Structural Optimization by Using the Stiffness Homogenization.

Transcription

1 Strutural Optimizatio by Usig th Stiffss Homogizatio. Ri Yogsop, Ri amhyok, Ri Cholji, Ri Cholsu ad Zhihua Ch Dpartmt of Mhais Egirig, im Il Sug Uivrsity, Pyogyag, DPR of ora Dpartmt of Miig Mahi, Chogji Miig ad Mtal Uivrsity, Chogji, DPR of ora Shool of Civil Egirig, iaji Uivrsity, iaji, PR of Chia,7 Corrspodig Author Abstrat h topology optimizatio usig th homogizatio mthod is to fid optimal layout of mirostruturs whih satisfis dsig dmad i th ighborhood of a arbitrary poit i a giv dsig domai. h homogizatio approah is to omput propr mhaial proprtis of thos mirostruturs i trms of itgral avragig by aptig th priodiity assumptio ad haratristi futio. I this papr, w dsribd a stiffss homogizatio thory that is abl to avoid th ompliatio of th homogizatio approah ad rdu th omputatio ost by osidrig th topology optimizatio pross i osist with th fiit lmt aalysis. h mthod a b applid mor flxibly to idividual or simultaous optimizatio of th topology, shap, siz ad matrial layout. W disussd a stiffss homogizatio durig mshig th spa strutur with diffrt kids of lmts i dtail. A optimal ritrio for th miimum wight dsig problm, as o of typial optimizatio dsig problms, was drivd ad th omputatioal algorithm was prstd. umrial rsults of a xampl wr ompard with prvious mthods, whih show th validatio of th mthod. ywords homogizatio mthod, topology optimizatio, stiffss homogizatio, strutur dsig Itrodutio I th strutural optimizatio dsig, th homogizatio mthod was first prstd from a ida that topology optimizatio problms of th strutur a b tratd i rlativly popularizd way as wll as sizig optimizatio problms i th lat 98's(Bdso ad ikuhi 988. h homogizatio dots th pross that th htrogous mdium ovrt to th homogous mdium of whih proprty is quivalt to its. I problms of th otiuum mhais iludig solid mhais, mathmatial osidratio is prformd udr th assumptio for homogity of mdium. h

2 most ommo tool dalig with ths problms is to study bhavior of mdium usig diffrtial ad itgral alulus, ad th fudamtal thiqu for mathmatial modlig ad aalysis ad rsarh of bhavior for otiuum is to solv th iitial ad boudary valu problm of th diffrtial quatio. But i th pratis, thr ar may ass of htrogous ad disrt mdium. Solvig suh problms usig mathmatial mthod is ommoly impossibl. For this raso, various homogizatio mthods to solv ths problms wr prstd from attmpts to us th paramtrs assumd as otiuous variabls for ovrtig htrogous-disrt mdium to homogous-otiuous o with proprtis quivalt to its(zhikov t al. 979; Marko ad hruslov 974; Makhvalov ad Paasko 99. I th homogizatio mthod, it is assumd that matrial at th ighborhood of a arbitrary poit i spa osists of a st of priodi mirostruturs i.. lls iludig hol. h rsultig mdium a b dsribd by fftiv marosopi matrial proprtis whih dpd o th gomtry of th basi ll, ad ths proprtis a b omputd by ivokig th formulas of homogizatio thory basd o th multisal mthod(pavliotis ad Stuart 8. h omputatios of ths fftiv proprtis play a ky rol for th topology optimizatio. hat is, th optimal dsig problm o htrogous-disrt domai i topologial optimizatio of th otiuum strutur by th matrial layout mthod is ovrtd to th optimal dsig problm o th homogous-otiuum domai i th optimal dsig by th homogizatio mthod As a rsult, th topology optimizatio problm o th homogizd domai a b solvd by dtrmiig paramtrs rprstig th hol s distributio ad siz usig sizig optimizatio mthod. I this ss, this optimizatio mthod is alld homogizatio mthod or i a broad ss, matrial layout mthod. Aftr this rsarh i th topology optimizatio fild, it attaid th highst stag of prosprity ad this mthod was applid to th pratial problms. Most of rsarhs for this mthod ar basd o th optimality ritria mthod(allair ; Rozvay t al. 994; Bdso ad Sigmud ; Hassai ad Hito 998, ad som of thm ar basd o th mathmatial programmig mthods(allair ;Yag ad Chuag 994. Morovr, oly th topology optimizatio problms was ivstigatd at first, but gradually th sizig, shap ad matrial layout optimizatio problms was ivstigatd(allair ; Bdso ad Sigmud ; Park 995. Espially, th matrial layout optimizatio basd o homogizatio approah hav also b xpadd to svral filds iludig a work for dtrmiig th strutur with th rquird odutivity by optimally distributig matrials with diffrt hat odutivitis(rodrigus ad Frads 995, pizoomposit atuator s optimal dsig(josma t al. 999; Emilio ad lli 998 ad so o. Bsids, thr wr may rsarhs i whih th homogizatio mthod was applid o th optimizatio of dyami problms(bdso ad Sigmud ; Diaz ad ikuhi 99 ad svral rsarhs for improvig th ovrg of th optimizatio pross was prstd. But th homogizatio mthod has

