Int J Adv Manuf Technol (2007) 32: 787 796 DOI 101007/s00170-005-0387-0 ORIGINAL ARTICLE Kong-Tan Zuo L-Png Chen Yun-Qng Zhang Jngzhou Yang Study of key algorthms n topology optmzaton Receved: 30 Aprl 2004 / Accepted: 22 Aprl 2005 / Publshed onlne: 21 March 2006 # Sprnger-Verlag London Lmted 2006 Abstract The theory of topology optmzaton based on the sold sotropc materal wth penalzaton model (SIMP) method s thoroughly analyzed n ths paper In order to solve complcated topology optmzaton problems, a hybrd soluton algorthm based on the method of movng asymptotes (MMA) approach and the globally convergent verson of the method of movng asymptotes (GCMMA) approach s proposed The numercal nstablty, whch always leads to a non-manufacturng result n topology optmzaton, s analyzed, along wth current methods to control t To elmnate the numercal nstablty of topology results, a convoluton ntegral factor method s ntroduced Meanwhle, an teraton procedure based on the hybrd soluton algorthm and a method to elmnate numercal nstablty are developed The proposed algorthms are verfed wth llustratve examples The effect and functon of the hybrd soluton algorthm and the convoluton radus n optmzaton are also dscussed Keywords Topology optmzaton MMA seres algorthms Hybrd soluton algorthm Numercal nstablty Convoluton ntegral factor method 1 Introducton Topology optmzaton s consdered to be one of the most challengng felds n structural optmzaton Topology optmzaton has been an nterestng research topc snce Bendsoe and Kkuch [1] ntroduced the homogenzaton K-T Zuo L-P Chen Y-Q Zhang Center for Computer-Aded Desgn, Huazhong Unversty of Scence & Technology, Wuhan, Hube, 430074, People s Republc of Chna e-mal: zuokt@hustcadcom J Yang (*) Center for Computer-Aded Desgn, The Unversty of Iowa, Iowa Cty, IA 52246, USA e-mal: yang@engneernguowaedu Tel: +1-319-3532249 Fax: +1-319-384-0542 method nto the feld After more than 20 years of development, researchers have made a great deal of progress n theores as well as n real engneerng applcatons Recent research n topology optmzaton has focused on such topcs as: materal nterpolaton methods, optmal algorthms sutable for topology optmzaton, methods to elmnate numercal nstabltes, and the applcaton of topology optmzaton n engneerng Materal nterpolaton used n topology optmzaton manly nclude the followng two types of methods: the homogenzaton method [1, 2], prmarly used n theory dervatons, such as the exstence of solutons, nvestgatng numercal nstabltes, etc; and the materal densty method, sometmes named the sold sotropc materal wth penalzaton model (SIMP) method, proposed by Mlenek and Schrrmacher [3], Sgmund [4], and Bendsoe and Sgmund [5] The SIMP method, whch s used n ths paper, has been accepted by most researchers and engneers, and has been successfully used n many engneerng felds At present, two types of optmzaton algorthms are used n topology optmzaton: the optmalty crtera (OC) approach and the method of movng asymptotes (MMA) seres approach The former [6, 2] s usually deduced by a Lagrange functon composed of obectve and constrant functons accordng to the Kuhn-Tucker condton Ths approach has good convergence because t s based on heurstc formulaton and does not need the dervatves of functons However, ts dsadvantage s that t has a narrow restrcton n the sngle-constrant condton and s dffcult to use n problems wth multple constrants The MMA approach [7 9, 12] can ft sngle and multple constrants condtons, so t has wder engneerng applcatons, such as the desgn of a complant mechansm, mcroelectromechancal systems (MEMS), and multdscplne optmzaton The dsadvantage of the MMA approach exsts n the bad convergence of calculaton The globally convergent verson of the method of movng asymptotes (GCMMA) has a better convergence as compared to the MMA approach However, the calculaton speed of GCMMA s not perfect A new hybrd algorthm based on MMA and GCMMA s proposed Ths hybrd algorthm combnes the
