Multi-objective optimization algorithms for finite element model updating

Transcription

1 Mult-obectve optmzato algothms fo fte elemet model updatg E. Ntotsos ad C. Papadmtou Uvesty of hessaly, Depatmet of Mechacal ad Idustal Egeeg Volos 38334, Geece emal: Abstact A mult-obectve optmzato method s peseted fo estmatg the paametes of fte elemet stuctual models based o modal esduals. he method esults multple Paeto optmal stuctual models that ae cosstet wth the measued modal data ad the modal esduals used to measue the dscepaces betwee the measued modal values ad the modal values pedcted by the fte elemet model. he elato betwee the mult-obectve detfcato method ad covetoal sgle-obectve weghted modal esduals methods fo model updatg s exploed. Computatoally effcet methods fo estmatg the gadet ad Hessas of the obectve fuctos wth espect to the model paametes ae poposed ad show to sgfcatly educe the computatoal effot fo solvg the sgle ad multobectve optmzato poblems. he poposed methods explot Nelso s fomulato fo the sestvty of the egepopetes wth espect to the paametes. heoetcal ad computatoal developmets ae llustated by updatg fte elemet models of a mult-spa efoced cocete bdge usg ambet vbato measuemets. I patcula, mult-obectve detfcato esults dcate that thee s wde vaety of Paeto optmal stuctual models that tade off the ft vaous measued modal quattes. Itoducto Stuctual model updatg methods (e.g. []) have bee poposed the past to ecocle mathematcal models, usually dscetzed fte elemet models, wth expemetal data. he estmate of the optmal model fom a paametezed class of models s sestve to ucetates that ae due to lmtatos of the mathematcal models used to epeset the behavo of the eal stuctue, the pesece of measuemet ad pocessg eo the data, the umbe ad type of measued modal o espose tme hstoy data used the ecoclg pocess, as well as the oms used to measue the ft betwee measued ad model pedcted chaactestcs. he optmal stuctual models esultg fom such methods ca be used fo mpovg the model espose ad elablty pedctos [], stuctual health motog applcatos [3-6] ad stuctual cotol [7]. Stuctual model paamete estmato poblems based o measued data, such as modal chaactestcs (e.g. [3-6]) o espose tme hstoy chaactestcs [8], ae ofte fomulated as weghted least-squaes poblems whch metcs, measug the esduals betwee measued ad model pedcted chaactestcs, ae buld up to a sgle weghted esduals metc fomed as a weghted aveage of the multple dvdual metcs usg weghtg factos. Stadad optmzato techques ae the used to fd the optmal values of the stuctual paametes that mmze the sgle weghted esduals metc epesetg a oveall measue of ft betwee measued ad model pedcted chaactestcs. Due to model eo ad measuemet ose, the esults of the optmzato ae affected by the values assumed fo the weghtg factos. he model updatg poblem has also bee fomulated a mult-obectve cotext that allows the smultaeous mmzato of the multple metcs, elmatg the eed fo usg abtay weghtg factos fo weghtg the elatve mpotace of each metc the oveall measue of ft. he mult-obectve paamete estmato methodology povdes multple Paeto optmal stuctual models

2 cosstet wth the data ad the esduals used the sese that the ft each Paeto optmal model povdes a goup of measued modal popetes caot be mpoved wthout deteoatg the ft at least oe othe modal goup. I ths wok, the stuctual model updatg poblem usg modal esduals s fst fomulated as a multobectve optmzato poblem ad the as a sgle-obectve optmzato wth the obectve fomed as a weghted aveage of the multple obectves usg weghtg factos. heoetcal ad computatoal ssues asg mult-obectve detfcato ae addessed ad the coespodece betwee the mult-obectve detfcato ad the weghted esduals detfcato s establshed. Emphass s gve addessg ssues assocated wth solvg the esultg mult-obectve ad sgle-obectve optmzato poblems. Fo ths, effcet methods ae poposed fo estmatg the gadets ad the Hessas of the obectve fuctos usg the Nelso s method [9] fo fdg the sestvtes of the egepopetes to model paametes. he poposed model updatg methodologes ae llustated by updatg a -shaped R/C bdge stuctue, usg ambet duced vbato measuemets. Model updatg based o modal esduals ( k) ( ) 0 Let { ˆ, ˆ k N D= ω R, =,, m, k =,, N } be the measued modal data fom a stuctue, D ( ) ˆ k cosstg of modal fequeces ω ad modeshape compoets at N 0 measued DOFs, whee m s the umbe of obseved modes ad N D s the umbe of modal data sets avalable. Cosde a paametezed class of lea stuctual models used to model the dyamc behavo of the stuctue ad let N θ R θ be the set of fee stuctual model paametes to be detfed usg the measued modal data. he obectve a modal-based stuctual detfcato methodology s to estmate the values of the N paamete set θ so that the modal data { ω ( ), ( ) d θ θ R, =,, m}, whee N d s the umbe of model degees of feedom (DOF), pedcted by the lea class of models best matches, some sese, the expemetally obtaed modal data D. Fo ths, let ˆ ω ( ) ˆ ( ) ( ) θ ω β θ L θ εω ( θ) = ad ε ( ) θ = () ˆ ω ˆ =,, m, be the measues of ft o esduals betwee the measued modal data ad the model pedcted modal data fo the -th modal fequecy ad modeshape compoets, espectvely, whee z = z z s the usual Euclda om, ad ( ) ˆ β θ = L( θ) / L ( θ) s a omalzato costat that guaates that the measued modeshape ˆ at the measued DOFs s closest to the model modeshape β( θ) L( θ ) N0 Nd pedcted by the patcula value of θ. he matx L R s a obsevato matx compsed of zeos ad oes that maps the model DOFs to the N obseved DOFs. Nd 0 I ode to poceed wth the model updatg fomulato, the measued modal popetes ae gouped to goups. Each goup cotas oe o moe modal popetes. he modal popetes assged the th goup ae detfed by the set g ( k), =,, ad k =,, wth ay elemet the set g ( k) s a tege fom to m. A elemet the set g ( k) wth k = efe to the umbe of the measued modal fequecy assged the goup, whle the elemets of the set g ( k) wth k = efe to the umbe of the measued modeshape assged the goup. Fo the th goup, a om J ( θ ) s toduced to measue the esduals of the dffeece betwee the measued values of the modal popetes volved the goup ad the coespodg modal values pedcted fom the model class fo a patcula value of the paamete set θ. he measue of ft a modal goup s the sum of the dvdual squae eos () fo ˆ( k )