3 drawbaks rquirig a lot of dsig variabls ad grat omputatioal ost yt. Bdso(989 had proposd SIMP (Solid Isotropi Matrial with Palizatio, whih itrodud matrial distributio dsity o th dsig domai of strutur i th rsarh of th topology optimizatio. h SIMP itrpolatio shm addrsss th itgr format of th origial sttig for th topology dsig problm. It ovrts this itgr problm to a sizig problm that fially rsults i pratial - dsigs. Aothr srious problm assoiatd with th - problm is that SIMP a lad to oxist of solutio of th problm. his ot oly is a srious thortial drawbak but also affts o th stimatio of ssitivity of omputatioal rsults to th fiss of th fiit lmt msh. As abov mtiod, th SIMP itrpolatio shm dos ot dirtly rsolv this problm, ad furthr osidratios ar i pla to assur a wll-posd distributd dsig problm that also is big i trms of msh fiss of th fiit lmt modl(hsu ad Hsu 5; Borrvall ad Ptrsso. Colligatig th rsarh rsults so far, w a fid that th homogizatio mthod was gralizd ad popularizd as a major mthod ovrtig th topology optimizatio problm to a matrial layout optimizatio or a sizig optimizatio problm. Also aordig to iras of th omputatio ability of omputrs, it is applid to th may optimal dsig problms for pratial struturs ad its appliatio fild is wid mor ad mor. Its appliatio objt ad fild is optimal dsig of stati strutur(allair ; Bdso ad Sigmud ; Hassai ad Hito 998; Hsu ad Hsu 5, topology optimizatio of a igvalu ad frquy rspos problm(bdso ad Sigmud ; Diaz ad ikuhi 99, topology dsig of struturs with a buklig load ostrait(vs t al. 995, topology optimizatio of thrmo-lasti struturs(rodrigus ad Frads 995, optimal dsig of otrolld struturs(ou ad ikuhi 996, Systmati sythsis of miro ompliat mhaisms(aathasursh t al. 99; Aathasursh t al. 994, optimal dsig of pizoomposit matrials, pizoltri trasdurs ad ltro-thrmal miro atuators(josma t al. 999; Emilio ad lli 998 ad so o. I this papr, th topology optimizatio pross usig stiffss homogizatio thory is disussd, i whih th priodi assumptio is rlaxd ad th omputatio ost is rdud without a ompliatd disussio as itgral avrag by itrodutio of th haratristi futio by simplifyig th homogizatio approah. h stiffss homogizatio mthod applis idividual ad simultaous optimizatio of sizig, shap, matrial layout ad topology. his mthod is mor ovit to apply to th omputatio produr ad strutural optimizatio pratis by usig th ommo mthods of th fiit lmt aalysis i all osidratio ad formulatig of ll ad lmt. hory of th stiffss homogizatio by usig ara (or volum rat of lmt. I this papr it is assumd that th strutural ompot of th dsig domai is