788 advantages of both MMA and GCMMA together and can be wdely used n complcated engneerng problems One mportant focus of topology optmzaton s developng a method to elmnate numercal nstablty, snce t may result n a non-manufacturable structure n engneerng Many methods, such as the permeter control method [16], densty slope control [17], and meshndependent flter [18], have been proposed to deal wth ths problem But most of them have some shortcomngs Ths paper frst revews the SIMP method n topology optmzaton Then, the MMA approach and GCMMA approach based on SIMP s appled to topology optmzaton, and a hybrd soluton algorthm based on both MMA and GCMMA s proposed To elmnate numercal nstablty n the topology optmzaton, a convoluton ntegral factor method s developed, and an teraton procedure ncludng the hybrd soluton algorthm and the convoluton ntegral factor method s presented Fnally, an llustratve example s used to dscuss the effect and functon of the hybrd algorthm and the convoluton ntegral factor method 2 SIMP method n topology optmzaton The general topology optmzaton problem of mnmal complance can be defned as: mnmze : C ¼ F T U subect to : V ¼ f V 0 ¼ XN x e v e V (1a) (1b) and E are the elastc moduluses of an element before and after optmzaton, respectvely, then E ¼ ðx e Þ p E 0 : If k 0 and k e are the element s ntal stffness and the after-optmzaton stffness matrces, respectvely, the followng relatonshp exsts: k e ¼ ðx e Þ p k 0 : The parameter p s a penalty factor, and t s mportant to penalze the mddle densty n order to decrease the number of mddle-densty elements and ensure that most element denstes are close to zero or one Gven the above precondtons, every element has only one desgn varable Compared to the homogenzaton method, the SIMP approach makes excellent progress on decreasng the number of desgn varables Another advantage of ths SIMP method s that the materal characterstc after the change can be wrtten as the exponent functon of the ntal element densty and the ntal materal characterstc, so that ths approach greatly smplfes the soluton of topology optmzaton Snce a relatonshp exsts n the dscrete element: V ¼ f V 0 ¼ PN x e v e ; where v e s the element volume after optmzaton, the topology optmzaton formulaton of the mnmal complance problem based on the SIMP approach wll be: mnmze : C ¼ U T KU ¼ XN subect to : V ¼ f V 0 ¼ XN F ¼ KU u e k e u e ¼ XN x e v e V ðx e Þ p u e k 0 u e F ¼ KU (1c) 0 < x mn x e x max (2) 0 x e 1 (1d) where C s the complance of the structure, F s the force vector, U s the dsplacement vector, K s the stffness matrx of the structure, V 0 s the ntal volume of the structure, V s the structure s volume after optmzaton, and f s the rato of the volume after optmzaton wth the ntal volume The volume constrant and equlbrum equaton of the structure are ncluded In the SIMP approach, the precondtons nclude: 1 The materal characterstcs, such as the elastc modulus, n a dscrete element are constant The desgn varable s the densty of the element, represented wth x e Ifρ 0 s the ntal element densty and ρ s the element densty after optmzaton, then ρ ¼ x e ρ 0 exsts 2 The materal characterstcs n an element should be changed wth the exponent of element densty If E 0 where xmn s the lower bound of the densty, whch s ntroduced to prevent sngularty of the equlbrum problem, and x max s the upper bound of the densty, u e s the element dsplacement, and N s the total number of dscrete elements 3 A new hybrd soluton algorthm for topology optmzaton 31 MMA and GCMMA approach based on the SIMP method The MMA approach, whch was ntally proposed by Svanberg [7, 8], s based on the frst-order Taylor seres expanson of the obectve and constrant functons Wth ths method, an explct convex subproblem s generated to approxmate the mplct nonlnear problem Because the subproblem s separable and convex, a dual approach or a prmal-dual nteror-pont method can be used to solve t
