Large Scale Topology Optimization Using Preconditioned Krylov Subspace Recycling and Continuous Approximation of Material Distribution

Transcription

1 Larg Scal Topology Optimization Using Prconditiond Krylov Subspac Rcycling and Continuous Approximation of Matrial Distribution Eric d Sturlr*, Chau L**, Shun Wang***, Glaucio Paulino** * Dpartmnt of Mathmatics, Virginia Tch, Blacksburg, VA **Dpartmnt of Environmntal and Civil Enginring, Univrsity of Illinois at Urbana-Champaign, Urbana, IL ***Dpartmnt of Computr Scinc, Univrsity of Illinois at Urbana-Champaign, Urbana, IL Abstract. Larg-scal topology optimization problms dmand th solution of a larg numbr of linar systms arising in th finit lmnt analysis. Ths systms can b solvd fficintly by spcial itrativ solvrs. Bcaus th linar systms in th squnc of optimization stps chang slowly from on stp to th nxt, w can significantly rduc th numbr of itrations and th runtim of th linar solvr by rcycling slctd sarch spacs from prvious linar systms, and by using prconditioning and scaling tchniqus. W also provid a nw implmntation of th 8-nod brick (B8) lmnt for th continuous approximation of matrial distribution (CAMD) approach to improv dsigns of functionally gradd matrials. Spcifically, w dvlop a B8/B8 implmntation in which th lmnt shap functions ar usd for th approximation of both displacmnts and matrial dnsity at nodal locations. Finally, w valuat th ffctivnss of svral solvr and prconditioning stratgis, and w invstigat larg-scal xampls, including functionally gradd matrials, which ar solvd with a spcial vrsion of th SIMP (solid isotropic matrial with pnalization) modl. Th ffctivnss of th solvr is dmonstratd by mans of a topology optimization problm in a functionally gradd matrial with 1.6 million unknowns on a fast PC. Kywords: Topology optimization, matrial distribution, fast solution schms, functionally gradd matrials, finit lmnts. INTRODUCTION Th dsird rsult of topology optimization is a domain whr ach lmnt ithr is void or contains matrial. Howvr, it is mathmatically difficult to work with intgr variabls, thus rlaxation is usually applid. By allowing intrmdiat matrial dnsity btwn 0 and 1, w can comput th dsird snsitivitis. To obtain a final solution without intrmdiat dnsitis, an intrmdiat dnsity is pnalizd and mad unconomical in trms of th stiffnss as a function of dnsity or volum. At an arly stag, th homognization mthod [1] was usd to driv th stiffnss of intrmdiat dnsitis from crtain configurations of microstructurs. Howvr, th final solution is not supposd to contain microstructurs. Thrfor, th drivation of th stiffnss for intrmdiat matrial dnsity basd on th homognization approach only srvs as a

2 mans of pnalization. Latr th Solid Isotropic Matrial with Pnalization (SIMP) approach [2] was proposd as a simplr way to driv th stiffnss for intrmdiat matrial dnsity (by intrpolation). In th finit lmnt stting, w us th nodal approach with continuous approximation of matrial distribution (CAMD). Th CAMD approach is thn xtndd to modl functionally gradd matrials (FGMs) with th so-calld FGM-SIMP modl [6]. Problm Statmnt in th Continuum Stting Th problm formulation for topology optimization using th CAMD approach is givn blow for th minimization of complianc subjct to a volum constraint. T min f u ρ [0,1] KE ( ) u= f p E = ( ρ( x)) E0 n such that ρ( x) = Niρ i= 1 m ρdv V Ω 0 = 1 whr n is th numbr of nods in ach lmnt, and m is th total numbr of lmnts. Morovr, K dnots th stiffnss matrix, which is a function of th dnsity distribution ρ, f is th load vctor, u is th displacmnt vctor, E dnots th matrix of lastic proprtis of th matrial, V0 is th total volum in us, and N i rfrs to th shap functions of th finit lmnt bing usd. Furthr dtails can b found in [ 6] (addrssing FGMs) and rlatd rfrncs in [3,7]. In cas th domain is functionally gradd, that is, th proprtis of th matrial in th domain vary in spac, th lasticity tnsor is a variabl with rspct to location. To handl matrial gradation th FGM-SIMP modl is usd [6]: H p E = ρ E ( 0 x ). For a simpl xponntially gradd matrial in 3D, th FGM-SIMP modl bcoms x y z E x = E α + β + γ. () i Functionally Gradd Matrial (FGM) Domain ( ) 0 0 Th CAMD approach is rcommndd to captur th gradint of FGM proprtis insid ach lmnt. KRYLOV SUBSPACE RECYCLING FOR SYMMETRIC MATRICES Th finit lmnt analysis in topology optimization rquirs th solution of a squnc of linar systms, which in most applications ar symmtric. In ach,

