Topological design of continuum structures subjected to forced vibration

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1 6 th Worl Congr of Structural an Multiicilinary Otimization Rio Janiro, 3 May - 3 Jun 5, Brazil Toological ign of continuum tructur ubjct to forc vibration Nil Olhoff, Jianbin Du Intitut of Mchanical Enginring, Aalborg Univrity, Aalborg, Dnmark, no@im.auc.k, j@im.auc.k. Abtract Thi ar al with th roblm of toological ign otimization of latic, continuum tructur without aming that ar ubjct to tim-harmonic, ign-innnt (or nnt) ynamic loaing with rcrib frquncy an amlitu. An imortant objctiv of uch a ign roblm i oftn to riv th ignfrqunci of th tructur a far away a oibl from th rcrib loaing frquncy in orr to avoi ronanc an ruc th vibration lvl of th tructur. Thi objctiv i imlmnt in thi ar by minimizing th ynamic comlianc of th tructur ubjct to th givn frquncy of th loaing, uing th volumtric niti of matrial in th finit lmnt in th amiibl ign omain a ign variabl, an alying th SIMP mol to naliz intrmiat matrial niti in th ign. Th total tructural volum, th bounary conition, an th matrial ar givn. Only loaing frqunci that ar mallr than th otimum valu of th con ignfrquncy of th corroning roblm for fr vibration, ar icu in thi ar.. Kywor: Toology otimization, ynamic comlianc, vibration lvl, loaing frquncy 3. Introuction Th mtho of toology otimization firt aar in th litratur in 988, an wa originally vlo for trmining th itribution of an latic matrial within an amiibl ign omain that ubjct to a givn tatic loaing yil th tifft oibl tructur for a rcrib amount of matrial, Bnø an Kikuchi [] an Bnø []. In comarion with uual izing an ha otimization, toology otimization imli mor ign from inc it allow nw hol an connction to b gnrat in th tructur an hrby ha th ciiv imact on th gain that can b achiv by otimization. Toology otimization i thrfor an imortant rrocing tool for izing an ha otimization ( Olhoff t al. [3]). During th lat ca, th mtho ha bn xtn to hanl vral othr ign objctiv an contraint. Toology otimization ha thrfor bcom a tanar tool for ynthi of art or whol tructur in th automotiv an aroac inutri, an it i raily raing into othr nginring ign icilin. Th rar i rfrr to th xhautiv txtbook by Bnø an Sigmun [4] an th rviw articl by Echnaur an Olhoff [5] for rcnt vlomnt an ublication. Lik th conct of comlianc in tatic tructural ign, ynamic comlianc i a ror global maur of th ynamic ron of a tructur. Problm of toology otimization with th objctiv of minimizing th ynamic comlianc (maximizing th ynamic tiffn) of tructur ubjct to tim-harmonic xtrnal loaing of givn frquncy an amlitu wr,.g., tui by Ma t al. [6] an Jog [7]. Toology ign ubjct to tranint xtrnal loaing wa tui by Min t al. [8], whr th ynamic comlianc i fin rlativ to a cifi tim intrval. A roblm that may ari in tructural toology otimization unr tim-harmonic ynamic loaing i that th tatic comlianc (corroning to th am loaing amlitu, but zro frquncy) may incra to a vry high lvl (.g. Tchrniak [9]) uring th roc whr th ynamic comlianc of th tructur i otimiz. In xtrm ca th tatic comlianc actually tn to infinity, which rflct that a iintgration of th tructur i bing crat uring th ign roc. Howvr, w hav foun that th ign objctiv of th ynamic roblm can b imlmnt along iffrnt otimization ath, an that it i oibl to avoi th roblm mntion abov by lction of th ror ath. Thu, in th rnt ar, an aroach i roo in which th tatic comlianc of th tructur i contrain or cra uring th roc of otimizing th ynamic rorti. An algorithm vlo for thi hanl th otimum ign roblm by a continuation