Single-loop system reliability-based topology optimization considering statistical dependence between limit-states

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1 Sruc Muldsc Opm 2011) 44: DOI /s RESEARCH PAPER Sngle-loop sysem relably-based opology opmzaon consderng sascal dependence beween lm-saes Tam H. Nguyen Junho Song Glauco H. Paulno Receved: 10 June 2010 / Revsed: 27 March 2011 / Acceped: 19 Aprl 2011 / Publshed onlne: 28 July 2011 c Sprnger-Verlag 2011 Absrac Ths paper presens a sngle-loop algorhm for sysem relably-based opology opmzaon SRBTO) ha can accoun for sascal dependence beween mulple lm-saes, and s applcaons o compuaonally demandng opology opmzaon TO) problems. A sngle-loop relably-based desgn opmzaon RBDO) algorhm replaces he nner-loop eraons o evaluae probablsc consrans by a non-erave approxmaon. The proposed sngle-loop SRBTO algorhm accouns for he sascal dependence beween he lm-saes by usng he marxbased sysem relably MSR) mehod o compue he sysem falure probably and s parameer sensves. The SRBTO/MSR approach s applcable o general sysem evens ncludng seres, parallel, cu-se and lnk-se sysems and provdes he gradens of he sysem falure probably o faclae graden-based opmzaon. In mos RBTO applcaons, probablsc consrans are evaluaed by use of he frs-order relably mehod for effcency. In order o mprove he accuracy of he relably calculaons for RBDO or RBTO problems wh hgh nonlneary, we nroduce a new sngle-loop RBDO scheme ulzng he second-order relably mehod and mplemen o he proposed SRBTO algorhm. Moreover, n order o overcome challenges n applyng he proposed algorhm o compuaonally demandng opology opmzaon problems, we ulze he mulresoluon opology opmzaon MTOP) T. H. Nguyen J. Song G. H. Paulno B) Dep. of Cvl and Envronmenal Engneerng, Unv. of Illnos, Urbana, IL 61801, USA e-mal: paulno@llnos.edu T. H. Nguyen e-mal: nguyen3@llnos.edu J. Song e-mal: junho@llnos.edu mehod, whch acheves compuaonal effcency n opology opmzaon by assgnng dfferen levels of resoluons o hree meshes represenng fne elemen analyss, desgn varables and maeral densy dsrbuon respecvely. The paper provdes numercal examples of woand hree-dmensonal opology opmzaon problems o demonsrae he proposed SRBTO algorhm and s applcaons. The opmal opologes from deermnsc, componen and sysem RBTOs are compared wh one anoher o nvesgae he mpac of opmzaon schemes on fnal opologes. Mone Carlo smulaons are also performed o verfy he accuracy of he falure probables compued by he proposed approach. Keywords Frs-order relably mehod Mulresoluon opology opmzaon Relably-based desgn opmzaon Second-order relably mehod Sngle-loop approach Sysem relably 1 Inroducon Topology opmzaon ams o fnd an opmal srucural layou under gven consrans hrough erave compuaonal smulaons. In he pas decades, a large number of sudes have been devoed o hs mporan research area of srucural opmzaon Bendsøe and Sgmund 2003). Topology opmzaon mehods have been successfully appled o a wde range of praccal engneerng problems Rozvany 2001; Bendsøe and Sgmund 2003). However, mos of he effors have been conduced n a deermnsc manner alhough unceranes n loads or maeral properes may resul n sgnfcan lkelhood of volang desgn consrans. In he curren sudy, hs approach s referred o as deermnsc opology opmzaon DTO). Recenly,

2 594 T.H. Nguyen e al. acve research has been performed o acheve opmal opologes wh accepable probably of sasfyng gven consrans. Ths approach s ofen ermed as relablybased opology opmzaon RBTO), and has been successfully appled o a varey of opology opmzaon problems Bae e al. 2002; Maue and Frangopol 2003; Allen e al. 2004; Jung and Cho 2004; Kangeal.2004; Kharmanda e al. 2004; Km e al. 2006; Gues and Igusa 2008; Rozvany 2008; Lógó e al. 2009; Luo e al. 2009;Cheneal. 2010). For example, Maue and Frangopol 2003)employed RBTO n he desgn of complan mcro-elecromechancal sysem mechansm MEMS). Jung and Cho 2004) appled RBTO o geomercally nonlnear srucures wh unceran loads and maeral properes. Addonally, Rozvany 2008) derved he analycal soluon for benchmark problems n probablsc opology opmzaon. Mos research effors on RBTO have been focused on sasfyng he probablsc consran gven for each falure mode. In hs paper, hs approach s referred o as componen relably-based opology opmzaon CRBTO). A generc formulaon for CRBTO problems s gven as follows mn f ρψ; d)) d s.. P[g ρψ; d), X) 0] P Kρψ; d)) u d = f d L d d U, = 1,..., n 1) where d R k s he vecor of deermnsc desgn varables; ρψ; d) s he maeral densy a he poson ψ R 2 or R 3 ha s generally deermned by a projecon funcon f p ) and he desgn varables,.e. ρψ; d) = f p d); f ) s he objecve funcon ha ofen descrbes he volume, complance, or dsplacemen of he srucure; X R m s he vecor of random varables represenng he unceranes n he problem; g ), = 1,...,n s he -h lm-sae funcon ha ndcaes volang a desgn consran gven n erms of volume, dsplacemen, or complance by s negave sgn,.e. g ) 0; P s he consran on he probably of he -h lm sae; K, u d and f respecvely denoe he sffness marx, dsplacemen vecor and load vecor n he equlbrum condon; and d L and d U are he lower and upper bounds on d, respecvely. For smplcy, he equlbrum condon and he bounds on he desgn varables wll be omed n he followng RBTO formulaons of he paper. The probably consran n 1) s descrbed n erms of eher he relably ndex RIA: Enevoldsen and Sørensen 1994) or he performance funcon,.e. he P - quanle of he lm-sae funcon PMA: Tu e al. 1999), whch s obaned by use of a srucural relably analyss mehod such as he frs-order relably mehod FORM). Whle mos research effors n he leraure have been focused on CRBTO, n ceran crcumsances, he probablsc consran should be gven on a sysem falure even,.e. a logcal or Boolean) funcon of mulple falure modes. For example, he falure of a opology desgn can be defned as an even ha a leas one of he poenal falure modes occurs. Ths s ermed as sysem relably-based opology opmzaon SRBTO). SRBTO nroduces addonal complexy o relably calculaons especally when componen evens are sascally dependen, or when he sysem even s no a seres.e. unon of evens) or parallel sysem.e. nersecon of evens). A generc formulaon forsrbtosasfollows. mn f ρψ; d)) d [ ] s.. PE sys ) = P g ρψ; d), X) 0 Psys, 2) k C k where PE sys ) s he probably of he sysem falure even; C k s he ndex se of he componens lm-saes) n he k-h cu-se; and Psys s he consran on he sysem falure probably. Any ype of sysem even may be consdered n SRBTO bu, for llusraon purpose, 2) shows a cuse sysem formulaon ha can descrbe seres, parallel, and cu-se sysems. A lmed number of sudes have been performed on SRBTO because calculaon of sysem probably and s parameer sensves nroduces addonal complexy o he opology opmzaon ha already requres hgh compuaonal cos. Recenly, SRBTO has been consdered for cases n whch all componen evens are sascally ndependen of each oher Slva e al. 2010). In hs case, he sysem falure probably and s parameer sensves can be obaned by algebrac calculaons of he componen probables and sensves. However, he lm-saes of SRBTO problems ofen show srong sascal dependence because of shared or correlaed random varables. In anoher recen research, SRBTO was appled o dscree srucures ha usually requre less compuaonal cos han connuum opology opmzaon Mogam e al. 2006). However, he dscree approach or so-called sze approach) canno change he srucural opology durng he soluon process, so he soluon wll have he same opology as he nal desgn Eschenauer and Olhoff 2001) whereas connuum opology opmzaon can opmze sze, shape and connecvy of he srucure. As an effor o overcome mpedmens o adopng SRBTO echnques n curren desgn pracce, hs sudy focuses on developng new SRBTO algorhms for connuum lnear elasc srucures ha can consder sascal dependence beween componen evens lm-saes). Frs, we nroduce an SRBTO procedure usng a marxbased sysem relably MSR) mehod Song and Kang 2009; Kang e al. 2011) o handle he sascal depen-