3 the coespodg modal popetes volved the modal goup. Specfcally, the measue of ft s gve by () J ( θ ) = εω ( θ) + ε ( ) θ g() g() he goupg of the modal popetes { ω( θ), ( θ ), =,, m} to goups ad the selecto of the measues of ft (esduals) J ( ),, ( ) θ J θ ae usually based o use pefeece. he modal popetes assged to each goup ae selected by the use accodg to the type ad the pupose of the aalyss. he afoemetoed aalyss accommodates geeal goupg schemes ad obectve fuctos. Fo demostato puposes, a specfc goupg scheme s ext defed by goupg the modal popetes to two goups as follows. he fst goup cotas all modal fequeces, wth the measue of ft J ( θ ) selected to epeset the dffeece betwee the measued ad the model pedcted fequeces fo all modes, whle the secod goup cotas the modeshape compoets fo all modes wth the measue of ft J ( ) θ selected to epesets the dffeece betwee the measued ad the model pedcted modeshape compoets fo all modes. Specfcally, the two measues of ft ae gve by = = (3) m m ω J = = J ( θ ) ε ( θ) ad ( θ) ε ( θ) he afoemetoed goupg scheme s used the applcato secto fo demostatg the featues of the poposed model updatg methodologes.. Mult-obectve detfcato he poblem of detfyg the model paamete values that mmze the modal o espose tme hstoy esduals ca be fomulated as a mult-obectve optmzato poblem stated as follows [0]. Fd the values of the stuctual paamete set θ that smultaeously mmzes the obectves y J J J = ( θ ) = ( ( θ),, ( θ)) (4) subect to equalty costas c( θ ) 0 ad paamete costas θlow θ θuppe, whee θ = ( θ,, θ Nθ ) Θ s the paamete vecto, Θ s the paamete space, y = ( y,, y) Y s the obectve vecto, Y s the obectve space, c( θ ) s the vecto fucto of costas, ad θ low ad θ uppe ae espectvely the lowe ad uppe bouds of the paamete vecto θ. Fo coflctg obectves J ( ),, ( ) θ J θ, thee s o sgle optmal soluto, but athe a set of alteatve solutos, kow as Paeto optmal solutos, that ae optmal the sese that o othe solutos the paamete space ae supeo to them whe all obectves ae cosdeed. he set of obectve vectos y = J ( θ ) coespodg to the set of Paeto optmal solutos θ s called Paeto optmal fot. he chaactestcs of the Paeto solutos ae that the esduals caot be mpoved ay goup wthout deteoatg the esduals at least oe othe goup. he multple Paeto optmal solutos ae due to modelg ad measuemet eos. Usg mult-obectve temology, the Paeto optmal solutos ae the o-domatg vectos the paamete space Θ, defed mathematcally as follows. A vecto θ Θ s sad to be o-domated egadg the set Θ f ad oly f thee s o vecto Θ whch domates θ. A vecto θ s sad to domate a vecto θ ' f ad oly f J ( θ ) J ( θ ') {,, } ad {,, } : J ( θ) < J ( θ ') (5)

