Plate/shell topological optimization subjected to linear buckling constraints by adopting composite exponential filtering function

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1 Acta Mech. Sn. 2016) 324): DOI /s RESEARCH PAPER Plate/shell topologcal optmzaton subjected to lnear bucklng constrants by adoptng composte exponental flterng functon Hong-Lng Ye 1 We-We Wang 1 Nng Chen 1 Yun-Kang Su 1 Receved: 29 June 2015 / Revsed: 4 September 2015 / Accepted: 7 October 2015 / Publshed onlne: 17 November 2015 The Authors) Ths artcle s publshed wth open access at Sprngerlnk.com Abstract In ths paper, a model of topology optmzaton wth lnear bucklng constrants s establshed based on an ndependent and contnuous mappng method to mnmze the plate/shell structure weght. A composte exponental functon CEF) s selected as flterng functons for element weght, the element stffness matrx and the element geometrc stffness matrx, whch recognze the desgn varables, and to mplement the changng process of desgn varables from dscrete to contnuous and back to dscrete. The bucklng constrants are approxmated as explct formulatons based on the Taylor expanson and the flterng functon. The optmzaton model s transformed to dual programmng and solved by the dual sequence quadratc programmng algorthm. Fnally, three numercal examples wth power functon and CEF as flter functon are analyzed and dscussed to demonstrate the feasblty and effcency of the proposed method. Keywords Topologcal optmzaton Plate/shell structure Lnear bucklng constrant Independent contnuous and mappng ICM) method Flter functon 1 Introducton Structural topology optmzaton s to fnd optmal materal layout wthn a gven desgn space, for a gven set of loads and boundary condtons such that the resultng layout meets a prescrbed set of performance targets. The B Hong-Lng Ye yehongl@bjut.edu.cn 1 College of Mechancal Engneerng and Appled Electroncs Technology, Bejng Unversty of Technology, Bejng , Chna essence of topology optmzaton les n searchng for the optmum path of transferrng loads, therefore the computatonal results of topology optmzaton are usually more attractve and more challengng than the results of crosssectonal and shape optmzaton. In the last decades, snce the landmark paper of Bendsøe and Kkuch [1], numercal methods for topology optmzaton of contnuum structures have been developed quckly n applcaton [2 4]. The classcal methods nclude the homogenzaton method [5,6], the varable densty method ncludng sold sotropc materal wth penalzaton model SIMP) and ratonal approxmaton of materal propertes RAMP) nterpolaton model) [7 10], evolutonary structural optmzaton ESO) [11 13], level set method [14 16], and so on. The plate/shell structure s popular for lghtweght constructons n natonal defense and cvl ndustres. However, t s shown from both research lterature and ndustral applcatons that plate/shell structures are prone to buckle. As bucklng affects the securty of the whole structure, t s necessary to address the structural stablty durng structure desgn. Bucklng topology optmzaton of plate/shell s to fnd optmal materal layout of plate/shell structure that meets the bucklng requrements. Although bucklng topology optmzaton s only n the phase of conceptual desgn n engneerng, the optmal results wll mpact the performance of the fnal structure sgnfcantly. Compared wth statc topology optmzaton, bucklng topology optmzaton s more complcated, and there are few nvestgatons up tll now. In 1995, lnear bucklng topology optmzaton of two-dmensonal structures had been studed by Neves et al. [17], the optmzaton results lay the foundatons for the non-lnear bucklng optmzaton. Meanwhle, Seo [18] studed the topology optmzaton of nner-wall stffener of cylndrcal contaners. The recprocal of crtcal bucklng load was adopted as an objectve functon, and the total