Construct the MMA/GCMMA form of ntal problem Choose a ntal calculaton pont Calculaton functon value and ther derves of obectve and constrants Construct MMA/GCMMA subproblem Solve MMA/GCMMA subproblem and fnd the approxmate soluton of ntal problem where: eðkþ f ðxþ ¼f x þ ðkþ Xn Xn ¼1 ð u ðkþ ¼1 p ðkþ ¼ 1; ; m x ðkþ p ðkþ þ q! ðkþ u ðkþ x x l ðkþ þ x ðkþ qðkþ l ðkþ Þ 789 (4b) Update the bound of desgn varables and non-monotonous parameters No [10, 13, 14] The solutons of a sequence of subproblems can converge towards the orgnal problem A general formulaton of nonlnear optmzaton can be wrtten as: mnmze : f 0 ðxþþa 0 z þ XM ¼1 subect to : f ðxþ a z y 0 x mn x x max y 0; z 0 ðc y þ 1 2 d y 2 Þ ¼ 1; ; m ¼ 1; ; n ¼ 1; ; m where the desgn varable x ¼ ðx 1 ; ; x n Þ T 2< n ; y ¼ ðy 1 ; ; y m Þ T 2< m ; z 2<; functons f 0, f 1,, f m are contnuous and dfferentable, x mn and x max are real numbers and satsfy x mn x max, and the coeffcents a 0, a, c 0, and d are postve real numbers The obectve and constrant functons are lnearly expanded at pont 1 u x or 1 x l to construct a subproblem The subproblem s then solved to approxmate the ntal problem The subproblem of Eq 3 can be expressed as: mnmze : e f ðkþ 0 ðxþþa 0 z þ XM ¼1 ðc y þ 1 2 d y 2 Þ subect to : e f ðkþ ðxþ a z y 0 ¼ 1; ; m α ðkþ x β ðkþ y 0; z 0 Convergence? Yes Output result The end Fg 1 Flow chart of the MMA/GCMMA approach ¼ 1; ; n ¼ 1; ; m (3) (4a) 2 x ðkþ @f þ x ðkþ þk ðkþ ; @x ¼ x ðþ k l ðþ k 2 @f (4c) x ðþ k þ k ðþ k @x p ðkþ ¼ u ðkþ q ðþ k n o ¼ max x mn ; 0:9l ðkþ þ 0:1x ðkþ ; n o (4d) ¼ mn x max ; 0:9u ðþ k þ 0:1x ðþ k α ðkþ β ðþ k @f þ x ðkþ @x ¼ max 0; @f x ðkþ ; @x @f x ðkþ @x ¼ max 0; @f (4e) x ðkþ @x (k) (k) The upper asymptotes u and lower asymptotes l should be updated wth teraton as follows: l ðkþ ¼ x ðkþ s ðkþ x ðk 1Þ l ðk 1Þ ; (4f) u ðkþ ¼ x ðkþ þ s ðkþ u ðk 1Þ x ðk 1Þ Accordng to the sgn of @f @x ðx ðkþ Þ; only one of p (k) and q (k) s not zero at the same tme Therefore, the approxmaton of the subproblem to the orgnal problem n MMA s monotonous The GCMMA approach, an extenson of the MMA approach, was proposed by Svanberg [9, 11] In GCMMA, p (k) and q (k) are not zero smultaneously, and a nonmonotonous parameter ρ (k) s ntroduced, so that the followng relatons exst: p ðkþ ¼ u ðkþ x ðkþ 2 @f @x ¼ x ðþ k l ðþ k 2 @f @x q ðþ k! þ! x ðkþ þ ρðkþ u ðkþ l ðkþ 2! þ ρ ðþ k! x ðkþ 2 u ðþ k ð Þ l k (5a)
790 The non-monotonous parameter ρ (k) can be updated as follows: ρ ðkþ1þ ¼ 2ρ ðkþ f g ðkþ x ðkþ1þ < f x ðkþ1þ (5b) ρ ðkþ1þ ¼ ρ ðkþ f g ðkþ x ðkþ1þ f x ðkþ1þ The startng pont of ρ (0) can be selected as: ρ ð0þ ¼ 0 f ρ ð0þ 0 ¼ 1 X n @f x ð0þ =@x 5n x max x mn ρ ð0þ ¼1 f ρ ð0þ > 0 (5c) Therefore, the GCMMA s a non-monotonous approxmaton of the ntal problem The soluton procedure of the MMA/GCMMA approach s shown n Fg 1 For mnmal the complance problem of topology optmzaton, we can construct the general formula n MMA/GCMMA as follows: mnmze : subect to : X N X N ðx e Þ p u T e k 0u e þ z þ 1000y 1 x e V 0 f y 1 0 0 x e 1 e ¼ 1; ; N y 1 0; z 0 It s obvous that the topology optmzaton problem s a specal case of the nonlnear programmng problem n Eq 3 Appylng the followng relaton to Eq 3: f 0 ¼ XN (6) ðx e Þ p u T e k 0u e ; f 1 ¼ XN x e V 0 f (7) and a 0 =1, c 1 =1000, d 1 =0, and a 1 =1, yelds the problem n Eq 6 The obectve and constrant functon values are then calculated, as well as ther dervatves In the SIMP approach, the dervatves can be calculated as follows: @f 0 ¼ pðx e Þ p 1 u T e @x k 0u e ; e @ 2 f 0 @x 2 ¼ pðp 1Þðx e Þ p 2 u T e k 0u e e @f 1 ¼ 1; @2 f 1 ¼ 0 @x e @x 2 e (8a) (8b) We can then construct the MMA/GCMMA