3 Sngle-loop sysem relably-based opology opmzaon 595 dence beween he lm-saes. The MSR mehod allows for accurae and effcen calculaon of sysem falure probably and s parameer sensves for general sysem problems ncludng seres, parallel, cu-se and lnk-se sysems. Second, we develop a new sngle-loop algorhm o mprove he accuracy of FORM-based RBTO by use of he second-order relably mehod SORM). Fnally, a recenly developed mulresoluon opology opmzaon MTOP; Nguyen 2010; Nguyen e al. 2010a) s negraed wh he SRBTO algorhm o enhance effcency n compuaonally demandng opology opmzaon problems. Ths approach uses hree dsnc meshes wh dfferen resoluons for fne elemens, densy and desgn varables n order o acheve hgh resoluon opmal desgns wh sgnfcanly reduced compuaonal coss. The remanng of hs paper s srucured as follows: Secon 2 descrbes he sngle-loop sysem relably-based opology opmzaon usng he marx-based sysem relably mehod; Secon 3 provdes a sngle-loop algorhm o enhance componen and sysem relably-based opology opmzaon by use of he second-order relably mehod; Secon 4 presens he mulresoluon opology opmzaon approach; Secon 5 provdes numercal examples of SRBTO; and fnally Secon 6 provdes summary and conclusons of he paper. 2 Sysem relably-based opology opmzaon usng marx-based sysem relably mehod In hs secon, we presen a sngle-loop formulaon for sysem relably-based opology opmzaon ha can accoun for sascal dependence beween lm-saes for general sysem falure evens. Afer a bref revew on exsng sngle-loop approaches for componen and sysem RBTO and mehods o accoun for sascal dependence, he new SRBTO formulaon usng marx-based sysem relably MSR) mehod s nroduced. 2.1 Sngle-loop componen and sysem relably-based opology opmzaon For CRBTO and SRBTO shown n 1) and2) respecvely, a nesed or double-loop approach has been ofen used, n whch each sep of he eraons for desgn opmzaon nvolves anoher loop of eraons for relably analyss. However, hs double-loop compuaon can be prohbve f he compuaonal cos for evaluang lm-sae funcons) durng he nner-loop search for he mos probable pon MPP) or desgn pon s expensve Yang e al. 2005). There have been acve research effors o overcome hs compuaonal challenge by decouplng he relably analyss and he desgn opmzaon loops Wu and Wang 1998; Royse e al. 2001; DuandChen2004; Lang e al. 2007, 2008; Shan and Wang 2008). For example, a sngle-loop approach Lang e al. 2007, 2008) replaces he nner-loop calculaons by an approxmae soluon obaned by he Karush Kuhn Tucker KKT) opmaly condon. As a resul, he double-loop opmzaon problem s convered no an equvalen sngle-loop problem. Ths sngle-loop approach was repored o have he accuracy comparable wh he double-loop approach and he effcency almos equvalen o ha of deermnsc opmzaon Lang e al. 2008). In hs sudy, we ulze hs sngle-loop approach for he RBTO formulaons. For he CRBTO problem n 1), he sngle-loop formulaon s gven as follows. mn d s.. f ρψ; d)) g P = g ρψ; d), xu )) 0, = 1,..., n where u = β ˆα )T ) ˆα = xg ρψ; d), xu))j x,u x g ρψ; d), xu))j x,u where g P g ); β u=ũ 3) s he P -quanle of he -h lm-sae funcon = 1 P ) s he arge generalzed) relably ndex where 1 ) denoes he nverse cumulave dsrbuon funcon CDF) of he sandard normal dsrbuon; J x,u s he Jacoban marx of he ransformaon from he sandard normal space o he orgnal random varable space,.e. x = xu); ˆα s he negave normalzed graden row) vecor of he -h lm-sae funcon evaluaed a he approxmae locaon for he performance funcon value ũ. Insead of searchng for he exac MPP a each sep of he desgn eraons, he sngle-loop approach obans an approxmae locaon for he performance funcon value ũ by solvng he sysem equaon gven by he KKT condon Lang e al. 2008). Then, he negave normalzed graden s scaled by he arge relably ndex β o deermne he locaon where g P s approxmaely evaluaed,.e. u. Smlarly, he SRBTO problem n 2) can be solved by a sngle-loop approach as follows. mn d,p f ρψ; d)) s.. g P = g ρψ; d), xu )) 0, = 1,..., n PE sys ; P ) = P g ρψ; d), X) 0 P k C k where u = β ˆα )T ) ˆα = xg ρψ; d), xu))j x,u x g ρψ; d), xu))j x,u sys u=ũ 4)

