STRESS CONSTRAINED THERMO-ELASTIC TOPOLOGY OPTIMIZATION WITH VARYING TEMPERATURE FIELDS VIA AUGMENTED TOPOLOGICAL SENSITIVITY BASED LEVEL-SET

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1 SRESS CONSRAINED HERMO-ELASIC OPOLOGY OPIMIZAION WIH VARYING EMPERAURE FIELDS VIA AUGMENED OPOLOGICAL SENSIIVIY BASED LEVEL-SE Shguang Dng, Gradua Sudn Krshnan Sursh, Profssor, Mchancal Engnrng, Unvrsy of Wsconsn, Madson, USA ABSRAC * Engnrng srucurs mus ofn b dsgnd o rss rmally nducd srsss. Sgnfcan progrss has bn mad on dsgn of such srucurs rough rmo-lasc opology opmzaon. Howvr, a compuaonally ffcn framwork o handl srss-consrand larg-scal problms s lackng. h man conrbuon of s papr s o addrss s lmaon. In parcular, a unfd opologcal-snsvy (S) basd lvl-s approach s prsnd n s papr for opmzng rmo-lasc srucurs subjc o non-unform mpraurs. h S flds for varous rmo-lasc objcvs ar drvd, and, o addrss mulpl consrans, an augmnd Lagrangan mod s dvlopd o xplor Paro opologs. Numrcal xampls dmonsra capably of proposd framwork o solv larg-scal dsgn problms. Comparson s mad bwn pur lasc problms, and s rmo-lasc counrpar, shddng lgh on nflunc of rmo-lasc couplng on opmzd opologs. Undr cagory of homognzaon, a ponrng mod (Rodrgus and Frnads ) was dvlopd o mnmz srucural complanc subjc o a volum consran whr D rmo-lasc srucurs wr rprsnd by mcro-vod modls. hrmal gradn was shown o hav a sgnfcanly dffrn mpac on fnal opologs compard o unform mpraur fld. Sgmund and orquao (Sgmund and orquao ) dsgnd composs w xrmal rmal xpanson coffcns by usng a r-phas opology opmzaon mod whch was proposd o fnd dsrbuon of ach maral phas by opmzng rmo-lasc proprs subjc o consrans of lasc symmry and volum consran. Jog (Jog ) xndd rmo-lasc complanc funcon o a nonlnar cas, whch was n mnmzd by usng a dnsy basd lnar pnalzaon mod and a prmr consran for rgularzaon.. INRODUCION Engnrng srucurs mus ofn b dsgnd o rss rmally and mchancally nducd srsss. As an llusrav xampl, consdr suppor srucur for a combusonxhaus sysm of a hgh Mach suprsonc arplan n Fgur. Durng flgh, cold ar s suckd no nl, mxd w ful and gnd wn combuson chambrs, w mpraur clmbng up o C, n rmxd w cool ar and xplld. h suppor srucurs ar rfor subjc o sgnfcan rmal gradns, gvng rs o rmally nducd srsss. In addon, suppor srucurs ar also subjc o mchancal loads from arodynamc ffcs, xhaus flow and prssurs from adjonng arfram srucurs. h opology opmzaon of such suppor srucurs s man focus of s papr. opology opmzaon has rapdly volvd from an acadmc xrcs no an xcng dscpln w numrous ndusral applcaons (Eschnaur and Olhoff ; Rozvany ). Applcaons nclud opmzaon of arcraf componns (Alonso ; Ksslr ), spaccraf moduls (Covrson-Carroll ), auomobls componns (Wang ), cas componns (Harzhm ), complan mchansms (Ananasursh al. ; Bruns and ororll ; Luo ; Nshwak ), c. Popular mods for rmo-lasc opology opmzaon nclud: homognzaon, Sold Isoropc Maral w Pnalzaon (SIMP), Raonal Approxmaon of Maral Proprs (RAMP), voluonary srucural opmzaon (ESO) and lvl-s. Fgur : (a) Aro-gas urbn ngn ; (b) combusonxhaus sysm ; (c) cross-scon vw and mpraur dsrbuon of suppor srucur; (d) opmzd suppor srucur o wdraw rmal srss Ovr las wo dcads, SIMP has volvd no on of mos popular opology opmzaon mods, du o s smplcy and gnraly, w applcaons rangng from fluds, srucural mchancs, and mul-physcs opmzaon (Sgmund ). Many ponrng rsarchrs hav appld SIMP o solv rmo-lasc dsgn problms as wll. In (L and Zhang ), srucural sran nrgy was mnmzd afr consdrng rmo-mchancal couplng whr snsvy was calculad va an adjon approach and opmzaon was solvd usng mod of movng asympos. Daon and Grandh (Daon and Grandh ) prsnd a dsgn scnaro for rsrand n shll srucur n a homognous Fgur crd: hps://arcurus.fls.wordprss.com///sdd_fsc_.jpg Fgur crd: hp://

2 rmal nvronmn whr radonal dsgn approach by accommodang rmal xpanson o lmna rmal srss was mpossbl. SIMP was s up and was shown a ypcal complanc mnmzaon n prsnc of rmal loadng dd no guaran favorabl dsgns. Yang and L (Yang and L ) mnmzd srucural dynamc complanc a rsonanc frquncs n a rmal nvronmn, ladng o concluson a fnal opologs wr srongly affcd by xcd mods and load locaons. In (Lu al. ), Lu compard wo rmo-lasc O formulaons: volum consrand complanc mnmzaon and wgh mnmzaon w dsplacmn consran whr nflunc of dffrn SIMP pnaly facors on rmal and mchancal flds wr sudd. Zhang (Zhang al. ) nvsgad dffrnc bwn wo mnmzaon objcvs n rmo-lasc O formulaons: man complanc and lasc sran nrgy rough snsvy analyss. A concp of load snsvy was nroducd o nrpr quanavly nflunc of rmal and mchancal loads on fnal opologs. Chn (Chn al. ) prsnd a unfd O algorm for mul-funconal D fn prodc srucurs whr snsvy a corrspondng locaons a dffrn componns ar rgulad o manan srucural prodcy. Mulpl objcvs wr smulanously opmzd rough a wghd avrag mod whr rmo-lasc couplng was also consdrd. Pdrsn (Pdrsn and Pdrsn a) qusond usag of rmo-lasc complanc as an objcv, and suggsd an alrna formulaon of mnmzng maxmum von Mss srss. Alough soluons provdd a sound orcal foundaon, on of challngs w SIMP s a maral nrpolaon xhbs zro slop a zro dnsy, ladng o parasc ffcs n rmo-lasc problms (Gao and Zhang ; Solp and Svanbrg ). By mployng a slghly dffrn maral nrpolaon schm, RAMP (Solp and Svanbrg ) was rpord o succssfully ovrcom s dfcncy w s supror prformanc ovr SIMP dmonsrad n (Gao and