subproblem based on Eq 6 and solve the subproblem to obtan the densty dstrbuton of the structure topology A sequence of solutons of the MMA/GCMMA subproblems gradually converges toward the global optmal of the densty dstrbuton 32 A new hybrd MMA GCMMA algorthm Some numercal tests [9] show that the GCMMA method s globally convergent, but ths has been proved only n sze optmzaton Applcatons and verfcatons for convergence and calculaton effcency of the GCMMA approach for large-scale topology optmzaton problems have not been reported It can be concluded mathematcally [15] that the optmzaton process can converge very quckly when a monotonous method, such as a one-order dervatve approxmaton, s used to approxmate a complcated problem at the desgn pont far away from the optmum, whereas the optmzaton process converges slowly f monotonous approxmaton s used at the desgn pont near the optmum On the other hand, when a nonmonotonous method, such as a second-order dervatve approxmaton, s used to approxmate a complcated problem, a stable and convergent soluton can be obtaned f the ntal desgn pont s properly selected, whereas calculaton may be slow f a bad ntal desgn pont far away from the optmum s used The topology optmzaton problem often exhbts hgh non-lnearty and a non-monotonous structural behavor The characterstc of the obectve and constrant functons s very complcated, and usually conssts of a non-convex and mplct form of desgn varables The number of desgn varables s very large MMA nvolves monotonous approxmaton to solve the problem, whereas GCMMA nvolves non-monotonous approxmaton Some research [12] has shown that t s mportant to select an approprate approxmaton scheme for calculatons The convergence process can be accelerated f a monotonous approxmaton s used for a monotonous structural response functon or f a non-monotonous approxmaton s used for a non-monotonous functon Oppostely, the convergence process may be slow f a monotonous approxmaton s used for a nonmonotonous structural response functon or a non-monotonous approxmaton s used for a monotonous functon Therefore, a non-monotonous GCMMA approach should be used to approxmate non-monotonous topology optmzaton At the same tme, the fast convergence characterstc of the MMA approach at a desgn pont far away from the optmum ought to be fully utlzed Some calculatons show that, when a smplex MMA approach s used n topology optmzaton, the calculaton can converge quckly and the obectve functons decrease rapdly at the frst few teratons However, the convergence and calculaton speed slow down, and, usually, some numercal oscllaton s exhbted when the desgn pont s near the optmum; fnally, the problem does not converge A smplex GCMMA approach used n topology optmzaton
791 can lead to a stable soluton, but the whole convergence process s very slow To deal wth large-scale and complex topology optmzaton, we propose a hybrd algorthm based on MMA and GCMMA, and provde a strategy for swtchng from the monotonous MMA approach to the non-monotonous GCMMA In our hybrd algorthm, the MMA approach s used when the desgn pont s far away from the optmum, where the quck convergence characterstc of the monotonous MMA approach can be fully utlzed After several teratons, the GCMMA approach s used when the desgn pont s near the optmum, where the non-monotonous characterstc of GCMMA can be fully utlzed In short, the hybrd algorthm has the combned advantages of MMA and GCMMA, and ensures calculaton effcency and convergence The flow chart of ths hybrd algorthm s llustrated n Fg 2 An mportant step n the hybrd algorthm s decdng when to swtch the optmzer from MMA to GCMMA As dscussed above, some numercal oscllaton s usually exhbted n topology optmzaton wth a smplex MMA approach when the desgn pont s near the optmum Therefore, the oscllaton value of the obectve