H ) where the left hand sde s the posteror probablty of the updatng parameters for the true model H, gven some data D, the pror probablty of the model parameters s θ H ), D θ, H ) s called the lkelhood of the model. The denomnator D H ) s called the margnal lkelhood, or the evdence, of the model where the parameters have been margnalzed out. Bayes theorem automatcally ncorporates the updatng of the (4) (3)

4 parameters by defnton; t updates the a pror probablty dstrbuton of the model parameters wth the lkelhood of the model explanng the measured data. In the FEM context the parameter varables that typcally govern the dynamcs of the fnte elements of the mathematcal model are the followng: Young s modulus, cross-sectonal areas, dampng coeffcent etc. These parameters n turn affect the stffness, mass and dampng matrces n equaton 1. In [11] the parameter estmaton context of the Bayesan framework was mplemented to obtan the posteror probablty dstrbuton of model parameters. By obtanng the posteror probablty of the updatng parameters the probablty dstrbuton of the modal propertes may then be calculated. Ths gves the researcher a quanttatve measure of the probable ranges of the modal and updatng parameters of the model. In ths paper the nterest s not n the posteror probablty of the updatng parameters per se but n the evdence of the chosen updatng model thus a model selecton problem although the algorthm mplemented provdes the posteror probabltes of the updatng parameters as a by-product. 3. Model Comparson In model comparson one dataset s observed and a number of possble models can be formulated. Bayesan nference provdes a platform for calculatng whch model s the most plausble from ths set. The posteror probablty of each model wthn a set of plausble models s gven by Bayes theorem: where H ) ( H D) = D H ) H ) D) P s the pror probablty of each model and D) (5) s the probablty of the data. Snce for all models the denomnator s ndependent of the model we may gnore t and the theorem then reduces to: P ( H D) D H ) H ) (6) The frst term on the rght of equaton 6 s the evdence from equaton 4. Gven that each model may be equally lkely to ft the data, the evdence term s the decdng factor on whch model s the most plausble for a partcular observed dataset. One standard method of comparng models n Bayesan analyss s Bayes factor defned as; Bayes Factor = D H ) = D H ) dθ D θ, H ) θ H ) dθ D θ, H ) θ H ) (7) So by calculatng each model s evdence we can quantfy ther ratos. In [1, 4 and 17] the concept of model selecton for engneerng structures was used. In [1] the posteror dstrbuton of the model was assumed to be Gaussan whch only works for certan types of models [1,, 4, 10 and 17]. The method proposed n ths paper does not place any assumpton of the form of the posteror dstrbuton of the model. In [17] a recently proposed MCMC type posteror probablty samplng algorthm (TMCMC from [4]) was mplemented. Ths algorthm estmates the model evdence by samplng the posteror probablty dstrbuton of the model by a sequence of non-normalzed ntermedate probablty functons. Ths algorthm suffers from the use of many free parameters; e.g. a varable to balance the samplng steps, the number of ntermedate probablty dstrbuton functons (PDF), the temperng parameter and the sample weghts [4, 17]. The algorthm proposed here s beleved to be smpler and to have fewer free-parameters. In the next secton we explan the Bayesan evdence. We then ntroduce an algorthm called nested samplng for effcently estmatng the model evdence that s theoretcally and practcally smpler than TMCMC. 3.3 Bayesan evdence From the prevous equaton each model s evdence s gven by equaton 8: = evdence = P D H ) dθ D θ, H ) θ H ) ( (8)

