Fast Computing Techniques for Bayesian Uncertainty Quantification in Structural Dynamics

Transcription

1 Fast Coputng Technques for Bayesan Uncertanty Quantfcaton n Structural Dynacs Costas Papadtrou and Dtra-Chrstna Papadot Dept. of Mechancal Engneerng, Unversty of Thessaly, Volos, Greece ABSTRACT A Bayesan probablstc fraework for uncertanty quantfcaton and propagaton n structural dynacs s revewed. Fast coputng technques are ntegrated wth the Bayesan fraework to effcently handle large-order odels of hundreds of thousands or llons degrees of freedo and localzed nonlnear actons actvated durng syste operaton. Fast and accurate coponent ode synthess (CMS) technques are proposed, consstent wth the fnte eleent (FE) odel paraeterzaton, to acheve drastc reductons n coputatonal effort when perforng a syste analyss. Addtonal substantal coputatonal savngs are also obtaned by adoptng surrogate odels to drastcally reduce the nuber of full syste re-analyses and parallel coputng algorths to effcently dstrbute the coputatons n avalable ult-core CPUs. The coputatonal effcency of the proposed approach s deonstrated by updatng a hgh-fdelty fnte eleent odel of a brdge nvolvng hundreds of thousands of degrees of freedo. Keywords: Bayesan nference; structural dynacs; coponent ode synthess; surrogate odels; HPC.. Introducton In structural dynacs, Bayesan nference [-3] s used for quantfyng and calbratng uncertanty odels based on vbraton easureents, as well as propagatng these odelng uncertantes n syste sulatons to obtan updated robust predctons of syste perforance, relablty and safety [4]. The Bayesan tools for dentfyng syste and uncertanty odels as well as perforng robust predcton analyses are Laplace ethods of asyptotc approxaton and ore accurate stochastc sulaton algorths, such as MCMC [5] and Transtonal MCMC [6]. These tools nvolve solvng optzaton probles, generatng saples for tracng and then populatng the portant uncertanty regon n the paraeter space, as well as evaluatng ntegrals over hgh-densonal spaces of the uncertan odel paraeters. A oderate to very large nuber of repeated syste analyses are requred to be perfored over the space of uncertan paraeters. Consequently, the coputatonal deands depend hghly on the nuber of syste analyses and the te requred for perforng a syste analyss. To relably update odels, hgh fdelty FE odel classes, often nvolvng a large nuber of DOFs, should be ntroduced to sulate structural behavor. For such large-order fnte eleent odels the coputatonal deands n pleentng asyptotc approxatons as well as stochastc sulaton technques ay be excessve. The present work proposes ethods for drastcally reducng the coputatonal deands at the syste, algorth and coputer hardware levels nvolved n the pleentaton of Bayesan tools. At the syste level, CMS technques [7] are ntegrated wth Bayesan technques to effcently handle large-order odels of hundreds of thousands or llons degrees of freedo and localzed nonlnear actons actvated durng syste operaton. Fast and accurate CMS technques are obtaned [8], consstent wth the FE odel paraeterzaton, to acheve drastc reductons n coputatonal effort. The CMS allows the repeated coputatons to be carred out n a sgnfcantly reduced space of generalzed coordnates. At the level of the TMCMC algorth, surrogate

2 odels are adopted to drastcally reduce the nuber of coputatonally expensve full odel runs. At the coputer hardware level, parallel coputng algorths are used to effcently dstrbute the coputatons n avalable ult-core CPUs [9]. 