4 he set of obectve vectos y = J ( θ ) coespodg to the set of Paeto optmal solutos θ s called Paeto optmal fot. he chaactestcs of the Paeto solutos ae that the modal esduals caot be mpoved ay modal goup wthout deteoatg the modal esduals at least oe othe modal goup. Specfcally, usg the obectve fuctos (3), all optmal models that tade-off the oveall ft modal fequeces wth the oveall ft the modeshapes ae estmated. he multple Paeto optmal solutos ae due to modellg ad measuemet eos. he level of modellg ad measuemet eos affect the sze ad the dstace fom the og of the Paeto fot the obectve space, as well as the vaablty of the Paeto optmal solutos the paamete space. he vaablty of the Paeto optmal solutos also depeds o the oveall sestvty of the obectve fuctos o, equvaletly, the sestvty of the modal popetes, to model paamete values θ. Such vaabltes wee demostated fo the case of two-dmesoal obectve space ad oe-dmesoal paamete space the wok by Chstodoulou ad Papadmtou []. It should be oted that the absece of modellg ad measuemet eos, thee s a optmal value ˆ θ of the paamete set θ fo whch the model based modal fequeces ad modeshape compoets match exactly the coespodg measued modal popetes. I ths case, all obectve fuctos J ˆ ( θ ),, J( ˆ θ ) take the value of zeo ad, cosequetly, the Paeto fot cossts of a sgle pot at the og of the obectve space. I patcula, fo detfable poblems [-3], the solutos the paamete space cosst of oe o moe solated pots fo the case of a sgle o multple global optma, espectvely. Fo o-detfable poblems [4-5], the Paeto optmal solutos fom a lowe dmesoal mafold the paamete space.. Weghted modal esduals detfcato he paamete estmato poblem s tadtoally solved by mmzg the sgle obectve J( θ; w) = wj ( θ ) (6) = fomed fom the multple obectves J ( θ ) usg the weghtg factos w 0, =,,, wth w. he obectve fucto = = J( θ ; w) epesets a oveall measue of ft betwee the measued ad the model pedcted chaactestcs. he elatve mpotace of the esdual eos the selecto of the optmal model s eflected the choce of the weghts. he esults of the detfcato deped o the weght values used. Covetoal weghted least squaes methods assume equal weght values, w = = w = /. hs covetoal method s efeed hee as the equally weghted modal esduals method..3 Compaso betwee mult-obectve ad weghted modal esduals detfcato Fomulatg the paamete detfcato poblem as a mult-obectve mmzato poblem, the eed fo usg abtay weghtg factos fo weghtg the elatve mpotace of the esduals J ( θ ) of a modal goup to a oveall weghted esduals metc s elmated. A advatage of the mult-obectve detfcato methodology s that all admssble solutos the paamete space ae obtaed. It ca be eadly show that the optmal soluto to the poblem (6) s oe of the Paeto optmal solutos. Fo ths, let ˆ θ be the global optmal soluto that mmzes the obectve fucto J( θ ; w) (6) fo gve w. he ths soluto s also a Paeto optmal soluto sce othewse thee would exst aothe

5 soluto, say θ ˆ, fo whch equato (5) wll be satsfed fo θ = ˆ θ ad θ = ˆ θ, that s, J ( ˆ θ ) J ( ˆ θ) {,, } ad {,, } : J ( ˆ θ ) < J ( ˆ θ ). As a esult of ths ad the fact that w 0, t s eadly deved usg the fom of J( θ ; w) (6) that J( ˆ θ ; w) < J( ˆ θ; w). he last equalty mples that θ ˆ, stead of ˆ θ, s the global soluto optmzg J( θ ; w), whch s a cotadcto. hus, solvg a sees of sgle obectve optmzato poblems of the type (6) ad vayg the values of the weghts w fom 0 to, excludg the case fo whch the values of all weghts ae smultaeously equal to zeo, Paeto optmal solutos ae alteatvely obtaed. hese solutos fo gve w ae deoted by ˆ( θ w). It should be oted, howeve, that thee may exst Paeto optmal solutos that do ot coespod to solutos of the sgle-obectve weghted modal esduals poblem [6]. he sgle obectve s computatoally attactve sce covetoal mmzato algothms ca be appled to solve the poblem. Howeve, a sevee dawback of geeatg Paeto optmal solutos by solvg the sees of weghted sgle-obectve optmzato poblems by ufomly vayg the values of the weghts s that ths pocedue ofte esults cluste of pots pats of the Paeto fot that fal to povde a adequate epesetato of the ete Paeto shape. hus, alteatve algothms dealg dectly wth the mult-obectve optmzato poblem ad geeatg ufomly spead pots alog the ete Paeto fot should be pefeed. Specal algothms ae avalable fo solvg the mult-obectve optmzato poblem. Computatoal algothms ad elated ssues fo solvg the sgle-obectve ad the mult-obectve optmzato poblems ae dscussed Secto 3. 3 Computatoal Issues Related to Model Updatg Fomulatos he poposed sgle ad mult-obectve detfcato poblems ae solved usg avalable sgle- ad mult-obectve optmzato algothms. hese algothms ae befly evewed ad vaous mplemetato ssues ae addessed, cludg estmato of global optma fom multple local/global oes, as well as covegece poblems. 3. Sgle-Obectve Idetfcato he optmzato of J( θ ; w) (6) wth espect to θ fo gve w ca eadly be caed out umecally usg ay avalable algothm fo optmzg a olea fucto of seveal vaables. hese sgle obectve optmzato poblems may volve multple local/global optma. Covetoal gadet-based local optmzato algothms lack elablty dealg wth the estmato of multple local/global optma obseved stuctual detfcato poblems [0,7], sce covegece to the global optmum s ot guaateed. Evoluto stateges (ES) [8] ae moe appopate ad effectve to use such cases. ES ae adom seach algothms that exploe bette the paamete space fo detectg the eghbohood of the global optmum, avodg pematue covegece to a local optmum. A dsadvatage of ES s the slow covegece at the eghbohood of a optmum sce they do ot explot the gadet fomato. A hybd optmzato algothm should be used that explots the advatages of ES ad gadet-based methods. Specfcally, a evoluto stategy s used to exploe the paamete space ad detect the eghbohood of the global optmum. he the method swtches to a gadet-based algothm statg wth the best estmate obtaed fom the evoluto stategy ad usg gadet fomato to acceleate covegece to the global optmum.