4 596 T.H. Nguyen e al. where P s he vecor of he arge falure probables, P, = 1,...,n. Noe ha n 4), he arge falure probables are reaed as desgn varables raher han predefned consran values as n 3). Ths s o conrol he sysem falure probably PE sys ) ndrecly n he sngleloop approach by conrollng he rad β = 1 P ), = 1,...,n of he spheres on whch he approxmae locaon for he performance funcon values are found. 2.2 Sysem relably-based opology opmzaon under sascal dependence When he lm-saes n an SRBTO problem are assumed o be sascally ndependen Slva e al. 2010), he sysem probably can be compued by algebrac calculaons of he probables of he ndvdual lm-saes because he probably of any nersecon can be compued by he produc of he ndvdual componen probables. However, f here exss sgnfcan sascal dependence beween lmsaes due o shared or correlaed random varables, one needs o use sysem relably analyss mehods ha can accoun for he dependence. In addon, he parameer sensves of he sysem falure probably would faclae he use of graden-based opmzaon algorhm. However, compuaon of he parameer sensves of a sysem falure probably s challengng when componen evens are sascally dependen or he sysem even s no a seres or parallel sysem. The auhors have recenly appled he marx-based sysem relably MSR) mehod o general relably-based desgn opmzaon problems Nguyen e al. 2010b). The curren sudy ams o use he MSR mehod for SRBTO. The MSR mehod Song and Kang 2009) compues he probably of a general sysem ncludng seres, parallel, cu-se and lnk-se sysem and s parameer sensves by sysemac marx calculaons. Consder a sysem even whose -h componen, = 1,...,n has wo dsnc saes, e.g. falure or survval. Then, he sample space can be subdvded no N = 2 n muually exclusve and collecvely exhausve MECE) evens, denoed by e j, j = 1,.., N. Then, any sysem even can be represened by an even vecor c whose j-h elemen s 1 f e j belongs o he sysem even and 0 oherwse. Le p j = Pe j ), j = 1,.., N denoe he probably of e j. Due o he e j s muual exclusveness, he probably of any general sysem even E sys,.e.pe sys ) s compued as he sum of he probables of e j s ha belong o he sysem even. Therefore, he sysem probably s compued by he nner produc of he wo vecors, ha s c T ps) f S s)ds dependen componens PE sys ) = s c T p ndependen componens 5) where p s he probably vecor ha conans p j s, j = 1,.., N; S denoes he random varables denfed as he sources of sascal dependence beween componens, ermed as common source random varables CSRVs). For a gven oucome of CSRVs, he componen evens are condonally ndependen of each oher, whch allows us o use he effcen procedure o consruc he probably vecor ha s applcable o ndependen componens Song and Kang 2009); ps) denoes he probably vecor consruced by use of he condonal falure probables of he lmsaes gven S = s,.e.p s) PE S = s) nsead of P PE );and f S s) s jon probably densy funcon PDF) of S. Marx-based procedures have been developed o consruc he vecors c and p effcenly; o compue condonal probables and componen mporance measures; and o evaluae parameer sensves of he sysem falure probably. The deals of hese procedures and mers of he mehod are summarzed n Song and Kang 2009). The mehod has been furher developed and successfully appled o varous sysem relably problems Kang e al. 2008, 2011; Song and Ok 2010; Lee e al. 2011). When CSRVs are no clearly shown as n Kang e al. 2008), one can denfy he source of sascal dependence beween lm-saes from he resuls of he componen relably analyses. For example, when he frs-order relably mehod FORM) s used for he componen relably analyses, he componen evens are descrbed as Z β, = 1,...,n, where Z and β respecvely denoe he sandard normal random varable and he relably ndex obaned by FORM. If Z, = 1,...,n, follow he generalzed Dunne Sobel DS) class correlaon model Dunne andsobel1955; SongandKang 2009), hey are represened n he form: 0.5 m m Z = 1 rk) 2 Y + r k S k, = 1,...,n 6) k=1 k=1 n whch Y, = 1,...,n and S k, = 1,...,m are uncorrelaed sandard normal random varables; and r k s are he coeffcens of he generalzed DS model ha deermne he correlaon coeffcen beween Z and Z j as ρ j = m k=1 r k.r jk ) for = j. Noe Z and Z j are condonally ndependen of each oher gven he oucome of CSRVs S k, = 1,...,m. Thus, he condonal probably of he -h componen even gven S = s s derved as P s) = PZ β s) = β k=1 m r ks k ) 7) 1 k=1 m r k 2 If a gven correlaon marx canno be descrbed exacly by a generalzed DS class, one can oban a generalzed DS

5 Sngle-loop sysem relably-based opology opmzaon 597 model wh he mnmum fng error for an approxmae denfcaon of CSRVs Kang e al. 2011). 2.3 Sngle-loop SRBTO algorhm usng MSR mehod The sngle-loop SRBTO usng he MSR mehod s formulaed as follows. mn d,p f ρψ; d)) s.. g P = g ρψ; d), xu )) 0, = 1,..., n PE sys ; P c T p s) f S s)ds Psys dependen ) = s c T p Psys ndependen where u =β ˆα )T ) ˆα = xg ρψ;d), xu))j x,u x g ρψ;d), xu))j x,u u=ũ 8) When lm-saes are sascally dependen, he sysem falure probably s defned as a funcon of desgn varables n P by consrucng ps) usngp s) n7) wh β replaced by β = 1 P ). For a case wh sascally ndependen lm-saes, he probably vecor p s consruced by use of P = β ).In8), we denoe he probably vecors by p and p s) o ndcae ha he probably vecors are consruced by use of β nsead of β. Inherng he mers of he MSR mehod, he proposed SRBTO/MSR approach can evaluae he probably of a general sysem even effcenly and accuraely wh sascal dependence consdered. Ths helps reduce he rsk of havng under- or over-conservave opmal desgns caused by naccurae sysem relably calculaons Nguyen e al. 2010b). The MSR mehod provdes he parameer sensves of PE sys ) wh respec o desgn varables so as o faclae he use of graden-based opmzaon algorhms for SRBTO. From 5), he sensvy of he sysem falure probably wh respec o a parameer θ can be compued as follows. PE sys ) c T ps) f S s)ds dependen = s θ 9) θ ndependen c T p θ Song and Kang 2009) developed an effcen marx procedure o consruc p/ θ and ps)/ θ from he parameer sensves of componen probables P / θ and P s)/ θ, respecvely. For example, one can oban componen-level parameer sensves usng he FORM Bjerager and Krenk 1989). Heren we derve he sensvy of P s) wh respec o he desgn varables n he proposed sngle-loop SRBTO, P, = 1,...,n so as o consruc p/ θ and ps)/ θ n 9) usng he aforemenoned marx procedure n Song and Kang 2009). The sensvy of P s) wh respec o P s derved as P s) P = P s) β β P = P s) β 1 ϕ β ) 10) n whch ϕ ) denoes he PDF of he sandard normal dsrbuon; and from 7), he sensvy wh respec o he arge relably ndex s derved as P s) β 1 = ϕ β m k=1 r ks k ) 11) 1 k=1 m r k 2 1 k=1 m r k 2 I s noed ha he paral dervave of P s) wh respec o d s zero wh P fxed). Therefore, for he consran PE sys ) Psys, s no necessary o evaluae he sensvy of PE sys ) wh respec o d. Nex, he sensves of g P = g ρψ; d), xu )) wh respec o he desgn varables are derved as follows. Frs, he sensves wh respec o desgn varables d are evaluaed as g ρψ; d), xu )) = d ρ g ρψ; d), xu )) ρ ρψ; d) d 12) where g ρ, x)/ ρ s compued for he gven lm-sae defnon, e.g. volume, complance and dsplacemen. For example, he adjon mehod Bendsøe and Sgmund 2003) may faclae he sensvy calculaon; and ρψ; d)/ d s obaned from he gven projecon funcon Nguyen e al. 2010a). The sensvy of g P wh respec o P s derved as g P P = [ u g ρψ; d), xu)) ] u u=u P = [ ] x g ρψ; d), xu)) J x,u u=u β P ˆα )T = 1 [ ϕ β ) x g ρψ; d), xu)) J x,u ]u=u ˆα )T 13) Noe ha hs paral dervave s approxmae because ˆα s assumed o be nsensve o he changes n P durng he desgn eraons.