2. Bayesan Uncertanty Quantfcaton and Propagaton Fraework Consder a class Μ of structural dynacs odels used to predct varous output quanttes of nterest f ( q Μ ) of a syste, where q s a set of paraeters n ths odel class that need to be estated usng experental data Dº { yˆ }. Followng a Bayesan forulaton [2-3] and assung that the observaton data and the odel predctons satsfy the predcton error equaton yˆ = f ( q Μ ) + e () where the error ter e ~ N(, S ) s a zero-ean Gaussan vector wth covarance SºS ( ) dependng on the paraeters q of the predcton error odel class Μ e, the updated dstrbuton p( q D, Μ ) of the augented paraeter set e q = ( q, q ), gven the data D and the cobned odel class Μ =(Μ, Μ ), results fro the applcaton of the Bayes e e theore as follows q e p( q D, Μ) = where pd ( q, Μ) p( q Μ) pd ( Μ) (2) -/2 S( q ) e pd ( q, Μ) = exp é - J( q; Μ ) ù (3) /2 (2 p) êë 2 úû s the lkelhood of observng the data fro the odel class, J q Μ = yˆ - f q Μ S q yˆ - f q Μ (4) T - ( ; ) [ ( )] ( )[ ( )] e s the easure of ft between the experental and odel predcted propertes, pq ( ) s the pror probablty dstrbuton of the odel paraeters based on prevous knowledge and/or user experence, and pd ( Μ ) s the evdence of the odel class. The Bayesan probablstc fraework can also be used to copare two or ore copetng odel classes and select the optal odel class based on the avalable data. Consder a faly Μ,,,, of alternatve, copetng, paraeterzed FE and predcton error odel classes, and let posteror probabltes P( Μ D ) of the varous odel classes gven the data D s [-] N q q Î R be the free paraeters of the odel class Μ. The P( Μ D) = pd ( Μ ) P( Μ ) pd ( Μ,, Μ ) k where P ( Μ ) s the pror probablty and pd ( Μ ) s the evdence of the odel class Μ. The optal odel class M best s selected as the one that axzes P( M D ) gven by (5). (5) For large enough nuber of easured data, the posteror dstrbuton of the odel paraeters n (2) can be asyptotcally approxated by a Gaussan dstrbuton [3] centered at the ost probable value ˆq of the odel paraeters wth covarance equal to the nverse of the Hessan h( q ) of the functon g( q; Μ) =- ln p( D q, Μ) = J( q ; Μ) + S( q) -ln p( q Μ ) (6) e 2 2

3 evaluated at the ost probable value ˆq of the odel paraeters. The ost probable value ˆq axzes the posteror probablty dstrbuton p( q D, Μ ) or, equvalently, nzes the functon g( q; Μ ). For odel selecton, an asyptotc approxaton based on Laplace s ethod s also used to gve an estate of the ntegral nvolved n the estaton of the evdence pd ( Μ ) n (2) and (5) []. For the case for whch analytcal expressons for the gradent of J ( q ; Μ ) wth respect to the structural odel paraeters q are avalable, coputatonally effcent gradent-based optzaton algorths can be used to obtan the optal value of the odel paraeters by nzng the functon g( q; Μ ). Specfcally, for lnear structural dynacs odels and experental data consstng of odal frequences and ode shapes, such analytcal expressons are avalable and can be coputed usng effcent adont technques (e.g. [2]). However, there are certan classes of nonlnear systes where such analytcal expressons or adont technques are not applcable or t s nconvenent to ntroduce wthn coercally avalable structural dynacs solvers. In such cases, non-gradent-based optzaton algorths can be used to obtan the ost probable value of the odel paraeters. Once the ost probable value has been coputed, the Hessan requred n the asyptotc approxaton can be estated usng ether hgher-order adont technques [2], f applcable, or usng nuercal dfferentaton technques. It should be noted that the asyptotc expresson s approxate. Moreover, even for large nuber of experental data, t ay fal to gve a good representaton of the posteror probablty dstrbuton n the case of ultodal dstrbutons. In addton, the asyptotc approxaton fals to provde acceptable estates for un-dentfable cases anfested for relatvely large nuber of odel paraeters n relaton to the nforaton contaned n the data. For ore accurate estates, one should use stochastc sulaton algorths (e.g. MCMC [5], Transtonal MCMC TMCMC [6], Delayed Reecton Adaptve Metropols DRAM [3]) to generate saples that populate the posteror probablty dstrbuton functon n (2) and then evaluate the ntegrals nvolved n propagatng uncertantes nto robust predctons. Aong the stochastc sulaton algorths avalable, the transtonal MCMC algorth [6] s one of the ost prosng algorths for selectng the ost probable odel as well as fndng and populatng wth saples the portance regon of nterest of the posteror probablty dstrbuton, even n the undentfable cases and ult-odal posteror dstrbutons. In addton, the TMCMC ethod yelds an estate of the evdence pd ( Μ ) of a odel class Μ, requred for odel class