6 3. Mult-Obectve Idetfcato he set of Paeto optmal solutos ca be obtaed usg avalable mult-obectve optmzato algothms. Amog them, the evolutoay algothms, such as the stegth Paeto evolutoay algothm [9], ae well-suted to solve the mult-obectve optmzato poblem. he stegth Paeto evolutoay algothm, although t does ot eque gadet fomato, t has the dsadvatage of slow covegece fo obectve vectos close to the Paeto fot [0] ad also t does ot geeate a evely spead Paeto fot, especally fo lage dffeeces obectve fuctos. Aothe vey effcet algothm fo solvg the mult-obectve optmzato poblem s the Nomal- Bouday Itesecto (NBI) method [0] whch poduce a evely spead of pots alog the Paeto fot, eve fo poblems fo whch the elatve scalg of the obectves ae vastly dffeet. Fo completeess ad fo the pupose of demostatg the mplemetato ssues asg mult-obectve stuctual model updatg, the dea of the NBI method s befly llustated geometcally wth the ad of () the two-dmesoal Paeto fot show Fgue. Fo ths, let ˆ θ, =,,, be the global optmal values of the paamete set that mmze the dvdual obectves J ( θ ), =,,, espectvely. he () () Paeto pots ˆ J J ( ˆ = θ ), show Fgue, deteme the locato of the boudaes of the Paeto fot the obectve space. hese edge pots of the Paeto fot ae estmated usg the sgle-obectve optmzato algothms outled Secto 3.. he utopa pot Jˆ = [ Jˆ ˆ,, J ], show Fgue, s toduced as the pot the obectve space wth coodates the dvdual mma ˆ ˆ( ) J = J ( θ ) of the () obectves. Let Φ be the matx wth the -th colum equal to the vecto J ˆ. he set of pots the obectve space that ae covex combatos of = ˆ() J Ĵ, obtaed by the pots { Φβ : β R, β =, β 0}, s efeed to as the Covex Hull of Idvdual Mma (CHIM). hese pots ae all pots alog the le segmet AB Fgue. he Paeto pots cosst of pots o the tesecto of the bouday Y of the obectve space Y ad the omal tatg fom ay pot the CHIM ad potg towads the og of the obectve space. J () J * J A Φβ Φ β + t+ J B J () * Fgue. Geometc llustato of NBI Method -dmesoal obectve space J A pot alog the Paeto fot ca be foud by solvg a sgle-obectve optmzato poblem. Gve the coodates β, Φ β epesets a pot o the CHIM ad Φ β + t, whee t R ad the omal to the CHIM, epesets the set of pots o the omal to the CHIM at the pot Φ β. he pot of tesecto

7 of the omal ad the boubay Y, closest to the og, s the global soluto of the commoly efeed as NBI optmzato poblem [0]: β subect to the costas max t θ, t * Φ β + t = J( θ) J (8) (7) Ay costas fom the ogal mult-obectve optmzato poblem (4) ca also be cosdeed by addg them as costas the NBI β optmzato poblem. By solvg the optmzato poblems NBI β fo vaous β values the set { β R : β =, β 0}, a potwse epesetato of the Paeto fot s effcetly costucted. he values of the paametes β ae selected so that a evely spead pots alog the CHIM ae obtaed, esultg to a evely spead pots alog the Paeto fot, depedetly of the scales of the obectve fuctos. Fo the two-dmesoal obectve space, ths s acheved by selectg the values of the compoet β of β = ( β, β) to be ufomly spaced the teval [0,] wth spacg legth δ = /( N ), whee N s the umbe of pots alog the CHIM cludg the edge pots. he fst compoet β s selected to satsfy β+ β =. Moe detals about the method, the selecto of β values fo moe tha two obectves, advatages ad dawbacks, ca be foud the ogal pape by Das ad Des [0]. It s also of teest to compae the computatoal tme volved fo estmatg the Paeto optmal solutos wth the computatoal tme equed covetoal weghted esduals methods fo estmatg a sgle soluto. hs estmate ca be made by otg that each Paeto optmal solutos s obtaed by solvg a sgle-obectve optmzato poblem NBI β. hus, ths computatoal tme s of the ode of the umbe of pots used to epeset the Paeto fot multpled by the computatoal tme equed to solve a sgle-obectve NBI β poblem fo computg each pot o the fot. Howeve, fo the NBI method, covegece ca be geatly acceleated by usg a good statg value fo the NBI β optmzato poblem close to the optmal value. hs s acheved by selectg the Paeto optmal soluto obtaed fom the cuet NBI β poblem to be used as statg value fo solvg the ext NBI β poblem. = 3.3 Fomulato fo gadets of obectves I ode to guaatee the covegece of the gadet-based optmzato methods fo stuctual models volvg a lage umbe of DOFs wth seveal cotbutg modes, the gadets of the obectve fuctos wth espect to the paamete set θ has to be estmated accuately. It has bee obseved that umecal algothms such as fte dffeece methods fo gadet evaluato does ot guaatee covegece due to the fact that the eos the umecal estmato may povde the wog dectos the seach space ad covegece to the local/global mmum s ot acheved, especally fo temedate paamete values the vcty of a local/global optmum. hus, the gadets of the obectve fuctos should be povded aalytcally. Moeove, gadet computatos wth espect to the paamete set usg the fte dffeece method eques the soluto of as may egevalue poblems as the umbe of paametes. he gadets of the modal fequeces ad modeshapes, equed the estmato of the gadet of J( θ ; w) (6) o the gadets of the obectves J ( θ ) (4) ae computed by expessg them exactly tems of the modal fequeces, modeshapes ad the gadets of the stuctual mass ad stffess matces wth espect to θ usg Nelso s method [9]. Specal atteto s gve to the computato of the gadets