functon can be used as a swtch condton The optmzer s swtched from MMA to GCMMA f the oscllaton s large enough The followng condton s checked as a threshold for swtchng from MMA to GCMMA The obectve nformaton of the current step and prevous two steps s used: f 0 x ðk 2Þ f0 x ðk 1Þ ðf 0 ðx ðk 2Þ Þþf 0 ðx ðk 1Þ ÞÞ=2 f 0 x ðk 1Þ f0 x ðkþ ðf 0 ðx ðk 1Þ Þþf 0 ðx ðkþ ÞÞ=2 < δ where δ s a small postve real number Convergence? Yes Output result The end No Defne the ntal desgn varables Fnte element analyss Update desgn varables wth MMA algorthm Numercal oscllaton? Yes Fnte element analyss Yes No Update desgn varables wth GCMMA algorthm Convergence? Output result The end Fg 2 Flow chart of hybrd MMA GCMMA algorthm No (9) 4 A method to elmnate numercal nstablty n topology optmzaton 41 Numercal nstablty n topology optmzaton Some numercal nstablty, such as checkerboard, mesh dependency, and porous materal, are often ncluded n topology optmzaton results, not only for the SIMP method but also for the homogenzaton method These phenomena often lead to a meanngless results because structures wth such flaws are dffcult to manufacture Porous materal means that many elements wth mddle densty are ncluded n the results, as shown n Fg 3a Results wth porous elements cannot be manufactured and are meanngless for engneerng purposes Checkerboard pattern, shown n Fg 3b, ndcates that the elements wth densty of ether 0 or 1 perodcally exst n the results Optmzaton results wth checkerboard pattern are also dffcult to manufacture and are meanngless from an engneerng standpont Mesh dependency means that the optmzaton results are correlated to the mesh densty A dfferent mesh densty may lead to a dfferent topology dstrbuton Normally, the fner mesh densty, then the greater the number of small structures ncluded n the results, as shown n Fg 3c e Results wth mesh dependency are also meanngless for engneerng applcatons 42 Current methods to elmnate numercal nstablty and ther shortcomngs Some of the reasons for numercal nstablty n topology optmzaton have been analyzed [20, 21] A few approaches [16, 17, 19] have been proposed to remove such nstablty Porous densty can be prohbted by ntroducng a penalty factor p nto the SIMP method Checkerboard pattern has some potental relaton to mesh dependency; a method desgned to avod checkerboard can often avod mesh dependency The research of Daz and Sgmund [20] demonstrates that checkerboard pattern has some connecton wth the element type A hgh-order-type element or non-conformng element can decrease or elmnate checkerboard pattern However, as the downsde, the calculaton cost ncreases wth a hgh-order element Some other methods have been used to elmnate checkerboard pattern and mesh dependency These technques nclude the permeter control method, densty slope control, and mesh-ndependent flter, etc These methods can be generally classfed nto two types The frst one s the global control approach, whch ncludes the permeter control method and the global gradent constrant method Ths approach can control and adust the global functon values and ther dervatves, but has lttle local effect Due to ths property, there are some locally superfluous structures n some engneerng problem results, such as very thn bars Furthermore, t s dffcult to determne the upper bound n permeter control and densty