Zhang ). Gao and Zhang (Gao al. ) proposd o pnalz rmal srss coffcn, whch s produc of rmal xpanson coffcn and Young s modulus, and RAMP was shown o b advanagous ovr SIMP. Pdrsn and Pdrsn (Pdrsn and Pdrsn ) compard SIMP, RAMP and an alrna wo paramr nrpolaon schm whr nflunc of nrpolaon on complanc snsvy analyss was sudd and snsvy of local von Mss srss was drvd for problms w a unform mpraur lvaon. Subsanal progrss has also bn mad n ESO (Munk al. ) whr lmns ar gradually rmovd from dsgn doman basd on r rlav sgnfcanc ordr, whl a BESO (Huang and X ) addrsss som of lmaons of ESO by prmng nsron of lmns. For rmolasc O, ESO was sn as on of arls approachs usd for solvng such dsgn problms. L adopd ESO o solv fully srssd rmo-lasc opology dsgn problms and lar xndd for rmal dsplacmn mnmzaon (L al. ). In (L al. ) ESO was ulzd o achv a mul-crron dsgn for srucurs n rmal nvronmn whr maral usag ffcncy was masurd by rmal srss lvls and ha flux dnsy. L (Qng L ) dvlopd an ESO procdur for dsgn cass w unform, non-unform and ransn mpraur flds subjc o sngl and mulpl ha loads. Rlav lmnal ffcncy dfnd n rms of rmal srss lvls was mployd o achv hghs ffcacy for maral usag. h lvl-s mod was dvlopd n (Oshr and San ) and nroducd o srucural opmzaon n (San ). Is prmary advanag ovr or O mods s a boundary s wll dfnd a all ms. In dscpln of rmo-lasc srucural dsgns, lvl-s mod was frs rpord by Xa and Wang (Xa and Wang ) whr a srucural man complanc was mnmzd w volum consrans. Snsvy analyss of connuum body was conducd w rspc o fr boundars whch wr smoond by a gomrc nrgy am durng opmzaon procss. In (Vrmaak al. ), a lvl-s basd framwork was dvlopd o sudy ffcs of ncludng maral nrfac proprs o rmo-lasc mul-phas srucurs. Fn maral nrpolaons w monoonc and nonmonoonc propry varaons wr ulzd o guaran maral proprs connu chang across nrfacs. Dng and Sursh (Dng al. ) xplod opologcal snsvy basd lvl-s mod o solv D srss consrand O problms subjc o homognous mpraur chang, whch was lar xndd o solv bucklng consrand problms n a rmal nvronmn (Dng and Sursh ). Dsp s advancs, wo rsarch gaps can b dnfd. Frs, an ffcn framwork o addrss larg-scal D rmolasc opology opmzaon problms, w mllons of dgrs of frdom, has no bn dmonsrad. Scond, mos suds ar lmd o unform mpraur scnaros. In s papr, a nw lvl-s mod s proposd o addrss s wo gaps. I combns a dscr approxmaon of opologcal snsvy w augmnd Lagrangan formulaon o solv spaally varyng mpraur problms subjc o a vary of consrans. h snsvy of rmally nducd p-norm srss s drvd, for frs m. Fnally, o addrss compuaonal challngs, assmbly-fr dflad fn lmn mod proposd n (Yadav and Sursh ) s xndd hr for ffcn larg-scal D rmo-lasc opmzaon.. ECHNICAL BACKGROUND. hrmo-elascy W rsrc our anon hr o wakly-coupld rmolasc problms whr mpraur nfluncs dsplacmns, bu no rvrs. Fn lmn analyss (FEA) of such problms ssnally rducs o posng and solvng wo lnar sysms of quaons (Hnarsk al. ). Frs, f mpraur fld s spaally varyng, on solvs rmal problm: K= q (.) hs s n followd by srucural problm: (.) s In abov wo quaons,

3 : mpraur fld d : Dsplacmn fld K : hrmal sffnss marx K : Srucural sffnss marx q : Ha flux f f s : Srucural load : hrmal load (.) h rmal load vcor n Equaon (.) s formd by assmblng load for ach fn lmn va (Daon and Grandh ): N f = ( f ) (.) whr w f Ω B D ε = Ω f = BDε dω (.) ε = α( ) Φ (.) : Nodal rmal load vcor for ach lmn : Elmn doman : Elmn sran-dsplacmn marx : Elmn lascy marx : Elmn rmal sran vcor α : hrmal xpanson coffcn : Armc avrag of nodal mpraur flds n on lmn : Rfrnc mpraur Φ : [ ] n D; [ ] n D : Each fn lmn N : oal numbr of fn lmns (.) Summng up conrbuons from all lmns, load du o rmal ffcs can b wrn as: whr H = f (.) N H= ( BDαΦ dω) (.) = Ω = (.) whr s a rfrnc mpraur vcor w sz of dgrs of frdom. h srsss ar oband by mulplyng maral nsor w dffrnc bwn oal sran and rmal sran (Daon and Grandh ): ( ) σ = D Bd ε (.) h complanc for a rmo-lasc sysm s dfnd as: J = ( f + f ) d = d Kd (.) m. hrmo-elasc opology Opmzaon Fndng a suabl opmzaon objcv n rmo-lasc O s mporan. Usng complanc as dsgn objcv could lad o a no srucur dsgn. Insad, mnmzaon of maxmum von Mss srss was argud as bng mor appropra (Pdrsn and Pdrsn b). Furr, was shown n (Zhang al. ) a rang rmo-lasc complanc and lasc sran nrgy as objcvs may lad o dffrn opologs. In s papr, objcv s o mnmz volum, subjc o bo complanc and srss consrans, avodng scnaro of no srucur dsgn. Furr, bo rmal and srucural loads ar consdrd roughou s papr. A rmo-lasc O problm can now b posd as: Mnϕ g ( d,, Ω) ; =,,..., m subjc o K= q s whr: ϕ : Objcv o b mnmzd Ω : opology o b compud (.) Ψ : Dsgn doman (.) g : Consrans m : Numbr of consrans In words, objcv s o fnd an opmal maral layou ( Ω ) wn dsgn doman ( Ψ ) such a quany of nrs ( ϕ ) s mnmzd whl consrans ( g ) ar sasfd. ypcal consrans nclud complanc, p-norm von Mss srss, bucklng safy facor, c. A spcal cas of Equaon (.) s volum mnmzaon problm w volum, complanc and srss consrans: Mn Ω Ω v J α J σ ασ subjc o f Ψ m K= q (.) As n any opmzaon problm, wll rmna f any of followng condons ar m: () volum fracon rachs v f, or () complanc rachs α ms nal valu, or () von Mss srss rachs α ms nal valu. Snc complanc and p-norm srss ar mposd as consrans, r snsvy analyss s prsnd hr. In addon, snsvy o dsplacmn was also drvd, and compard agans drvaon n (Bndsø ) for corrcnss.