8 ad the Hessas of the obectve fuctos fo the pot of vew of the educto of the computatoal tme equed. Aalytcal expessos fo the gadet of the modal fequeces ad modeshapes ae used to ovecome the covegece poblems. I patcula, Nelso s method [9] s used fo computg aalytcally the fst devatves of the egevalues ad the egevectos. he advatage of the Nelso s method compaed to othe methods s that the gadet of egevalue ad the egevecto of oe mode ae computed fom the egevalue ad the egevecto of the same mode ad thee s o eed to kow the egevalues ad the egevectos fom othe modes. Fo each paamete the set θ ths computato s pefomed by solvg a lea system of the same sze as the ogal system mass ad stffess matces. Nelso s method s also exteded Secto 3.4 to compute the secod devatves of the egevalues ad the egevectos. he computato of the gadets ad the Hessa of the obectve fuctos s show to volve the soluto of a sgle lea system, stead of N θ lea systems equed usual computatos of the gadet ad Nθ ( Nθ + ) lea systems equed the computato of the Hessa. hs educes cosdeably the computatoal tme, especally as the umbe of paametes the set θ cease. he expessos fo the fst devatves of the obectve fuctos ae ext peseted. Summazg, Nelso s method [9] specalzed fo symmetc mass ad stffess matces computes the devatves of the -th egevalue ad egevecto wth espect to a paamete θ the paamete set θ fom the followg fomulas ad whee θ ω = ( K ωm ) θ * * = ( I M) A F M (9) (0) A = K ω M () A F, = = ( I M )( K ωm ) θ M( θ ) K( θ ) M M ( θ) =, K K ( θ) = θ θ Fo otatoal coveece, the depedece of seveal vaables o the paamete set θ has bee dopped. * Fo a matx A efeg to the fomulato fo the -th mode, A s used to deote the modfed matx deved fom the matx A by eplacg the elemets of the k -th colum ad the k -th ow by zeoes ad the ( k, k ) elemet of A by oe, whee k deotes the elemet of the modeshape vecto * wth the hghest absolute value. Also, the vecto b s used to deote the modfed vecto deved fom b eplacg the k -th elemet of the vecto b by zeo. Moe detals ca be foud the wok by Nelso [9]. he gadet of the squae eo ε ( θ ) s gve by ω ε ( θ) ε ( θ) ε ( θ) ω ω ω ω = = ( K ωm ) θ ω θ ω () (3) (4)

9 ad the gadet of the squae eo ε ( θ ) s gve by ε ( θ) ϕ = ε θ = ε θ L θ θ θ [ ϕ ( )] [ ( )] ϕ ϕ ϕ (5) Substtutg (0) to (5), the gadet of the squae eo ε ( θ ) s smplfed to ε ( θ) * = x F, z M θ (6) whee F, s gve (), z [ ( )] ad x s gve by the soluto of the lea system of equatos = ε θ L (7) A X * = D (8) wth D = ( I M ) L ϕ ε ( ϕ θ) ad X eplaced by x. he system of equatos (8) ca be vewed as the adot system fo the model updatg optmzato poblem based o modal esduals. It should be oted that fo the specfc obectve fuctos ε ω ( θ ) ad ε ϕ ( θ ) gve by (), the afoemetoed expessos fo the gadets of the obectve fuctos smplfy futhe. Specfcally, usg () ad otg that ε ( θ) L = ε ( θ) ϕ = 0, oe eadly obtas that whee ε ( θ) εω ( θ) = ˆ ω ω ω (9) ε ( θ) = e ( θ) β (0) ϕ ϕ ϕ ˆ βl e ϕ = () ˆ ϕ z = 0 ad s gve by the equato D D = L β e ϕ ( θ ) () he computato of the devatves of the squae eos fo the modal popetes of the -th mode wth espect to the paametes θ eques oly oe soluto of the lea system (8), depedet of the umbe of paametes θ. Fo a lage umbe of paametes the set θ the above fomulato fo the gadets of the mea eos modal fequeces ad the modeshape compoets () ae computatoally vey effcet ad fomatve. he depedece o θ comes though the tem K ωm ad the tem fomulato s futhe smplfed. M. Fo the case whee the mass matx s depedet of θ, M = 0 ad the It should be oted that fo the specal case of lea depedece betwee the global mass ad stffess N matces o the paametes the set θ, that s, M( θ ) M θ 0 M θ = + ad =

10 K( θ ) K θ K θ N = +, the gadets of ( ) 0 = M θ ad K ( θ ) ae easly computed fom the costat matces M 0, K0, M ad K, =,, N θ. I ode to save computatoal tme, these costat matces ae computed ad assembled oce ad, theefoe, thee s o eed ths computato to be epeated dug the teatos volved optmzato algothms. Fo the geeal case of olea depedece betwee the global mass ad stffess matces o the paametes the set θ, the matces M ad K volved the fomulato (see (3)) ca be obtaed umecally at the elemet level ad assembled to fom the global matces. 3.4 Fomulato fo Hessa of obectves A smla aalyss to that followed Nelso s method [9] fo computg the fst devatve ca also be followed fo computg the secod devatves of the egevalues ad the egevectos, esultg the followg expessos fo the secod devatves ad whee ad g ω = g θ θ, = ( I M) A G d θ θ * *, (3) (4) G = ( I M ) g (5) A A K M λ M λ M, = + + λ θ θ θ θ θ θ θ θ θ θ θ θ M M M d, = M (7) θ θ θ θ θ θ θ θ he Hessa of the obectve fuctos ε ω ( θ ) ad ε ϕ ( θ ) ca be eadly computed fom the secod devatves of the egevalues ad the egevectos, espectvely. Specfcally, the (, ) elemet of the Hessa of ε ( θ ) s obtaed by dffeetatg (4) wth espect to θ, esultg he (, ) esultg ω ε ( θ) ε ( θ) ε ( θ) ω ω ω ω ω ω = + θ θ ( ω ) θ θ ω θ θ ε ( θ) ε ( θ) = + ( ) elemet of the Hessa of ω ω [ ( ) ][ ( ) ] K ωm K ωm g ω ω (6) (8) ε ( θ ) s obtaed by dffeetatg (5) wth espect to θ, ϕ ε ( θ) θ ϕ ϕ ϕ ϕ = [ ϕ ( )] [ ( )] ϕ ε ϕ θ + ϕ ε ϕ θ θ θ θ θ θ (9)

