Bayesian Uncertainty Quantification and Propagation in Large-Order Finite Element Models using CMS Techniques

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1 EACS 5 th Euopean Confeence on Stuctual Contol Genoa, Italy 8- June Pape No. # 9 Bayesan Uncetanty Quantfcaton and Popagaton n Lage-Ode Fnte Element Models usng CMS Technques Costas PAPADIMITRIOU*, Dmta-Chstna PAPADIOTI Unvesty of Thessaly, Depatment of Mechancal Engneeng Pedon Aeos, Volos, Geece costasp@uth.g, dxpapadot@uth.g ABSTRACT Bayesan nfeence s used fo quantfyng and calbatng uncetanty models n stuctual dynamcs based on vbaton measuements, as well as popagatng these modellng uncetantes n stuctual dynamcs smulatons to acheve updated obust pedctons of system pefomance, elablty and safety. The Bayesan tools fo dentfyng system and uncetanty models as well as pefomng obust pedcton analyses ae Laplace methods of asymptotc appoxmaton and/o moe accuate stochastc smulaton algothms (e.g. MCMC). These tools nvolve solvng optmzaton poblems, geneatng samples fo tacng and then populatng the mpotant uncetanty egon n the paamete space, as well as evaluatng ntegals ove hgh-dmensonal spaces of the uncetan model paametes. They eque a modeate to vey lage numbe of epeated system analyses to be pefomed. Consequently, the computatonal demands depend hghly on the numbe of system analyses and the tme equed fo pefomng a system analyss. Component mode synthess (CMS) technques ae ntegated wth Bayesan technques to effcently handle lage-ode models of hundeds of thousands o mllons degees of feedom and localzed nonlnea actons actvated dung system opeaton. Fast and accuate CMS technques ae poposed, consstent wth the fnte element (FE)model paametezaton, to acheve dastc eductons n computatonal effot. Keywods: Bayesan nfeence, uncetanty quantfcaton, dynamcs, component mode synthess. INTRODUCTION. Bayesan Model Updatng Bayesan technques ae used fo quantfyng and calbatng uncetanty models n stuctual dynamcs based on vbaton measuements, as well as popagatng these modelng uncetantes n stuctual dynamcs smulatons to acheve updated obust pedctons of system pefomance, elablty and safety (Papadmtou et al. ). The Bayesan model dentfcaton technques ae next pesented usng expementally estmated modal popetes. Let N D = {ˆ ω, ϕˆ R, =,, n} be the estmated modal fequences ˆ ω and modeshape components ˆ φ at N measued DOFs, whee n s the numbe of obseved modes. Consde a paametezed FE model class Μ ( m) and let ( m ) ( m) N θ R θ be the fee paametes of the model class, ( m) whee N θ s the numbe of paametes n θ ( m ). Let Π θ ( m ) Μ ( m ( ; ) ) = ω θ ( m ) Μ ( m ) ϕ θ ( m { ( ; ), ( ) ; Μ ( m ) N ) R } be the pedctons of the modal fequences * Coespondng autho

2 ( m) and modeshapes fom a patcula model n the model class Μ, wth ϕ ϕ θ ( m ) Μ ( m ) = φ θ ( m ( ; ) ( ) ; Μ ( m ) L ), whee φ ( θ ; Μ ) the complete modeshape and N N L R selects the N measued DOFs fom the N DOFs of the FE model. In Bayesan nfeence the pobablty dstbuton of the model paametes θ ( m) s updated based on the avalable measuements. The fomulaton stats by buldng a pobablstc model that chaactezes the dscepancy between the model pedctons Π( θ ; Μ ) obtaned fom a ( m) ( e) patcula value of the model paametes θ and the coespondng data D. Let Μ be a famly of pobablty model classes fo the dscepancy tems, that depend on a set of pedcton eo ( e) paametes θ. The Bayesan appoach to model calbaton deals wth updatng the values of the paamete set θ = θ ( m ( ), θ ( e ) ) assocated wth the stuctual model paametes and the pedcton eo paametes. A pobablty dstbuton πθ ( Μ ) s assgned a po to ncopoate subectve ( ) ( ) po nfomaton