792 a Porous materal b Checkerboard pattern c (100 240 elements) d (120 288 elements) e (140 336 elements) Fg 3 a Porous materal b Checkerboard pattern Mesh dependency: c 100 240 elements; d 120 288 elements; e 140 336 elements slope control methods If the permeter bound s too tght, there may be no soluton of the optmzaton problem, whle the method shows no strength f the constrant s too loose In addton, some extra constrant should be added to the optmzaton problem f the global approach s used, whch may lead to some convergence and numercal problems, although some advanced large-scale mathematcal programmng algorthms can solve problems wth extra constrants The second type s the local control approach, whch ncludes the densty slope control method and mesh-ndependent flter, n whch a local threshold s defned to flter the local varable The local approach may lack some global nformaton that needs to be fltered The mesh-ndependent flter s easly appled and has a good convergence wthout extra constrants added nto the optmzaton problem However, ths method s based on heurstc crtera The densty slope control method ntroduces a large number of extra constrants nto optmzaton, and, thus, t s computatonally prohbtve for practcal applcatons 43 A new method to guarantee numercal stablty As mentoned above, some shortcomngs exst n current methods to erase numercal nstablty However, It s very mportant to choose an approprate method to ensure that the results are manufacturable for engneerng purposes In theory, checkerboard pattern and porous materal can be seem as local sngulartes of numercal calculaton To elmnate these sngular numercal problems, a specal convoluton ntegral factor H (2r 1) (2r 1), n whch r, named the convoluton radus, s ntroduced to convolute the desgn varable and smooth or even the local sngularty n the desgn doman Ths process can elmnate checkerboard pattern and mesh dependency effectvely wth an approprate convoluton radus n matrx H The convoluton ntegral factor can be defned as follows: 2 3 1 1 1 1 1 H ð2r 1Þð2r 1Þ ¼ 3 1 r 1 r 1 r 1 1 1 r 1 r r 1 1 4r 3 r 1 r 1 r 1 r 1 1 6 4 7 5 1 1 1 1 1 (10a) 2 3 r ¼ 2 yelds H 33 ¼ 1 1 1 1 4 1 2 15 (10b) 10 1 1 1 2 3 1 1 1 1 1 r ¼ 3 yelds H 55 ¼ 1 1 2 2 2 1 1 2 3 2 1 35 6 7 4 1 2 2 2 15 1 1 1 1 1 (10c)
793 The desgn varable x k e (m, n) can be treated wth the convoluton ntegral factor as: Defne desgn doman, non-desgn doman, load and constrant Mesh the structure bx e k ð; Þ ¼ X þr 1 þr 1 X m¼ rþ1 n¼ rþ1 x e k ðm; nþ Hðm þ r; n þ rþ (11) Intalze element desgn varables Calculate stff matrx and dsplacement Calculate obectve and senstvty The dervatve of obectve @C @x ð; Þ can also be treated e wth the convoluton ntegral factor as: þr 1 @bc @x e ð; Þ ¼ X þr 1 X m¼ rþ1 n¼ rþ1 @C @x e ðm; nþ Hðm þ r; n þ rþ (12) No extra constrants are ntroduced nto the optmzaton wth convoluton ntegral factor method Therefore, ths convoluton ntegral factor method s convenent for calculatng and programmng Ths method can be used for a varety of problems 5 Soluton procedure of topology optmzaton The soluton procedure of the topology optmzaton based on the SIMP method, hybrd soluton algorthm, and the convoluton ntegral factor method can be wrtten as follows: 1 Defne desgn doman, non-desgn doman, constrants, loads, etc Elements relatve denstes n the desgn doman should vary wth teraton procedure, whle those n the non-desgn doman are of fxed value, whch are set as a fxed value of ether 0 or 1 2 Mesh the structure, calculate the elements stffness matrces when the relatve densty of all elements s set as 1 3 Intalze desgn varables, set every element n the desgn doman to a predetermned relatve densty value 4 Calculate materal characterstcs of every element, such as Young s modulus, calculate the elements stffness matrces, assemble the stffness matrces of the entre structure, and calculate the dsplacement of nodes 5 Calculate the obectve and constrant functon value, and calculate ther dervatves Treat the desgn varables wth the convoluton ntegral factor method 6 Update the desgn varables usng the hybrd soluton algorthm 7 Check the convergence of the result Return to step 4 f the result does not converge, whle contnue to step 8 and fnsh the teraton f the result converges The relatve dfference of the desgn varable n Eq 13 or that of the obectve n Eq 14 can be used as the udgement