4 In spcal cas whn r s no volum consran, and r s no sac loadng, opmzaon wll rmna o a vod dsgn.. PROPOSED MEHOD In ordr o solv abov problm, w addrss snsvy analyss n Scon.. hn, n Scon., w dscuss how on can drcly us snsvy flds as a lvl-s o carry ou opology opmzaon. Fnally, n Scon., w xplcly addrss consrans rough augmnd Lagrangan formulaon.. Snsvy Analyss L Q b any quany of nrs n a rmo-lasc opmzaon problm; Q can r b an objcv, or a consran. h followng quaons ar drvd o compu snsvy of Q w rspc o a opologcal chang for problms w spaally varyng mpraur flds. h snsvy of Q w rspc o a opologcal dsgn varabl x s dnod by: Q Q (.) x h drvavs of global sffnss marx wll b dnod by: K K' (.) x By assumng quany of nrs Q s a funcon of mpraur and dsplacmn d, s snsvy fld can b xprssd as: ( ) ( ) Q = Q d + Q (.) d By akng drvav of Equaon (.) w rspc o dsgn varabl x, w oban: = K K (.) - Subsung Equaon (.) no Equaon (.): Q d Q = K ( HΔ K d) - (( dq) K H+ ( Q) ) K K - ( ) (.) For as of compuaon, wo adjons λ and ω ar nroducd as follows: w Kλ = d Q (.) Kω = ( Hλ Q) (.) Equaon (.) can n b smplfd as: = + + Q λ HΔ ω K λ Kd (.) For clary, on can xprss Equaon (.) as: Q = Q Q (.) Q Q s = λ H Δ + s = λ Kd ω K (.) Obsrv a, as opology volvs, snsvy n Equaon (.) can ak r a posv or a ngav valu. Whl s non-monoonc bhavor can pos challngs for radonal monoonc approxmaon mods, can b proprly approxmad w non-monoonous approachs lk globally convrgn vrson of MMA (GCMMA) and gradnbasd MMA (GBMMA) (Bruynl and Duysnx ). In s papr, w mploy opologcal snsvy basd lvl-s mod whch s provn o b robus and ffcn, as llusrad lar rough numrcal xprmns. Furr, obsrv a adjons λ and ω dpnd on quany of nrs Q. hr spcfc nsancs of Q ar consdrd blow. K d + Kd = f + f (.) s If on assums xrnal srucural load f s s ndpndn of dsgn varabls, s snsvy can b droppd from Equaon (.): d = K ( f Kd ) (.) On or hand, snc rmal load f dpnds on dsgn varabl x, chang n rmal load du o a opologcal chang can b calculad by akng drvav of Equaon (.): f = HΔ + H ' (.) By subsung Equaon (.) and (.) no Equaon (.), w hav: ( Q) ( ' d K H Δ K d) (( dq) K H+ ( Q) ) + Q = (.) If on also assums a rmal flux n Equaon (.) s ndpndn of a opologcal chang. hn, akng drvav of Equaon (.) gvs us: Dsplacmn If quany of nrs s a dsplacmn a a pon a, n, followng noaons proposd n (Bndsø and Sgmund ): Q d a =d (.) hn, gradns dfnd n Equaon (.) and (.) can b found as: ( ) Q = d d = d ( ) Q d = = d (.) (.) In ordr o compu rm of d, w ak drvav of Equaon (.) w rspc o mpraur fld : d ( f + fs ) d = = K (.) Snc srucural load f s s ndpndn of mpraur fld, w hav:

5 d f ( H ( )) = = = K K K H (.) Subsung Equaon (.) back o Equaon (.), w hav: = Q K H (.) hn, subsung Equaon (.) and (.) o Equaon (.) and (.), wo adjons can b xprssd as: Kλ = (.) K ω = ( H λ K H ) (.) From Equaon (.) and (.), s asy o xprss adjons as: λ = K (.) ω = λ HK H K K (.) Subsung Equaon (.) and (.) back o Equaon (.) lads o: d ' = λ ( Kd HΔ ) + ω K a - = ( K ) ( HΔ K d) + (.) ' ( λ H H K )( K K ) whr by subsung Equaon (.) and (.) no Equaon (.), w hav: - d ' = ( K ) ( HΔ K d) + a ( K H+ HK )' (.) Equaon (.) s dncal w on n (Bndsø and Sgmund ). A dald comparson bwn wo can b found n Appndx. Complanc If quany of nrs s complanc, adjons ar gvn by: = = λ K f d (.) K ω= Hd Q (.) hrfor, afr subsung Equaon (.) and (.) back o Equaon (.), complanc snsvy can b smplfd o: = + J ω K d HΔ d Kd (.) P-norm von Mss Srss If quany of nrs s p-norm srss, a s: whr: / p p Q ( σ ) (.) ( σ σ ) + ( σ σ) +... σ = ( σ σ) +... ( σ σ + σ σ + σ σ ) (.) hn, adjon λ dfnd n Equaon (.) s gvn by Kλ = Q g (.) d whr g dfns rgh-hand sd of s adjon problm: p g = p ( σ ) p g (.) whr g s assmbld from lmnal vcor g whch s dfnd by: ( σ - σ )( F,: - F,: ) + p ( σ - σ)( F,: - F,: ) + g = p ( σ ) ( σ - σ)(,: -,: ) (.) F F + ( σ,: + σ,: + σ,: ) F F F F DB (.) = whr B s lmn sran-dsplacmn marx dfnd n Equaon (.); plas s (Sursh and akalloozadh ) for dals. o accoun for scond rm on rgh hand sd of Equaon (.), w nroduc anor adjon ξ by: K ξ = Q g (.) whr