6 598 T.H. Nguyen e al. 3 Improvng accuracy of componen and sysem relably-based opology opmzaon Ths secon nroduces new sngle-loop approaches o mprove he accuracy of relably calculaons n componen and sysem RBTO problems wh hghly nonlnear lm-sae funcons. 3.1 Accuracy n FORM-based relably-based desgn and opology opmzaon As shown n 2), 4) and8), he sysem relably analyss durng an SRBTO employs he resuls from he componen relably analyses on he gven lm-saes. Therefore, he accuracy of SRBTOs n sasfyng he probablsc consran on he sysem even,.e. PE sys ) Psys, depends on ha of he componen relably analyses. The naccuracy of he FORM-based relably-based desgn and opology opmzaon has been repored n he leraure Mogam e al. 2006; Royseeal.2006; McDonald and Mahadevan 2008; RahmanandWe2008; Slva e al. 2010; Lee e al. 2010). Some sudes have been conduced o mprove he accuracy. For example, Royse e al. 2006) employed he frs-order approxmaon for falure probably and hen used hgher-order relably approxmaons or Mone Carlo smulaons o adjus parameers o mprove he accuracy of sysem relably-based desgn opmzaon. Lee e al. 2010) proposed o use he MPP-based dmenson reducon mehod Xu and Rahman 2005) nhe SRBDO framework. Mos of he sngle-loop SRBDO and SRBTO approaches Lang e al. 2007; McDonald and Mahadevan 2008; Nguyen e al. 2010b; Slva e al. 2010) also employ he FORM for componen probably analyses, whch poenally resuls n unconservave or non-opmal soluons when he lm-sae funcons are hghly nonlnear. For example, f a lm-sae funcon s defned n erm of he complance under unceran loads, he funcon s a quadrac funcon of he random varables represenng he uncerany n he loads. Therefore, he lnear approxmaon by FORM may cause sgnfcan errors n componen relably analyses, and hus also n sysem relably calculaons. As an effor o apply he sngle-loop approach o a wde range of opology opmzaon problems, we propose a mehod o enhance he accuracy of he falure probables calculaed durng he sngle-loop componen and sysem RBTO. 3.2 Sngle-loop componen relably desgn and opology opmzaon wh mproved accuracy Frs, le us consder he sngle-loop CRBTO approach n 3). A each sep of he desgn eraons, he approxmae locaon for he performance funcon value u s obaned by scalng he negave normalzed graden vecor ˆα evaluaed a he pon obaned from he KKT condon, u = ũ by he arge relably ndex β,.e.u = β ˆα )T.The valdy of hs approxmae locaon for he performance funcon value u s checked a he fnal sep of he desgn eraons. We modfy hs procedure o mprove he accuracy of he sngle-loop approach. Insead of fndng he approxmae locaon for he performance funcon value on he surface of he sphere wh he fxed radus β,he radus s updaed a each sep of he desgn eraons by he rao of β o he relably ndex mproved based on he curvaures a he approxmae locaon for he performance funcon value u of he prevous sep. The formulaon of he proposed scheme s as follows. mn d s.. f ρψ; d)) g P = g ρψ; d), xu )) 0, = 1,..., n a he k-h sep: u = β k) ˆα )T β β k) k = 1 = β oherwse ˆα = β k 1)SORM) β k 1) xg ρψ; d), xu))j x,u x g ρψ; d), xu))j x,u ) u=ũ 14) where β k) s he arge relably ndex used o fnd he approxmae locaon for he performance funcon value a he k-h sep of he eraons; and β k 1)SORM) s he relably ndex of he k 1)-h sep whch was mproved based on he curvaures of he lm-sae funcon a he approxmae locaon for he performance funcon value as follows ) β k 1)SORM) = 1 P k 1)SORM) P k 1)SORM) = β k 1) ) m ) j=1 1 + κ j β k 1) n whch P k 1)SORM) denoes he falure probably esmaed by use of he approxmae locaon for he performance funcon value u = u a he k 1)-h sep by he concep of he second-order relably mehod SORM; Breung 1984; Der Kureghan 2005); and κ j j = 1,...,m 1) denoe he prncpal curvaures around he approxmae locaon for he performance funcon value a he k 1)-h sep, u = u. Fgure 1 shows he acual MPP u of he gven desgn and he approxmae locaon for he performance funcon value u where he mproved relably ndex β k 1)SORM) s compued usng Breung s

7 Sngle-loop sysem relably-based opology opmzaon 599 β G = G p G= 0 u =βαˆ u ~ ˆα u where P SORM) s he vecor of he componen falure probables by Breung s formula 15) a he approxmae locaon for he performance funcon values, u, = 1,..., n; p SORM) s) andp SORM) denoe he probably vecor consruced by use of he Breung s formula relably ndexes β SORM) nsead of β ;andũ s obaned by use of he KKT condon usng β. The only change from 8) s ha he probably vecor s consruced by use of he SORM-based relably ndexes nsead of he FORM relably ndexes a he approxmae locaon for he performance funcon values. 4 Mulresoluon Topology Opmzaon MTOP) Fg. 1 Acual MPP u ) and approxmae locaon for he performance funcon value u ) formula n 15). The convergence of he radus β k) ndcaes ha he SORM-based relably ndex approaches he arge relably ndex β. Compared o smlar echnques n he leraure Royse e al. 2006; Rahman and We 2008; Lee e al. 2010), our sudy focuses on mplemenaonofsormnohesngle-loop RBTO. In hs sudy, he mproved CRBTO by SORM n 14) sermedas SORMbased CRBTO. I should be noed ha oher relably analyss mehods han SORM, such as mporance samplng mehod or dmenson reducon mehod can be used for he updang rule n he proposed approach f necessary. 3.3 Sngle-loop sysem relably-based desgn and opology opmzaon wh mproved accuracy The sngle-loop SRBTO/MSR n 8) s also mproved by enhancng he accuracy of componen relably analyss resuls ha are used for sysem relably analyses. The formulaon of he SORM-based SRBTO/MSR s as follows mn d,p s.. f ρψ; d)) g P = g ρψ; d), xu )) 0 = 1,..., n P E sys ; P SORM)) c T p SORM) s) f S s)ds Psys dependen = s c T p SORM) P ndependen sys where u = β. ˆα )T ) ˆα = xg ρψ; d), xu))j x,u x g ρψ; d), xu))j x,u u=ũ 16) A man challenge n performng RBTOs for realsc problems s he hgh compuaonal cos whch s nhered from deermnsc opology opmzaon. The maeral dsrbuon mehod Bendsøe 1989) s ofen used n opology opmzaon. Ths mehod raserzes he doman va he densy of pxels/voxels, and hus ofen requres a large number of desgn varables, especally n hree-dmensonal applcaons. Mos of he research effors o overcome hs challenge focused on fne elemen analyss ha consues he domnan compuaonal cos n opology opmzaon. For example, researchers make use of powerful compung resources such as parallel compung Borrvall and Peersson 2001; Evgrafov e al. 2008), approxmaon procedure Amr e al. 2009), or fas erave solvers Wang e al. 2007; Amr e al. 2010). These sudes employ he same level of resoluons for fne elemen mesh and he desgn mesh durng opmzaon process. In order o oban hgh resoluon opology desgns wh a relavely low compuaonal cos, we hereby propose o employ a recenly developed mulresoluon opology opmzaon approach Nguyen e al. 2010a) for SRBTO problems. In hs secon, he MTOP approach s nroduced and furher developed o nclude paern symmery and paern repeon consrans. 4.1 MTOP formulaon To llusrae he MTOP approach, le us consder a mnmum complance opology opmzaon problem: mn f ρψ; d)) = Cρψ; d), u d ) = f T u d d s.. V ρψ; d)) = ρψ; d)dv V s 17) where Cρ, u) = f T u d s he complance of he connuum; V ρ) s he oal volume; and V s s he prescrbed volume consran. A desrable soluon of opology opmzaon

