Element connectivity parameterization method for the stress based topology optimization for geometrically nonlinear structure

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1 0 th World Congrss on Structural and Multidisciplinary Optimization May 9-4, 03, Orlando, Florida, USA Elmnt connctivity paramtrization mthod for th strss basd topology optimization for gomtrically nonlinar structur Gil Ho Yoon School of Mchanical Enginring, Hanyang Univrsity, Rpublic of Kora, Abstract This rsarch dvlops a novl computational approach for th strss-basd topology optimization mthod (STOM) minimizing volum subjct to locally dfind strss constraints of gomtrically nonlinar structur in th framwor of th lmnt connctivity paramtrization (ECP) mthod. It is a classical but difficult nginring problm to constraint th local strss constraints in topology optimization. In th dnsity basd topology optimization mthod, rcntly som succssful optimization mthods hav bn dvlopd for linar lastic structur. Nvrthlss bing no rsarch to considr static failur constraint in topology optimization for gomtrically nonlinar structur, this rsarch dvlops a novl computational approach for th STOM for gomtrically nonlinar structur. In addition to th strss singularity issu, th many constraint issu and th highly nonlinar bhavior issu of th local strss constraints, th so calld unstabl lmnt issu should b proprly addrssd for gomtrically nonlinar structur. To rsolv this issu ffctivly, this rsarch adopts th ECP mthod which dos intrpolat and optimiz th connctivitis among solid finit lmnts. It is also found that in addition to th unstabl lmnt, th strss singularity issu diffrnt to that of th dnsity basd TO ariss in th ECP mthod. By invstigating th singularity bhavior in dtail, a nw qp-rlaxation mthod suitabl for th ECP mthod can b dvlopd. To show th validity of th prsnt ECP mthod with th modifid qp-rlaxation, som two dimnsional TO problms ar solvd.. Kywords: strss-basd topology optimization; gomtrically nonlinar structur; lmnt connctivity paramtrization mthod 3. Introduction This prsntation dvlops a novl topology optimization (TO) mthod to addrss th classic, but still unsolvd, challnging TO problm of minimizing volum subjct to local strss constraints that ar dfind at th cntr of vry finit lmnt of a gomtrically nonlinar structur[,,3,4,5,6]. Hr it is namd as strss-basd topology optimization (STOM) for gomtrically nonlinar structur and th dtails ar rportd in [] (Th most part of this papr is tan from th rfrnc []). In this ara of rsarch, on of th most challnging TO problms continus to b to minimiz volum subjct to local strss constraints, to prvnt static or dynamic failurs [,3,4,5,6]. For rigorous considration of local strss constraints in TO, many studis hav dmonstratd that a numbr of thortical and practical issus rmain to b rsolvd, such as th singular bhavior of strss with rspct to th dnsity dsign variabls of TO, th highly nonlinar bhavior of local strss constraints, and th trad-off btwn local and global strss constraints. In addition to ths complx issus, from an nginring point of viw th unstabl lmnt issu, i.., flippd lmnt and ngativ ara lmnt, which is oftn obsrvd at finit lmnts with low stiffnss valus, should b proprly addrssd for TO problms for gomtrically nonlinar structur[7]. To our bst nowldg, thr has bn littl rlvant rsarch rlatd to th STOM problm for gomtrically nonlinar structur bfor th prsnt study. To rigorously conduct this study on th STOM without th unstabl lmnt issu, this rsarch applis th lmnt connctivity paramtrization (ECP) mthod to paramtriz connctivitis among solid finit lmnts, as shown in Figur..