2 650 H.-L. Ye et al. mass of stffener was constraned to a prescrbed value. Later, Neves et al. [19] presented a two-scale asymptotc method for the lnearzed elastc stablty analyss. Topology optmzaton of the perodc mcrostructures s carred out based on the local bucklng nstabltes n the perodc boundary condtons PBC). Combnng the lnearzed elastc bucklng model wth the nverse homogenzaton and an egenvalue bucklng analyss wth Floquet-bloch wave theory [20], the mnmum crtcal bucklng stran s obtaned and maxmzed wth the PBC havng a constant volume fracton of materals. In 2002, Ramm et al. [21,22] constructed the topology optmzaton model wth lnear bucklng constrants based on the SIMP method to study the nfluence of geometrcal nonlnear behavor to topology optmzaton desgn. In 2004, the shell s topologcal optmzaton under lnear bucklng response usng SIMP power-law penalzaton of stffness was gven to acheve the dscrete topology [23]. In 2009, Lund [24 26] studed the bucklng topology optmzaton of lamnated mult-materal composte shell structures by ntroducng nterpolaton functons, whch s from the SIMP approach. In 2012, Browne [27] studed the method of solvng the large-scale quadratc programmng problem, and the method s appled to the topology optmzaton problem usng complance and bucklng as constrants wth the mnmum structural weght as objectve. In 2013, Lndgaard [28] studed the statc geometry nonlnear structure topology optmzaton of nstablty to maxmze the bucklng crtcal load. Up tll now, dfferent optmzaton methods have been used to solve the bucklng structure topology optmzaton problems, however, there s no unform vald method to deal wth the bucklng topology optmzaton of plate/shell. Ths s so snce buldng the bucklng topology optmzaton model s more complex and dffcult than statc topology optmzaton, and calculatons for senstve analyss are enormous. In ths paper, we nvestgate bucklng topology optmzaton based on ndependent, contnuous and mappng ICM) method, proposed by Su [29] for skeleton and contnuum structural topology optmzaton n The topologcal varables are ndependent of desgn varables such as sectonal szes, geometrcal shape, densty or Young s modulus of materal. Flter functons are used to map the changng process of topologcal desgn varables from dscrete to contnuous and back to dscrete. The smooth model for structural topology optmzaton s establshed and solved by the tradtonal algorthms n mathematcal programmng. The ICM method has been manly used to study statc and dynamc topology optmzaton [30 32]. We extend ths method and do n depth research for the bucklng problem. A model of topology optmzaton for the lghtest plate/shell structures wth the crtcal bucklng load constrants s constructed. Usually, a power functon PF) s selected as the flter functon n the past, and we select a composte exponental functon CEF) as the flterng functon to complete the changng process of desgn varables. The optmal results wth two dfferent flter functons are compared by numercal examples. Ths paper s organzed as follows. In Sect. 2, a bucklng topology optmzaton model of plate/shell structure based on the ICM method s establshed. In Sect. 3, a strategy for solvng the bucklng optmzaton model s ntroduced. An optmal algorthm to solve the mathematcal optmzaton problem s gven. In Sect. 4, the program flow of the optmzaton algorthm s charted. Three numercal smulatons are presented n Sect. 5. In Sect. 6, conclusons are gven. 2 Establshment of lnear bucklng topology optmzaton model 2.1 Lnear bucklng analyss of plate/shell structures Structural bucklng wdely exsts n practcal engneerng structure. Bucklng s a mathematcal nstablty, leadng to a falure mode. As the appled load s ncreased on a structure by a small amount beyond the crtcal load, the structure deforms nto a buckled confguraton. Further load wll cause sgnfcant and somewhat unpredctable deformatons, possbly leadng to complete loss of the structural load-carryng capacty. The nterpretaton of ths result s that for P < P cr j, the structure