condton of a convergent result: max ðx kþ1 Þ max ðx k Þ max ðx k Þ <" (13) C kþ1 C k <" (14) C k 8 Output the desgn varables, obectve value and constrant value, and fnsh the calculaton A flow chart of the above calculaton s shown n Fg 4 6 Numercal example Update desgn varables wth hybrd soluton algorthm Treat desgn varables wth convoluton ntegral factor method Convergence? Yes Output result The end Fg 4 Flow chart of topology optmzaton The followng example llustrates the effect and functon of the hybrd soluton algorthm and the convoluton ntegral factor method proposed n ths paper A two-dmensonal smply supported beam wth unt thckness s clamped at the rght end and supported at left end subect to a downward unt force P at the rght center, and can be approxmated as a plane-stress problem, as shown n Fg 5a The formula n Eq 14 s used to udge the convergence The beam s meshed nto 80 60 quadrangle elements wth four nodes The obectve s to mnmze the complance subect to a volume fracton constrant of 05 wth an equlbrum equaton constrant 61 Effect and functon of the hybrd algorthm Results wth the MMA, GCMMA, and hybrd algorthms are compared and lsted n Fg 5b d,e g, and h, No
794 respectvely The convoluton ntegral factor method s used to elmnate numercal nstablty and the convoluton radus s set as 2 Fgure 5b d are the results wth the smplex MMA approach on the 100th, 400th, and 800th teratons Some checkerboard and porous elements are exhbted n the results, and the calculaton cannot converge by the end of the 800th teraton Ths means that the smplex MMA approach s not helpful to obtan a manufacturable result Fgure 5e g show the results wth the smplex GCMMA approach on the 100th, 400th, and 800th teratons The calculaton s stable and converges, but very slowly, so that a perfect and clear topology structure cannot be obtaned by the end of 800th teraton 80 60 a P=1 b c d e f g h Fg 5 a Smply supported beam under sngle load b d Results of MMA on 100th, 400th, and 800th teratons, respectvely e g Results of GCMMA on 100th, 400th, and 800th teratons, respectvely h Results of hybrd algorthm on 100th, 200th, and 400th teratons, respectvely k Curves for complance versus teraton numbers
795 Fg 5 (contnued) Fgure 5h are the results wth the hybrd algorthm on the 100th, 200th, and 400th teraton The calculaton s stable and can converge very quckly A perfect and manufacturable topology result can be obtaned by as early as the 400th teraton Fgure 5k shows the convergence curves of obectve functon about the three dfferent algorthms It shows that the obectve functon wth MMA decreases very quckly at the frst several teratons, whle some numercal oscllaton s exhbted on the subsequent teratons, and the soluton s dffcult to converge The calculaton wth GCMMA s stable, but the obectve functon decreases very slowly The optmzaton wth the hybrd algorthm converges very quckly and exhbts no numercal oscllaton 62 Effect and functon of the convoluton radus The convoluton radus s ndependent of mesh sze An approprate convoluton radus s helpful to elmnate checkerboard pattern and porous materal n the result Fgure 6a d shows the effect of dfferent convoluton raduses wth the hybrd soluton algorthm used n calculaton Fgure 6a shows the topology result wth convoluton radus r=1, and many checkerboard patterns can be seen Fgure 6b,c are the topology results wth r=2 and 3, respectvely As can be seen, the checkerboard patterns have dsappeared and a perfect topology s found Fgure 6d shows the topology result wth r=4, and some porous materal can be seen Fgure 6 allows us to conclude that an approprate convoluton radus can beneft n the producton of a manufacturable result, so, a proper range of convoluton radus 2 r 3 s suggested 7 Concluson Topology optmzaton wth the sold sotropc materal wth penalzaton (SIMP) method has strong