3 optimization stp, th algorithm updats th dnsity of ach lmnt (th topology), and th changs in th dsign variabls tnd to b small from on optimization stp to th nxt. This holds spcially towards th nd of th optimization procss, whn th topology is convrging. Hnc, th optimization lads to small changs from on linar systm to th nxt, and crtain proprtis of th solution of on systm, or th sarch spac gnratd for on systm, rmain usful for subsqunt systms. First, th solution of on systm can b usd as an initial guss of th nxt systm to rduc th initial rsidual. Scond, an approximat invariant subspac drivd from th Krylov spac gnratd for on linar systm can b usd to improv th convrgnc rat solving th nxt linar systm. Othr subspacs may also b usd for rcycling [5], and ths tchniqus may gratly improv th convrgnc rat of Krylov solvrs if an appropriat subspac is chosn. This is th basic ida of Krylov subspac rcycling [5]. For symmtric systms, w adapt th MINRES algorithm [4] for Krylov subspac rcycling. By xploiting th symmtry of th matrix, w mak th itration chapr and th rcycling schm mor ffctiv. Furthr dtails can b found in rfrnc [8]. PRECONDITIONING AND SCALING For Krylov subspac mthods applid to symmtric or Hrmitian systms th ratio btwn th absolut largst and smallst ignvalus, which is th condition numbr of th matrix, govrns an uppr bound on th convrgnc rat. Th linar systms arising from larg-scal finit lmnt simulations in physics and nginring ar gnrally ill-conditiond. In topology optimization, th ill-conditioning is significantly xacrbatd by th wid rang of lmnt dnsitis. To rmdy this ill-conditioning, w rscal th stiffnss matrics such that th diagonal cofficints ar all th sam, which is th cas for a problm with homognous dnsity. W propos to rscal th stiffnss matrics K by multiplying with a diagonal matrix on both sids, ~ 1/ 2 1/ 2 K = D KD, whr D is th diagonal of K. Th importanc of such scaling and why it hlps is xplaind for an idalizd 1D problm in [8]. To furthr improv th conditioning and rduc itrations, w apply an incomplt Cholsky prconditionr with zro fill-in to th xplicitly rscald systm, ~~ ~ 1/ 2 1/ 2 T K = D KD LL. Figur 1 compars th condition numbrs of four matrics, th original stiffnss matrix K, th diagonally scald stiffnss matrix K ~, and both matrics multiplid with thir rspctiv incomplt Cholsky prconditionrs. Th rscaling significantly rducs th condition numbr, and also improvs th ffctivnss of th incomplt Cholsky prconditioning. Not that applying th Cholsky prconditionr to th stiffnss matrix without first scaling lads to a condition numbr that is wors than that of th diagonally scald systm (without a Cholsky prconditionr).

4 FIGURE 1. Condition numbrs for (1) th original stiffnss matrics, K, (2) thir incomplt Cholsky prconditiond forms, L 1 KL T, (3) th diagonally scald stiffnss matrics, K ~, and (4) thir incomplt Cholsky prconditiond forms, T L ~ 1~ KL ~ NUMERICAL RESULTS In this sction, w giv numrical rsults to illustrat th ffctivnss of our rcycling MINRES mthod and prconditioning as wll as two larg-scal dsign xampls with th CAMD schm. Krylov Subspac Rcycling To tst th ffctivnss of our rcycling MINRES solvr, w solv a cantilvr bam that is subjct to a point load at th cntr of th fr surfac. Th domain is discrtizd using brick lmnts and continuous distribution of matrial dnsity insid ach lmnt (B8/B8). Figur 2 shows a comparison of itration counts and run tims for th standard and rcycling MINRES (RMINRES) solvrs for svral paramtrs choics (s [8]). Without rcycling, th tim to solv ach linar systm in th optimization problm is roughly constant. Aftr an initial phas, this tim is about 80% highr than th tim to solv linar systms using th RMINRES solvr with th bst paramtr choic. Our rcycling mthod xploits th slow variation of th linar systms, and thus rducs th numbr of itrations significantly. Larg-Scal Dsign with CAMD Approach To dmonstrat th ffctivnss of th itrativ solvr and th CAMD approach, w solv a dsign problm of approximatly 1.6 million unknowns on a singl procssor PC. Th smooth approximation of matrial dnsity using th CAMD approach for larg problms lads to a highr fidlity solution with smoothr boundary surfacs. Figur 3 shows th final dsign in a homognous matrial..