5 dθ wth a model havng a pror Ths equaton may be nterpreted as follows; gven a unt of parameter space probablty of P θ H ) over ths space, the posteror probablty dstrbuton of the model over the same space ( wll depend on how the model parameters ft the data D, H ) P θ. In ths paper the lkelhood functon s defned ( as the normalzed exponental error between measured and real natural frequences (the error gven n equaton 1) at the measured modes as n [Ref.13]. The model posteror probablty wll be hghest at the most probable parameter set whch occupes a small regon of the orgnal parameter space. In smple models an ntally smaller, by defnton, pror parameter space would almost be fully utlzed to ft the data as the posteror probablty would occupy a large porton of the pror parameter space resultng n a correspondngly large evdence value. Complex models have larger parameter spaces because they have many free parameters that allow them to ft almost any data and ths often results n over-fttng [, 10]. Ths larger parameters space s wasted n accountng for the resultant, relatvely narrow posteror densty. The Bayesan nference paradgm thus automatcally penalzes complex models wthout the need for a model regularzaton term often ncorporated n non-bayesan approaches [, 10, 19]. Contnung wth the evdence formulaton, equaton 8 can be re-wrtten n the followng form: Z = L( θ ) π ( θ ) dθ (9) Analytcally evaluatng ths ntegral may be dffcult or mpossble f the product of the pror and lkelhood s not smple. Ths often happens when for example the parameter space s hgh dmensonal whch requres calculatng multdmensonal lkelhoods. The most popular approach to approxmatng such ntegrals has been to apply numercal technques such as mportance samplng and thermodynamc ntegraton but these confront the problem that the pror parameters are dstrbuted n regons where the lkelhood functon s not hghly concentrated [,10,16]. Other recent samplng technques for example TMCMC n [4] that are based on MCMC paradgm can be used but these tend to only sample the peaks of the posteror dstrbuton whch can under sample most of the narrow best-ft regons. Recently Skllng n [18] proposed an algorthm, nested samplng, whch s able to estmate ntegrals of the form shown n equaton 9 effcently. The algorthm works by transformng the multdmensonal parameter space ntegral to a one dmensonal one where classcal numercal approxmaton technques of estmatng the area under the functon can be appled. The algorthm has been successfully appled and further modfed n a number of recent astronomy and cosmology papers [7, 14, 15, 0] 4. Samplng 4.1 Nested samplng Nested samplng s a Monte Carlo but not a Markov Chan method of samplng. The man dea behnd the method s to dvde the pror parameter space nto equal mass unts and to order these by model lkelhood. The total pror mass s denoted X and each unt n ths pror mass s P ( θ H ) dθ = π ( θ ) dθ = dx. The lkelhood functon s wrtten P ( D θ, H ) = L( θ ) = L( X ) n ths space. Ths formulaton transforms the lkelhood nto a functon of a one-dmenson parameter. The evdence ntegral can now be wrtten: evdence = Z = L θ ) π ( θ ) dθ = ( L dx The algorthm then supposes lkelhoods can be evaluated at all 1 0 X so that L = L( X ), where < <... < X < X < X 0 X N 1 0 = (10) X s a sequence of values that decrease from 1 to 0 such that 1as llustrated n the bottom mage of fgure 1. The top mage shows a number of samples and the lkelhood so-contours n the -dmensonal posteror probablty parameter space.

6 Fgure 1: Samplng from a D parameter space by nested samplng Such a one dmensonal ntegral functon, equaton 10, can easly be estmated by any numercal method such as the trapezod rule: N Z = 1 N Z = Z = Lb = 1 (11) 1 where b = ( 1 +1 ) X X and L s the lkelhood at that sample. In the context of fnte element updatng, the algorthm acheves the above approxmaton n the followng manner: 1. Sample N updatng parameters from the pror probablty dstrbuton. Evaluate ther lkelhoods.. From the N samples select the sample wth the lowest lkelhood ( L ). 3. Increment the evdence by = L Z ( 1 +1) X X. 4. Dscard the sample wth the lowest lkelhood and replace t wth a new pont from wthn the remanng pror volume [ 0, X ]. The new sample must satsfy the constrant L new > L 5. Repeat steps -4 untl some stoppng crteron s reached. Ths could be the desred precson on the evdence or some teraton count (Max n our algorthm). For further detals on nested samplng see [18]. The next secton presents the experments performed on dfferent fnte element models and the evaluaton of ther evdences. 4. Model defnton and comparson In FEM a mathematcal model ( H n secton 3) s defned by the quantty and locaton of free-parameters (updatng parameters). Here fnte element models were desgned to have a dfferent number of free-parameters at dfferent postons along the same structure, effectvely creatng dfferent mathematcal models for the same structure. In mathematcs the number of free-parameters s related to the complexty of a mathematcal model and to advocate Occam s razor; a model wth fewer parameters s preferred. Ths s useful n not only determnng whch parameters need to be updated but also how many can be deemed suffcent to approxmate the measured data well. To determne how well dfferent models compare n fttng the data, Jeffrey s scale [10], shown n table 1, was used.