selecton [,4], based on the saples generated by the algorth. The saples generated at the fnal stage of the TMCMC algorth can further be used for estatng the probablty ntegrals encountered when nterested n robust predctons of varous perforance quanttes of nterest. Specfcally, consder an output quantty q of nterest n structural dynacs sulatons. Posteror robust predctons of q are obtaned by takng nto account the updated uncertantes n the odel paraeters gven the easureents D. Let pq ( q, Μ ) be the condtonal probablty dstrbuton of q gven the values of the paraeters. Usng the total probablty theore, the posteror robust probablty dstrbuton pq ( DΜ, ) of q, takng nto account the odel Μ and the data D, s gven by [4] pq ( D, Μ) = ò pq ( q, Μ) p( q D, Μ ) dq (7) as an average of the condtonal probablty dstrbuton pq ( q, Μ ) weghtng by the posteror probablty dstrbuton p( q D, Μ ) of the odel paraeters. Let also Gq ( ) be a functon of the output quantty of nterest q. A posteror robust perforance easure of the syste gven the data D s EGq [ ( ) D, Μ)] = ò Gq ( ) p( q D, Μ ) dq (8) The evaluaton of the ult-densonal ntegrals n (7) and (8) cannot be perfored analytcally. Asyptotc approxatons are gven n [4]. Alternatvely, stochastc sulaton ethods can be convenently used to estate the ( ) ntegral fro the saples q, =,, N, generated fro the posteror probablty dstrbuton p( q D, Μ ). In ths case, the ntegrals (7) and (8) can be approxated by

4 N () f( q D, Μ)» å f( q q, Μ ) (9) N = and E Gq DΜ Gq () N ( ) [ ( ), )]» å ( ) N = respectvely, where q º q( q ) ( ) ( ) 3. Fast Coputng Technques for Large Order Fnte Eleent Models 3.. Coponent Model Synthess for Paraeter Estaton n Structural Dynacs At the syste level, dynac reducton technques such as CMS can be pleented wth Bayesan uncertanty quantfcaton and propagaton fraework n order to allevate the coputatonal burden assocated wth each odel run n the re-analyses requred n the optzaton and stochastc sulaton ethods. CMS technques have been successfully eployed for odel reducton n optzaton and stochastc sulaton algorths nvolved n odel updatng [5-6]. CMS technques [7] dvde the structure nto coponents wth ass and stffness atrces that are reduced usng fxednterface and constraned odes. Dvdng the structure nto coponents and reducng the nuber of physcal coordnates to a uch saller nuber of generalzed coordnates certanly allevates part of the coputatonal effort. However, at each teraton or TMCMC saplng pont one needs to re-copute the egen-proble and the nterface constraned odes for each coponent. Ths procedure s usually a very te consung operaton and coputatonally ore expensve that solvng drectly the orgnal atrces for the egenvalues and the egenvectors. It was recently shown [8] that for certan paraeterzaton schees for whch the ass and stffness atrces of a coponent depend lnearly on only one of the free odel paraeters to be updated, often encountered n fnte eleent odel updatng forulatons, the full re-analyses of the coponent egen-probles are avoded. The egenpropertes and the nterface constraned odes as a functon of the odel paraeters can be coputed nexpensvely fro the egenpropertes and the nterface constraned odes that correspond to a nonal value of the odel paraeters. Specfcally let D be the set of structural coponents that depend on the -th paraeter q. Consder the case for whch ( the stffness atrx of a coponent s ÎD depends lnearly q and the ass atrx s ndependent of q,.e. s ) ( s) K = K q ( s ) ( s ) and M = M. It can be readly derved that the stffness and ass atrces of the Crag-Bapton reduced syste adts the representaton N q CB CB CB CB CB = + q and =, = K, ˆ CB ˆ CB, Kˆ Kˆ å Kˆ Mˆ Mˆ () where the coeffcent atrces ˆ CB K and M n the expanson () are assebled fro the coponent stffness and ass atrces. It s portant to note that the assebled atrces K ˆ CB, K ˆ CB and M ˆ CB of the Crag-Bapton reduced syste, n the expanson () are ndependent of the values of q. In order to save coputatonal te, these constant atrces are coputed and assebled once and, therefore, there s no need ths coputaton to be repeated durng the teratons nvolved n