11 Substtutg (4) to (9) ad usg (8), the Hessa ca be fally smplfed to ε ( θ) ε θ ε θ L d (30) * = L [ ( )] ( ) [ ( )] L x I M g θ θ θ θ It should be oted that fo the specfc obectve fuctos ε ω ( θ ) ad ε ϕ ( θ ) gve by (), the afoemetoed expessos fo the Hessa of the obectve fuctos smplfy futhe. Specfcally, usg () ad otg that ε ( θ) L = ε ( θ) ϕ = 0, oe eadly obtas that, ε ( θ) ω = 4 ( ω ) ˆ ω ( ) ( ˆ )( ˆ ϕ ) ϕ ε ϕ θ β ϕ ϕ βϕ ϕ β ϕ I = ˆ ϕ ϕ (3) (3) ad ε ( θ ) θ θ (30) smplfes to ε ( θ) * * * * * = ( z F, )( z F, ) β L F, XXF, x ( I M) g θ θ ˆ L whee ad z s gve by the soluto of the lea system (8) wth D ( ) ( ˆ = I M L βl ) X s gve by (8) wth D = ( I M ) L. It should be oted that oly the last tem (8) ad the last tem (33) deped explctly o the devatves / θ. Numecal esults suggest that the Hessa of ε ω ( θ ) ad ε ( θ ) ca be adequately appoxmated the fom (8) ad (33), gog the cotbuto fom the last tems (8) ad (33). hus the Hessa of ε ω ( θ ) ad ε ( θ ) ca be computed fom the soluto of the system (8), estmates of the egevalues ad egevectos of the mode, ad the sestvtes K ad M of the global stffess ad mass matces wth espect to the paametes θ. Summazg, t should be oted that the computato of the fst ad secod devatves of the squae eos fo the modal popetes of the -th mode wth espect to the paametes θ eques oly the solutos of the lea system (8), depedet of the umbe of paametes θ. Fo a lage umbe of paametes the set θ, the above fomulato fo the gadets ad Hessa of the mea eos modal fequeces ad the modeshape compoets () ae computatoally vey effcet ad fomatve. (33) 4 Applcato he poposed famewok has bee appled to a -shaped R/C bdge (Fgue a) of Egata Odos motoway whch cosses Nothe Geece the east-west decto. he s located at Polymylos ad has bee stumeted wth specal aay of 4 acceleometes. he espose to ambet exctato caused by taffc ad wd has bee systematcally motoed. he modal detfcato usg these ambet vbatos esulted the elable estmato of the fst eght modes. o mplemet the model updatg techques, a appopate paametc fte elemet model of the bdge s cosdeed usg theedmesoal two-ode beam-type fte elemets to model the deck, the pes ad the beags. hs model s show Fgue b ad has 038 degees of feedom. he ete smulato s pefomed wth the

12 COMSOL Multphyscs [] modelg evomet. A thee paamete model class s employed ode to demostate the applcablty of the poposed methodologes, ad pot out ssues assocated wth the mult-obectve detfcato. he fst paamete θ accouts fo the stffess of the elastomec beags at the abutmets, the secod paamete θ accouts fo the stffess of the deck, whle the thd paamete θ 3 accouts fo the stffess of the pes. he omal fte elemet model coespods to values of θ= θ= θ3=. he paametezed fte elemet model class s updated usg the thee modal fequeces ad modeshapes obtaed fom opeatoal modal aalyss ad the two modal goups wth modal esduals gve by (3). (a) Fgue : (a) Vew of the Polymylos bdge, (b) Fte elemet model. (b) he esults fom the mult-obectve detfcato methodology ae show Fgue 3. Fo each model class ad assocated stuctual cofguato, the Paeto fot, gvg the Paeto solutos the twodmesoal obectve space, s show Fgue 3a. he o-zeo sze of the Paeto fot ad the o-zeo dstace of the Paeto fot fom the og ae due to modelg ad measuemet eos. Specfcally, the dstace of the Paeto pots alog the Paeto fot fom the og s a dcato of the sze of the oveall measuemet ad modelg eo. he sze of the Paeto fot depeds o the sze of the model eo ad the sestvty of the modal popetes to the paamete values θ [6]. Fgues 3b-d show the coespodg Paeto optmal solutos the thee-dmesoal paamete space. Specfcally, these fgues show the poecto of the Paeto solutos the two-dmesoal paamete spaces ( θ, θ ), ( θ, θ 3) ad ( θ, θ 3) J Paeto Solutos w= θ Paeto Solutos w= J θ θ Paeto Solutos w= θ Paeto Solutos w= θ θ Fgue 3: Paeto fot ad Paeto optmal solutos the (a) obectve space ad (b-d) paamete space