8 600 T.H. Nguyen e al. specfes he densy a every pon n he doman as eher 0 vod) or 1 sold). However, snce s mpraccal o perform such an neger opmzaon, he problem s relaxed such ha he densy can have any value beween 0 and 1. For example, n he Sold Isoropc Maeral wh Penalzaon SIMP) approach Bendsøe 1989; Rozvany e al. 1992), he consuve marx s parameerzed usng sold maeral densy as follows Dψ) = ρψ; d) p D 0 18) where D 0 s he consuve marx of he maeral n he sold phase, correspondng o he densy ρψ; d) = 1; and p s he penalzaon parameer. To preven sngulary of he sffness marx, a small posve lower bound, e.g. ρ mn = 10 3, s placed on he densy. Usng he penalzaon parameer p > 1, he nermedae densy approaches eher 0 vod) or 1 sold). In he convenonal elemenbased approach, he densy of each elemen s represened by one value ρ e. In hs case, he global sffness marx K n 1) s expressed as N el N el K = K e ρ e ) = B T Dρ e )B d 19) e e=1 e=1 where K e ρ e ) s he sffness marx of he elemen e; B s he sran-dsplacemen marx of shape funcon dervaves; e denoes he doman of he elemen e; anddρ e ) s he consuve marx n he elemen deermned by he densy ρ e. Dfferen from he convenonal approaches ha use he same mesh for fne elemen analyss and desgn Fg. 2a), he MTOP approach ulzes hree dfferen meshes: a relavely coarse f ne elemen FE) mesh o perform he analyss, a fne desgn varable mesh o perform he op- Dsplacemen a Densy Desgn varable Fg. 2 Elemen-based and MTOP elemens: a Q4/U; and b MTOP Q4/n25/d25 b mzaon, and a fne densy mesh o represen maeral dsrbuon and compue he sffness marces. The densy mesh s fner han he fne elemen mesh so ha each fne elemen consss of a number of densy elemens subelemens). Whn each densy elemen, he maeral densy s assumed o be unform. For example, Fg. 2a showsa convenonal elemen-based approach Q4/U elemen whle Fg. 2b shows an MTOP Q4/n25/d25 elemen where n25 and d25 respecvely ndcae ha he number of densy elemens and desgn varable per a Q4 elemen s 25. The MTOP approach needs a scheme o oban he elemen sffness marx from correspondng densy elemens and desgn varables. The sffness marx s compued as he summaon of he negraon of he sffness negrand over each densy elemen, whch has unform densy. As a resul, he formulaon for he sffness marx negraon s expressed as follows. K e = B T DB d = e = n ρ ) p =1 I = B T DBJ dξdη ) B T D 0 BJdA 0 = n ρ ) p I =1 B T D 0 BJdA 0 20) where ξ and η denoe he nrnsc coordnaes n he nerval [ 1,1]; J s he Jacoban; A 0 s he area/volume of each densy elemen n he reference doman 0 ;andρ s he densy n he -h densy elemen. The soluon of he opmzaon problem n 17) by a graden-based opmzer would requre he compuaon of sensves of objecve funcon and consran. The sensves of he complance and he volume wh respec o desgn varables are derved as follows. C = d n ρ V = d n ρ C ρ ρ d n = ρ u T K ρ u ρ d n V ρ ρ d n 21) where he sensvy K/ ρ can be derved from 20) and 21) Bendsøe and Sgmund 2003; Nguyen e al. 2010a). The sensves of he densy wh respec o desgn varables depend on he defnon of he maeral densy funcon. The MTOP approach ulzes a projecon mehod Gues e al. 2004) o compue he densy of each densy elemen from he desgn varables va a projecon funcon f p.). The projecon mehod also provdes he mesh ndependence and mnmum lengh scale for he opology desgn Nguyen e al. 2010a). For example, f a lnear projecon mehod s employed, he unform densy of a densy

9 Sngle-loop sysem relably-based opology opmzaon 601 elemen, ρ s compued as he weghed average of he desgn varables n he neghborhood,.e. n S ρ = d n w r n ), n S w r n ) r mn r n f r n r mn where w r n ) = r mn 0 oherwse 22) where d n denoes he n-h desgn varable; S s he subdoman correspondng o he -h densy elemen; and r n s he dsance from he pon assocaed wh desgn varable d n o he cenrod of he -h densy elemen,.e. r n = ψ n ψ n n whch ψ n and ψ n are he coordnaes of he pon assocaed wh desgn varable d n and S, respecvely. Here s assumed ha he change of maeral densy occurs over he physcal radus r mn, whch s ndependen of mesh. Usng he projecon funcon wh a mnmum lengh scale, he mesh ndependen soluon s obaned. In hs sudy, he mehod of movng asympoes MMA; Svanberg 1987) s used as he graden-based opmzer. The MTOP approach s used n all numercal examples of he relably-based opology opmzaon n hs paper. Varous wo- and hree-dmensonal problems demonsraed ha he MTOP approach can acheve hgh resoluon opmal opologes wh relavely low compuaonal cos n comparson o he convenonal elemen-based approach Nguyen 2010, Nguyen e al. 2010a). The approach can promoe hgh-resoluon opology opmzaon n varous problems ncludng bomedcal problems, e.g. opmal desgn of cranofacal segmenal bone replacemens Suradhar e al. 2010). 4.2 Paern symmery and paern repeon n MTOP The opology opmzaon approach s usually appled o concep desgn of srucures. Due o some praccal desgn consrans or demands, hese srucures may requre paern symmery and/or paern repeon n he desgn. For example, paern symmery and repeon have been successfully ncorporaed no he opology opmzaon of funconally graded maeral n wo-dmensonal srucures Almeda e al. 2010). In hs sudy, we mplemen paern symmery and repeon condons no he framework of he mulresoluon opology opmzaon. Because he desgn varables are separaed from he analyss model n he MTOP framework, we can choose a basc se of desgn varables and map o he whole doman o sasfy he paern symmery and/or paern repeon condon. Fgure 3 llusraes he mappng schemes o gan paern symmery and repeon n he opmal desgn. paern symmery paern repeon Fg. 3 Desgn varables mappng for paern symmery and paern repeon 5 Numercal examples In hs secon, he proposed SRBTO/MSR procedure and he SORM-based mprovemen on CRBTO and SRBTO/MSR are demonsraed by numercal examples. In all numercal examples, MTOP s used for compuaonal effcency. Frs, a wo-dmensonal brdge example demonsraes he mpac of sascal dependence beween he lm-saes n SRBTO, whch can be aken no accoun by he SRBTO/MSR approach. Second, a hree-dmensonal cube example shows he mprovemen n he accuracy of he SORM-based RBTOs over he radonal FORMbased RBTOs. Thrd, a hree-dmensonal buldng example demonsraes ha he SORM-based SRBTO approach can be appled o compuaonally demandng opology opmzaon problems wh paern repeon scheme by use of he MTOP approach. For smplcy, all he quanes are gven dmensonless. 5.1 Two-dmensonal brdge Consder a wo-dmensonal brdge desgn n a doman of and hckness of 0.05 as shown n Fg. 4. The objecve of he opmal desgn s o mnmze he volume of he srucure under consrans on he dsplacemens a seleced locaons. The soropc maeral s assumed o have Young s modulus E 0 of and Posson s rao ν of 0.3. The mnmum lengh scale r mn = 1.25, and penalzaon parameer p = 3 are employed. These maeral properes are hereby assumed o be deermnsc snce he unceranes n maeral properes usually have mnmal mpacs on relably-based opmal opologes for a srucure under lnear elasc behavor. Sochasc loads are appled a nne locaons on a non-desgnable layer wh hckness of wo) 250 non-desgnable layer F 1 F 2 F 3 F 4 Fg. 4 Confguraon of wo-dmensonal brdge example F 5 F 4 F 3 F 2 F 1