g dfns rgh-hand sd of s adjon quaon a and s assmbld from lmnal vcor g : W p ( ( ) ) p g = σ p a g (.) [ g, g, g, g, g, g, g, g ] g = (.) g a = p ( σ ) ( σ - σ )( G - G ) + ( σ - σ )( G - G ) + ( σ - σ)( G - G ) + ( σg + σg + σg ) p (.) whr componns G ar dfnd va: G = α DΦ (.) whr D and Φ wr dfnd n Equaon (.). Onc adjons λ and ξ ar solvd, y can b pluggd no Equaon (.) o compu adjon ω. From r, snsvy of p-norm von Mss srss can b asly oband. I should b nod a snsvy analyss for problms w unform mpraur chang s jus a spcal cas of s drvaon, whch can b asly oband by droppng mpraur varaon rm from Equaon (.).. Lvl-S Paro racng A smpl approach o xplong abov snsvy flds n opology opmzaon s o kll msh-lmns w low valus. Howvr, s wll lad o nsably. Alrnaly, snsvy fld can b usd o nroduc hols va an auxlary lvl-s (Allar al. ). In s papr, w ra snsvy fld as a lvl-s, as dscrbd nx (also s (Amsuz and Andra )). o llusra, consdr pur srucural problm llusrad n Fgur a. For xampl, on can compu lasc

6 complanc snsvy fld from Equaon (.) by sng mpraur chang o. h rsulng lasc complanc snsvy fld s llusrad n Fgur b whr fld has bn normalzd., for xampl llusras Paro-opmal opologs for a D srucural problm. Fgur : (a) a srucural problm, and (b) corrspondng opologcal snsvy fld for lasc complanc whr hgh capurs magnud of fld. In Paro algorm (Amsuz and Andra ), opologcal snsvy fld llusrad n Fgur b srvs drcly as a lvl-s. h opmzaon sars a a full dsgn doman (w no addonal hols),.., a a volum fracon of.. hn, a small ncrmnal volum sp s akn; s s nalzd o., bu s dynamcally modfd, as xpland blow. Usng opologcal lvl-s, a cung plan dfnd by a τ paramr τ ; on can dfn a nav opology Ω pr: τ Ω = {( xyz,, ) Q > τ} (.) τ In or words, doman Ω s s of all pons whr snsvy fld xcds a gvn valu τ (Sursh ). h τ valu s drmnd from ncrmnal volum sp. For xampl, an nducd doman w % volum fracon s llusrad n Fgur. Fgur : h Paro-opmal curv and opmal opologs for a D srucural problm. An mporan faur of fxd-pon raon s a allows for rnroducon of prvously dld maral wn dsgn doman. hs faur s xpland n (Krshnakumar and Sursh ), and s llusrad rough Fgur whr a canlvr bam s opmzd. I can b obsrvd a maral rmovd a on volum fracon (markd n box ) s rcovrd a a lar sp. Fgur : (a) opologcal snsvy and lvl-s, and (b) nducd (nav) opology. τ Obsrv a opology Ω s only nav; fn lmn analyss and snsvy compuaons ar rcompud ovr nw opology unl convrgnc, ladng o fxd- pon algorm dscussd n (Céa al. ), (Norao al. ), (Sursh ). hs consss of followng r sps: τ () solv fn lmn problm ovr Ω () r-compu opologcal snsvy, and () fnd a nw valu of τ for dsrd volum fracon as shown n Fgur. Onc algorm has convrgd, nx volum sp s akn. If algorm fals o convrg, volum sp sz s rducd (plas s (Céa al. ) for dals). Fgur : Fxd pon raon nvolvng r quans. hs rsuls n a srs of Paro-opmal opologs, and concp can b asly gnralzd o D (Novony ); Fgur Fgur : (a) a ypcal canlvr bam problm, (b) volvng opologs w dffrn volum fracons.. Consran Handlng W now consdr consrans. Spcfcally, consdr rmo-lasc O problm posd arlr n Equaon (.). h consrans can b combnd w objcv funcon o dfn augmnd Lagrangan (Nocdal and Wrgh ): m L( d, Ω; γ, µ ) ϕ+ L ( d, Ω; γ, µ ) (.) = whr µ g + γ ( g) µ + γg > (, ;, ) L d Ω γ µ = (.) µ / γ µ + γg whr

7 L : Augmnd Lagrangan L : Auxlary Lagrangan µ : Lagrangan mulplrs γ : Pnaly paramrs (.) By usng snsvy dfnon n Equaon (.), gradn of augmnd Lagrangan s gvn by: whr m L = ϕ + L (.) =. h complanc s now compud ovr nw opology. If complanc has convrgd, n opmzaon movs o nx sp, ls gos o sp.. h currn volum fracon s s o ( v v). If arg volum fracon has no bn rachd, opmzaon rurns o sp o rpa raons; ls, rmna raon and x.. Sp-sz s rducd; chck f volum dcrmn s blow rshold. If no, opmzaon rurns o Sp-; ls, rmna raon. ( µ γ g ) g µ γ g L + + > = µ + γg (.) h snsvy of objcv and ach of consrans can b compud usng Equaon (.). h Lagrangan mulplrs and pnaly paramrs ar nalzd o an arbrary s of posv valus. hn augmnd Lagrangan s mnmzd usng, for xampl, conjuga gradn mod. In vry raon, Lagrangan