Complexity Control in the Topology Optimization of Continuum Structures

Transcription

1 E. L. Cardoso Grupo de Mecânca Aplcada Departamento de Engenhara Mecânca UFRGS Rua Sarmento Lete, Porto Alegre, RS. Brazl J. S. O. Fonseca Grupo de Mecânca Aplcada Departamento de Engenhara Mecânca UFRGS Rua Sarmento Lete, Porto Alegre, RS. Brazl Complexty Control n the Topology Optmzaton of Contnuum Structures A general mesh ndependent flter as a mean to control the complexty of topology optmzaton desgned structures s dscussed. A new mesh-ndependent flter, appled over the move-lmts of the sequental lnear programmng s proposed, and t s shown that ts use allevates common problems n the contnuum topology optmzaton, lke checkerboardng, mesh dependency, as well as effects assocated to non-structured meshes, lke numercal ansotropy. The structural optmzaton formulaton adopted n ths work s the mnmzaton of a penalzed functon of the volume, wth constrants on the complance of each load case. Aspects of ths penalzed objectve functon are dscussed, and several numercal examples are shown. Keywords: Topology optmzaton, flterng, gradent control, complexty control Introducton The man objectve of ths work s to present a methodology to control the complexty of structures desgned by topology optmzaton. Ths control s attractve because the cost of a mechancal part depends on ts complexty, and therefore, the economy acheved n materal can be easly overrun wth the ncrease of the complexty of the part. Addressng ths ssue, ths work proposes the use of a general gradent control technque (flterng), whch can be used n arbtrary meshes. Also, t s known that restrctng the spatal varaton (gradent) of the densty makes the problem well-posed (Bendsøe, 995), prevents the appearance of the checkerboard and, as the complexty of the topology s related to the number of holes (transton from vod to fll), the gradent control also controls the complexty. Studes about gradent controls (spatal varaton of the desgn varable) have been done to avod the checkerboard and to assure the exstence of solutons (Nordson, 983; Swan and Kosaka, 997; Peterson and Sgmund, 998; Fonseca and Kkuch, 998; Cardoso and Fonseca, 999 and Bourdn, 2). Some references (Díaz and Sgmund, 995 and Jog and Haber, 996) suggested that the checkerboard s assocated to the nterpolaton order of dsplacement and densty felds n the fnte element soluton, lke n the Stokes problem. Due to ths reason, some researchers use hgh order fnte elements wth the usual constant nterpolaton for the densty. Other approaches are the use of gradent controls, lke dgtal magng flterng or the strcter slope control proposed by Nordson, 983, and mplemented for the contnuum problem by Peterson and Sgmund 998 and Cardoso and Fonseca, 999. The man advantages of usng gradent controls over hgh order fnte elements are the complexty control, the fact that the set of the admssble desgns s closed (Bendsøe, 995), and computatonal effcency. The man objectve of the topology optmzaton problem s to fnd a materal dstrbuton that extremzes a gven functonal (objectve functon) subjected to a set of constrants. The evoluton of engneerng lead to the necessty of effcent methodologes to desgn mechancal parts and structures, thus savng materal and tme. Ths s the reason why topology optmzaton s becomng a very mportant research feld. The basc goal of topology optmzaton, the materal dstrbuton, s acheved by a consstent parameterzaton of the materal propertes n each part of the desgn doman. When dealng Paper accepted July, 23. Techncal Edtor: Edgar Nobuo Mamya wth sotropc materals, a natural queston s whether there exsts or not materal n a gven pont, whch leads to a dscrete problem. It s well-known that ths nteger parameterzaton leads to numercal problems, assocated to the non-unqueness of solutons (Ambroso and Buttazzo, 993 and Bendsøe, 995). To relax the nteger parameterzaton of the materal dstrbuton, Bendsøe and Kkuch, 988, ntroduced the parameterzaton of the materal dstrbuton by means of the homogenzaton method (Hassan and Hnton, 998a and 998b), whch enlarges the space of admssble solutons makng the problem well-posed. The homogenzaton method s smple and useful, however t ncreases the number of desgn varables used n the optmzaton problem, as t requres dervng a model for the dependence of the materal propertes wth respect to the geometrcal parameters of the cell. A dfferent approach to the non-unqueness of the soluton to the nteger program s to reduce the admssble set by the ntroducton of permeter constrants, proposed by