2 (a) Dsign variabl Wa lins Solid lmnt Wa lmnt Strong lins Patch Dsign variabl SIMP approach I-ECP approach (b) (c) Figur : Comparison of th modling approachs of th SIMP (solid isotropic matrial with pnalization) approach and th intrnal ECP (lmnt connctivity paramtrization) mthod. Compard with th dnsity-basd TO mthod, topological volutions in th dsign domain ar diffrntly modld and rprsntd in th ECP mthod, as shown in Figur [7]. In th lmnt dnsity basd TO mthod, a dsign domain is usually discrtizd by th finit lmnts. And th lmnts matrial proprtis ar intrpolatd, such as Young s modulus, thrmal conductivity, and lctric conductivity, dpnding on th physics of intrst, with rspct to th dnsity dsign variabls of th SIMP mthod or with rspct to th microstructur dsign variabls of th homognization mthod, as illustratd in Figur (b). By changing th matrial proprtis of th finit lmnts for ithr void or solid domains, topological modifications can b simulatd without altrations of th manifold gomtry of an FE modl. By finding th matrial proprtis that minimiz an objctiv subjct to svral constraints, th dnsity-basd TO mthod can provid innovativ layouts for nginrs. Howvr, thr is th critical complication of th unstabl lmnt in applying th dnsity-basd TO mthod for a gomtrically nonlinar structur whos strss and strain masurs ar th nd Piola-Kirhoff strss and th Grn-Lagrangian strain, with th Nwton-Raphson mthod as a solution mthod. As largr displacmnts ar allowd, unstabl lmnts having ngativ aras mostly at void rgions ar invitably obsrvd. For ths flippd lmnts with non-positiv aras, th tangnt stiffnss matrix assmbld with thir stiffnss matrics oftn bcoms a non-positiv matrix. To handl this complx issu, svral numrical tchniqus and various optimization mthods hav bn proposd in th framwor of th dnsity-basd approach. Among th most rcnt contributions, a nw intrpolation mthod calld th lmnt connctivity paramtrization (ECP) mthod, illustratd in Figur (c), was proposd. In th ECP mthod, th matrial proprtis of th finit lmnts ar not intrpolatd, but thir connctions ar intrpolatd, as shown in Figur (c). To account for th topological changs of th ECP mthod, zro-lngth lins ar usd to connct th nods of th discrtizing finit lmnts. In short, bcaus th ECP mthod can handl unstabl lmnts ffctivly, this rsarch adopts th intrnal ECP mthod for th STOM applid to gomtrically nonlinar structur. A nw qp-rlaxation tchniqu for th ECP mthod A nw vrsion of th qp-rlaxation mthod is rquird for th STOM for gomtrical nonlinar structur. Among many rlaxation mthods, th qp-rlaxation mthod in which th pnalty valus of th constitutiv matrics for forward analysis and snsitivity analysis ar diffrnt has bn widly and succssfully usd for TO of linar structurs. In th qp-rlaxation mthod, by mploying diffrnt pnalty valus of th constitutiv matrics for th analyss in th SIMP mthod, it sms that void rgions having wa stiffnss valus and xprincing xcssiv distortions with non-zro strss valus bcom non-favord, from an optimization point of viw. Th ECP mthod dos not intrpolat th matrial proprtis of th finit lmnts with rspct to th dsign variabls. Thrfor, from th viwpoint of th SIMP mthod, it can b rgardd that th pnalization factor of th dsign variabl of discrtizing lmnts is st to 0. Although th ECP mthod dos not hav th singularity issu in th forward analysis, w should intrpolat th constitutiv matrix in th snsitivity analysis procss with rspct to th dsign variabls to obtain physically accptabl layouts. In addition, w found that th qp-rlaxation mthod of th dnsity-basd approach plays a partial rol as a rgulation mthod that prvnts th strss-basd topology optimization from having no-structur. A main finding of this rsarch is that th ECP mthod also rquirs som matrial intrpolation in th snsitivity analysis for th STOM that minimizs volum subjct to local strss