on the uncetanty n the values of these paametes, whee m e Μ = {Μ,Μ } ncludes the stuctual and pedcton eo model classes. The updated dstbuton p( θ D, Μ ) of the set θ, gven the data D and the model class Μ, esults fom Bayes theoem as follows pd ( θ, Μ) π( θ Μ) p( θ D, Μ) = () pd ( Μ) whee pd ( θ, Μ ) s the lkelhood of obsevng the data fom the model class and pdμ ( ) s the evdence of the model class Μ gven by the mult-dmensonal ntegal p( Μ D) = p( D θ, Μ) π( θ Μ ) dθ () Θ ove the space of the uncetan model paametes. Assumng that the pedcton eos ae ndependent Gaussan zeo-mean andom vaables wth vaance σ, the lkelhood s eadly obtaned n the fom (Chstodoulou and Papadmtou 7) N( n+ ) pd ( θ, Μ) = exp J( θ ; Μ ) N ( n+ ) (3) ( πσ σ ) whee J ( θ ; Μ ) gven by n n ω θ Μ ω αϕ( θ ; Μ ) ϕˆ [ ( ; ) ˆ J( θ ; Μ ) = + (4) n = ωˆ n = ϕˆ epesents the measue of ft between the expementally obtaned modal data and the modal data ( m) pedcted by a patcula model n the class M, and s the usual Eucldan nom.. Bayesan Model Selecton The Bayesan pobablstc famewok can also be used to compae two o moe competng model classes and select the optmal model class based on the avalable data. Consde a famly Μ Fam = {Μ, =,, μ}, of μ altenatve, competng, paametezed FE and pedcton eo model classes, and let θ R be the fee paametes of the model class Μ. The posteo pobabltes P( Μ D) of the vaous model classes gven the data D s (Beck and Yuen 4) pd ( Μ) P( Μ) P( Μ D) = (5) pd ( ΜFam) whee P( Μ ) s the po pobablty and pd ( Μ ) s the evdence of the model class Μ. The optmal model class M s selected as the one that maxmzes P( M D) gven by (5). best

3 BAYESIAN TOOLS The Bayesan tools fo dentfyng uncetanty models and pefomng obust pedcton analyses ae Laplace methods of asymptotc appoxmaton and stochastc smulaton algothms.. Asymptotc Appoxmatons Fo lage enough numbe of expemental data, the posteo dstbuton of the model paametes n () can be asymptotcally appoxmated by a Gaussan dstbuton (Beck and Katafygots 998) / h( θˆ ) (, ) exp ( ˆ T ) ( ˆ)( ˆ p θ D Μ θ θ h θ θ θ) (6) / ( π) centeed at the most pobable value ˆθ of the model paametes wth covaance equal to the nvese of the Hessan h( θ ) of the functon g( θ; Μ) = ( NN /)[ σ J( θ; Μ) + ln σ ln π( θ Μ) evaluated at the most pobable value ˆθ. Such appoxmaton eques the computaton of the most pobable value ˆθ and the Hessan h( θ ˆ). The asymptotc expesson (6) s appoxmate. Moeove, even fo lage numbe of expemental data, t may fal to gve a good epesentaton of the posteo pobablty dstbuton n the case of multmodal dstbutons. In addton, the asymptotc appoxmaton fals to povde acceptable estmates fo un-dentfable cases manfested fo elatvely lage numbe of model paametes n elaton to the nfomaton contaned n the data. Fo model selecton, an asymptotc appoxmaton based on Laplace s method s also used to gve an estmate of the ntegal n () appeang n (5) (Papadmtou & Katafygots 4): ˆ ˆ N / ( ) [ ( ) J πθ θ Μ J θ pd ( Μ ) c ( π ) (7) det h ( ˆ θ ) whee ˆ θ mnmzes the functon g( θ ; Μ) and h ( ˆ θ ) s the Hessan of g( θ ; Μ) evaluated at ˆ θ.. Stochastc Smulaton Algothms Fo moe accuate estmates, one should use stochastc smulaton algothms (e.g. MCMC, Tanstonal MCMC TMCMC, Delayed Reecton Adaptve Metopols - DRAM) to geneate samples that populate the posteo pdf n () and then evaluate the ntegal (). Among the stochastc smulaton algothms avalable, the tanstonal MCMC algothm (Chng & Chen 7) s one of the most pomsng algothms fo selectng the most pobable model as well as fndng and populatng wth samples the mpotance egon of nteest of the posteo pdf, even n the undentfable cases and mult-modal posteo pobablty dstbutons. In addton, the TMCMC