Ambroso and Buttazzo, 993 and further developed by Beckers, 997. The power-law or SIMP s a smpler approach to relax the space of admssble solutons wthout ncreasng the number of desgn varables (Bendsøe and Sgmund, 999). It has been used as an alternatve to the full homogenzaton, where the materal propertes are parameterzed by p E = ñ E, ρ () where E s the nterpolated fourth order elastc consttutve tensor, ρ s the materal densty or materal fracton, E s the fourth order tensor relatve to the materal propertes of the base materal and p s used to adjust the degree of nonlnearty of ths equaton. Ths approach s a contnuous nterpolaton of the materal propertes wth respect to the densty (amount of materal) n each pont of the desgn doman. Ths contnuous parameterzaton relaxes the orgnal nteger problem, enlargng the desgn space. Recently, Bendsøe and Sgmund, 999, assocated ths parameterzaton to the use of sotropc composte mcrostructures. Wth ths result, t s clear that the use of the SIMP approach enlarges the desgn doman lke the use of the homogenzaton, but t s also clear that the spaces obtaned are dfferent, because the space spanned by non-sotropc mcrostructures s larger than the space spanned by ther sotropc counterparts. Due to ths reason, one can acheve better (extreme) results usng the homogenzaton nstead of SIMP (Park 995 and Bendsøe, 995). When the goal s to obtan a mechancal part made of sotropc materal, the use of contnuous parameterzaton, lke J. of the Braz. Soc. of Mech. Sc. & Eng. Copyrght 23 by ABCM July-September 23, Vol. XXV, No. 3 / 293

2 Complexty Control n the Topology Optmzaton of Contnuum Structures homogenzaton and SIMP, leads to the problem of ntermedate (composte) materals n some regons of the desgn doman. These regons are mportant and, sometmes, cannot be dscarded n a smple threshold post processng procedure. To ths end, one should obtan a pure nteger desgn after relaxng the orgnal problem wth dscrete materal parameterzaton. Usng the SIMP approach, one can start wth a untary exponent n Eq. () and ncrement the exponent as the optmzaton s carred on. Ths process, a contnuaton approach, enlarges the desgn space by usng a contnuous parameterzaton and, after that, reduces ths space usng penalzaton. Ths s a well-known result of the mxture theory, n whch a lnear relaton s an unattanable upper-bound and thus can lead to extreme desgns, lke the one obtaned wth homogenzaton. In fact, the mxture of two materals has a nonlnear relaton between the propertes of each materal and the amount of each materal n the mxture. Startng wth a untary exponent n the SIMP relaton provdes the upper bound and thus we are lookng for a soluton n a larger desgn space. Ths soluton s extreme, but contans large areas wth ntermedate materal. Penalzng the consttutve relaton, the stffness of the ntermedate materal s no longer attractve and, for some value of the exponent, t s possble to acheve an almost black and whte desgn. In ths work, a lnear relaton between the propertes of the base materal and the materal n each pont of the desgn doman s used. The objectve functon used s a penalzed functon of the amount of materal (volume) of the structure. Ths objectve functon s used because we are concerned wth materal reducton for a fxed complance. The objectve functon used s not related to the flterng technque proposed, and can be used wth any other knd of gradent control or constrants. Gradent Controls Gradent controls are used to constran the spatal varaton (gradent) of the desgn varable. As shown by Bendsøe, 995, constranng the spatal gradent of the densty makes the H norm ( ( ) ) /2 2 = + (2) Ω 2 H ρ ρ bounded (H < + ), makng t possble to prove that the problem s well-posed and has (at least) one soluton. Ths control can be acheved usng dgtal magng flterng technques or an addtonal set of constrants n the optmzaton formulaton (strct gradent control). Although the strct gradent control allows a better control over the gradent of the denstes, t s too costly, due to the large number of addtonal constrants mposed to the optmzaton algorthm. Flterng means that an approxmated gradent constrant s beng mposed, usng some procedure related to mage manpulaton, whch s faster, but not so precse. Flterng has been used n dfferent forms, and the results obtaned show that t can control the checkerboard and, n some cases, control the complexty of the topology. In a general form, flterng means applyng a mathematcal operator to some value lke the densty, the gradent of the objectve functon or any other varable. Therefore, f one wants