13 It s obseved that a wde vaety of Paeto optmal solutos ae obtaed fo dffeet stuctual cofguatos that ae cosstet wth the measued data ad the obectve fuctos used. he Paeto optmal solutos ae cocetated alog a oe-dmesoal mafold the thee-dmesoal paamete space. Compag the Paeto optmal solutos, t ca be sad that thee s o Paeto soluto that mpoves the ft both modal goups smultaeously. hus, all Paeto solutos coespod to acceptable compomse stuctual models tadg-off the ft the modal fequeces volved the fst modal goup wth the ft the modeshape compoets volved the secod modal goups. he effectveess of the aalytc expessos fo the gadets ad the Hessa of the obectve fuctos (3) volved o the soluto of the model updatg poblem has bee vestgated by updatg the fte elemet model of the Polymylos bdge usg smulated modal data. Specfcally, thee paametezed model classes wee updated usg smulated modes ad applyg the Newto ust-ego o-lea optmzato method [3] ad the BFGS quas-newto method [5]. he paametezed model classes that wee updated cluded a lmted umbe of 3, 5 ad 7 paametes. Fo the 3-paamete model class the fst paamete accouts fo the stffess of the elastomec beags at the abutmets, the secod paamete accouts fo the stffess of the deck, whle the thd paamete accouts fo the stffess of the pes. Fo the 5-paamete model class the exta two paametes wee toduced to model the stffess that was assumed depedet fo the left ad ght beags of the bdge ad the two colums at the cetal pe. Fo the 7-paamete model class the exta two paametes wee toduced to model the stffess of the beags that was assumed depedet alog the logtudal ad tasvese decto of the bdge. he effectveess of the poposed optmzato schemes s vestgated compag the covegece ad the computatoal tme fo each method ad fo each paametezed model class. A compaso betwee the optmzato methods coceg covegece (umbe of teatos) ad computatoal tme s peseted able. he values able efeed to the BFGS medum-scale optmzato algothm show that ths optmzato scheme s supeo fo the soluto of the specfc poblem fo all model classes. he umbe of teatos equed fo the Newto ust-ego lage-scale optmzato method usg the aalytc expessos of the Hessa matx s of the same ode of magtude as the umbe of teatos equed fo the BFGS method. he computatoal tme fo the Newto ust-ego method has bee slghtly ceased due to the exta computatos equed to fom the aalytcal Hessa. he umbe of teatos equed fo the Newto ust-ego optmzato method usg fte dffeece appoxmatos of the Hessa has cease about 50% as compaed wth the teatos equed fo the ust-ego method usg aalytc Hessa expessos. Futhemoe, the computatoal tme fo the ust-ego optmzato method usg fte dffeece appoxmatos of the Hessa has ceased by oe ode of magtude. Fally, t should be oted that wthout povdg the aalytc expessos fo the gadets of the obectve fucto the algothms peset covegece poblems fo all cases. 3 paametes model 5 paametes model 7 paametes model Optmzato method tme tme tme Iteatos Iteatos (m) (m) (m) Iteatos BFGS ust-ego (appoxmate Hessa usg fte dffeece) ust-ego (aalytc Hessa) able : Compaso betwee computatoal tme ad umbe of teatos 5 Coclusos Model updatg algothms wee poposed to chaacteze ad compute all Paeto optmal models fom a model class, cosstet wth the measued modal data ad the oms used to measue the ft betwee the measued ad model pedcted modal popetes. Computatoal algothms fo the effcet ad elable

14 soluto of the esultg mult- ad sgle-obectve optmzato poblems wee peseted. he algothms ae classfed to gadet-based, evolutoay stateges ad hybd techques. he Nomal Bouday Itesecto method, patcula, s used as the gadet-based method to solve the mult-obectve optmzato. Effcet algothms ae toduced fo educg the computatoal cost volved estmatg the gadets of the obectve fuctos. Specfcally, a fomulato equg the soluto of the adot ege-poblem s peseted, avodg the explct estmato of the gadets of the egevalues ad the egevectos. he adot method s also exteded to cay out effcetly the estmato of the Hessa of the obectve fuctos, avodg the explct estmato of the Hessa of the egevalues ad egevectos. he computatoal cost fo estmatg the gadets s show to be depedet of the umbe of stuctual model paametes. he methodology s patculaly effcet to system wth seveal umbe of model paametes ad lage umbe of DOFs whee epeated gadet evaluatos ae computatoally qute tme cosumg. Gadet-based optmzato algothms such as the BFGS algothm ad the Newto ust Rego algothm avalable Matlab, explot the poposed aalytcal gadets ad Hessas estmates ode to sgfcatly educe the computatoal tme. I patcula, algothms usg fte dffeece appoxmatos of the gadets o eve Hessas ae show to pefom pooly fo modal-based fte elemet model updatg applcatos. he effectveess of the poposed optmzato algothms fo fte elemet model updatg by povdg the aalytc expesso fo the gadets ad Hessa matx of the obectve fuctos was demostated usg ambet measuemets fom a efoced cocete bdge. Ackowledgemets hs eseach was co-fuded 75% fom the Euopea Uo (Euopea Socal Fud), 5% fom the Geek Msty of Developmet (Geeal Secetaat of Reseach ad echology) ad fom the pvate secto, the cotext of measue 8.3 of the Opeatoal Pogam Compettveess (3 d Commuty Suppot Famewok Pogam) ude gat 03-ΕΔ-54 (PENED 003). hs suppot s gatefully ackowledged. Refeeces [] J.E. Motteshead, M.I. Fswell, Model updatg stuctual dyamcs: A suvey, Joual of Soud ad Vbato, Vol. 67 (993), pp [] C. Papadmtou, J.L. Beck, L.S. Katafygots, Updatg obust elablty usg stuctual test data, Pobablstc Egeeg Mechacs, Vol. 6 (00), pp [3] C.P. Ftze, D. Jeewe,. Kefe, Damage detecto based o model updatg methods, Mechacal Systems ad Sgal Pocessg, Vol., No. (998), pp [4] A. eughels, G. De Roeck, Damage detecto ad paamete detfcato by fte elemet model updatg, Achves of Computatoal Methods Egeeg, Vol., No. (005), pp [5] M.W. Vak, J.L. Beck, S.K. Au, Bayesa pobablstc appoach to stuctual health motog, Joual of Egeeg Mechacs (ASCE), Vol. 6 (000), pp [6] E. Ntotsos, C. Papadmtou, P. Paetsos, G. Kaaskos, K. Peos, Ph. Pedkas, Bdge health motog system based o vbato measuemets, Bullet of Eathquake Egeeg (008), pess [7] K.V. Yue, J.L. Beck, Relablty-based obust cotol fo uceta dyamcal systems usg feedback of complete osy espose measuemets, Eathquake Egeeg ad Stuctual Dyamcs, Vol. 3, No. 5 (003), pp [8] J.L. Beck, L.S. Katafygots, Updatg models ad the ucetates- I: Bayesa statstcal famewok, Joual of Egeeg Mechacs (ASCE), Vol. 4, No. 4 (998), pp