tchniqu whr th loaing frquncy i quntially incra from a ufficintly low initial valu u to it rcrib valu. Numrical xaml ar rnt to montrat th valiity of th aroach. A an xtnion rlativ to traitional toology otimization, our tuy alo inclu ca whr th ynamic loaing i ign-nnt, i.., ca whr both th location an irction of th loaing chang a th tructural toology chang. Th mtho vlo by Hammr an Olhoff ([], []; alo Du an Olhoff [], [3]) ar mloy to hanl th ign roblm aociat with uch kin of loaing. At th n of th ar om illutrativ otimum toology rult ar rnt, both for ca of ign-innnt an ign-nnt loaing. 4. Formulation of toology ign roblm for minimization of ynamic comlianc Th roblm of otimizing th toology of a continuum tructur (without aming) for minimum valu of th intgral ynamic tructural comlianc can b formulat in a icrt form a follow:

2 min { C ρ = P T U } ( K ω M) U = P, N E = Subjct to: ρ V V < ρ ρ,, ( V = αv ), ( =, L, N Th ynamic comlianc i qual to th work on by th xtrnal ynamic forc againt corroning ilacmnt, an i roortional to th xrion givn for C in (), whr U not th ilacmnt amlitu vctor of th tay-tat, timharmonic vibration, an P not th loa magnitu vctor of th xtrnal loaing. Th harmonic xtrnal loaing vctor (t) with th iω givn xcitation frquncy ω can thn b xr a t iω t ( t) =P an th ilacmnt ron vctor a a ( t) = U. Th ymbol K an M rrnt th N imnional tructural tiffn an ma matric, whr N i th numbr of DOF. Th ymbol ρ i th volumtric matrial nity of lmnt an i th ign variabl in th roblm. Th ymbol α not th volum fraction of availabl matrial an i givn by V /V, whr V i th volum of th amiibl ign omain an V i th rcrib total volum of availabl matrial. Th ymbol N E rrnt th total numbr of finit lmnt. It i not that th ynamic tiffn K fin a K = K ω M may b ngativ finit whn th loaing frquncy ω ha a high valu,.g. highr than th funamntal ignfrquncy of th tructur. In thi ca, th rouct of th amlitu of th xtrnal loaing vctor an th ilacmnt vctor of th tructural ron may attain a ngativ valu, an in orr to inclu thi oibility in our roblm formulation, w u th abolut valu of th abov rouct a th ynamic comlianc fin in our roblm, Eq. (). Th firt quation lit in th contraint of roblm () i th ynamic tat quation an can b olv in a irct way by Gau limination or by uing a tchniqu of moal uroition. 4.. Th SIMP Matrial mol Th SIMP (Soli Iotroic Microtructur with Pnalty) introlation mol (,.g., Bnø []; Rozvany, t al. [4]; Bnø an Sigmun [5]) i u hr togthr with a filtring tchniqu (,.g., Sigmun [6]), in orr to avoi chckrboar formation an nncy of otimum toology olution on finit lmnt mh-rfinmnt. Thu, th finit lmnt laticity matrix E i xr in trm of th lmnt volumtric matrial nity ρ, ρ, an th nalization owr,, a E ( ρ ) = ρ E, () whr E i th laticity matrix of a corroning lmnt with th fully oli latic matrial th tructur i to b ma of. By analogy with (), for a vibrating tructur th finit lmnt ma matrix may b xr a q M ( ρ ) = ρ M, (3) whr M rrnt th lmnt ma matrix corroning to fully oli matrial, an th owr q. Aart from xction brifly icu in th following ction, normally q = i chon. Th global tiffn matrix K an ma matrix M for th finit lmnt ba tructural ron analy bhin th otimization can now b calculat by K = N E N E = = ). q ρ K, M = ρ M, (4) whr K i th tiffn matrix of a finit lmnt fully oli matrial, an N E not th total numbr of finit lmnt in th amiibl ign omain. 