method yelds an estmate of the evdence n () of the model class Μ based on the samples geneated by the algothm. The samples ( θ ), =,, N geneated at the fnal stage of the algothm can futhe be used fo estmatng the pobablty ntegals (Papadmtou et al ) encounteed n obust pedcton of vaous pefomance quanttes of nteest..3 Computatonal Issues The asymptotc appoxmatons and the stochastc smulaton algothms, nvolve solvng optmzaton poblems, geneatng samples fo tacng and then populatng the mpotant egon n the uncetan paamete space, as well as evaluatng ntegals ove hgh-dmensonal spaces of the uncetan model paametes. They eque a modeate to vey lage numbe of epeated system analyses to be pefomed ove the space of uncetan paametes. Consequently, the computatonal demands depend hghly on the numbe of system analyses and the tme equed fo pefomng a 3

4 system analyss. The poposed Bayesan technques eque a lage numbe of FE model smulatons to be caed out whch mposes sevee computatonal lmtatons on the applcaton of the technque. Fo FE models nvolvng hundeds of thousands o even mllon degees of feedom and localzed nonlnea actons actvated dung system opeaton these computatonal demands fo epeatedly solvng the lage-scale egen-poblems and the gadent of the egensolutons may be excessve. The obectve of ths wok s to examne the condtons unde whch substantal eductons n the computatonal effot can be acheved usng dynamc educton technques such as CMS. Dvdng the stuctue nto components and educng the numbe of physcal coodnates to a much smalle numbe of genealzed coodnates cetanly allevates pat of the computatonal effot. Howeve, n each teaton one needs to e-compute the egen-poblem and the nteface constaned modes fo each component. Ths pocedue s usually a vey tme consumng opeaton and computatonally moe expensve that solvng dectly the ognal matces fo the egenvalues and the egenvectos. It s shown n ths study that fo cetan paametezaton schemes fo whch the mass and stffness matces of a component depend lnealy on only one of the fee model paametes to be updated, often encounteed n FE model updatng fomulatons, the epeated solutons of the component egen-poblems ae avoded, educng substantally the computatonal demands n FE model updatng fomulatons, wthout compomsng the soluton accuacy. 3 INTEGRATION OF COMPONENT MODE SYNTHESIS TECHNIQUES In CMS technques (Cag & Bampton 965), a stuctue s dvded nto seveal components. Fo each component, the unconstaned DOFs ae pattoned nto the bounday DOFs, denoted by the subscpt b and the ntenal DOFs, denoted by the subscpt. The bounday DOFs of a component ae common wth the bounday DOFs of adacent components, whle the ntenal DOFs of a component ae not shaed wth any adacent component. The stffness and mass matces of a ( s ) ( s ) ( s ) ( s ) ( s) n n ( s) n n component s ae K and M. Wthout loss of genealty, we lmt the fomulaton to stffness matces that depend lnealy on the model paametes θ and constant mass = +, M ( ) M = matces,.e. K( θ) K K, θ θ =, whee M, K and 4 K, =,,, ae constant matces. The lnea epesentaton mples a smla epesentaton at component level,.e. ( s) ( s) ( s), = ( s) ( s) M K = K + K θ (8) and M =. Consde the case fo whch the stffness matx of a component s popotonal to a sngle paamete n θ. Let S be the set of components that depends on the -th vaable θ n the paamete set θ. Due to (8), the stffness matx of the component n s S take the fom ( s) ( s) K = K θ (9) ( s ) It can be shown that the matx Λ of the kept egenvalues and the matx of the egenvectos kk Φ k of the component fxed-nteface modes ae gven wth espect to the paamete θ n the fom ( s ) ( s ) ( s ) and kk kk θ k k ( s ) ( s ) Λ and k ( s ) ( s) ( s) ( s) ( s) k M k kk Λ = Λ Φ = Φ () whee the matces Φ ae solutons of the followng egen-poblem K Φ = Φ Λ () whch s ndependent of the values of θ. The subscpt k denotes the kept modes of a component. Also the constaned modes, gven by ( s ) ( s ) ( ) ( ) ( ) [ s s [ s K Ψ = K = K K () b b b, N θ