remans stable. For P > P cr j, the structure s unstable and buckles. P cr j s the crtcal load for bucklng. Usually, once the form of structure s establshed, ts bucklng wll have a varety of modes and multple crtcal loads. The structure wll not work before the mode reaches hgherorder bucklng mode, so we just care about the frst-order crtcal load of bucklng mode. In ths paper, the lnear elastc and pre-bucklng of contnuum structure s consdered. Assumng the structure to be perfect wth no geometrcal mperfectons, stresses are proportonal to the loads,.e., stress stffness depends lnearly on the load, dsplacements at the bucklng confguraton are small, and the load s ndependent of the dsplacements. The lnear bucklng problem can be represented as [28,29] K + λ j G ) u j = 0, j = 1, 2,...,J), 1) where K and G denote the structural stffness matrx and geometrc stffness matrx, respectvely, λ j s the j-th egenvalue,.e., bucklng crtcal load factor and u j descrbes the correspondng egenvector, j denotes the j-th order of the modal. 2.2 Descrpton of the flter functon based on the ICM method Flter functon s the key strategy of the ICM method. It dentfes the correspondng element or subdomans of geometrc

3 Plate/shell topologcal optmzaton subjected to lnear bucklng constrants quantty or physcal quantty, such as the element weght, the element stffness matrx, and the element geometrc stffness matrx. Dscrete desgn varables can be mapped to contnuous varables by flter functon and nversed back to dscrete varables. For the bucklng topology optmzaton model, we defne element weght, the element stffness matrx, and the element geometrc stffness matrx as follows w = f w t ) w 0, k = f k t ) k 0, 2) g = f g t ) g 0, where t s the topology varable value of the -th element. w, k, g denotes the element weght, stffness matrx, and geometrc stffness matrx of -th element n the optmal process, respectvely. And w 0, k0, g0 represent the ntal element weght, stffness matrx, and geometrc stffness matrx of the -th element, respectvely. f w t ), f k t ), and f g t ) are the flter functons of the element weght, element stffness matrx, and the element geometrc stffness matrx. In addton, the element weght, element stffness matrx, element geometry stffness matrx, and element qualty matrx are changed by takng advantage of flter functons. These physcal quanttes of every element change a lot when the structural topology changes, and then the flter functons n the formulaton can lead to convergence. Furthermore, flter functons nfluence the speed of convergence and the stablty of the soluton of the optmal process. Several types of flter functon are suggested n the ICM method [33]. Among whch, the PF s used frequently as follows f t ) = t α, α 1. 3) Here, α s a gven postve constant. Now, we ntroduce a new flter functon CEF to take the place of PF, and t s as follows f t ) = et /γ e 1/γ, γ > 0, 4) where γ s a gven postve constant. In Sect. 5, the performances of the two types of flter functon are compared. From Eqs. 2), 3), and 4), the specfc expressons of PF and CEF n the model of bucklng topology optmzaton are gven : f w t ) = t α w, f k t ) = t α k, f g t ) = t α g, f w t ) = e t /γ w )/ ) e 1/γ w, f k t ) = e t /γ k )/ ), f g t ) = e t /γ g )/ ) e 1/γ g. 5) It should be ponted out that these parameters of flter functons can be determned by usng the least squares method or numercal experments, see Refs. [29 31]. 