optzaton or TMCMC saplng algorths for odel updatng due to the changes n the values of the paraeter vector q. Ths s an portant result whch saves substantal coputatonal effort snce t avods (a) re-coputng the fxednterface and constraned odes for each coponent, and (b) asseblng the reduced atrces fro these coponents. The forulaton guarantees that the reduced syste s based on the exact coponent odes for all values of the odel paraeters. The aforeentoned forulaton can readly be extended to treat the ore general case n whch the stffness and ass ( s) ( s) ( s) ( s) atrces of a coponent depends nonlnearly on a sngle paraeter, that s K = K f( q ) and M = M rq ( ), where f ( q ) and rq ( ) are scalar nonlnear functons of a paraeter. In ths case the lnear representaton () s no longer applcable for such coponents. However, the reduced stffness and ass atrces for each coponent as a functon of the

5 odel paraeters can be readly obtaned fro the egen-propertes and constraned nterface odes obtaned fro a sngle analyss for a nonal value of the coponent paraeter q. It turns out that substantal coputatonal savngs arse fro the fact that the re-analyses of the fxed-nterface and constraned odes for each coponent requred at each teraton or TMCMC sapng pont s copletely avoded. In ths general case, the reduced stffness and ass coponent atrces have to be re-assebled n order to derve the Crag-Bapton reduced syste atrces. Slar to the lnear case [8], the coputatonal deands n FE odel updatng forulatons are agan substantally reduced wthout coprosng the soluton accuracy Surrogate Models At the level of the TMCMC algorth, surrogate odels can be used to reduce the coputatonal te by avodng the full odel runs at a large nuber of saplng pont n the paraeters space. Ths s done by explotng the functon evaluatons that are avalable at the neghbour ponts fro prevous full odel runs n order to generate an estate at a new saplng pont n the paraeter space. Surrogate odels are well-suted to be used wth MCMC algorths, ncludng the TMCMC algorth [6]. The krgng technque [7] s used to approxate the functon evaluaton at a saplng pont usng the functon evaluatons at neghbor ponts n the paraeter space. To ensure a hgh qualty approxaton, certan condtons are posed n order a surrogate estate be accepted. Specfcally, the estate s accepted based on a nu nuber of neghbour desgn ponts that depend on the denson of the uncertan paraeter space. The surrogate pont has to belong to the convex hull of the desgn ponts so that an nterpolaton s perfored, whle extrapolatons are prohbted. The neghbour desgn ponts are selected as the ones closest to the surrogate estate and also wthn the hyper- ellpse of the TMCMC proposal covarance atrx scaled to nclude the nu nuber of desgn ponts. The estate s also accepted based on local optalty condtons for the selected surrogate schee, guaranteeng that the error n the surrogate estate provded by the krgng technque s saller than a user-defned value. Detals of the ntegraton of the krgng technque wthn the TMCMC algorth can be found n [9]. An order of agntude reducton n the nuber of full odel runs nvolved n TMCMC algorth has been reported whch results n addtonal coputatonal savngs Parallel Coputng Algorths Ipleented At the coputer hardware level, hgh perforance coputng (HPC) technques can be used to reduce the coputatonal te. Most MCMC algorths nvolve a sngle Markov chan and are thus not parallelzable. In contrast, the TMCMC algorth nvolves a large nuber of ndependent Markov chans that can run n parallel. Thus, the TMCMC algorth s very-well suted for parallel pleentaton n a coputer cluster [9]. Specfcally, parallelzaton s actvated at every stage of the TMCMC algorth explotng the large nuber of short, varable length, chans that need to be generated startng fro the leader saples deterned fro the TMCMC algorth at the partcular stage. Statc and dynac schedulng schees can be convenently used to optally dstrbute these chans n a ult-host confguraton of coplete heterogeneous coputer workers. The statc schedulng schee dstrbutes the chans n the workers usng a weghted round-robn algorth so that