3 constraints []. 4. Strss-basd topology optimization formulation To dal with thir bing too many local constraints dfind at vry finit lmnt, a rgional schm basd on th sorting algorithm and th p-norm approximation of (3) ar also mployd for th ECP mthod, as follows: NE Minimiz V( γ) = v ( γ : Filtrd dnsity ) γ subjct to max max () max RN γ ( γ) with th dnstiy filtr max c PN () PN / ( ) p p ( ) (3) c / max, c (4) PN σ tt tt tt tt S S S S 0 x 0 y 0 y 0 z tt tt tt tt tt S S S S S 0 z 0 x 0 xy 0 yz 0 xz 6 / (5) whr th volum of th -th finit lmnt is v. Convntional notations ar usd to indicat th nominal and shar strss componnts of th finit lmnt in th abov formulation. Hr, th dsign variabls assignd to th NE lmnts of th dsign domain ar dnotd as γ with which th lin stiffnss valus ar intrpolatd. It is worth noting that th dsign variabls or th snsitivity valus can b filtrd as in th SIMP approach. To cop with th many constraints issu, th locally dfind strss constraints ar aggrgatd by th rgional strss constraints with th strss p-norm of th -th rgion. Bcaus thr ar discrpancis btwn th max PN p-norm valu and th ral maximum valu, th corrction factor c was introducd, which is th ratio btwn th strss p-norm and th ral maximum strss valu in th prvious - optimization ation at th th max, rgion. Hr, w should mphasiz that th abov updating of th corrction factor, c, is huristic. In othr words, th indx ordr of finit lmnts for ach rgion ar bing subjct to chang during an optimization, and th contributing lmnts for ach rgion ar also diffrnt, that is, a non-diffrntiabl condition of an optimization. Howvr as th optimization convrgs, for most cass, th indx ordr for ach rgion bcoms almost stabl and fixd. In th cas of gomtrical nonlinar analysis, this non-diffrntiabl condition can caus a significant dtrioration of an optimization history. Thus, w updat th corrction factor vry fw ations,.g., th 0th ation. Th maximum allowabl von Miss strss valu,, is providd by nginrs to constrain th -th strss valu,. From a grat dal of rlvant rsarch, it is nown that, for stabl convrgnc in th lmnt dnsity mthod, th constitutiv matrics for th static analysis and th snsitivity analysis ar formulatd diffrntly, as follows: n Th lmnt dnsity mthod (forward analysis): C γc0 (6) n s Th lmnt dnsity mthod (snsitivity analysis): SC γ C0 (7) 3

4 whr th pnalization factors for th forward analysis and snsitivity analysis ar n and n, rspctivly, and s many rsarchrs suggst that thir valus should b btwn 3 to 5 and 0.5, rspctivly. As xplaind bfor, a main ida of th ECP mthod is that th constitutiv matrix of th discrtizing lmnts rmains a constant matrix. Forward analysis in th ECP mthod: C C (8) S 0 Bcaus discrtizing lmnts rmain solid and th connctivitis among finit lmnts ar dfind by th zro-lngth lins, th sid ffcts of th lmnt dnsity mthod can b rsolvd. And it is our concrn that th constant constitutiv matrix can still b usd for a stabl convrgnc of th original STOM for gomtrical nonlinarity. Howvr, in this rsarch, our many numrical tsts rvald that th original ECP mthod cannot ovrcom th singularity issu: Th strss constraints should not b considrd whn th associatd lmnts ar modld for th void rgion and th no-structur without any solid domain should b rmovd from th candidats of optimal solutions. Thrfor, th prsnt study has dtrmind that th ECP mthod still nds th following intrpolation of th constitutiv matrix with rspct to th dsign variabl assignd to th -th finit lmnt, as follows: Snsitivity analysis of th ECP mthod: n _ γ ECP s S_ ECP 0 C C (9) To dnot that th abov constitutiv matrix and th pnalization factor ar usd for th snsitivity analysis of th ECP mthod, S_ECP is insrtd bfor th intrpolatd constitutiv matrix, and th associatd pnalization factor is dnotd by n. Thn it bcoms an important issu to dtrmin how to choos this pnalization factor ECP _ s for stabl convrgnc in strss-basd topology optimization for gomtrically nonlinar structur. Aftr furthr analysis, this rsarch dtrmind th following rlationship for th pnalization factors of th lmnt dnsity-basd approach and th ECP mthod for stabl convrgnc. n n 0 n s ECP _ s Th lmnt dnsity Th ECP mthod basd approach (0) 5. Optimization xampls For th topology optimization xampl, th wll-studid L-shapd structur is considrd. Th dsign domain is 0. m by 0. m and is discrtizd by QUAD lmnts of 0.00 m by 0.00 m. To simulat th uppr right hol, th corrsponding dsign variabls ar st to th lowr bound. Th top lin is clampd, and a downward structural 3 3 forc of 0 N and 40 0 N ar applid at th 5 nods to rmov th strss concntration, as dpictd in Figur (a) and (b), rspctivly. Thn w minimiz th mass usag, subjct to th local von-miss strss of th nd PK strss valus with th diffrnt strss limits in Figur. As shown, dpnding on th strss limitations, th diffrnt dsigns can b obtaind. With th largr forc, th right vrtical bars of th dsign bcom inclind du to th gomtrical nonlinarity. 4