10 602 T.H. Nguyen e al. Fg. 5 The resuls of wo-dmensonal brdge example: a DTO μ F = 10 5, volfrac = 39.07%); b FORM-based CRBTO μ F = 10 5, volfrac = 48.64%); c FORM-based SRBTO/MSR μ F = 10 5, volfrac = 47.70%); and d FORM-based SRBTO/MSR μ F = , volfrac = 16.66%) a he boom of he brdge as shown n Fg. 4. A symmerc loadng condon s assumed, so he nne loads are modeled by use of fve random varables. Each of he fve random varables s assumed o follow a Gaussan dsrbuon wh he mean μ F ) of 100,000 and coeffcen of varaon rao of he sandard devaons o he means) of 1/6. All he fve random varables are assumed o be uncorrelaed. The consrans on he dsplacemens a he locaons of he appled forces are descrbed by he lm-sae funcons g ρ, F) = d 0 d ρ, F), = 1,.., 5 23) where ρ denoes he vecor of he elemen denses; F s he vecor of he fve random varables represenng appled forces; d ρ, F) s he vercal dsplacemen a he -h locaon predced by a fne elemen analyss; and d 0 s he lm on he dsplacemen. In hs example, he dsplacemen lms are gven as {d 0} =1,..,5 = {1.25, 1.50, 1.75, 2.00, 2.25}. Because of he symmery condons, only a half of he doman s aken no he analyss model wh MTOP elemens Q4/n9/d9). Frs, a deermnsc opology opmzaon DTO) s performed wh he loads equal o he gven mean values. Ths s performed by 1) excep ha he probablsc consrans are replaced by deermnsc ones,.e. g ρ, F) 0. The correspondng opmal desgn s shown n Fg. 5a. The volume fracon volfrac) of he opmal desgn,.e. he rao of he opmal volume o ha of he orgnal doman s 39.07%. Nex, a FORM-based CRBTO s conduced as n 3) wh all he relably ndex arges β = 2orP = ). The opmal opology shown n Fg. 5b has he volume fracon of 48.64%. Ths opmal volume s hgher han ha by he DTO snce he opology ha avods he falure under he mean loads s expeced o have sgnfcanly hgher probably o volae he consrans han he gven arge falure probably. Afer he CRBTO opmzaon s compleed, he probably ha a leas one of he consrans s volaed.e. seres sysem) s esmaed by he MSR mehod as P sys = Nex, a FORM-based SRBTO/MSR s performed for he seres sysem even wh he arge sysem falure probably P sys = , whch was chosen o be he same as he sysem falure probably of he opmal opology by he CRBTO n order o compare he opmal opologes by CRBTO and SRBTO ha have he same sysem falure probably. The SRBTO opmal opology, whch s dfferen from hose by DTO and CRBTO, sshownnfg.5c volume fracon of 47.70%). Anoher SRBTO s performed wh he mean values of he random loads reduced o 25% Fg. 5d) o nvesgae he mpacs of he load nensy on he opmal opology. Table 1 shows he componen and sysem falure probables by Mone Carlo smulaons MCS) for he opmal desgns by he CRBTO and SRBTO n order o verfy he accuracy of he FORM-based RBTO procedures n hs example. The resuls confrm ha he FORM-based RBTO desgns provde falure probables ha are compable wh he arge probables on componen evens CRBTO) and sysem even SRBTO). Ths s because he lm-sae funcons n hs example are lnear funcon of he unceran loads and he random varables are assumed o follow Gaussan dsrbuons. Thus, he mprovemen schemes proposed n Secon 3 s no needed. Table 1 Two-dmensonal brdge example: verfcaon of falure probables of CRBTO and SRBTO desgns by MCS 10 6 mes, c.o.v = 0.005) CRBTO MCS on CRBTO desgn SRBTO/MSR MCS on SRBTO desgn P P P P P P sys

11 Sngle-loop sysem relably-based opology opmzaon 603 The effcency of MTOP over he convenonal elemenbased approach s nvesgaed by usng he SRBTO problem above. To oban a smlar level of resoluon usng he elemen-based approach, s necessary o use Q4/U elemens. Afer 50 eraons, he compuer run me of he elemen-based approach s abou wo mes more han he MTOP Q4/n9/d9 approach. The relave effcency s furher ncreased as we am a a hgher level of resoluon. More deals on comparson of compuaonal coss are found n Nguyen 2010; Nguyen e al. 2010a). In hs example, he volume fracon from SRBTO 47.70%) s farly close o ha of he CRBTO 48.64%) ha gves he same sysem falure probably. Ths mgh gve an mpresson ha s no necessary o perform SRBTO consderng he addonal compuaons for he sysem falure probably. However, SRBTO s sll preferred for RBTO problems when probablsc consran s gven on he sysem falure even for he followng reasons. Frs of all, he probablsc consrans on ndvdual lm-saes ha would sasfy he gven consran on he sysem falure probably are no known a pror. In hs numercal example, we chose he consran on he sysem falure probably n SRBTO as he sysem falure probably of he resul of he CRBTO jus for comparson purpose. Second, n usng CRBTO formulaon for solvng SRBTO problems, all he componen arge falure probables are ofen gven equal manly because he acual componen falure probables of an opmal desgn ha would sasfy he sysem consran are no known. Inroducng such unform arge componen falure probables ofen makes he SRBTO problems more consraned han necessary, whch may lead o non-opmal soluons Nguyen e al. 2010b). Fnally, n SRBTO, one can denfy he relave conrbuon of each lm-sae o he sysem probably based on componens probables of he opmal desgn or by use of he componen mporance measures by he MSR mehod Song and Kang 2009; Nguyen e al. 2010b). Accordng o he componen falure probables of he opmal desgns, he mporance rankng of he lm-saes s as follows: 4 mos mporan) leas mporan). The effecs of he mean values, coeffcen of varaons, and he correlaons beween random varables F s on he opmal opologes are also nvesgaed. For smplcy, all he loads are assumed o have he same mean values μ F ), coeffcens of varaon c.o.v), and correlaon coeffcens ρ j ). The SRBTO problem s solved agan wh he same arge sysem probably of whle he mean values, coeffcens of varaon and correlaon coeffcens are vared.frs,fg.6aand b show ha he ncrease n mean values from o ) and coeffcens of varaon from 0.01 o 0.50) resuls n he ncrease n volume fracons of he opmal opologes. Nex, he mpacs of changes n he correlaon coeffcens from 0.00 o a opmal volume fracon b opmal volume fracon c opmal volume fracon sascal dependence consdered sascal dependence gnored c.o.v = 1/6 ρ j = mean µ F ) 10 5 µ F = 10 5 ρ j = coeffcen of varaon c.o.v.) µ F = 10 5 c.o.v = 1/ coeffcen of varaon ρ j ) Fg. 6 Impac on FORM-based SRBTO resuls volume fracon) by changes n a mean values c.o.v = 1/6, ρ j = 0.0); b coeffcens of varaon μ F = 100,000, ρ j = 0.0); and c correlaon coeffcens μ F = 100,000, c.o.v = 1/6) of he load random varables 0.90) of he load random varables are shown n Fg. 6c. I s seen ha posve correlaon among he random loads resuls n hgher volume fracons,.e. more conservave desgn. Ths s because posvely correlaed loads ncrease he dsplacemens, and hus he falure probables. Therefore, n hs problem, f he posve correlaon s gnored, he RBTO may lead o an unsafe desgn. Also presened n each plo are he resuls wh he sascal dependence beween lm-saes gnored,.e. each jon probably can be compued by he produc of he componen probables Nguyen 2010; Slva e al. 2010). As shown n Fg. 6, desgns become more conservave han necessary when sascal dependence s gnored. Ths s because he falure