4.. Localiz vibration mo With valu aign to an q a tat abov, alication of th SIMP mol for roblm of toology otimization with rct to ynamic tructural ron for rcrib, low xcitation frqunci may la to th occurrnc of uriou, localiz vibration mo. Th localiz mo may occur in ub-rgion of th ign omain with low valu of th matrial nity (.g. ρ.), whr th ratio btwn th tiffn (with, ay, = 3 in th introlation formula) an th ma (with q = ) i vry mall. To liminat th uriou mo w hav t th ma vry low via a high valu of th nalization owr in ub-rgion with low matrial nity (Prn [7]; Tchrniak [9]). Thu, th introlation formula (3) for th finit lmnt ma matrix wa moifi a ρm, ( ρ >.) M ( ρ ) =, (5) r ρ M, ( ρ.) whr th nalization owr r i chon to b about r = 6, i.., much largr than th nalization owr for th tiffn, which i kt unchang at a valu about = Snitivity analyi Th nitivity of th objctiv function C in roblm () with rct to th ign variabl ρ i givn by T T T C = P T U = ign( P U) ( P U + P U ), (6) whr rim not artial rivativ with rct to ρ, an ign( ) i th ign function. Th nitivity P of th loa vctor will b ()

3 3 zro if it i ign-innnt, othrwi it can b hanl uing th mtho crib by Hammr an Olhoff ([], []), an alo by Du an Olhoff [], [3]. Th nitivity U of th ilacmnt vctor i givn by ( K ω M) U = f P ( K M ω ) U, (7) whr th nitiviti of th tiffn an ma matric can b irctly obtain from th SIMP matrial mol, i.. Eq. (4). Th vctor f i known a th uo loa an i fin by th trm on th right-han i of Eq. (7). Inta of olving Eq. (7), th ajoint mtho (.g. Tortorlli an Michalri [8]) may b u to calculat th nitivity of th objctiv function in a mor fficint mannr, which giv th following rult T T C ( P T U)(U P U ( K M = ign ω ) U). (8) Accoringly, th otimality conition for roblm () can b xr in th following form by man of th mtho of Lagrang multilir, T T T ign( P U)(U P U ( K ω M ) U) + ΛV =, (9) whr Λ i th Lagrang multilir corroning to th matrial volum contraint, an th i contraint for ρ hav bn ignor. Th otimization roblm () can b olv by uing th wll-known MMA mtho (Svanbrg [9]) or an otimality critrion mtho,.g. th fix oint mtho, a vi in Chng an Olhoff []. 5. Numrical xaml 5. Toology ign of a D inlt ubjct to hyroynamic rur loaing Thi xaml illutrat otimum toology ign of a ingl-matrial inlt for flui flow. Th flui flow in th channl of th initial inlt i a hown in Fig. (a), an i aum to xrt a uniform hyroynamic rur loaing of givn frquncy an amlitu on th innr wall of th inlt. Th hyroynamic rur i a ign-nnt loaing an it i trat a crib in th ar (Du an Olhoff []). Th matrial of th inlt i iotroic with Young moulu E = 7, Poion ratio υ =.3 an th cific ma γ m = (SI unit ar u throughout). Th availabl matrial volum fraction i t a 4%. Th ign objctiv i to minimiz th ynamic comlianc of th inlt. Fig. (a) how th amiibl ign omain an th initial loaing bounary. Fig. (b-) how otimiz toologi an th aociat loaing bounari for thr givn loaing frqunci ω = (tatic loaing), ω = 8 an ω =. Th loaing frqunci ar all lowr than th otimum obtainabl valu Ω ot = 38 of th funamntal ignfrquncy of th inlt. It how that whn th loaing frquncy i incra from to 8, th ha of th inlt i lightly chang whil th otimum toology of th inlt rmain th am (Fig., (c)). Whn th loaing frquncy i incra u to, both th toology an th ha of th inlt ar chang (Fig. ()). A a comarion, Fig. 3 how th otimum toology of th inlt (for th am matrial volum fraction a abov) whn it rform fr vibration at th maximum valu Ω ot = 38 of it funamntal ignfrquncy. Th funamntal ignfrquncy of th initial ign of th inlt (with uniformly itribut matrial ovr th amiibl ign omain, Fig. (a)) i 455. Figur. Otimiz toologi (4% volum fraction) an loaing bounari of D inlt for thr iffrnt loaing frqunci. (a) Amiibl ign omain, loaing an uort conition. ω =. (c) ω = 8. () ω =.