2.3 Mathematcal model of bucklng topology optmzaton Based on the ICM method, the optmal model to mnmze the structural weght subjected to the lnear bucklng constrants s as follows fnd t E N, make W = N w mn, s.t. P cr j P cr j, j = 1, 2,, J), 0 t 1, = 1, 2,, N), where t denotes the vector of topologcal desgn varables, W s the structural weght, and w s the element weght of structure, P cr j presents the crtcal bucklng load, P cr j s the lower lmt bucklng crtcal load, and J and N denote the total number of the bucklng modes and number of elements. The relatonshp between crtcal load and external load P can be expressed as P cr j = λ j P. 7) Then the bucklng crtcal load factor s used as constrants n the optmal model. The bucklng topology optmal model 6) can be transformed as follows fnd t E N, make W = N w mn, s.t. λ j λ j, j = 1, 2,, J), 0 t 1, = 1, 2,, N), where λ j s the lower lmt bucklng crtcal load factor. In order to solve the optmal model, the recprocal of flter functon wth stffness matrx s used as a desgn varable as follows x = 1 / f k t ). 9) Therefore, the topologcal desgn varable s expressed as t = fk 1 / ) 1 x. 10) Then Eq. 2) can be transformed nto Eq. 11) w = f w fk 1 / ) ) 1 x w 0, k = 1 / ) x k 0, 11) / ) ) g = f g 1 x g 0. f 1 k 6) 8)

4 652 H.-L. Ye et al. Wth the ntroducton of flterng functons and the recprocal of flter functon of stffness matrx, the optmzaton model 8) s wrtten as fnd x E N, make W = N f w fk 1 / ) ) 1 x w 0 mn, s.t. λ j x) λ j, j = 1, 2,, J), 1 x x, = 1, 2,, N), 3 Strategy for solvng the bucklng optmzaton model 3.1 Desgn senstvty analyss 12) To estmate the desgn senstvty, we have to consder the dervatve of the egenvalue n Eq. 1). The egenvalues can be expressed usng the Raylegh quotent: λ j = ut j Ku j u T j Gu. 13) j The dervatve of the egenvalue s gven as follows [ λ j u T / ) j K x u j + λ j u T / ) ] j G x u j = x u T j Gu, 14) j K = x x 2 k 0 = k, 15) x G = βx ) f g x x )) 1 fk 1 g 0 = βx ) g x x. 16) β can be obtaned accordng to dfferent type of flter functon. When PF s selected as flter functon, β x ) = f g t ) f k t ) f g t ) f k t ) = α g. 17) α k And CEF acts as flter functon, β x ) = γ k e t /γ k ) 1 γ g e t /γ g )et γg γ 1 k ). 18) Therefore, Eq. 14) s deduced as λ j x = U j + λ j β x ) V j V j x. 19) Here, U j = 0.5u T j k u j and V j = 0.5u T j g u j represent the stran energy and geometrc stran energy for the -th element n j-th mode, respectvely. V j = u T j Gu j s the structural geometrc stran energy n the j-th mode. 3.2 Explct approxmaton of bucklng constrants As the constrant s mplct about desgn varables, we make t explct by usng the frst order Taylor expanson: λ j x) λ j x v)) + / λ j x x x v)). 20) Here, the superscrpt v s the number of teratons. Take Eq. 20) nto Eq. 21), we have λ j x) λ j x v)) + 1 A j x v) x A j, 21) where A j = U j+λ j βx )V j V. j Then the bucklng constrants n model 12) can be expressed as 1 A j We set x v) c j = A j 1 x v) x λ λ j x v)) + A j. 22), d j = λ λ j x v)) + A j. So the bucklng constrants can be smplfed to the followng form c j x d j. 23) 3.3 The standardzaton of the objectve functon In order to obtan an explct objectve, the second-order Taylor expanson s used. When PF s selected as a flter functon, the structural weght can be wrtten Here, w 0 x α w/α k a x 2 + b x ). 24) α α + 1) a = 2 x ) α+2 w0, b α α + 2) = x ) α+1 w0, / α = α w αk. When CEF acts as flter functon, the structural weght can be wrtten

5 Plate/shell topologcal optmzaton subjected to lnear bucklng constrants [ e 1/γ k ) /x + 1 ] γ k /γ w e 1/γ w w 0 where, a = 1 2 γ k γ w b = γ k γ w w 0 x v) [ γk + 1 γ w w 0 x v) [ γk + 2 γ w ) 4 e 1/γ w ) ) ) 3 e 1/γ w ) ) a x 2 + b x ), ) γ k γw 2 x v) + 1 ] + 2x v) ) γ k γw 2 x v) + 1 ] + 3x v),. 25) Therefore, the standard quadratc programmng model for Eq. 12) can be obtaned as follows fnd x E N, make W = N a x 2 ) + b x mn, s.t. c j x d j, j = 1, 2,...,J), 1 x x, = 1, 2,...,N). 