5 ae constant ndependent of the values of the paamete θ. It should be noted that only a sngle component analyss s equed to estmate the fxed-nteface and constaned modes, ndependent ( ) ( ) ( ) ( ) of the values of θ. The component s mass and stffness matces ae Mˆ s = Ψ st M s Ψ s and ˆ ( s ) ( st ) ( s ) ( s ) ( ) ( ) K = Ψ K Ψ. It s staghtfowad to vefy that ˆ s ˆ s ( ) K = K θ, whee ˆ s K s a constant ( ) ( ) matx gven by ˆ s s ( ) ( ) ( ) K kk = Λ kk, ˆ s ˆ s T s ( ) ( ) ( ) ( ) ( ) Kkb = Kbk = kb and ˆ s s s s s Kbb = Kbb [ K Kb Kb, ( s) ( s) ndependent of the model paametes θ. Also, usng the fact that M = M s constant, the ( ) ( ) ( ) ( ) ( ) educed matx ˆ s st s s ˆ s M = Ψ M Ψ M s also constant. The assembled Cag-Bampton stffness matx ˆ CB nq nq K and mass matx ˆ CB nq nq M fo the educed set of ndependent genealzed coodnates qt () s F[ (),, ( Ns ) Ns F s[ ( s) s= ˆ CB K Kˆ Kˆ Kˆ = = (3) F[ (),, ( Ns ) Ns F s[ ( s) s= n n n n ˆ CB M = Mˆ Mˆ = Mˆ (4) N N Fo N matces A R,, A N R, the mathematcal opeatos F[ M,, M N and F s[ A s T ae defned as follows F[ A,, AN = S blockdag( A,, AN) S, whee blockdag( A,, A N ) s a block dagonal matx havng as dagonal blocks the matces ( A,, A N ) and F[ s As = F[ n,,,,,, n A s s n whee s+ nn R denotes a matx of zeoes. Intoduce the ndex set Σ = { s,, sn θ } to contan the stuctual components that depend on a paamete n the set θ. Then the set Σ = {,, N s } Σ contans the component numbes that the popetes ae constant, ndependent of the values of the paamete set θ. Usng the afoementoned analyss, the stffness matx of the educed system admts the epesentaton ˆ CB ˆ CB ˆ CB, θ = CB ˆ CB ˆ CB K = K + K (5) ˆ CB and Mˆ = M, whee K and K, ae assembled fom the component stffness matces by Kˆ K ˆ K K ˆ CB ( s) ˆ CB ( s ) = F s[ and, = F s [ (6) s Σ s S The sum n the second of (6) consdes that moe than one components s S depend on θ. Solvng the egen-poblem assocated wth the educed matces M ˆ CB and K ˆ CB ˆ CB ˆ CB K Q = M Q Λ (7) Nk Nk one obtans the etaned modal fequences n Λ = dag( ω ) and the coespondng mode nq Nk shapes Q of the educed system, whle the physcal mode shapes ae ecoveed as follows Φ = Sˆ Ψ SQ= LQ o φ ˆ = Lq (8) whee ˆ N np S maps the genealzed coodnates of each stuctual component to the physcal coodnates of the stuctue, Q= [ q ˆ,, ˆ q m s the matx of mode shapes fo the educed system, () ( N ) np n s q Ψ = blockdag[ Ψ,, Ψ, and L = Sˆ Ψ S s constant, ndependent of the paametes θ. The matces K ˆ CB and K ˆ CB, n (5) ae ndependent of the values of θ. In ode to save computatonal tme, these constant matces ae computed and assembled once and, theefoe, thee s no need ths computaton to be epeated dung the teatons nvolved n optmzaton and 5

6 stochastc smulaton algothms. At each teaton step nvolved n model updatng fo whch the value of the paamete set θ changes, ths pocedue saves sgnfcant computatonal tme snce t avods (a) e-computng the fxed-nteface and constaned modes, and (b) assemblng the educed matces fom these components. It should be noted that the modal fequency and mode shape esduals have the same exactly fom as n (4) wth φ ( θ ) eplaced by qˆ ( θ ) and L eplaced by ˆ L S S = Ψ. Avalable Bayesan uncetanty quantfcaton and popagaton softwae can thus be eadly used to handle the paamete estmaton usng the educed mass and stffness matces by ust eplacng the ognal egenvalue poblem