mulplrs ar updad as follows: µ + = max{ µ + γ g ( xˆ ),}, =,,,..., m (.) k k k whr xˆ k s local mnmum a currn k raon. h pnaly paramrs ar updad va: γ γ k k+ k k + mn(,) mn(,) = k k+ k ηγ k g > ς g g max(, ) mn(,) mn(,) ς g (.) whr < ς < and η >, ς =. and η = (Nocdal and Wrgh ). Radrs ar rfrrd o (Dng and Sursh ) for dals.. Proposd Algorm Fnally, proposd algorm procds as follows (s Fgur ):. h doman s dscrzd usng fn lmns (hr D hxahdral lmns). h opmzaon sars a a volum fracon of.. h volum dcrmn v s s o.. h nal valus of Lagrangan mulplr and pnaly numbr ar s as and.. h rmal problm (f ncssary) and srucural problm n Equaons (.) and (.) ar solvd.. h consran valus ar calculad, and opmzaon paramrs (mulplr and pnaly) ar updad.. If any of consrans s volad, algorm procds o sp-, ls, procds o sp-.. h snsvs ar calculad for ach of consrans, and augmnd lmn snsvy fld s compud.. rang augmnd snsvy fld as a lvl-s; a nw opology w a volum fracon of ( v v ) s xracd. Fgur : An ovrvw of algorm.. NUMERICAL EXPERIMENS In s Scon, w dmonsra ffcacy of proposd algorm rough numrcal xprmns. h dfaul paramrs ar as follows: h maral s assumd o b sl,.., lasc modulus s E = GPa, Posson rao s ν =., o coffcn of rmal xpansonα =. / C and conducvy coffcn s W/(m K). h rfrnc mpraur s o C. Unlss orws nod, p-norm valu usd for compung p-norm srss s. -nodd hxahdral lmns ar usd for D FEA. All xprmns wr conducd usng C++ on a Wndows -b machn w followng hardwar: Inl I CPU quad-cor runnng a.ghz w GB of mmory. h dsrd volum fracon s., unlss orws nod. In or words, opmzaon rmnas f consrans ar volad or f dsrd volum fracon of. s rachd. h numrcal xprmns ar organzd as follows. Scon. s a bnchmark xampl o sudy ffcvnss

8 of proposd mod for unformly lvad mpraur; bo complanc and srss domnad problms ar consdrd. In Scon., anor bnchmark xampl s consdrd o sudy ffc of spaally varyng mpraur. In Scon., a cas-sudy nvolvng a flang subjc o a unform mpraur ncras s consdrd. Fnally, n Scon., a cas sudy s consdrd whr srucur s subjc o mpraur gradn flds. Imporan conclusons ar drawn for ach of xampls.. Bnchmark: B-clampd bam w a pon load h am of s xprmn s wo-fold: () llusra proposd algorm for a bnchmark problm (Rodrgus and Frnads ), () show mpac of mpraur varaons on fnal opology. h srucur s llusrad n Fgur (Rodrgus and Frnads ), uns ar n mrs, load s N, cknss s.m, and srucur s also subjc o a homognous mpraur ncras, spcfd blow. Snc cknss s small, problm can b modld as plan-srss (Rodrgus and Frnads ). Howvr, s modld hr n D, and doman s mshd w, hxahdral lmns. Fgur : (a) Opmzd opology, and (b) srss dsrbuon of complanc-consrand O. In ypcal mplmnaons of SIMP, volum fracon rmans fxd, whl complanc connuously dcrass. On or hand, n Paro volum fracon dcrass as Paro curv s racd, and corrspondng complanc for dcrasng volum fracon, ncrass on can xpc. h raon hsory s llusrad n Fgur whr rlav complanc and volum fracon ar plod agans numbr of FEAs xcud. Obsrv a, a ach volum fracon, aks ~ raons o convrg o opmal dsgn, and s can b obsrvd as a sar-cas ffc n Fgur. Fgur : h b-clampd srucur w a cnral pon load. Complanc Formulaon (Sff Dsgns) W frs consdr complanc-consrand rmo-lasc O problm Mn Ω Ω. Ψ J J subjc o s : Spcfd (.) Fgur : Iraon hsory of complanc and srss for problm n Fgur. Nx, w consdr mpac of mpraur chang on fnal opology. h arg volum fracon was s o. and fnal opologs ar llusrad n Fgur for a mpraur chang rangng from o C o + o C. As on can obsrv, fnal opology s a srong funcon of mpraur chang, spcally for a posv chang. Obsrv a volum consran s oponal ( allows opmzaon o sop a a dsrd volum fracon). Also obsrv a rmal problm n Equaon (.) nd no b consdrd snc mpraur ncras s prscrbd. If mpraur ncras s C, opmzd opology for a. volum fracon s llusrad n Fgur whr fnal complanc and srss ar. and. ms r nal valus, rspcvly. h compuaonal m s sconds, nvolvng FEAs; opology s dncal o on oband n (Rodrgus and Frnads ). h fnal complanc of srucur w % volum fracon s almos wc nal complanc of srucur w % volum fracon, whl srss has no ncrasd sgnfcanly.