3.4 Soluton of the optmzaton model 26) As the number of desgn varables s much bgger than that of the constrants, we deduce the dual model to decrease the number of desgn varables as follows n order to reduce the amount of calculaton. fnd z E J, make Φ z) = 2 1 zt H z + C T z mn, 27) s.t. z j 0, j = 1, 2, 3,..., J ), where z s the desgn varable vector of the dual model, Φ z) s the objectve functon, and Φz) = mnlx, z)), Lx, z) = a x 2 + b x + H jk = I a C j C k / 2a, J N ) z j c j x d j, j=1 C k = / ) N C k x + b 2a C k x + d k. I a In ths paper, the convergence crteron s chosen as follows W = W v+1) W v) ) / W v+1) ε. 28) W v+1) and W v) are the current teraton and the prevous teraton of structural weght. ε s the convergence precson, ε = Dscrete degree of topologcal desgn varables In order to measure the dscrete degree of topologcal desgn varables, we use M nd [34] as a crteron, and t s gven M nd = 4T 1 T ) N 100 %, 29) where T s the topologcal varable value for the -th element and N s the total number of the elements. Followng Eq. 29), f the topologcal varable value s 0 or 1, then M nd s 0; f the topologcal varable values s 0.5, then M nd s 1. The closer the topologcal varable value to 0 or 1, the smaller s the value of M nd and the better the optmal result. 4 Program flow of optmzaton algorthm The numercal procedures are developed by the PCL toolkt n the MSC. Patran software platform. We use MSC.Nastran to analyze the numercal soluton of Eq. 1). The correspondng program flow as shown below Step 1 Buld fnte element model by usng MSC. Patran; Step 2 Input ntal optmal parameters and set up optmal model; Step 3 Make bucklng analyss by usng MSC. Nastran; Step 4 Calculate and extract the crtcal bucklng factor and stran energy; Step 5 Input parameters of the optmal algorthm; Step 6 Solve the dual optmzaton model 27) by the dual sequence quadratc programmng DSQP); Step 7 Judge convergence of the optmal results. If the structural weght satsfes the formula 28), then the program s termnated. Otherwse, update desgn varables x and topology desgn varables t, then go to step3.

6 654 H.-L. Ye et al. Fg. 1 Base structure Fg. 4 Iteratve hstory curve of crtcal bucklng load Fg. 2 Topology confguraton wth CEF flter functon Fg. 3 Topology confguraton wth PF flter functon n Ref. [34] 5 Numercal examples In ths secton, three examples of topology optmzaton of sngle materal plate/shell structures are gven. All the materal s sotropc wth Young s modulus E = MPa, Posson s rato μ = 0.3. In the ntal desgn, the avalable materal s unformly dstrbuted over the admssble desgn doman. The structures are meshed by four-node 2D plate/shell fnte elements. The specfc boundary condton and force form are shown n the followng three examples. Example 1 As shown n Fg. 1, the base structure s a plane elastc body wth sze mm 3, and mass densty ρ = kg/mm 3. The dstrbuted compresson load at the top edge s 100 N/mm. Two corners of the bottom edge are fxed. The bucklng constrant value s 100 N n Ref. [34]. The regon whch ncludng the above two layers of the unt s a non-desgn and should be mantaned. The base structure s weght s 0.73 kg. The topology confguraton of the structure wth CEF flter functon s gven n Fg. 2. It s smlar to the topology confguraton wth PF as n Ref. [34] as shown n Fg. 3.The teratve hstory curve of bucklng load and structural weght are shown n Fgs. 4 and 5. The optmal structural weght Fg. 5 Iteratve hstory curve of structural weght wth CEF s kg and the teratve number s 36, as the optmal structural weght wth PF s kg and the teratve number s 51. From the pont vew of structural weght and teraton, the optmal results wth CEF s better that of PF. Example 2 As shown n Fg. 6, the base structure s a plane elastc body and the mass densty s ρ = kg/m 3. The forces P = N are located on the mdpont of the top and bottom boundary. Four corners of the structure are fxed. The base structure s weght s 0.73 kg. After fnte element analyss, the frst-order bucklng load factor of the structure s λ 1 = The crtcal bucklng load s 1600 N, and the bucklng load 1300 N s defned as constrant value. The topology confguratons of the structure wth dfferent flter functons before and after dscreton are gven n Fg. 7. In addton, the frst-order bucklng modal of optmal structure s computed as shown n Fg. 8. The teratve hstory of bucklng load and structural weght wth dfferent flter functons are depcted n Fgs. 9 and 10. We can see that the crtcal bucklng loads of the optmal structures satsfy the bucklng constrant. From Fg. 10, we can see a clear dffer-