4 mshd with lmts, ah of lmts is mad of th mirostrutur, that is, ifiitly small lls with hols ad th shap of ll ad lmt is triagl or ttrahdro. Of ours, th highr ordr s lmt a b usd ad hol s troid is plad i th troid of th ll i whih it blogs ad hol s shap is gomtrially similar to that ll.. Stiffss homogizatio by usig first ordr pla triagl lmt. First, lt s osidr th liar triagl pla lmt. Fig.. Gomtry modl (a dsig domai, b lmt, ll I th lls ad lmts, th hol s siz is ompltly dtrmid with a paramtr by usig ara rat: S ( S whr S is th total ara of ll or lmt ad S is ara of hol i th ll or th lmt. h haratristis for th lls is labld ad for th lmts is labld. h stiffss matrix of th lls for th liar lmt is rdud by th hol rat as ompard with that without hol ad is formulatd as followig: t B D B ds ( S whr t B D B ds, S 4

5 D D D D D, D t is thikss of th lmt(or th ll ad B is strai matrix of th lmt h stiffss distributio of th pla triagl lmt, whih is assumd to b mad of th ifiitly may lls, a b rprstd i th form itrpolatd from th stiffsss pr uit ara at th ods. ( i i j j m m whr i, j ad m ar od umbrs of th lmt rsptivly, ( i, j, m is th shap futio rprstd by th ara oordiats, ad ( i, j, m is assumd to b otiuous i th lmt. h stiffss matrix of th pla triagl lmt is; S ds i ( i j ( j m ( m (4 S hat is; whr i j m S (- is th stiffss of lls without voids, S is th stiffss matrix of th i j m lmt mad of th lls without hol ad (- is avrag ara rat. Aordigly, th stiffss of th triagl lmt is rprstd as followig (5 ( i j m S, hat is, th lmt s stiffss is rdud by th hol s rat as ompard 5

6 with that without hol i lls. Cosidrig t B D B S, th w a rwrit as followig quatios: [ t ( ]B D B S (6 t B [ D ( ] B S (7 t B D B S ( ] (8 ( [ From Equatio (6-(7, it is foud that this problm a b osidrd as th homogizatio of th thikss, matrial ostat, ara ad tir stiffss ad i th futur, sizig, shap, topology ad matrial layout optimizatio a b idividually or simultaous prformd.. Stiffss homogizatio by usig ttrahdro lmt. I th spa problm, mshig with ttrahdro lmt, B D B dv (9 V whr B, D ar strai ad lasti matrix of th spa problm, rsptivly. a V b V V C Fig.. gomtry modl(a.dsig domai, b.lmt,.ll 6

7 whr By usig abov similar mthod ( ( i j m V ad 4 From Equatio ( w a also obtai: B [ D ( ] B V ( B DB [ V ( ] ( ( I drivig abov quatios, w usd followig th itgral formulas i ara ad volum oordiats o triagl ad ttrahdro domai. S a b a! b!! i j mds S, ( a b! V d ds 6V a i b j m p a! b!! d! ( a b d!. h stiffss homogizatio for th aisotropi matrial. If th strutur is mad of th orthotropi matrial ad w osidr th orthotropi proprty i th stiffss homogizatio mthod, w disuss th triagl of lls ad lmts at th loal oordiats of whih axs oiid with th orthotropi axs. 7

8 Fig.. Gomtrial modl for orthotropi matrial (a.dsig domai, b.lmt,.ll h stiffss matrix of lls is drivd at th loal oordiat systm as followig; whr ~ ˆ ~ t B D B ds ( S is th ll s stiffss matrix without void i th loal oordiat systm, B is th strai matrix at th loal oordiat systm ad th orthotropi lasti matrix at th loal oordiat systm D is: D D D D D. D I th triagl lmt full of suh lls, if th lmt siz is ough small, th loal oordiat axs of th ir lls ar qual to ah othr. Assumig that th stiffss distributio of th lmt is itrpolatd by th stiffsss of lls at ods as ˆ ˆ i i ˆ,th stiffss matrix of th triagular lmt is; j j ˆ m m ˆ ˆ ds S S ~ i ~ ( i j ~ ( j m ( m 8

9 ~ S i j m (-. (4 Cosidrig ~ S ~ i j m ad, th th homogizd stiffss matrix of lmt at th loal oordiat systm is rprstd as followig;. ˆ ~ ( hat is, th lmt s stiffss is rdud by th ll void s rat with that without hols. as ompard If th agl btw loal oordiat s x axis ad global oordiat s x axis is, t B D B S t B D B S hrfor th homogizd lmt stiffss matrix at th global oordiat systm is; ( whr th trasformatio matrix of oordiat is, si, os, ad B is th strai matrix at th global oordiat systm. hat is, rplaig th lasti matrix D ad th strai matrix homogizd lmt s stiffss matrixs at th loal oordiat systm with 9 B i th ad B, w obtai th homogizd lmt s stiffss matrixs at th global oordiat systm..4 Stiffss homogizatio by usig high ordr lmt. I th fiit lmt aalysis by usig high ordr lmt, th stiffss homogizatio mthod a b also usd. Usig abov mthod, w a fid that all homogizd stiffss matris for high ordr lmts hav th form as D