12 604 T.H. Nguyen e al. probably of a seres sysem s overesmaed when sascal dependence s gnored. Fgure 6b and c show ha he effec of sascal dependence on he opmal desgns ncreases as he coeffcens of varaon or he correlaon coeffcens of he random loads ncrease. 5.2 Three-dmensonal cube Ths numercal example s o demonsrae he mproved accuracy of he proposed SORM-based RBTO mehods. The objecve of opmzaon s o mnmze he volume n a cube doman shown n Fg. 7 whle sasfyng deermnsc or probablsc consrans on he complances for mulple load cases. One corner s fxed n all hree drecons whle he oher corners are resrced n he vercal drecon only. The soropc maeral s assumed o have Young s modulus of E 0 =1,000 and Posson s rao of ν =0.3. A cube wh edge lengh L = 24 s dvded no B8/n125/d125 MTOP elemens wh a oal of 216,000 densy elemens. The mnmum lengh scale r mn = L/10, and penalzaon parameer p = 3 are employed. The srucure s subjeced o hree random loads appled a fve locaons as shown n Fg. 7. F 1 denoes he magnude of he force a he cener whle F 2 and F 3 represen he loads a he mdpons beween he cener and he four corner pons of he op face. F 1,F 2 and F 3 are assumed o be normal random varables wh he mean values 100, 0 and 0, and wh he sandard devaons 10, 30 and 40, respecvely. Lm-saes are defned on he complances caused by wo load combnaons F 1 = F 1, F 2 ) and F 2 = F 1, F 3 ) as follows. g ρ, F ) = C C ρ, F ) = C u T F, = 1, 2 24) where C = 120) s he hreshold value on he complance; C ρ, F ) s he complance correspondng o he load case Fg. 8 Opmal opologes by a DTO volfrac = 6.3%); b SORMbased CRBTO σf 1 ) = 10, volfrac = 24.4%); c SORM-based SRBTO σf 1 ) = 10, volfrac = 22.3%); and d SORM-based SRBTO σf 1 ) = 20, volfrac = 23.9%) F ;andf s he global force vecor assembled based on he load case F. The followng hree opology opmzaon problems are nvesgaed: 1) Deermnsc Topology Opmzaon DTO) usng he mean values of he loads wh deermnsc consrans g ρ, F ) 0; 2) CRBTO wh probably consrans P1 = P 2 = ,.e. relably ndexes β 1 = β 2 = 2.0; and 3) SRBTO wh he sysem lm-sae E sys ={g 1 ρ, F 1 ) 0) g 2 ρ, F 2 ) 0)} wh L F 3 F F 3 2 F 1 F 2 F 1 L L volume fracon DTO CRBTO SRBTO eraon Fg. 7 Three-dmensonal opology opmzaon of a cube Fg. 9 Convergence hsores of opology opmzaons of a hreedmensonal cube

13 Sngle-loop sysem relably-based opology opmzaon 605 = , whch s gven so as o mach he sysem falure probably of he opmal opology of he CRBTO. Fgure 8 shows he opmal opologes by DTO Fg. 8a), SORM-based CRBTO Fg. 8b), and SORM-based SRBTO Fg. 8c). The volume fracon of DTO s lower han CRBTO and SRBTO because he rsk of hgh complance caused by he load unceranes s gnored. I s noeworhy ha, wh he same sysem falure probables, he volume fracon of CRBTO s 10% hgher han SRBTO. Ths s because CRBTO approach assgnng fxed consrans on ndvdual componens) s generally more consraned Psys han SRBTOs assgnng a consran on sysem even, no on he ndvdual componens) a he same level of sysem falure probably Nguyen e al. 2010b). Fgure 8d shows he resul of he SRBTO wh he sandard devaon of F 1 ncreased o 20 n order o see he mpac of he load varably on he opmal opology. In summary, s seen from Fg. 8 ha he opmal opology s affeced sgnfcanly by he load varably and he falure even defnons on he opmal opology of a srucure. The convergence hsores of he opmzaons are shown n Fg. 9. The proposed sngle-loop SORM-based CRBTO a opmal volume fracon 0.7 FORM-based CRBTO SORM-based CRBTO b componen probably MCS on FORM-based MCS on SORM-based consran on P 1 C 1 ) sandard devaon σ F 1 ) sandard devaon σ F 1 ) c componen probably MCS on FORM-based MCS on SORM-based consran on P 2 C 2 ) d relably ndex 2.15 σ F 1 )= 50 β k)sorm β 1 k) β sandard devaon σ F 1 ) eraon k) e 2.15 σ F 1 )= 50 β 2 relably ndex β 2 k)sorm β 2 k) eraon k) Fg. 10 CRBTOs wh sandard devaon of load F 1 : a volume fracon of opmal desgns; b falure probables of he frs lm-sae; c falure probables of he second lm-sae; d relably ndex convergence for he frs lm-sae wh sandard devaon σ F 1 ) = 50; and e relably ndex convergence for he second lm-sae wh sandard devaon σ F 1 ) = 50

14 606 T.H. Nguyen e al. and SRBTO show smlar raes of convergence, whch are also comparable o ha of DTO. The sysem falure probably of he opmal opology found by SORM-based SRBTO/MSR, P sys = s verfed by a farly close esmae by MCS, P sys = mes, c.o.v = 0.005). In order o demonsrae he mproved accuracy of he SORM-based sngle-loop CRBTO mehod, he resuls are compared wh hose by he FORM-based CRBTO wh he componen probably arges P1 = P 2 = Fgure 10a shows he dfference n he volume fracons of he opmal desgns. Mone Carlo smulaons MCS: 10 6 mes, c.o.v = 0.005) are performed o fnd he componen falure probables of he opmal opologes by he FORMbased and SORM-based CRBTOs. The resuls n Fg. 10b and c show ha he componen probables of SORMbased CRBTOs are farly close o he arge probables whle he FORM-based CRBTOs show sgnfcan errors especally when he random loads have large varably. In addon, Fg. 9d and e show he convergence hsores of he relably ndexes of he frs and second lm saes, respecvely, for he sandard devaon load F 1, σf 1 ) = 50. I s observed ha he relably ndex mproved by curvaures a he approxmae locaon for he performance funcon value β k)sorm converges o he componen arge relabl- a b sysem probably opmal volume fracon FORM-based SRBTO SORM-based SRBTO sandard devaon σ F 1 ) MCS on FORM-based MCS on SORM-based consran on P sys sandard devaon σ F 1 ) Fg. 11 SRBTOs wh sandard devaon of load F 1 : a volume fracons of opmal desgns; and b sysem falure probables P q q/2 L L a L L P 5L P 2,q 2 P 1,q 1 Load case 2 4 symmery axes P 2,q 2 P 2,q 2 P 3,q 3 Load case 1 P1,q 1 Load case 3 L/12 10L/12 P 3,q 3 P 3,q 3 b P 1,q 1 θ P 1,q 1 P 2,q 2 P 3,q 3 L/12 Fg. 12 Buldng core example: a doman of he opology opmzaon; and b load cases y ndex afer a few desgn eraons, whch allows for more accurae assessmen of he falure probably assessmen of he desgn n achevng he arge relably. The accuracy of he SORM-based sngle-loop SRBTO mehod s also nvesgaed. The FORM-based and SORMbased SRBTO are performed wh he sysem probably arge of whle sandard devaon of load F 1 s vared from 10 o 60. Fgure 11a compares he volume fracons by he FORM and SORM-based SRBTOs. I s seen ha he FORM-based SRBTO provdes unconservave desgns due o he naccuracy n relably calculaons. The resuls of Mone Carlo smulaons MCS: 10 6 mes; c.o.v = 0.005) n Fg. 11b show ha he proposed SORM-based SRBTO provdes mproved accuracy n predcng he sysem falure probably. In general, he compuaonal cos for he SORM-based approach s more expensve han he FORM-based approach snce addonal cos s requred for calculang he curvaures around he approxmae locaon Table 2 Three-dmensonal buldng example: sascal parameers of he load random varables and consran on he complances Load cases P qa op) C Mean c.o.v Mean c.o.v Case Case Case