4 4 Figur 3. Otimum toology of th D inlt (for 4% volum fraction) obtain by maximizing th funamntal ignfrquncy of fr vibration of th inlt. Th maximum funamntal ignfrquncy i Ω ot = 38. By comaring th otimum toologi in Fig. an 3 w fin om intrting fatur. Thu, whn th loaing frquncy i much lowr than th otimum funamntal ignfrquncy of th tructur, th rulting toology ( Fig. (c)) obtain by th ynamic ign of th rnt ar i imilar to th tatic ign that utain th amlitu of th loaing at zro frquncy, Fig., which imli that th ynamic ign i ominat by th atial itribution of th amlitu of th xtrnal loaing vctor. Howvr, if th loaing frquncy i clor to th valu of th otimum funamntal ignfrquncy of th tructur, th ign i ominat by th ynamic rquirmnt, an riv th funamntal ignfrquncy of th tructur a far away a oibl from th rcrib loaing frquncy. For an intrmiat valu of th loaing frquncy, th otimum toology of th inlt i a kin of comromi btwn th loaing amlitu ominat ign an th ignfrquncy ominat ign (cf. Fig. (b-) an Fig. 3). 5. Minimum ynamic comlianc ign of a lat-lik tructur Thi xaml concrn otimum toology ign of a lat-lik tructur with uort conition a hown in Fig. 4(a). Th finit lmnt mol of th tructur conit of 6 (3 ) 8-no 3D brick lmnt with Wilon incomatibl ilacmnt mol to imrov rciion. A tim-harmonic, concntrat tranvr xtrnal loa (t) = Pcoωt i ali to th cntr of th lat. Th ign objctiv i to minimiz th ynamic comlianc of th lat for a rcrib loaing frquncy ω = ω = 8 an a volum fraction of 5% for th givn oli matrial, which ha th Young moulu E =, Poion ratio υ =.3 an th cific ma γ m = 78. Th funamntal ignfrquncy of th lat in th initial ign ( Fig. 4(a)) i Ω = 6.6, i.., l than th givn loaing frquncy. Minimization of th ynamic comlianc riv th ign away from th ronanc oint which imli a continual cra of Ω a hown in Fig. 4. A a rult, th tatic comlianc of th tructur incra vry quickly (Fig. 4(c)). Fig. 4() how that at itration t 3, th lat ha bcom vry wak at th two fix uort. Thi inicat cration of a rigi boy vibration mo in aociation with th firt ignfrquncy, an that th tructur cannot ffctivly utain th tatic loa aociat with ω =. Frqunci Prcrib loaing frquncy ω = 8 Firt ignfrquncy Ω (a) Itration numbr Dynamic an tatic comlianc.5 x -6 Static comlianc C (ω = ) Dynamic comlianc C for ω = ω = Itration numbr (c) () Figur 4. (a) Amiibl ign omain (a = 3, b = an c =.) with loaing an uort conition. Itration hitory for th firt ignfrquncy Ω of th lat (Ω < ω = 8). (c) Itration hitori for th ynamic an tatic tructural comlianc (th lattr corron to th am loaing amlitu but frquncy ω = ). () Matrial itribution at itration t 3.

5 5 Frqunci Firt ignfrquncy Prcrib loaing frquncy ω = ω = 8 6 Initial loaing frquncy ω Itration numbr (a) Ω Structural comlianc Dynamic comlianc 5 x Dynamic comlianc C for loaing frquncy ω = ω = 8 Static comlianc C (ω = ) Itration numbr 6 x -7 ω -- Prcrib loaing frquncy 5 4 ω = ω = 8 3 ω = 6 ω = Itration numbr (c) ω = 8 () Figur 5. (a), Itration hitori for th firt ignfrquncy of th lat, th loaing frquncy, an th ynamic an tatic comlianc. (c) Otimum toologi (5% volum fraction) for ω =8. () Itration hitori (t 5 to 3) of th ynamic comlianc of th lat ubjct to four iffrnt loaing frqunci. (a) ω = (tatic loaing) ω = 6 (c) ω = 8 () ω = Figur 6. Otimiz toologi of th lat-lik tructur for four iffrnt loaing frqunci. In orr to avoi uch a tatically wak ign, a mor xint aroach i aot for olution of th abov roblm. Thu, w tart out th ign roblm with a valu ω = ω of th loaing frquncy that i lowr than th firt ignfrquncy Ω of th initial ign, an w thn quntially incra ω u to it originally rcrib valu ω = ω = 8 (Fig. 5(a)). In th convrg rult, a tructur with minimiz ynamic comlianc an imrov tatic tiffn i now obtain ( Fig. 5(b,c)). Fig. 5() how th itration hitori of th ynamic comlianc of th tructur unr th am loaing conition a Fig. 4(a) but with four iffrnt rcrib loaing frqunci ω = (tatic loaing), ω = 6, ω = 8 an ω =. Th otimum ynamic comlianc incra a th rcrib loaing frquncy incra. Fig. 6(a-) giv th otimum toologi of th lat-lik tructur corroning to th four rcrib loaing frqunci. A th loaing frquncy incra, th otimum toology of th tructur i graually chang from th ign that i ominat by th tatic bhaviour of th tructur (i.. th ign mainly n on th atial itribution of th loaing amlitu) to th on that i ominat by th ynamic bhaviour of th tructur (i.. frquncy ron). To illutrat thi oint, w may rform th toology otimization tak of maximizing th funamntal ignfrquncy of fr vibration of th lat. Hrby, w fin that th otimum valu of th funamntal ignfrquncy i Ω ot = 7.6, an th corroning toology a wll a th itration hitory of th funamntal ignfrquncy ar hown in Fig. 7(a) an.