10 ( At this tim, w rprstd th stiffss distributios of diffrt high ordr lmts by th orrspodig shap futios i th ara or volum oordiats ad usd itgral formula i th ara or volum oordiat systm. For xampl, shap futios for th sod ordr pla triagl lmt ar prstd by th ara oordiats,, as (, ( 4 4 (,, hus; ( ( 45 6 (5 6 Ad for th third ordr pla triagl lmt th shap futio is as blow ( (, ( ( 9 ( (, 4 ( (, 6 ( (, 8 ( 9 9 (, 7 hrfor 9 ( ( (6 Ad for sod ordr ttrahdro lmt; (, (, (, 4 4( 4,

11 hrfor 5 4, 6 4, , 9 4 4, 4 4 ( 4 ( (7 From abov all, w a fid that th avrag of th void s rat of ll multiplid by som wights for lls at th vrtis, ods o dgs ad ir ods of lmt is orrspodig to th stiffss rdutio of lmt s. If i for vry lls is th sam as, th. h wight offiit is th sam as i th followig tabl. triagl ttrahdro ordr od at th dg ir first sod third /4 9/ first sod -/ I this way, by usig th ara or volum rat orrspodig to th hols siz, w a obtai th homogizd stiffss matrix by usig th itrpolatio s mthod of th fiit lmt mthod ad a asily solv th optimal dsig problm. Problm formulatio for optimal dsig ad optimal ritrio. ow, w shall formulat th miimum volum dsig problm by usig th stiffss homogizatio mthod ad disuss its solvig mthod.. Optimizatio Modl h miimum volum dsig problm is to fid th strutur with miimum volum(or wight udr th rstritio of th global stiffss. hat is, th problm is to fid satisfyig th followig quatios. V ( t ( S mi (8 U U u u U U u u (9 u P (

12 whr ( Ulik i th ovtioal optimal dsig problms, all th sizig, shap, topology ad matrial layout optimizatio problms i th stiffss homogizatio mthod is rdud to th problm whih is fidig optimal dsig paramtr whih is th ara rat(volum rat haratrizig hol s siz of lls. I th abov problms, V, U : th giv volum ad strai rgy haratrizig global stiffss i th strutur rsptivly. Ad th quality rstritio u P is th stiffss quatio, amly, th rstritio of th stat quatio. Also, for ovi, w assum that is th lmt s stiffss matrix whih is xpadd i aorda with th dimsio of th strutur s stiffss matrix.. Optimality ritrio Lt s driv th problm s optimal ritrio; Lagrag s quatio is ostrutd as, u V ( u u U μ u P L ( whr, μ ar Lagrag s urtai multiplirs. From a ssary oditio for th optimal solutio of th oliar programmig problm, th strutur s stat at th optimal poit, amly, optimizatio ritrio is drivd. L Usig th oditio at th optimal poit,th u hrfor μ u u μ ( L Ad usig th oditio at th optimal poit (=,,,, th u u V u u μ u u μ (

13 V U (=,,, (4 Hr, osidrig, th th quatio ( U u u u u boms to U u u, whih prsts th lmt s strai rgy wh th lmt has o hol. Summig up th two sids of Equatio (4 for all lmts rptivly, V U (=,,, Multiplyig ( to th two sids of th abov quatio ad Equatio (4, limiatig urtai multiplir ad osidrig from th oditio of th optimal poit; u u U,th w obtai th qualizatio ritrio about th dsity of th strai rgy at th optimal poit as V ( U ( V ( U (=,,, (5 whr U is lmt s strai rgy obtaid by th fiit lmt aalysis. From Equatio (5 w a fid that th strutur of th optimal siz, shap, topology ad matrial layout is th strutur i whih lmt s ara (volum rat is dtrmid so that dsity of th lmt s strai rgy is qualizd to th dsity of th strutur s global strai rgy. 4 Algorithm h optimality ritrio a b rwritt as V ( U ( V ( (=,,, U