7 Plate/shell topologcal optmzaton subjected to lnear bucklng constrants Fg. 6 Base structure Fg. 9 Iteratve hstory of the bucklng load wth dfferent flter functons Fg. 7 Topology confguraton wth dfferent flter functons before and after dscreton. a and b before dscreton wth PF and CEF, c and d after dscreton wth PF and CEF Fg. 10 Iteratve hstory of structural weght wth dfferent flter functons Table 1 Dstrbuton of topologcal desgn varables The range of topologcal values Element number PF CEF 0,0.1] Fg. 8 The frst-order bucklng modal of optmal structure wth dfferent flter functon. a PF. b CEF 0.1,0.2] ,0.3] ence wth PF and CEF n the structural weght and teratve number. The optmal structural weght wth CEF s lghter and the number of teratons s less than that of PF. The dstrbutons of topologcal desgn varables are lsted n Table 1. The dscretzaton of topologcal desgn varables s evaluated by usng Mnd. We can fnd that Mnd wth PF and CEF are % and 7.25 %. Therefore, the optmal result wth CEF s better than that of PF from the vew of dscretzaton of topologcal desgn varables. 0.3,0.4] ,0.5] ,0.6] ,0.7] ,0.8] ,0.9] ,1] Total element numbers Mnd % 7.25 % Example 3 As shown n Fg. 11, the base structure s a part of the cylndrcal shell, the generatrx s 260 mm, arc length s 520 mm, and the radus s 5000 mm. The force P = N s located on the center of the cylndrcal shell along the radal drecton. After fnte element analyss, the frst-order bucklng load factor of the structure s λ1 = The crtcal bucklng load s 48.7 N, and the crtcal bucklng load 40 N s defned as the bucklng constrant value.

8 656 H.-L. Ye et al. Fg. 11 Base structure The ntermedate results and optmal topology confguraton of the structure wth dfferent flter functons are ndcated n Fg. 12. The teratve hstory curve of bucklng load and structural weght wth PF and CEF s gven n Fgs. 13 and 14. The performances of topologcal optmzaton wth dfferent flter functons are gven n Table 2. From Fg. 13, we get that the optmal structure wth PF and CEF all satsfy the bucklng constrant. The optmal structural weght wth CEF s lghter than that of PF as shown n Fg. 14. From the above three examples, we can see that the objectve weght) wth CEF s apparent lghter than that of PF. We can also fnd that the optmal result wth CEF has the best performance from the pont of vew of teratve number. Fg. 13 Iteratve hstory of the bucklng load wth dfferent flter functons The dstrbuton of optmal topologcal values show that the percentages of M nd wth CEF s lower than that of PF, so the CEF flter functon has the best performance from the vewpont of dscreteness. 6 Concluson In ths paper, a bucklng topologcal optmal model of plate/shell structure s establshed based on the ICM method. Fg. 12 The ntermedate results and optmal topology confguraton of the structure wth dfferent flter functons. a The ntermedate optmal results wth PF. b The ntermedate optmal results wth CEF

9 Plate/shell topologcal optmzaton subjected to lnear bucklng constrants Fg. 14 Iteratve hstory of structural weght wth dfferent flter functons Table 2 Optmal results wth dfferent flter functons Iteratve number Bucklng constrant/n Weght/kg PF CEF CEF s selected as a flter functon to recognze the desgn varables, as well as to mplement the changng process of desgn varables from dscrete to contnuous and back to dscrete. Explct formulatons of bucklng constrants are gven based on two dfferent flter functons, frst-order Taylor seres expanson by extractng structural stran and structural knetc energy from the results of structural modal analyss. The program based on DSQP for solvng the optmal model s developed on the platform of MSC. Patran & Nastran. Three numercal examples of contnuum structure show that clear and stable confguratons