to smooth the spatal varaton of the desgn varable, an operator that prevents fast varatons of the fltered varable should be appled. The smplest operator s the average mean of the value of a gven patch of elements. The expresson for a general flter s (, ) (, ) (, ) g xy = f xy hxy (3) where g s the fltered value of f, due to the cooluton operator h. Flters are classfed by the way the neghbor elements are consdered: fxed grd, where only neghbors that share nodes and/or sdes of the central element are taken nto account, and spatal flters, where neghbors nsde a gven spatal neghborhood are consdered. As the man objectve of the flters s constranng the spatal varaton of the densty (frequency), from now on only lowpass flters are consdered. Avodng frequences hgher than the transton from vod to fll, n the dstance of two elements, makes t possble to avod the checkerboard, and changng the frequency of the flter, t s possble to control the complexty of the topology. For smooth (low frequency) flters, there should be no checkerboards nether tny renforcements. For very weak (hgh frequency) flters, there should be tny renforcements and maybe checkerboards. Therefore, the flter can be used as a complexty control. Recently, Bourdn, 2, studed a modfed verson of the tradtonal mnmum complance formulaton, where the densty feld s regularzed by the applcaton of a cooluton operator. The exstence of solutons and the coergence of the fnte element approxmatons are establshed. Fxed Grd Flters Fxed grd flters are the smplest ones. Bascally, the only dfference among these flters s the way the average value s consdered. Another advantage of ths type of flter s the closed relaton to the mage manpulaton flterng technques, where each element corresponds to a pxel (or a voxel). The man dsadvantage of fxed grd flters s ts dependency on the fnte element dscretzaton, whch lmts ts best performance to structures meshes. The frst attempt to avod the checkerboard was made by Bendsøe and Kkuch (Bendsøe, 995), usng a four elements patch, prevously used for ncompressble problems. It s a fxed grd flter, and t s appled over the densty feld. Other very smple fxed grd flter s the 3x3 neghborhood flter, 2 H = b b b 2 ( b + 2) b b where H s the dscrete cooluton operator. The weghtng factor b can be selected wthn the nterval [, + ], and smaller the b, weaker s the flter. Ths flter s usually appled over the densty feld, over some senstvty feld, wth the form 3 3 G(, j) = F( m, n) H( m+ C, n+ j C), (5) m= n= where C=(c+)/2 s used, c=3 for a 3 x 3 flter and m and n are the (relatve) centrodal postons of each neghbor element. The flter proposed by Fonseca and Kkuch, 998, can be seen as the frst attempt to mplement a strct gradent control n the topology optmzaton of contnuum structures. Ths flter s a fxed grd flter, where the weght factors are evaluated n advance to mpose, approxmately, a gradent constrant. The flter s appled over the upper and the lower move lmts of the SLP, and usng ths flter, one can control the complexty of the result. All of these flters are restrcted to structured meshes made of rectangular fnte elements. One very nterestng fxed grd flter s the one proposed by Swan and Kosaka, 997, where node and sde neghbors are selected. Each class of neghbor has a fxed weght, and the volume of the neghbor s taken nto account n the averagng process. The expresson for ths flter s (4) 294 / Vol. XXV, No. 3, July-September 23 ABCM

3 nen nen av + W av + W a V aˆ, j j 2 j j j= j= = nen nen V + WV j+ W2Vj j= j= where a s the varable beng fltered, â s the fltered value, W and W 2 are weghtng factors and V s the volume of an element of the mesh. Usng ths flter, one can have dfferent fnte elements n the same mesh, whch s a great advance when compared to other fxed grd flters. Swan and Kosaka flter s nadequate to mpose strct bounds on the spatal varaton of the densty feld; however t s effectve n preventng checkerboards and keepng some control on the complexty of the desgn. Spatal Flters A serous drawback of fxed grd flters, as the one proposed by Swan and Kosaka, s that ts area of nfluence s spanned by fnte elements neghborhood. Thus, refnng the mesh reduces the spatal area of nfluence of the flter, and leads to more complex desgns. Therefore, t s necessary to ntroduce spatal flters to avod mesh dependency. Another advantage of the spatal flters over most of the fxed grd flters s that controllng the area of nfluence of the spatal flter, t s possble to control the complexty of the topology. Ths control s not a very strct control. The