5 FIGURE 2. Comparison of itration counts and runtims btwn th standard MINRES solvr and th rcycling MINRES (RMINRES) solvr for svral paramtrs choics. FIGURE 3. A cantilvr bam dsign solvd using th prconditiond, rcycling MINRES solvr and th CAMD approach; msh siz: B8/B8 lmnts; total numbr of unknowns: approximatly 1.6 million. Lft: final dsign of an homognous matrial. Right: Two cross-sctions from th final dsign. Larg-Scal Dsign in a Functionally Gradd Domain W also implmntd th cantilvr bam dsign for a simpl xponntially gradd matrial in 3D using th FGM-SIMP modl, dscribd arlir. In this xampl, th matrial is only gradd along th hight of th bam with non-homognity cofficint β = 2 h ( α = γ = 0 ), whr h is th hight of th bam. Th configuration of th problm is th sam as in th prvious xampl. Th rsults shown in Figur 4 rfr to a non-symmtric structural configuration, which illustrats th ffct of th matrial gradation in th dsign domain.

6 FIGURE 4. A cantilvr bam solvd in th FGM domain using th prconditiond, rcycling MINRES solvr and th CAMD approach; msh siz: B8/B8 lmnts; total numbr of unknowns: approximatly 1.6 million. Lft: final dsign of FGM bam. Right: two cross-sctions from th final dsign. CONCLUSIONS As suggstd by th xampls in this papr, th us of topology optimization is moving from concptual dsigns towards final dsigns that can b usd for fabrication. This volution of th tchnology can b achivd by combining mor accurat modling (CAMD) with fficint solution schms for larg-scal problm (Krylov subspac rcycling and prconditioning). ACKNOWLEDGEMENTS W acknowldg th (1) Midwst Structural Cntr, supportd by th U.S. Air Forc Rsarch Laboratory Air Vhicls Dirctorat undr contract numbr FA , (2) UIUC Cntr for Procss Simulation and Dsign via grant NSF DMR , (3) UIUC Matrials Computation Cntr via grant NSF DMR , and (4) Vitnam Education Foundation (VEF) by providing a Fllowship to Mr. Chau L. REFERENCES 1. M. P. Bndsø, N. Kikuchi, Gnrating Optimal Topologis in Structural Dsign Using a Homognization Mthod in Computr Mthods in Applid Mchanics and Enginring, 71: (2006). 2. M. P. Bndsø, O. Sigmund, Matrial Intrpolation Schms in Topology Optimization in Archivs of Applid Mchanics, 69: (1999). 3. J. Mackrl, Topology and Shap Optimization of Structurs using FEM and BEM: A bibliography ( ) in Finit Elmnts in Analysis and Dsign, 39: (2003). 4. C. C. Paig and M. A. Saundrs, Solution of Spars Indfinit Systms of Linar Equations in SIAM Journal on Numrical Analysis, 12: (1975). 5. M. L. Parks, E. d Sturlr, G. Macky, D. D. Johnson, S. Maiti, Rcycling Krylov Subspacs for Squncs of Linar Systms, in SIAM Journal on Scintific Computing, 28(5): (2006) 6. G. H. Paulino, E. C. N. Silva, Dsign of Functionally Gradd Structurs Using Topology Optimization in Matrials Scinc Forum, : (2005). 7. G. I. N. Rozvany, Aims, Scop, Mthods, History and Unifid Trminology of Computr-Aidd Topology Optimization in Structural Mchanics in Structural and Multidisciplinary Optimization, 21: (2001). 8. S. Wang, E. d Sturlr, G. Paulino, Larg-scal Topology Optimization using Prconditiond Krylov Subspac Mthods with Rcycling in Intrnational Journal for Numrical Mthods in Enginring, 69: (2007).