can be obtaned by usng the ICM method. We also fnd that confguratons computed wth PF and CEF are smlar. But we can see that the optmal result wth CEF has the better performance from the pont of vew of optmal objectve, teratve numbers, and dscrete degree. Acknowledgments The project was supported by the Natonal Natural Scence Foundaton of Chna Grants , ). Open Access Ths artcle s dstrbuted under the terms of the Creatve Commons Attrbuton 4.0 Internatonal Lcense ons.org/lcenses/by/4.0/), whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded you gve approprate credt to the orgnal authors) and the source, provde a lnk to the Creatve Commons lcense, and ndcate f changes were made. References 1. Bendsøe, M.P., Kkuch, N.: Generatng optmal topologes n structural desgn usng a homogenzaton method. Comput. Methods Appl. Mech. Eng. 71, ) 2. Eschenauer, H.A., Olhoff, N.: Topology optmzaton of contnuum structures: A revew. Appl. Mech. Rev. 54, ) 3. Deaton, J.D., Grandh, R.V.: A survey of structural and multdscplnary contnuum topology optmzaton: Post Struct. Multdscp. Optm. 49, ) 4. Rozvany, G.I.N.: Ams, scope, methods, hstory and unfed termnology of computer-aded topology optmzaton n structuralmechancs. Struct. Multdscp. Optm. 21, ) 5. Bendsøe, M.P., Sgmund, O.: Topology Optmzaton: Theory, Methods and Applcatons, 2nd edn. Sprnger, Berln 2003) 6. Hassan, B., Hnton, E.: A revew of homogenzaton and topology optmzaton homogenzaton theory for meda wth perodc structure. Comput. Struct. 69, ) 7. Cao, M.J., Ma, H.T., We, P.: A novel robust desgn method for mprovng stablty of optmzed structures. Acta Mech. Sn. 31, ) 8. Zhang, H., Lu, S.T., Zhang, X.: Topology optmzaton of 3D structures wth desgn-dependent loads. Acta Mech. Sn. 26, ) 9. Sgmund, O.: A 99 lne topology optmzaton code wrtten n Matlab. Struct. Multdscp. Optm. 21, ) 10. Bendsøe, M.P., Sgmund, O.: Materal nterpolaton schemes n topology optmzaton. Arch. Appl. Mech. 69, ) 11. Xe, Y.M., Steven, G.P.: A smple evolutonary procedure for structural optmzaton. Comput. Struct. 49, ) 12. Xe, Y.M., Steven, G.P.: Evolutonary Structural Optmzaton. Sprnger, Berln 1997) 13. Rozvany, G.I.N., Quern, O.M., Gaspar, Z., et al.: Weght ncreasng effect of topology smplfcaton. Struct. Multdscp. Optm. 25, ) 14. Luo, J., Luo, Z., Chen, S., et al.: A new level set method for systematc desgn of hnge-free complant mechansms. Comput. Methods Appl. Mech. Eng. 198, ) 15. Wang, M.Y., Chen, S., Wang, X.: Desgn of multmateral complant mechansms usng level-set methods. J. Mech. Des. 127, ) 16. Wang, M.Y., Wang, X., Guo, D.: A level set method for structural topology optmzaton. Comput. Methods Appl. Mech. Eng. 192, ) 17. Neves, M.M., Rodrgues, H., Guedes, J.M.: Generalzed topology crteron desgn of structures wth a bucklng load crteron. Struct. Optm. 10, ) 18. Seo, Y.-D., Youn, S.-K., Yeon, J.-H., et al.: Topology optmzaton of nner-wall stffener for crtcal bucklng loads of cylndrcal contaners. In: Multdscplnary Analyss and Optmzaton Conference, 30 August-1 September. Albany: New York 2004) 19. Neves, M.M., Sgmund, O., Bendsøe, M.P.: Topology optmzaton of perodc mcrostructures wth a penalzaton of hghly localzed bucklng modes. Int. J. Numer. Methods Eng. 54, ) 20. Neves, M.M.: Analyss and contnuum topology optmzaton of perodc solds wth lnearzed elastc bucklng crteron. In: IUTAM Symposum on Topologcal Desgn Optmzaton of Structures, Machnes and Materals: Status and Perspectves, pp ) 21. Ramm, E., Kemmler, R.: Stablty and Large Deformatons n Structural Optmzaton. Warsaw 2002) 22. Kemmler, R., Lpka, A., Ramm, E.: Large deformatons and stablty n topology optmzaton. Struct. Multdscp. Optm. 30, ) 23. Zhou, M.: Topology optmzaton for shell structures wth lnear bucklng. Comput. Mech. b0105, ) 24. Lund, E.: Bucklng topology optmzaton of lamnated multmateral composte shell structures. Compos. Struct. 91, )

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