14 U ( V ( (=,,, (6 V U Our purpos is to gt solutio, of th oliar algbrai simultaous Equatio (6. Equatio (6 is alulatd by usig th simpl itratio shma as ( k ( k ( k U ( V ( ( k (=, (k=,, (7 V U tak h algorithm is as followig:. Iitializ th dsig paramtrs (k = vrywhr i th domai. Comput (=~ for k = (i gral ass, ( k ( k ( k U ( by solvig th stat quatio, ( u P. k (k u, V ( ad. Comput ( k usig th itratio shma (7.. Rpat th itratio loop for k = k+, util th ovrg is ahivd. Hr, th ovrg oditio is h dsig paramtrs ar optimal solutios. V ( k k V (, ( k ( k or ( ( (=~ k = Figur 4 shows th abov algorithm. Obtaiig optimal dsig paramtr (=~ whih satisfy th abov oditio (=,, th th optimal struturs about vry lmts a b obtai by usig followigs; Siz ad topology optimizatio; t t (8 4

15 Shap optimizatio; S S Matrial layout optimizatio; D (9 D ( If Poisso's ratio s hag is gltd, istad of Equatio (, E E ( Simultaous optimizatio of th siz, shap ad topology; V V ( Simultaous optimizatio iludig matrial layout; ( Limits of th idividual paramtrs ar osidrd i th omputatio pross. Wh th simultaous optimizatio is prformd, th paramtr with strogr limit is first dtrmid or th paramtrs ar dtrmid i aorda with wights spifid by dsigr, osidrig thir importa. 5

16 5 Exampls Fig. 4. Optimizatio algorithm Cosidr a thi bam i a pla strss stat(th out-of-pla displamt is rstraid, whih is a stadard tstig problm(allair for vrifyig th optimal dsig mthod abov disussd. hikss m, P= k E. Pa, E.5 Pa U =.46E- m Fig. 5. h boudary oditio about th plat bam problm h strutur is mshd by 6 liar plai triagl lmts. 6

17 Aordig to th algorithm as rfrrd abov, first is foud ad o or svral limits is osidrd wh all fial paramtrs is dtrmid. Figur 6 shows th strss stat of th iitial strutur. Fig. 6. h strss stat of th iitial strutur. Figur 7 shows that th hkrboard pattr is gratd i th optimizatio prossig. At this tim i th rpat pross o-ovrgd or impossibl strutur is obtaid. o ovrom problms of th gratio of th gry domai, hkrboard phomo ad so o, th prvious mthods(allair ; Hsu ad Hsu 5 suh as avragig, filtrig ad palizig ar usd. Fig. 7. hkrboard pattr. h obtaid rsults is show i abl togthr with piturs. First ad sod piturs i th tabl show th shap optimizatio rsult at th diffrt limits ad third pitur shows th topology optimizatio rsult. Forth pitur is th alulatio rsult of th simultaous siz ad shap optimizatio udr th maximum thikss limit ad fifth is th marosopi matrial layout optimizatio usig two kid of matrials. h sixth shows th simultaous optimizatio rsult of th matrial layout ad topology. As show i this xampl of pla problm, th purposd optimal strutur is obtaid with % of th rpat umbr ad muh lss omputatio ost as ompard with th prvious mthod. abl shows th thr dimsioal topology optimizatio rsults as ompard with th prvious homogizatio mthod s rsults. h omputio xampl is rprstd as ompard with th sam objts as i litratur(hsu ad Hsu 5. h mthod s validity for th pratial problms of th svral struturs is vrifid through th rsults of th thr dimsioal topology optimizatio basd o th stiffss homogizatio mthod. abl. h alulatio rsult of -D problm. 7

18 Optimal strutur optimizatio iitial Homogizatio Mthod(Allair Et..588 itratio ( limit.9.44 itratio ( limit.8.4 itratio ( 4.4 itratio 5 ( t max = 5.m itratio ( E, E itratio ( out dg: 4 Pa i: Pa 8