6 6 Funamntal ignfrquncy Itration numbr (a) Ω ot = 7.6 Figur 7. (a) Otimum toology (5% volum fraction) aociat with th maximum valu Ω ot = 7.6 of th funamntal ignfrquncy. Itration hitory of th funamntal ignfrquncy of th lat. Finally lt u conir a ca with a rcrib valu of th loaing frquncy,.g. ω = ω = 5, which i highr than th otimum valu of th funamntal ignfrquncy Ω ot = 7.6 of th lat. Thn, a icu arlir, to nur a raonabl tatic tiffn of th ign, w introuc an ur boun 8 C C = 5 for th tatic comlianc C in th formulation of th roblm. Th otimum toology rult for thi roblm i hown in Fig. 8. Figur 8. Otimum toology (5% volum fraction) for ω =5 with an ur boun on th tatic comlianc, i.. C C ). Fig. 9(a) how th itration hitori of th ynamic comlianc of th lat ubjct to th highr loaing frquncy (ω = ω = 5) an four iffrnt ur boun contraint on th tatic comlianc C (aociat with th am loaing amlitu but zro frquncy). Th grah how that th otimum ynamic comlianc cra a th ur boun contraint on C i incra. In Fig. 9, itration hitori ar hown for minimum comlianc toology ign of th lat ubjct to a givn ur boun contraint on 7 th tatic comlianc ( C C =.5 ) for four iffrnt high loaing frqunci. Th grah how that for th high loaing frquncy ign, th ynamic comlianc of th tructur cra a th rcrib loaing frquncy i incra. Thi fatur i ooit to that obtain by minimum comlianc toology ign ubjct to rcrib lowr or mium loaing frqunci hown in Fig. 5(). A a concluion, variation of th minimum ynamic comlianc with rct to iffrnt loaing frqunci ar ict in Fig. (a), an Fig. rnt th tatic comlianc aociat with th minimum ynamic comlianc ign ubjct to iffrnt rcrib loaing frqunci. Dynamic comlianc of tructur x -7 Dynamic comlianc C for ω = ω = 5 an iffrnt ur boun on C C =.5x -7 C = x -7 C = 3x -7 C = 4x Itration numbr Itration numbr (a) Figur 9. (a) Itration hitori of th ynamic comlianc of th lat ubjct to a high loaing frquncy (ω = ω = 5 > Ω ot = 7.6) an four iffrnt ur boun contraint on th tatic comlianc C, i.. C C. Itration hitori of th ynamic comlianc of th lat ubjct to a givn ur boun contraint on C ( C C 7 =.5 ), for four iffrnt loaing frqunci Dynamic comlianc of tructur 6 x Dynamic comlianc C for ω = ω = 3 ω = ω = 5 ω = ω = 8 ω = ω = with contraint -7 C <= C =.5x ω = 3, ω = 5, ω = 8 an ω =, all of which ar highr than th otimum valu of th funamntal ignfrquncy Ω ot.