7 Table 1: Jeffrey's factor scale Z / Z log ( Z / Z ) loge ( Z / Z ) log 10 ( Z / Z ) Evdence aganst model H 1 to 3. 0 to to 1. 0 to 0.5 Weak 3. to to to to 1 Substantal 10 to to to to Strong > 100 > 6.6 > 4.6 > Decsve > 1000 > 10 > 7 >3 Beyond reasonable doubt 5. Experments A smple unsymmetrcal H-beam, wth sx degrees of freedom, prevously used n [13], s modeled. The measured natural frequences of ths structure occur at; 53.9Hz, 117.3Hz, 08.4Hz, 54Hz and 445Hz whch correspond to modes 7, 8, 10, 11 and 13 respectvely. To valdate that model evdence calculaton can reveal the most plausble fnte element model(s), four randomly desgned models of one beam structure are developed and the evdence of each s calculated. The example assumes no a pror knowledge of whch updatng parameters should be chosen s avalable. The obectve s then to determne from evdence ratos, the most probable model from ths random set. The structure was modeled usng the SDT verson 6.0 Matlab toolbox. Each model used standard sotropc materal propertes and Euler Bernoull beam elements to approxmate the beam sectons of the structure. It s assumed that the Young s modulus of some elements s not certan and was thus consdered a varable to be updated n the partcular updatng parameters. 5.1 Unsymmetrcal H beam Fgures and 3 llustrate a sketch of a 600mm long unsymmetrcal alumnum structure wth the left vertcal beam of 400mm and the rght beam of 00mm. It s dvded nto 1 elements. Each cross-sectonal area s 9.8mm by 3. mm (see [13] for more detals on the structure and expermental set-up). The structure s elements are numbered to clarfy the readng of table, where two sets of models are shown; model 1[A-C] and model [A-C] Fgure : Model 1A of a 1 Element Unsymmetrcal H beam. 3. The updatng parameters were sampled from a Gaussan pror probablty dstrbuton wth a mean of 7.x10 10 N.m - and an exponental nverse varance of 4.0e-0 for Young s modulus values between 6.8x10 10 N.m - and 8.0x10 10 N.m - for alumnum. The number of samples, N, was set to 100 (to sample the dstrbuton well) and the samplng algorthm stoppng crteron s expermentally set to a maxmum of 50 teratons. Model 1A n table s the modeled structure such that the elements labeled n fgure are set to a constant value of Young s modulus for alumnum (7.x10 10 N.m - ). The Young s modulus values for elements labeled are sampled from a pror wth a Gaussan probablty dstrbuton as mentoned above. Model 1B s the opposte settng where now the labeled parameters are fxed and the parameters are sampled from a Gaussan pror dstrbuton. All the elements that are free or updated n each model are lsted under the headng Parameter label n table. The set of updated parameters does not necessarly have to be fxed; all parameters can be vared smultaneously as shown by models 1C and C n table. Fgure 3 llustrates model A from the second set of models.

8 Table : Updated parameters and model evdences Model Log(Evdence) Quantty of parameters Parameter label 1A.188 ± ,3,5,10,11 1B ± ,4,6,7,8,9,1 1C 1.15 ± All A ± ,5,6,7,8,9,10 B.188 ± ,3,4,11,1 C 1.74 ± All Fgure 3: Model A of a 1 Element Unsymmetrcal H beam. Table 3 shows the results usng Jeffrey s scale. Models 1A and A are very dfferent; model 1A s a better model for ths data than model A. It s nterestng to note that the models wth the least number of updatng parameters produced the best evdences and ths s because t s relatvely smple and n accordance wth Occam s razor whch states that the least complex model that best descrbes the data s the correct one. Model 1A and B are shown to be relatvely smlar whle there s strong evdence aganst model A from model 1B. So evdence calculaton can provde a mechansm for elmnatng poor models from the onset. It also provdes a platform to determne salent parameters to consder n the updatng process. Table 3: Bayes factors for Unsymmetrcal beam model H p / H q Bayes Factor Jeffrey s Scale Evdence aganst model H q ( p q Z / Z ) log ( Z / Z ) 1A/1B Beyond reasonable doubt 1A/A Beyond reasonable doubt 1A/B 1 0 Weak 1B/ A Strong 6. Concluson e In ths paper we have ntroduced the model selecton concept to the problem of fnte element model updatng. It was argued that plausble model evdences should be calculated before models can be updated. A recently proposed method of comparng mathematcal models was ntroduced and mplemented n the fnte element model context. Ths algorthm, nested samplng, effcently calculates the Bayesan evdence of a model and provdes the posteror probablty dstrbuton of the model parameters as a by-product. A smple beam structure wth a number of random mathematcal model formulatons, defned by the number and poston of updatng parameters, s used as proof-of-concept for the algorthm n ths doman. Clear dstnctons between the models are evdent from ther relatve evdences.

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