9 Fgur : h fnal opologs for dffrn mpraur varaons for problm n Equaon (.). Obsrv from Fgur a f mpraur varaon s posv, complanc monooncally ncrass; on or hand, f mpraur varaon s ngav, complanc frs dcrass, and n ncrass. On possbl rason s a whn mpraur dcrmn s small (.g. C ), comprssv rmal load parly cancls nsl srucural load, rsulng n a smallr complanc valu. h rlav magnuds of rmal and mchancal loads ar summarzd n abl. By comparng cass w mpraur varaons of = o C and = o C whch hav clos magnuds of rmal loads, nflunc of rmally nducd xpanson and conracon on fnal opologs can b clarly sn. abl : Load raos for dffrn mpraur varaons ( C ) f / f s Srss Formulaon (Srong Dsgns) W pos a srss domnad rmo-lasc O as follows: Mn Ω Ω. Ψ σ σ subjc o s : Spcfd (.) Fgur : h fnal opologs for dffrn mpraur varaons.. Bnchmark: Dsrbud load B-clampd bam h am of s xprmn s o sudy mpac of nonunform mpraur on fnal opology. W onc agan consdr b-clampd bam bu w a dsrbud load as llusrad n Fgur (Rodrgus and Frnads ). h dmnson of s bam s.m.m.m and dsrbud load s P = Pa. Onc agan problm s modld n D, and doman s mshd w, hxahdral lmns. Smlar o prvous xprmn, mpraur s unformly lvad by C. h rsulng opology w. volum fracon s llusrad n Fgur whr s fnal complanc and srss qual o. and. ms r nal valus, rspcvly. h compung m was sconds nvolvng FEA. h ncrasd compung m s du o addonal adjon FEA a nds o b prformd. Fgur : h b-clampd srucur w a dsrbud load. Complanc Formulaon (Sff Dsgns) h spcfc problm bng consdrd hr s: Fgur : (a) Fnal opmzd opology, and (b) srss dsrbuon of srss-consrand O. Comparng Fgur and Fgur, can b obsrvd a: complanc and srss domnad O lad o slghly dffrn opologs, and opology n Fgur has lowr complanc whl opology n Fgur and has lowr srss, as xpcd. h fnal opologs for dffrn mpraur varaons ar llusrad n Fgur. As on can obsrv, opologs ar sgnfcanly dffrn from os n Fgur. Mn Ω Ω. Ψ J J subjc o s K= q (.) If mpraur s unformly lvad by C, rsulng opology and srss dsrbuon ar llusrad n Fgur. hs s conssn w rsuls n (Rodrgus and Frnads ). h rsulng complanc and srss valus ar. and. ms corrspondng nal valus.

10 Fgur : Fnal opology (a) and srss dsrbuon (b) whn srucur n Fgur s subjc o unform mpraur rs. Nx, w consdr mpac of spaally varyng mpraur on opmal dsgns. Spcfcally, w ncrasd mpraur on lf dg by C, and on rgh dg by C,.., avrag chang n mpraur s C. For rmal conducon analyss, mpraur on lf dg s fxd a C whl mpraur on rgh dg s fxd a C.h fnal opology and s srss dsrbuon ar shown n Fgur whr asymmry s du o spaal rmal gradn. Comparson bwn Fgur and Fgur hghlghs mporanc of accounng for spaally dsrbud mpraur profls. Comparng rsuls n Fgur w Fgur, s clar: frsly, wo opmzaon problms lad o dsnc opologs; also, a sam fnal volum fracon, a complanc mnmzaon lads o a lowr complanc rsul whl a srss mnmzaon lads o a lowr srss valu.. Cas sudy: Flang h purpos of s scon s o show robusnss of proposd algorm for a non-rval applcaon. In parcular, a rmo-lasc O problm ovr a flang s sudd n s scon. Flangs ar commonly usd, for xampl, o fasn pps and ral-jons, and y ar ofn subjc o mpraur changs. h dmnsons of flang and boundary condons ar llusrad n Fgur. h flang s fxd a wo bol cnrs, and a vrcal forc of N s appld as shown. For FEA,, hxahdral lmns ar usd o dscrz dsgn doman, rsulng n, DOF. Fgur : Fnal opology (a) and srss dsrbuon (b) whn srucur s subjc o a spaally mpraur gradn w rsulng complanc and srss ar as larg as. and. ms r nal valus. Srss Formulaon (Srong Dsgns) Nx a srss domnad problm s consdrd for abov problm n Fgur : Mn Ω Ω. Ψ σ σ subjc o s K= q (.) On lf dg r was no mpraur chang,.., = C was s as rmal boundary condon on lf dg, and on rgh dg mpraur was = C. h fnal opology and srss dsrbuon rsuls ar llusrad n Fgur. hr fnal complanc and srss ar. and. ms r nal valus. Obsrv srong asymmry n srss-domnad problm. Fgur : Flang srucur and dmnsons (un: m). h spcfc rmo-lasc O problm consdrd hr s: Mn Ω Ω. Ψ J J σ.σ subjc o s (.) Frs, a pur lasc problm (.., zro rmal load n Equaon (.)) s consdrd. h rsulng opology s llusrad n Fgur and fnal consran valus ar shown n abl whr opmzaon rmnas du o acv srss consran dnfd w a "box". Fgur : Fnal opology (a) and srss dsrbuon (b) of srss-consrand O subjc o spaally mpraur gradn. Fgur : op vw and boom vw of fnal opology for pur lasc flang problm. abl : Consrans and rsuls for problm n Fgur.