0. 0.04 0. 0.04 4.0 0 N 4.0 0 N 4.0 0 N 4.0 0 N 4.0 0 N 4.0 0 N E 6 0 N/m, 0.

5 N N N N N N E 6 0 N/m, by 00 QUAD lmnts (a) =0.8 N/m, Mass: 0.9 % N N N N N N 5

6 (b) = 35 N/m, Mass: 9.03 % Figur :. L-shapd structur and optimization rsults. (a) A problm dfinition (RN=8, n intr =3, / structur 0 9 l =, max diagonal l / 0 min diagonal th maximum strss limit =0.8 N/m, and (c) an optimizd dsign with 6.7% mass usag for th maximum strss limit = N/m. structur -9 =, radius of filtr: tims th finit lmnt), (b) an optimizd dsign with 0.9% mass usag for 6. Conclusions This papr dvlops th strss-basd topology optimization (STOM) of minimizing volum subjct to local von-miss strss constraints of gomtrically nonlinar structur using th lmnt connctivity paramtrization (ECP) approach. Th ECP mthod is ffctiv in solving TO of gomtrically nonlinar structur bcaus it paramtrizs th lin s stiffnss valus for th connctivity among finit lmnts (). W found that th singularity issu, th highly nonlinar constraint issu, and th many constraints issu still ndd to b solvd. In particular, th singularity issu in th ECP mthod ndd to b proprly dfind from a dsign variabl paramtrization point of viw: w ndd to dtrmin whthr intrpolation of th constitutiv matrix for th snsitivity analysis is rquird with rspct to th dsign variabls. By comparing th Cauchy s and th nd PK strss bhaviors of som intuitiv structurs, this papr dmonstrats that th ECP mthod additionally rquirs th matrial intrpolation with th spcial pnalization factor satisfying th drivd condition, n n n, in th ECP _ s s calculation of th strss for locally dfind strss constraints. 7. Rfrncs [] S.J. Moon and G.H. Yoon, A nw dvlopmnt of th qp-rlaxation mthod of lmnt connctivity paramtrization to achiv strss-basd topology optimization for gomtrically nonlinar structur, in rviw, 03. [] M.P. Bndso and O. Sigmund, Topology Optimization: Thory, Mthods and Applications, Springr-Vrlag, Brlin, 003. [3] M. Bruggi, P. Duysinx, Topology optimization for minimum wight with complianc and strss constraints, Struct Multidiscip O, 46 (0) [4] M. Bruggi, On an altrnativ approach to strss constraints rlaxation in topology optimization, Struct Multidiscip O, 36, 5-4,008. [5] S.H. Jong, S.H. Par, D.H. Choi, G.H. Yoon, Topology optimization considring static failur thoris for ductil and brittl matrials, Computr & Structurs, 0-,6-3,0. [6] C. L, J. Norato, T. Bruns, C. Ha, D. Tortorlli, Strss-basd topology optimization for continua, Struct Multidiscip O, 4,605-60, 00. [] G.H. Yoon, Y.Y. Kim, Elmnt connctivity paramtrization for topology optimization of gomtrically nonlinar structurs, Int J Solids Struct, 4, ,

TOPOLOGY DESIGN OF STRUCTURE LOADED BY EARTHQUAKE. Vienna University of Technology

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