wth the egenvalue poblem (8) of the educed system matces and also eplacng the matx L of zeos and ones by the constant matx L = Sˆ Ψ S. It should be ponted out that the sgnfcant savngs asng patly fom the educton of the sze of the egenvalue poblem n the CMS technque and patly fom the fact that the estmaton of the component fxed-nteface modes and the constaned nteface modes need not to be epeated fo each teaton nvolved n the optmzaton. The computatonal savngs depend on the sze of the educed system. Ths sze s contolled by the numbe of fxed nteface modes needed to descbe the defomaton of the component as well as the numbe of nteface DOFs fo each component. Howeve, the numbe of nteface DOFs may be lage compaed to the numbe of the fxed nteface modes. The nteface DOFs may contol the sze of the educed mass and stffness matces. Futhe educton n the genealzed coodnates can be acheved by eplacng the nteface DOFs by a educed numbe of constant nteface modes fomed by a educed bass. Selectng the educed bass to be constant, ndependent of θ, the fomulaton sgnfcantly smplfes. The educed bass can be kept constant at each teaton nvolved n the optmzaton algothm o updated evey few teatons n ode to mpove convegence and mantan accuacy. 4 APPLICATION The computatonal effcency and accuacy of the CMS technque fo FE model updatng s demonstated usng smulated data fom the Metsovo bdge. Detaled FE models ae ceated usng 3-dmensonal tetahedon quadatc Lagange FEs to model the whole bdge. An exta coase mesh s chosen to pedct the lowest modal fequences and mode shapes of the bdge. The model has 97,636 FEs and 563,586 DOFs. Fo demonstaton puposes, the bdge s dvded nto ffteen physcal components wth eght ntefaces between components as shown n Fgue. Each deck component conssts of seveal 4-5m deck sectons. The tallest pe also conssts of seveal sectons. The sze of the elements n the exta coase mesh s the maxmum possble one that can be consdeed, wth typcal element length of the ode of the thckness of the deck coss-secton. The cut-off fequency ω c s ntoduced to be the hghest modal fequency that s of nteest n FE model updatng. In ths study the cut-off fequency s selected to be equal to the th modal fequency of the nomnal model..e. ω c = 4.55 Hz. The effectveness of the CMS technque as a functon of the numbe of modes etaned fo each component s next evaluated. Fo each component t s selected to etan all modes that have fequency less than ωmax = ρωc, whee the ρ values affect computatonal effcency and accuacy of the CMS technque. Repesentatve ρ values ange fom to. The total numbe of ntenal DOFs pe component befoe the model educton s appled ae shown n Fgue a. The numbe of modes etaned pe components fo vaous ρ values s also gven n Fgue a. Fo the case ρ = 8, a total of 76 ntenal modes ae etaned fo all 5 components. The total numbe of DOFs of the educed model s 8,35 whch consst of 76 fxed nteface genealzed coodnates and 8,49 constant nteface DOFs fo all components. It s clea that a two odes of magntude educton n the numbe of DOFs s acheved usng CMS. 6

7 Fgue - Components of FE model of Metsovo bdge. Fgue b shows the factonal eo between the modal fequences computed usng the complete FE model and the ones computed usng the CMS technque as a functon of the mode 5 numbe fo ρ =, 5 and 8. It s seen that the eo fo the lowest modes fall below fo ρ = 8, 4 fo ρ = 5 and 3 fo ρ =. A vey good accuacy s acheved fo the case of ρ =. Fgue - (a) Numbe of DOFs pe component of FE model; (b) Factonal modal fequency eo between pedctons of the full and educed model. It s thus obvous that a lage numbe of genealzed coodnates fo the educed system ases fom the nteface DOFs. A