7 7 Minimum ynamic comlianc C.5 x -7.5 Dynamic ign without contraint on C Dynamic Dign with contraint C <= C C =.5x Ω ot Static comlianc C of minimum ynamic comlianc ign.5 x -7.5 Dynamic ign without contraint on C Dynamic ign with contraint C <= C C =.5x Ω ot 5 5 Loaing frquncy ω Loaing frquncy ω (a) Figur. (a) Minimum ynamic comlianc C v. iffrnt loaing frqunci. Static comlianc C (corron to th am loaing amlitu but zro frquncy) aociat with th ign in Fig. (a) v. iffrnt loaing frqunci. Not that if th rcrib loaing frquncy i clo to or highr than th otimum valu Ω ot = 7.6 of th funamntal ignfrquncy for th corroning roblm of fr vibration of th lat, an ur boun contraint C C i rcrib for th tatic comlianc in orr to avoi obtaining a tatically too wak tructur from th ynamic ign. 6. Concluion Problm of tructural toology otimization with th objctiv of minimizing th ynamic comlianc (maximizing th intgral ynamic tiffn) of continuum tructur ubjct to tim-harmonic forc vibration with rcrib frquncy an amlitu of th ynamic loaing ar tui in thi ar. Th frquncy ω of th loaing i aum to b mallr than th otimum valu of th con ignfrquncy of th corroning roblm with fr vibration. Th rult how that th ign objctiv of minimizing th ynamic comlianc yil a tructur who ignfrqunci of fr vibration ar gnrally far from th givn xcitation frquncy ω of th ynamic loaing, which imli fficint avoianc of ronanc hnomna an ruction of th vibration lvl of th tructur. It i foun that th ign objctiv of th forc vibration roblm may b imlmnt along iffrnt otimization ath accoring to iffrnt lvl of th xtrnal xicitation frquncy ω. For ca whr th loaing ha a lowr or mium valu of ω, th minimum ynamic comlianc ign roc may b rivn by a continuation tchniqu whr th loaing frquncy i quntially incra from a ufficintly low initial valu u to it rcrib valu, ω. Thi rocur livr th ir rult that th otimum tructur i aociat with minimum ynamic comlianc ubjct to th rcrib loaing frquncy, an alo imli an ffctiv imrovmnt (cra) of th tatic comlianc of th tructur. Thi ar alo rval that if th loaing frquncy ω i incra from a lowr valu u to th otimum valu Ω ot of th funamntal ignfrquncy of th corroning fr vibration roblm, thn th otimum toology of th tructur i graually chang from a ign that i ominat by th tatic bhaviour of th tructur (i.., th ign mainly n on th atial itribution of th loaing amlitu) to a ign that i ominat by th ynamic bhaviour of th tructur (i.., it frquncy ron). Finally, if th loaing ha a high valu of th of xcitation frquncy (i.., omwhat blow or abov th otimum valu Ω ot of th funamntal ignfrquncy of th corroning fr vibration roblm), w hav foun it xint to introuc an ur boun contraint on th tatic comlianc in orr to maintain a raonabl tatic tiffn of th ign. 7. Rfrnc [] M.P. Bnø an N. Kikuchi. Gnrating otimal toologi in tructural ign uing a homognization mtho. Com. Mth. Al. Mch. Eng., 988, 7: 97-4 [] M.P. Bnø. Otimal ha ign a a matrial itribution roblm. Struct. Otim., 989, : 93- [3] N. Olhoff, M.P. Bnø an J. Ramun. On CAD-intrgrat tructural toology an ign otimization. Comut. Mth. Al. Mch. Engng., 99, 89: [4] M. P. Bnø an O. Sigmun. Toology Otimization: Thory, Mtho an Alication, Sringr, 3 [5] H. Echnaur an N. Olhoff. Toology otimization of continuum tructur: A rviw. Al. Mch. Rv.,, 54(4): [6] Z.D. Ma, N. Kikuchi an H.C. Chng. Toological ign for vibrating tructur. Comut. Mtho. Al. Mch. Engrg., 995, : 59-8 [7] C.S. Jog. Toology ign of tructur ubjct to rioic loaing. J. Soun an Vibration,, 53(3): [8] S. Min, N. Kikuchi, Y.C. Park, S. Kim an S. Chang. Otimal toology ign of tructur unr ynamic loa. Struct. Otim., 999, 7: 8-8 [9] D. Tchrniak. Toology otimization of ronating tructur uing SIMP mtho. Int. J. Numr. Mth. Engng.,, 54: 65-6 [] V.B. Hammr an N. Olhoff. Toology otimization with ign nnt loa. In: Proc. WCSMO-3, Thir Worl Congr of Structural an Multiicilinary Otimization, Buffalo, NY, May 7-, 999. [] V. B. Hammr an N. Olhoff. Toology otimization of continuum tructur ubjct to rur loaing. Struct. Multiic. Otim.,, 9:

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