mesh dependency s allevated wth the use of these flters, as for a constant area of nfluence, refnng the mesh ncreases the number of elements consdered, avodng renforcements smaller than some scale. Fgure llustrates the concept of spatal flter. In ths fgure, the element beng fltered has a neghborhood, selected by the radus of the flter. Elements wth centrods nsde the area of nfluence of the flter (crcle) are consdered n the averagng process. (6) where s the number of neghbors. As can be seen n the prevous equatons, ths flter does not take nto account the sze of the neghbors, makng t adequate to be used wth regular meshes. Sgmund, 997, uses a spatal flter on the gradents of the optmzaton problem. The flter proposed by Sgmund has the followng form ˆ f f = Wjρ j. ρ ρ k j = ρ kwj j = As dscussed before, ths flter s mesh ndependent, but does not take nto account the sze of the neghbors. In order to use a spatal flter wth any knd of mesh t s proposed a spatal verson of the flter proposed by Swan and Kosaka. In ths flter, the weghts are evaluated by the relatve poston of the neghbor to the central element. So, there are two possbltes, usng an average weght or usng ndvdual weghts for each neghbor. From these two possbltes, two flter defntons are proposed: the frst flter wll be referenced as AWSF (average weght spatal flter) where av + W a V j j j = aˆ =, V + WVj j = W R R j (9) () max j Rmax = () j = and the second formulaton as IWSF (ndvdual weght spatal flter) av + j j j j= aˆ =, V + WaV W V j j j = (2) where Fgure. Concept of spatal flter. The central element (darker) has an area of nfluence (crcle) whch spans the elements wth centrod nsde the crcle (gray). by The smplest spatal flter s the lnear flter, wth weghts gven { max } W = R R, j CV, CV = j I R R (7) max j j where R max s the radus of the flter, R j s the dstance between the centrods of elements and element j and CV s the set contanng the neghbors of the element. The average value s gven by W R R. max j j = (3) Rmax In the above equatons, a s the value of the varable beng fltered, â s the fltered value, V s the volume of the element, W s a weghtng factor, W s an averaged value of the weghtng factors and s the number of elements n CV. Ths flter can be appled to any varable, lke densty feld or the gradent feld. In ths work, the flter s appled over the upper and lower move lmt felds of the SLP. Dong ths, the densty feld and gradent felds are not dsturbed n an artfcal way. The only functon of the flter t to gude the possble values obtaned by the SLP at each teraton. j j j = aˆ =, j = Wa W j (8) Strct Gradent Controls Strct gradent controls have been used snce Nordson, 983. The approach proposed by Nordson was n the opposte drecton of the soluton proposed by Cheng and others. Whle Cheng, 98, relaxed the set of possble solutons, usng mcrostructures, J. of the Braz. Soc. of Mech. Sc. & Eng. Copyrght 23 by ABCM July-September 23, Vol. XXV, No. 3 / 295

4 Complexty Control n the Topology Optmzaton of Contnuum Structures Nordson restrcted ths set, restrctng the possble varaton of the thckness, and also solved the ll-posed problem of thckness optmzaton of a plate. After that, Bendsøe, 995, studed ths restrcton, assocatng t to bounds on the H norm. Therefore, even f the goal of usng a flter s to control the checkerboard, t also controls the spatal varaton of the densty, boundng ts norm. Usng ths nformaton, Fonseca and Kkuch, 998, proposed a fxed grd flter whose weghts are evaluated to restrct the spatal varaton of the densty. Dong ths, one can restrct the set of admssble solutons (by boundng the norm), and also control the complexty of the soluton. Petersson and Sgmund, 998, proposed a strct gradent control, where the varatons of the denstes are constraned drectly n the SLP problem, as a new set of constrants. Ths gradent control s very expensve, but allows a very strct constrant on the H norm, controllng the checkerboard and the complexty of the topology. Ths gradent control was mplemented for regular meshes. Cardoso and Fonseca, 999, proposed a more general mplementaton for ths gradent control, allowng ts use wth non regular meshes. The large number of constrants ntroduced to bound the varaton of the densty fled from each element to ts neghbors make ths approach too expensve for practcal applcatons. Also, the use of flterng technques s much cheaper and leads to smlar results Volume Mnmzaton wth Complance Constrant Clearly the effects of a strct gradent control and the effect of flterng are the same: constran the spatal varaton of the