15 Sngle-loop sysem relably-based opology opmzaon 607 Fg. 13 Buldng core opmal opologes hree-dmensonal and sde vews): a DTO volfrac = 21.93%; b SRBTO volfrac = 28.15% P sys = 0.05); andc SRBTO volfrac = 22.25% Psys = 0.85) for he performance funcon value Breung 1984; Der Kureghan 2005); however, he ncrease of compuaonal me n he numercal examples was moderae whle he SORM-based approach allows for sgnfcan mprovemen n accuracy for hghly nonlnear relably problems. The resuls n Fgs. 10a and11a show he volume fracons of he opmal desgns ncrease sgnfcanly as he load varably ncreases. I s because he varably of random load ncreases he uncerany of he complance and hus he probably of volang gven consrans. 5.3 Three-dmensonal buldng The proposed SRBTO/MSR mehod and he MTOP approach allow for sysem relably-based opmzaon for large-scale srucural opologes. In hs example, he SORM-based SRBTO employng he MTOP approach s appled o desgn he srucural opology of a buldng core subjeced o horzonal loads. The objecve of he opmzaon s o mnmze he volume under he consran on sysem falure even defned n erms of he complances for mulple load cases. Fgure 12a shows he doman of he opology wh he dmensons of L L 5L L/12 n whch L/12 represens he hckness of he core L = 24). The doman s dvded no B8/n125/d125 MTOP elemens, whch resuls n a oal of 2,640 brck elemens and 330,000 densy elemens. The four corners of he doman are non-desgnable regons whch are shown as black areas n Fg. 12b. Young s modulus E 0 of 10 6, Posson s rao ν of 0.3, he mnmum lengh scale r mn = L/10, and penalzaon parameer p = 4are employed. In hs example, he buldng core s desgned wh four symmerc axes: x, y and wo dagonal drecons dash-dolnes n Fg.12b). We consder hree load cases as shownnfg.12b. In he frs load case, he unceran pon loads P 1 ) and he unceran dsrbued loads lnearly varyng from q 1 /2oq 1 along he hegh as shown n Fg. 12a) are appled wh he angle of θ = 45 dagonal drecon). The second and hrd load cases have he angle of θ = 0 x drecon), and θ = 90 y drecon), respecvely. Durng he fne elemen analyses, for smplcy, he dsrbued load s convered o he equvalen pon loads appled a he fne elemen nodes along he hegh of he buldng. All sx random varables {P 1, P 2, P 3, q 1, q 2, q 3 } are assumed o Table 3 Three-dmensonal buldng example: componen and sysem probables by SRBTO/MSR and MCS 10 6 mes) P 1 P 2 P 3 P sys Case I ρ same = 0.50 ρ dff = 0.25 SRBTO/MSR MCS c.o.v = 0.005) Case II ρ same = 0.50 ρ dff = 0.25 SRBTO/MSR MCS c.o.v = 0.001) Case III ρ same = 0.90 ρ dff = 0.45 SRBTO/MSR MCS c.o.v = 0.004) The changes from he defaul case are shown n bold

16 608 T.H. Nguyen e al. opmal volume fracon SRBTO ρsame =0.50, ρdff =0.25) DTO arge sysem falure probably Fg. 14 Opmal volume fracons wh arge sysem falure probably Nex, we vary he sysem probably arge P sys from 0.01 o 0.85 Case II). Fgure 14 shows he volume fracons of he opmal desgns for he range. I s seen ha he decrease of he arge probably.e. more conservasm) ncreases he volume fracons of he opmal desgns. The volume fracon of he SRBTO converges o ha of DTO as he arge probably ncreases. For example, he arge sysem probably of 0.85 resuls n he volume fracon of 22.25%, whch s only 1.4% dfferen from DTO 21.93%). Even hough hese wo opmal volume fracons are farly close o each oher, s noeworhy ha he opmal opology of SRBTO P sys = 0.85) n Fg. 13c s dfferen from ha of DTO n Fg. 13a. The opology opmzaon problem s solved agan usng paern repeon consrans along he vercal drecon. Ths ype of paern repeon consran s ncluded for boh DTO and SRRBTO n hs numercal example. The number of paern repeons along he vercal drecon denoed by m) s vared from 1 o 12 o nvesgae he mpac of hese consrans on he opmal opologes. The opmal opologes by DTO and SRBTO wh Psys = 0.05) are shown n Fg. 15a and b, respecvely. Fgure 15 demonsraes sgnfcan mpacs of he paern repeon follow normal dsrbuons. Table 2 provdes he means and he coeffcens of varaon c.o.v) of he random varables and he correspondng consrans gven on he complances of he sysem. These load random varables are assumed o be correlaed wh correlaon coeffcen ρ same = 0.50 when hey belong o he same load case and he correlaon coeffcen ρ dff = 0.25 for he loads from dfferen load cases. Frs of all, he opmzaon problem s solved whou paern repeon consrans Case I). The deermnsc opology opmzaon s performed usng he mean values of he loads Fg. 13a) and he SORM-based SRBTO s conduced wh he arge sysem probably P sys = 0.05 on he seres sysem even of complance lm-saes deermned for he hree load cases Fg. 13b). The DTO volfrac 21.93%) and SRBTO volfrac 28.15%) resuled n sgnfcanly dfferen opologes. The hgher volume fracon n he SRBTO opology mples he mporance of consderng he unceranes n he loads for buldng srucures. The componen and sysem probables of he opmal opologes by SRBTO/MSR and MCS 10 6 mes, c.o.v = 0.005) are shown n Table 3, whch confrms he accuracy of he SORM-based SRBTO. The componen probables of , , and help denfy he relave mporance rankng of he hree consrans as m=3 m=6 m=10 m=12 a m=3 m=6 m=10 m=12 b Fg. 15 Buldng core opmal opologes wh paern repeon: a DTO; and b SRBTO ρ same = 0.50, ρ dff = 0.25, P sys = 0.05)