futhe educton n the numbe of genealzed coodnates fo the educed system can be acheved by etanng only a facton of the constaned nteface modes. Fo each nteface, t s selected to etan all modes that have fequency less than ωmax = νωc, ν s use and poblem dependent. Results ae shown n Fgue b fo ν = and 5. It can be seen that the 3 factonal eo fo the lowest modes of the stuctue fall below fo ν = 5. The numbe of modes etaned fo dffeent ν values s gven n Table. In patcula, the value of ν = 5 and ρ = 5 gves accuate esults and the numbe of etaned ntefaces modes fo all ntefaces s 54. The educed system has 55 DOFs fom whch genealzed coodnates ae fxed-nteface modes fo all components and the est 54 genealzed coodnates ae constaned nteface modes. The numbe of genealzed coodnates s dastcally educed. Table - Numbe of ntenal and bounday DOFs. Total DOFs Ognal Reduced Reduced Reduced ν = 8 & ρ = 8 ν = 5 & ρ = 5 ν = & ρ = Bounday 8, Intenal 554, Total 56,

8 The computatonal tme needed to estmate the lowest modal popetes usng CMS wth ρ 8 s fve tmes less than the tme equed to solve the complete FE model. Reducng the constaned nteface modes ( ν 5 ), the computatonal tme educes by thee to fou odes of magntude. It s thus obvous that CMS s expected to dastcally educe the computatonal effot n Bayesan uncetanty quantfcaton and popagaton famewok wthout sacfcng n accuacy. 5 CONCLUSIONS Component mode synthess methods wee pesented to substantally educe the computatonal effot equed n the Bayesan tools fo uncetanty quantfcaton and popagaton n stuctual dynamcs. Explotng cetan schemes often encounteed n fnte element model paametezaton, the mass and stffness matces of the educed system ae shown to depend lnealy on the model paametes wth the mass and stffness senstvty matces to be assembled once and to eman constant dung the teaton pocess. The only tme consumng opeaton left s assocated wth the soluton of the egen-poblem of the educed system, avodng the expensve estmaton of the component egen-poblems at each teaton. The methodology s patculaly effcent fo lagescale fnte element models whee the soluton of the component egen-poblem may be a computatonally demandng opeaton. Futhe computatonal savngs can be acheved by adoptng suogate models to substantally speed-up computatons, and paallel computng algothms to effcently dstbute the computatons n avalable GPUs and mult-coe CPUs. ACKNOWLEDGEMENTS Ths eseach has been co-fnanced by the Euopean Unon (Euopean Socal Fund - ESF) and Geek natonal funds though the Opeatonal Pogam " Educaton and Lfelong Leanng" of the Natonal Stategc Refeence Famewok (NSRF) - Reseach Fundng Pogam: Heacletus II. Investng n knowledge socety though the Euopean Socal Fund. REFERENCES [ Beck, J.L. & Katafygots, L.S Updatng models and the uncetantes. I: Bayesan statstcal famewok. J. of Engneeng Mechancs, ASCE 4(4): [ Beck, J.L. & Yuen, K.V. 4. Model selecton usng esponse measuements: Bayesan pobablstc appoach. J. of Engneeng Mechancs, ASCE 3(): 9-3. [3 Chng J. and Chen Y.C. (7). Tanstonal Makov Chan Monte Calo method fo Bayesan updatng, model class selecton, and model aveagng. J. of Eng. Mech., ASCE 33: [4 Chstodoulou, K. and Papadmtou, C. (7). Stuctual dentfcaton based on optmally weghted modal esduals. Mechancal Systems and Sgnal Pocessng :4-3. [5 Cag J., R.R and Bampton, M.C.C Couplng of substuctues fo dynamc analyss. AIAA J. 6(7): [6 Papadmtou, C. and Katafygots, L.S. 4. Bayesan modelng and updatng. In Engneeng Desgn Relablty Handbook, Nkolads N., Ghocel D.M., Snghal S. (Eds), CRC Pess. [7 Papadmtou, C., Beck, J.L. and Katafygots, L.S.. Updatng obust elablty usng stuctual test data. Pobablstc Engneeng Mechancs 6():3-3. 8