applcatons n engneerng A hybrd soluton algorthm based on the method of movng asymptotes (MMA) and the globally convergent verson of the method of movng asymptotes (GCMMA) can be successfully used n topology optmzaton, and calculaton effcency and convergence can be effectvely mproved The convoluton ntegral factor method s an effectve way to treat desgn varables An approprate convoluton radus can lead to a manufacturable result The work deeply dscussng the calculaton effcency and convergence of optmalty crtera (OC), MMA, GCMMA, and hybrd algorthms n topology optmzaton s currently beng undertaken a r=1 b r=2 c r=3 d r=4 Fg 6 a r=1 b r=2 c r=3 d r=4
796 Acknowledgments The authors are thankful for the fnancal support from the Natonal 973 Proect and the Natonal 863 Proect The assstance on the MMA/GCMMA algorthms from Professor K Svanberg s gratefully acknowledged References 1 Bendsoe MP, Kkuch N (1988) Generatng optmal topologes n structural desgn usng a homogenzaton method Comput Methods Appl Mech Eng 71(2):197 224 2 Hassan B, Hnton E (1999) Homogenzaton and structural topology optmzaton: theory, practce and software Sprnger, London 3 Mlenek HP, Schrrmacher R (1993) An engneerng approach to optmal materal dstrbuton and shape fndng Comput Methods Appl Mech Eng 106:1 26 4 Sgmund O (1994) Desgn of materal structures usng topology optmzaton PhD thess, Department of Sold Mechancs, Techncal Unversty of Denmark, Denmark 5 Bendsoe MP, Sgmund O (1999) Materal nterpolatons n topology optmzaton Arch Appl Mech 69(9 10):635 654 6 Zhou M, Rozvany GIN (1991) The COC algorthm, part II: topologcal, geometry and generalzed shape optmzaton Comput Methods Appl Mech Eng 89:197 224 7 Svanberg K (1987) The method of movng asymptotes a new method for structural optmzaton Int J Numer Methods Eng 24:359 373 8 Svanberg K (1999) The MMA for modelng and solvng optmzaton problems In: Proceedngs of the Thrd World Congress of Structural and Multdscplnary Optmzaton, Buffalo, New York, May 1999 9 Svanberg K (1999) A new globally convergent verson of the method of movng asymptotes Techncal report, Department of Mathematcs, Royal Insttute of Technology, Stockholm, Sweden 10 Svanberg K (1998) Two prmal-dual nteror-pont methods for the MMA subproblems Techncal report TRITA-MAT-1998-0S12, Department of Mathematcs, Royal Insttute of Technology, Stockholm, Sweden 11 Svanberg K (1995) A globally convergent verson of MMA wthout lnesearch In: Rozvany GIN, Olhoff N (eds) Proceedngs of the Frst World Congress of Structural and Multdscplnary Optmzaton, Goslar, Germany, May/June 1995, pp 9 16 12 Bruyneel M, Duysnx P, Fleury C (2002) A famly of MMA approxmatons for structural optmzaton Struct Multdscpl Optm 24(4):263 276 13 Fleury C (1979) Structural weght optmzaton by dual methods of convex programmng Int J Numer Methods Eng 14:1761 1783 14 Fleury C, Brabant V (1986) Structural optmzaton: a new dual methods usng mxed varables Int J Numer Methods Eng 23(3):409 428 15 Lu WX (1994) Mechancal optmal desgn Tsnghua Unversty Press, Beng, Chna 16 Haber RB, Jog CS, Bendsøe MP (1996) A new approach to varable-topology shape desgn usng a constrant on permeter Struct Optm 11:1 12 17 Petersson J, Sgmund O (1998) Slope constraned topology optmzaton Int J Numer Method Eng 41(8):1417 1434 18 Sgmund O, Petersson J (1998) Numercal nstabltes n topology optmzaton: a survey on procedures dealng wth checkerboards, mesh dependences and local mnma Struct Optm 16(1):68 75 19 Zhou M, Shyy YK, Thomas HL (2001) Checkerboard and mnmum member sze control n topology optmzaton Struct Multdscpl Optm 21(2):152 158 20 Daz AR, Sgmund O (1995) Checkerboard patterns n layout optmzaton Struct Optm 10(1):40 45 21 Jog CS, Haber RB (1996) Stablty of fnte element models for dstrbuted parameter optmzaton and topology desgn Comput Methods Appl Mech Eng 130(3 4):203 226