the nuber of lkelhood evaluatons s arranged to be the sae for each coputer worker. The statc schedulng schee s coputatonal effcent when the coputatonal te for a lkelhood evaluaton s the sae ndependently of the locaton of saple n the paraeter space as well as when surrogate estates are not actvated. The dynac schedulng schee s ore general, ensurng a ore effcent balancng of the loads per coputer worker n the case of varable run te of lkelhood functon evaluatons and unknown nuber of surrogates actvated durng estaton. Specfcally, each worker s perodcally nterrogated at regular te ntervals by the aster coputer about ts avalablty and saples fro TMCMC chans are subtted to the workers on a frst coe frst serve bass to perfor the lkelhood functon evaluatons so that the dle te of the ultple workers s nsed. Detals of the parallel pleentaton of the TMCMC algorth are gven n [9]. 4. Applcaton on Fnte Eleent Model Updatng of a Brdge The effcency of the proposed fast coputng tools n the Bayesan fraework s deonstrated by updatng a FE odel of the Metsovo brdge (Fg. a) usng sulated odal data. A detaled FE odel of the brdge s created usng 3-densonal tetrahedron quadratc Lagrange FEs. An extra coarse esh, chosen to predct the lowest 2 odal frequences and ode shapes of the brdge, results n a nu 97,636 FEs and 562, DOF. The sze of the eleents n the extra coarse esh s the axu possble one that can be consdered, wth typcal eleent length of the order of the thckness of the deck crosssecton.

6 Fg. (a) Metsovo brdge, (b) Coponents of FE odel of the brdge Let w be the cut-off frequency whch represents the hghest odal frequency that s of nterest n FE odel updatng. c Heren, the cut-off frequency s selected to be equal to the 2 th odal frequency of the nonal odel..e. w = 4.55 Hz. For c deonstraton purposes, the brdge s dvded nto nne physcal coponents wth eght nterfaces between coponents as shown n Fg. b. For each coponent t s selected to retan all odes that have frequency less than w = rw, where the ax c r values affect coputatonal effcency and accuracy of the CMS technque. The total nuber of nternal DOFs before the odel reducton s appled and the nuber of odes retaned for varous r values are gven n Table. For the case r = 8, a total of 286 nternal odes out of the 558,8 are retaned for all 9 coponents. The total nuber of DOFs of the reduced odel s 3,586 whch consst of 286 fxed nterface generalzed coordnates and 3,3 constrant nterface DOFs for all coponents. It s clear that a two orders of agntude reducton n the nuber of DOFs s acheved usng CMS. Table also shows the fractonal error between the odal frequences coputed usng the coplete FE odel and the ones coputed usng the CMS technque for r = 2, 5 and 8. It s seen that the error fall below.2% for r = 8,.7% for r = 5 and.% for r = 2. A very good accuracy s acheved for the case of r = 5. Table Nuber of DOF and percentage odal frequency error for the full (unreduced) and reduced odels Full Model Reduced Model (Retaned Modes) r = 8 r = 5 r = 2 r = 8 n = 2 r = 5 n = 2 r = 2 n = 2 Internal DOF 558, Interface DOF 3,3 3,3 3,3 3, Total DOF 562, 3,586 3,4 3, Hghest Percentage Error [%] For the specfc applcaton, a large nuber of generalzed coordnates for the reduced syste arses fro the nterface DOFs. A further reducton n the nuber of generalzed coordnates for the reduced syste can be acheved by retanng only a fracton of the constraned nterface odes [8]. For each nterface, t s selected to retan all odes that have frequency less than w = nw, where n s user and proble dependent. Results are coputed for n = 2. The nuber of nterface ax c odes retaned s gven n Table. It can also be seen that the fractonal error for the lowest 2 odes of the structure fall below.2% for n = 2. In partcular, the value of n = 2 and r = 5 gves suffcently accurate results and the nuber of retaned nterfaces odes for all nterfaces s 36. The reduced syste has 46 DOFs fro whch generalzed coordnates are fxed-nterface odes for all coponents and the rest 36 generalzed coordnates are constraned nterface odes. Usng CMS, the nuber of generalzed coordnates s drastcally reduced.