11 Inal Fnal Rsuls Consrans J J J =.J σ.σ σ =.σ Fnal volum & m (sc) V =. m = (sc) Fnal load rao f = f hn, rmal ffc s addd; w mak srucur subjc o a unform mpraur lvaon of C. h opmzd opology, compud n FEAs, s llusrad n Fgur. Or rsuls ar summarzd n abl ; s problm rmnad du o an acv complanc consran. Alough rmal load s small compard o srucural load, as nod n four column of abl, s has a sgnfcan ffc on fnal opology. s Fgur : op vw and boom vw of fnal opology of flang subjc o a unform mpraur rs. abl : Consrans and rsuls for problm n Equaon (.). Inal Fnal Fnal volum Fnal load Consrans Rsuls & m (sc) rao J J J = J V =. f σ.σ = (sc) σ =.σ fs =.. Cas sudy: Exhaus sysm Nx w consdr ngn xhaus-washd srucur, usd n a low obsrvabl suprsonc arcraf; s was frs sudd by J. Daon (Daon and Grandh ). Du o low radar obsrvably rqurmn, ngn and xhaus sysm ar burd nsd arcraf. Bcaus of spac rsrcon, xhaus sysm s fxd ono arcraf skns; rmal xpanson s rfor lmd. In ordr o rduc nfrard dcably, ho xhaus gas s coold wn xhaus duc. A smplfd xhaus sysm concpon w s dmnsons (un: mr) s llusrad n Fgur whr srucur s fxd a lf and rgh nds, and fxurs. A mpraur a nak s assumd C and coold down o C a oupu nozzl. For FEA, doman s mshd w, hxahdral lmns, rsulng n,, DOF. Whl srucur s spaally fxd a fxurs, mpraur s s as C and C a lf and rgh nds, rspcvly. Fgur : Concpual xhaus sysm (op) and s dmnsons (down). h spcfc rmo-lasc O problm solvd hr s: Mn Ω J.J σ.σ subjc o s K= q (.) h fnal opology s llusrad n Fgur. Opmzaon rsuls ar summarzd n abl. On rmnaon, complanc consran s acv and maxmum p-norm srss s rducd. Fgur : Sd vw (lf) and fron vw (rgh) of opmzd xhaus abl : Consrans and rsuls for problm n Equaon (.). Inal Fnal Rsuls Fnal volum Consrans & m (h) J.J J =.J V =. σ.σ σ =.σ =.. CONCLUSIONS h man conrbuon of s papr s a comprhnsv mod for solvng consrand rmo-lasc O problms. For rmal scnaro w complx mpraur flds, snsvy o complanc and p-norm srss ar drvd. Augmnd Lagrangan mod was usd for mul-consrand

12 rmo-lasc O. h assmbly-fr FEA mod was mplmnd for acclraon. As numrcal xprmns rval, mpac of bo unform mpraur varaons and spaally rmal gradns on fnal opologs can b sgnfcan for cran problms. Fuur work wll focus on ncludng or consrans ncludng gn-mods and larg-dformaon bucklng n rmolasc O analyss. Acknowldgmns h auors would lk o ank suppor of Naonal Scnc Foundaon rough grans CMMI-, CMMI-, and IIP-, and DOE rough ARPA-E ARID gran. Prof. Krshnan s a CEO of ScAr, LLC, whch has lcnsd Paro chnology rpord n s publcaon, rough Wsconsn Alumn Rsarch Foundaon. Appndx In ordr o prov corrcnss of snsvy analyss n s papr, dsplacmn snsvy n Equaon (.) s compard w s counrpar n (Bndsø and Sgmund ). As shown n Chapr.. of (Bndsø and Sgmund ) snsvy of a wakly-coupld rmo-lasc problm can b drvd as blow. Whr h fn lmn quaons ar gvn for wo sysms: H = f (.) Kd = f (.) = (.) h forc vcor on rgh hand sd of Equaon (.) s sum of rmal load and dsgn-ndpndn mchancal load: f = f + f (.) s If w hav an nrss n dsplacmn a a gvn pon a. Usng adjon mods, quaon can b formd as followng: d = d + ( H f ) + ( Kd f ) (.) a hn, snsvy w rspc o lmn psudo-dnsy can b shown as: d ' = ( H' f ') + ( Kd ' f') (.) a whr wo adjons ar dfnd as: K = (.) f H = = ( K ) H (.) Pluggng wo adjons no Equaon (.) and smplfyng: d a ' = ( H' f ') + ( Kd ' f') (.) = ( HK + K H )' + ( K )( H' Kd ' ) whr Equaon (.) s shown dncal o Equaon (.). Rfrncs Allar, G., Jouv, F., and oadr, A. M. (). Srucural Opmzaon usng Snsvy Analyss and a Lvl-s Mod. Journal of Compuaonal Physcs, (),. Alonso, J. J. (). Arcraf dsgn opmzaon. Mamacs and Compurs n Smulaon, (),. Amsuz, S., and Andra, H. (). A nw algorm for opology opmzaon usng a lvl-s mod. Journal of Compuaonal Physcs,,. Ananasursh, G. K., Koa, S., and Ganchandan, Y. (). A modcal approach o dsgn of complan mcromchansms. Sold Sa Snsor and Acuaor Workshop,. Bndsø, M. P. (). opology opmzaon ory, mods and applcaons. Sprngr Vrlag, Brln Hdlbrg. Bndsø, M., and Sgmund, O. (). opology Opmzaon: hory, Mods and Applcaon. Sprngr. Bruns,. E., and ororll, D. A. (). opology opmzaon of non-lnar lasc srucurs and complan mchansms. Compur Mods n Appld Mchancs and Engnrng, ( ),. Bruynl, M., and Duysnx, P. (). No on opology opmzaon of connuum srucurs ncludng slf-wgh. Srucural and Muldscplnary Opmzaon, (),. Céa, J., Garrau, S., Gullaum, P., and Masmoud, M. (). h shap and opologcal opmzaon conncon. Compur Mods n Appld Mchancs and Engnrng, (),. Chn, Y., Zhou, S., and L, Q. (). Mulobjcv opology opmzaon for fn prodc srucurs. Compurs & Srucurs, ( ),. Covrson-Carroll, V. H. (). Opmal mul-objcv low-rus spaccraf rajcors. Compu. Mods Appl. Mch. Eng.,,. Daon, J., and Grandh, R. V. (). Sffnng of hrmally Rsrand Srucurs va hrmolasc opology Opmzaon. Honolulu, Hawa. Daon, J., and Grandh, R. V. (). opology Opmzaon of hrmal Srucurs w Srss Consrans. Boson, MA. Dng, S., and Sursh, K. (). Mul-consrand opology opmzaon va opologcal snsvy. Srucural and Muldscplnary Opmzaon, (),. Dng, S., and Sursh, K. (). opology opmzaon undr lnar rmo-lasc bucklng. Charlo, NC. Dng, S., Sursh, K., and Joo, J. (). Srss-Consrand hrmo-elasc opology Opmzaon: A opologcal Snsvy Approach. ASME, Buffalo, NY, USA. Eschnaur, H. A., and Olhoff, N. (). opology opmzaon of connuum srucurs: A rvw. Appld Mchancs Rvw, (),. Gao,., Xu, P., and Zhang, W. (). opology Opmzaon of hrmo-lasc Srucurs w Mulpl