19 abl. -D topology optimizatio rsults xampl homogizatio mthod(hsu ad Hsu 5 stiffss homogizatio mthod 6 Colusios As abov-mtiod, th stiffss homogizatio mthod proposd for strutural optimizatio is a kid of matrial layout mthod, i whih th ompots of th spa strutur ar orrspodig to ah lmt ad th stiffss matrix is stimatd aordig to th mirostrutur hol s rat. 9

20 h strutur optimizatio by th stiffss homogizatio mthod maks it possibl to us th thory ad mthod of th fiit lmt mthod i all th prosss of homogizatio, strutur aalysis ad optimizatio i a uifid way. h drivatio of th optimality ritrio for miimum volum dsig problm ad umrial xampl of th two ad thr dimsioal problms shows that th thory for th strutural optimizatio by th stiffss homogizatio is asily xtd to th diffrt kids of optimizatio problms ad pratis. Rfrs Allair, G. : Shap optimizatio by th homogizatio mthod. Hidlbrg, w York: Sprigr Brli, Aathasursh, G..; ota, S.; Giahadai, Y. 99: Systmati sythsis of miroompliat mhaisms prlimiary rsult. Prodigs of th hird atioal Cofr o Applid Mhaisms ad Robotis. Ciiati Ohio 8 Aathasursh, G..; ota, S.; ikuhi,. 994: Stratgis for systmati sythsis of ompliat MEMS. Prodigs of th 994 ASME Bdso, M. P. 989: Optimal shap dsig as a matrial distributio problm. Strut. Optim., 9- Bdso, M. P.; ikuhi,. 988: Gratig optimal topologis i strutural dsig usig a homogizatio mthod. Comput. Mth. Appl. Mh. Egrg. 7, 97 4 Bdso, M. P.; Sigmud, O. : opology optimizatio: hory, mthods ad appliatios. Brli, Hidlbrg, w York: Sprigr Borrvall,.; Ptrsso,J : Larg-sal topology optimizatio i D usig paralll omputig. Comput. Mth. Appl. Mh. Egrg. 9, 6 69 Diaz, A. R.; ikuhi,. 99: Solutios to shap ad topology igvalu optimizatio problms usig a homogizatio mthod. It. J. um. Mth. Egrg. 5, Emilio, C; lli, S 998: Dsig of pizoomposit matrials ad pizoltri trasdurs usig topology optimizatio. Hassai, B.; Hito, E. 998: A rviw of homogizatio ad topology

21 optimizatio. Part III: opology optimizatio usig optimality ritria. Comput. Strut. 69, Josma, J.; Sigmud, O.; Bouwstra, S. 999: Multi dgrs of frdom ltro-thrmal miroatuators. RASDUCERS 99, 7 75 Makhvalov,.; Paasko, G. 99: Homogizatio: avragig prosss i priodi mdia Mathmatis ad its appliatios. Vol. 6, luwr Aadmi Publishrs. Marko, V.; hruslov, E. 974: Boudary valu problms i domais with a fi graid boudary. (i Russia aukova Dumka, iv Mig - Hsiu Hsu ; Yh - Liag Hsu 5: Itrprtig thr- dimsioal strutural topology optimizatio rsults. Comput. Strut. 8, 7 7 vs, M. M.; Rodrigus, H. C.; Guds, J. M. 995: Gralizd topology dsig of struturs with a buklig load ritrio. Strut. Optim., 7-78 Ou, J. S.; ikuhi,. 996: Optimal dsig of otrolld struturs. Strut. Optim., 9-8 Park, Y.. 995: Extsios of Optimal Layout Dsig Usig th Homogizatio Mthod. h Uivrsity of Mihiga Pavliotis, G. A.; Stuart, A. M. 8: Multisal Mthods: Avragig ad Homogizatio, xts i Applid Mathmatis 5. Sprigr Rodrigus, H. C.; Frads, P. 995: A matrial basd modl for topology optimizatio of thrmolasti struturs. It. J. um. Mth. Egrg. 8, Rozvay, G. I..; Zhou, M.; Sigmud, O. 994: opology optimizatio i strutural dsig. Advas i Dsig Optimizatio Yag, R. J.; Chuag, C. H. 994: Optimal topology dsig usig liar programmig. Comput. Strut. 5, Zhikov, V.; ozlov, S.; Oliik, O. 979: Homogizatio ad G-ovrg of diffrtial oprators. Russ. Math. Surv. 4, 65 47