7 For deonstraton purposes, the FE odel s paraeterzed usng fve paraeters assocated wth the odulus of elastcty of one or ore structural coponents shown n Fgure b. Specfcally, the frst two paraeters q and q 2 account respectvely for the odulus of elastcty of the per coponents 3 and 7 of the brdge. The paraeter q 3 accounts for the odulus of elastcty of the coponents and 2 of the deck, the paraeter q 4 accounts for the coponents 4 and 5, whle the paraeter accounts for the coponents 6 and 8. The coponent 9 s not paraeterzed. q 5 The estaton of the paraeter values and ther uncertantes of the FE odel s based on odal frequences and ode shapes. Sulated, nose contanated, easured odal frequences and ode shapes are generated by addng a % and 3% Gaussan nose to the odal frequences and odeshape coponents, predcted by the nonal non-reduced FE odels. The added Gaussan nose reflects the dfferences observed n real applcatons between the predctons fro a odel of a structure and the actual (easured) behavor of the structure. 38 sensors are placed on the brdge to ontor vertcal and transverse acceleratons. The easured data contan the values of the ten lowest odal frequences and odeshapes. The odel paraeters are ntroduced to scale the nonal values of the propertes that they odel so that the value of the paraeters equal to one corresponds to the nonal value of the FE odel. The odel updatng s perfored usng the stochastc sulaton algorth TMCMC wth the followng settngs of the TMCMC paraeters: tolcov =., b =.2 and saples per TMCMC stage [6]. The nuber of FE odel runs for the fve-paraeter odel class depends on the nuber of TMCMC stages whch was estated to be 9. The resultng nuber of FE odel re-analyses are 9,. The parallelzaton features of TMCMC [9] were also exploted, takng advantage of the avalable four-core ult-threaded coputer unt to sultaneously run eght TMCMC saples n parallel. For coparson purposes, the coputatonal effort for solvng the egenvalue proble of the orgnal unreduced FE odel s approxately 39 seconds. Multplyng ths by the nuber of 9, TMCMC saples and consderng parallel pleentaton n a fourcore ult-threaded coputer unt, the total coputatonal effort for the odel class s expected to be of the order 7 days. In contrast, for the reduced-order odels for r = 8, the coputatonal deands for runnng the odel class are reduced to approxately 3 hours (759 nutes), whle for the reduced-order odels for r = 8 and n = 2 these coputatonal deands are drastcally reduced to 4 nutes. It s thus evdent that a drastc reducton n coputatonal effort for perforng the structural dentfcaton based on a set of ontorng data s acheved fro approxately 7 days for the unreduced odel class to 4 nutes for the reduced odel classes correspondng to r = 8 and n = 2, wthout coprosng the predctve capabltes of the proposed paraeter estaton ethodology. Ths results n a factor of over 5 reducton n coputatonal effort. It should be noted that krgng technque further reduces the coputatonal effort by approxately one order of agntude so that the updatng of the 562, DOF fnte eleent odel, requrng 9, odel runs, can be perfored n 2 nutes whch s a rearkable reducton n coputatonal effort. 5. Conclusons Asyptotc approxatons and stochastc sulaton algorths (e.g. the TMCMC algorth) used n Bayesan odel uncertanty quantfcaton, calbraton and propagaton requres a large nuber of FE odel sulaton runs. For large sze FE odels wth hundred of thousands or even llon DOFs and localzed nonlneartes, the coputatonal deands nvolved n the optzaton or TMCMC saplng algorths ay be excessve. Drastc reductons can be acheved at the syste, algorth and coputer equpent level. At the syste level, CMS technques that explot certan schees often encountered n FE odel paraeterzaton are shown to be effectve n copletely avodng the large nuber of egenproble re-analyses wthn the coponents or nterfaces, requred durng the applcaton of the optzaton or TMCMC saplng algorths. Thus paraeterzaton consstent CMS technques result n drastc reducton of the coputatonal effort. At the level of the algorth, surrogate odels are well adapted to the TMCMC algorth for sgnfcantly reducng the nuber of full odel runs requred, wthout sacrfcng the accuracy n the surrogate estates. At the coputer hardware level, parallel coputng algorths are also very well suted to be used wth TMCMC algorth to effcently dstrbute the coputatons n avalable ult-core CPUs. Applcaton of the fraework to uncertanty calbraton of a structural odel usng vbraton easureents was ephaszed n ths work. The ethod has also been successfully appled to structural health ontorng for dentfyng the locaton and severty of daage [8]. The fast coputng technques pleented wthn the Bayesan fraework can also be used for updatng robust odel-based predctons and relablty gven ontorng data.