15 [9] R.B. Nelso, Smplfed calculato of egevecto devatves, AIAA Joual, Vol. 4, No. 9 (976), pp [0] Y. Haalampds, C. Papadmtou, M. Pavldou, Mult-obectve famewok fo stuctual model detfcato, Eathquake Egeeg ad Stuctual Dyamcs, Vol. 34, No. 6 (005), pp [] K. Chstodoulou, C. Papadmtou, Stuctual Idetfcato Based o Optmally Weghted Modal Resduals, Mechacal Systems ad Sgal Pocessg, Vol. (007), pp [] L.S. Katafygots, eatmet of Model Ucetates Stuctual Dyamcs, echcal Repot EERL9-0, Calfoa Isttute of echology, Pasadea, CA. (99). [3] L.S. Katafygots, J.L. Beck, Updatg models ad the ucetates. II: Model detfablty, Joual of Egeeg Mechacs (ASCE), Vol. 4, No. 4 (998), pp [4] L.S. Katafygots, C. Papadmtou, H.F. Lam, A pobablstc appoach to stuctual model updatg, Iteatoal Joual of Sol Dyamcs ad Eathquake Egeeg, Vol. 7 (998), pp [5] L.S. Katafygots, H.F. Lam, agetal-poecto algothm fo mafold epesetato udetfable model updatg models, Eathquake Egeeg ad Stuctual Dyamcs, Vol. 3, No. 4 (00), pp [6] K. Chstodoulou, E. Ntotsos, C. Papadmtou, P. Paetstos., Stuctual model updatg ad pedcto vaablty usg Paeto optmal models, Comput. Methods Appl. Mech. Egg. (008), do:0.06/.cma [7] A. eughels, G. De Roeck, J.A.K. Suykes, Global optmzato by coupled local mmzes ad ts applcato to FE model updatg, Computes ad Stuctues, Vol. 8, No. 4-5 (003), pp [8] H. G. Beye, he theoy of evoluto stateges, Bel, Spge-Velag (00). [9] E. Ztzle, L. hele, Mult-obectve evolutoay algothms: A compaatve case study ad the stegth Paeto appoach, IEEE asactos o Evolutoay Computato, Vol. 3 (999), pp [0] I. Das, J.E., J., Des, Nomal-Bouday Itesecto: A ew method fo geeatg the Paeto suface olea mult-ctea optmzato poblems, SIAM Joual of Optmzato, Vol. 8 (998), pp [] E. Ntotsos, Ch. Kaakostas, V. Lekds, P. Paetsos, G. Nkolaou, C. Papadmtou, Stuctual Idetfcato of Egata Odos Bdges based o Ambet ad Eathquake Iduced Vbatos, Bullet of Eathquake Egeeg (008), pess. [] COMSOL AB (005) COMSOL Multphyscs Use s Gude. [ [3].F. Colema, Y. L, A Iteo, ust Rego Appoach fo Nolea Mmzato Subect to Bouds, SIAM Joual o Optmzato, Vol. 6 (996), pp [4].F. Colema, Y. L, O the Covegece of Reflectve Newto Methods fo Lage-Scale Nolea Mmzato Subect to Bouds, Mathematcal Pogammg, Vol. 67, Numbe (994), pp [5] C.G. Boyde, he Covegece of a Class of Double-Rak Mmzato Algothms, Joual Ist. Math. Applc. (970), Vol. 6, pp [6] R. Fletche, A New Appoach to Vaable Metc Algothms, Compute Joual (970), Vol. 3, pp [7] D. Goldfab, A Famly of Vaable Metc Updates Deved by Vaatoal Meas, Mathematcs of Computg (970), Vol. 4, pp [8] D.F. Shao, Codtog of Quas-Newto Methods fo Fucto Mmzato, Mathematcs of Computg (970), Vol. 4, pp