13 Marals Undr Mass Consran. Compu. Sruc., (C),. Gao,., and Zhang, W. (). opology opmzaon nvolvng rmo-lasc srss loads. Srucural and Muldscplnary Opmzaon,,. Harzhm, L. (). A rvw of opmzaon of cas pars usng opology opmzaon II-opology opmzaon w manufacurng consrans. Srucural and Muldscplnary Opmzaon,,(),. Hnarsk, R. B., Ignaczak, J., Noda, N., Sum, N., and angawa, Y. (). hory of Elascy and hrmal Srsss: Explanaons, Problms and Soluons. Sold Mchancs and Is Applcaons, Sprngr. Huang, X., and X, Y. M. (). A nw look a ESO and BESO opmzaon mods. Srucural and Muldscplnary Opmzaon, (),. Jog, C. (). Dsrbud-paramr opmzaon and opology dsgn for non-lnar rmolascy. Compur Mods n Appld Mchancs and Engnrng, (),. Ksslr, E. (). Muldscplnary dsgn analyss and mul-objcv opmsaon appld o arcraf wng. WSEAS ransacons on sysms and Conrol and Cybrncs, (),. Krshnakumar, A., and Sursh, K. (). Hng-Fr complan mchansm dsgn va opologcal Lvl- S. Journal of Mchancal Dsgn, (). L, D., and Zhang, X. (). opology Opmzaon of hrmo-mchancal Connuum Srucur. Canada. L, Q., Svn, G. P., Qurn, O. M., and X, Y. M. (). Srucural opology dsgn w mulpl rmal crra. Engnrng Compuaons, (),. L, Q., Svn, G. P., and X, Y. M. (). Dsplacmn mnmzaon of rmolasc srucurs by voluonary cknss dsgn. Compur Mods n Appld Mchancs and Engnrng, ( ),. Lu, X., Wang, C., and Zhou, Y. (). opology opmzaon of rmolasc srucurs usng gudwgh mod. Scnc Chna chnologcal Scncs, (),. Luo, Z. (). Complan mchansm dsgn usng mulobjcv opology opmzaon schm of connuum srucurs. Srucural and Muldscplnary Opmzaon,,. Munk, D. J., Vo, G. A., and Svn, G. P. (). opology and shap opmzaon mods usng voluonary algorms: a rvw. Srucural and Muldscplnary Opmzaon,. Nshwak, S. (). opology Opmzaon of Complan Mchansms usng Homognzaon Mod. Inrnaonal Journal for Numrcal Mods n Engnrng,,. Nocdal, J., and Wrgh, S. (). Numrcal Opmzaon. Sprngr. Norao, J. A., Bndsø, M. P., Habr, R. B., and ororll, D. A. (). A opologcal drvav mod for opology opmzaon. Srucural and Muldscplnary Opmzaon,,. Novony, A. A. (). opologcal-shap Snsvy Mod: hory and Applcaons. Sold Mchancs and s Applcaons,,. Oshr, S., and San, J. A. (). Frons Propagang w Curvaur-dpndn Spd: Algorms Basd on Hamlon-Jacob Formulaons. J. Compu. Phys., (),. Pdrsn, P., and Pdrsn, N. L. (a). Srng opmzd dsgns of rmolasc srucurs. Srucural and Muldscplnary Opmzaon, (),. Pdrsn, P., and Pdrsn, N. L. (b). Srng opmzd dsgns of rmolasc srucurs. Srucural and Muldscplnary Opmzaon, (),. Pdrsn, P., and Pdrsn, N. L. (). Inrpolaon/pnalzaon appld for srng dsgn of D rmolasc srucurs. Srucural and Muldscplnary Opmzaon, (),. Qng L, Y. M. X., Gran P.Svn. (). hrmolasc opology opmzaon for problms w varyng mpraur flds. Journal of hrmal Srsss, (),. Rodrgus, H., and Frnads, H. (). A maral basd modl for opology opmzaon of rmolasc srucurs. Inrnaonal Journal for Numrcal Mods n Engnrng,,. Rozvany, G. I. N. (). A crcal rvw of sablshd mods of srucural opology opmzaon. Srucural and Muldscplnary Opmzaon, (),. San, J. (). Srucural boundary dsgn va lvl s and mmrsd nrfac mods. Journal of Compuaonal Physcs, (),. Sgmund, O. (). A ln opology opmzaon cod wrn n Malab. Srucural and Muldscplnary Opmzaon, (),. Sgmund, O., and orquao, S. (). Dsgn of marals w xrm rmal xpanson usng a r-phas opology opmzaon mod. Journal of Mchancs and Physcs of Sold,,. Solp, M., and Svanbrg, K. (). An alrnav nrpolaon schm for mnmum complanc opology opmzaon. Srucural and Muldscplnary Opmzaon,,. Sursh, K. (). A -ln Malab cod for Paro-opmal racng n opology opmzaon. Srucural and Muldscplnary Opmzaon, (),. Sursh, K. (). Effcn Gnraon of Larg-Scal Paro-Opmal opologs. Srucural and Muldscplnary Opmzaon, (),. Sursh, K., and akalloozadh, M. (). Srss- Consrand opology Opmzaon: A opologcal Lvl- S Approach. Srucural and Muldscplnary Opmzaon, (),. Vrmaak, N., Mchalds, G., Parry, G., Esvz, R., Allar, G., and Bréch, Y. (). Maral nrfac ffcs on opology opmzaon of mul-phas srucurs usng a lvl s mod. Srucural and Muldscplnary Opmzaon,.

14 Wang, L. (). Auomobl body rnforcmn by fn lmn opmzaon. Fn Elmns n Analyss and Dsgn, (),. Xa, Q., and Wang, M. Y. (). opology Opmzaon of hrmolasc Srucurs Usng Lvl S Mod. Ro d Janro, Brazl. Yadav, P., and Sursh, K. (). Larg Scal Fn Elmn Analyss Va Assmbly-Fr Dflad Conjuga Gradn. J. Compu. Inf. Sc. Eng, (), --. Yang, X., and L, Y. (). Srucural opology opmzaon on dynamc complanc a rsonanc frquncy n rmal nvronmns. Srucural and Muldscplnary Opmzaon, (),. Zhang, W., Yang, J., Xu, Y., and Gao,. (). opology opmzaon of rmolasc srucurs: man complanc mnmzaon or lasc sran nrgy mnmzaon. Srucural and Muldscplnary Opmzaon, (),.