8 Acknowledgeents Ths research has been co-fnanced by the European Unon (European Socal Fund - ESF) and Greek natonal funds through the Operatonal Progra "Educaton and Lfelong Learnng" of the Natonal Strategc Reference Fraework (NSRF) - Research Fundng Progra: Arstea. REFERENCES [] Beck, J.L., Katafygots, L.S., Updatng odels and ther uncertantes. I: Bayesan statstcal fraework, ASCE Journal of Engneerng Mechancs 24(4), , 998. [2] Beck, J.L., Bayesan syste dentfcaton based on probablty logc, Structural Control and Health Montorng 7(7), , 2. [3] Yuen, K.V., Bayesan Methods for Structural Dynacs and Cvl Engneerng, John Wley & Sons, 2. [4] Papadtrou, C., Beck, J.L., Katafygots, L.S., Updatng robust relablty usng structural test data, Probablstc Engneerng Mechancs 6(2), 3-3, 2. [5] Metropols, N., Rosenbluth, A.W., Rosenbluth, M.N., Teller, A.H., Teller, E., Equaton of state calculatons by fast coputng achnes, The Journal of Checal Physcs 23(62), 87-92, 953. [6] Chng, J., Chen, Y.C., Transtonal Markov Chan Monte Carlo ethod for Bayesan updatng, odel class selecton, and odel averagng, ASCE Journal of Engneerng Mechancs 33, , 27. [7] Crag Jr., R.R, Bapton, M.C.C., Couplng of substructures for dynac analyss, AIAA Journal 6(7), , 965. [8] Papadtrou, C., Papadot, D.C., Coponent Mode Synthess Technques for Fnte Eleent Model Updatng, Coputers and Structures, DOI:.6/.copstruc.22..8, 22. [9] Angelkopoulos, P., Papadtrou, C., Kououtsakos, P., Bayesan Uncertanty Quantfcaton and Propagaton n Molecular Dynacs Sulatons: A Hgh Perforance Coputng Fraework, The Journal of Checal Physcs 37(4), 22 DOI:.63/ [] Beck, J.L., Yuen, K.V., Model selecton usng response easureents: Bayesan probablstc approach, ASCE Journal of Engneerng Mechancs 3(2), 92-23, 24. [] Papadtrou, C., Katafygots, L.S., Bayesan odelng and updatng, In Engneerng Desgn Relablty Handbook, Nkolads N., Ghocel D.M., Snghal S. (Eds), CRC Press, 24. [2] Ntotsos, E., Papadtrou, C., Mult-obectve optzaton algorths for fnte eleent odel updatng, ISMA28 Internatonal Conference on Nose and Vbraton Engneerng, Leuven, , 28. [3] Haaro, H., Lane, M., Mra, A., Saksan, E., DRAM: Effcent adaptve MCMC, Statstcs and Coputng 6, , 26. [4] Muto, M., Beck, J.L., Bayesan updatng and odel class selecton usng stochastc sulaton, Journal of Vbraton and Control 4, 7 34, 28. [5] Goller, B., Stochastc Model Valdaton of Structural Systes, Ph.D. Dssertaton, Departent of Engneerng Mechancs, Unversty of Innsbruck, 2. [6] Goller, B., Brogg, M., Calv, A., Schueller, G.I., A stochastc odel updatng technque for coplex aerospace structures, Fnte Eleents n Analyss and Desgn 47(7), , 2 [7] Lophaven, S.N., Nelsen, H.B., Sondergaard, J., Dace, a atlab krgng toolbox, Techncal Report,IMM-TR-22-2, DTU, DK-28 Kgs Lyngby Denark, 22. [8] Castaner, M.P., Tan, Y.-C., Perre, C., Characterstc constrant odes for coponent ode synthess. AIAA Journal 39(6), 82-87, 2.