desgn varable. Controllng the varaton of the densty feld s actually a control on the spatal varaton of the materal propertes. Therefore t makes no sense to adopt a nonlnear relaton of the materal propertes wth respect to the densty. For example, usng a strct gradent control does not allow - desgns, so "gray" areas are always created (areas wth ntermedary denstes). To further penalze the appearance of gray areas, Petterson and Sgmund, 998, ncreased the penalty exponent of the consttutve relaton up to 5. Dong ths, the result s not related to the spatal varaton of the base materal, especally for a very smooth spatal varaton of the densty. It s practcally mpossble to buld a part wth large areas of ntermedate materal, even though t s possble to assocate ths SIMP "gray" materal to a partcular sotropc mcrostructure. Consequently, one can use a slghtly dfferent approach. Instead of penalzng the materal propertes, one can penalze the cost (objectve) functon. Therefore, f one wants to mnmze the amount of materal used n a partcular desgn, constranng the complance of the part, the followng formulaton can be used Mn Volume T S. t. f u F k =.. nlc (4) k k lmk where F lmk s the maxmum complance for a partcular load case k, and nlc s the number of load cases. Consderng the followng relaton between the volume and the desgn varable V ( ρ) = ρdω, (5) Ω the objectve functon and the set of constrants are coex, so the soluton may be unque, dependng on the materal nterpolaton. If a lnear relaton (upper bound) between the effectve propertes and the base materal s used, ths problem s coex and has a unque soluton. As ths parameterzaton represents an upper bound for the parameterzed materal propertes, the result usually has large extents of gray areas, due to the artfcally hgh stffness of the ntermedate materal. For ths reason, some researches penalze the stffness of ntermedate denstes usng a nonlnear relaton (SIMP). If nstead of penalzng the materal propertes, the cost of the ntermedate denstes s penalzed, we also avod large areas of ntermedate denstes n the result. The followng relaton n ( ρ, ) = ρ Ω, = (,] V n d n (6) Ω also penalzes the ntermedate denstes. Ths relaton makes the cost (volume) larger for ntermedate denstes, so the optmzer tres to avod ntermedate denstes. As can be seen, ths new objectve functon s non-coex, so usng n dfferent than one makes the problem non-coex, lke when penalzng the consttutve relaton. Therefore, a contnuaton approach can be used. The orgnal coex problem (no penalzaton) s taken as the startng pont. Once n the global mnma (t depends on the optmzer) of the desgn set (whch also depends on the gradent control) the problem s penalzed, changng the objectve functon. Ths new objectve functon s noncoex, so the problem wll coerge to some local mnma, whch contans fewer ntermedate denstes, as the penalzaton prevents t. If the coex formulaton s changed drectly to a hghly noncoex problem, the change n the result can be dramatc, so t s a good dea changng slghtly the non coexty of the objectve functon. Dong ths, t s also possble to estgate the nfluence of the non-coexty of the objectve functon n the complexty of the result. Ths s the same behavor encountered n the modfcaton of the consttutve relaton, where the exponent of the consttutve relaton can be related to some specal knd of mcrostructure and the penalzaton of the objectve functon can be related to some economc features, lke the prce of the ntermedate denstes (so constructng the sotropc mcrostructure can be related to an ncrease n the cost of the process). An addtonal penalzaton term can be consdered n Eq. (6), lke the one used by Fernandes, Guedes and Rodrges, 999, resultng n n ( ρ, ) = ρ Ω+ β ρ( ρ), = (, ], β (7) V n d n Ω Ω whch can be used to further penalze the desgn. The algorthm starts wth a coex objectve functon (n= and β =), changes to a non-coex objectve functon (n< and β =) and, f needed, makes t even more non-coex ( β >). Ths process can be automatc, changng the parameters after the coergence of each level. The need for addtonal penalzaton appears when the gradent of the desgn varable s constraned, so large gray areas, correspondng to the transtons from fll to vod, can be reduced. Mathematcal Programmng Among several optmzers, the sequental lnear programmng was chosen, due to ts smplcty and because the relable publc doman SLATEC lbrary s freely avalable (Hanson and Hrbert, 98). The lnear approxmaton for the proposed problem s Mn ne dv V ( ρ) + ( ρ ρ ) d ρ = St.. ne df df ρ Flm F ( ) k k ρ + ρ dρ = dρ (8) max α, α ρ ρ mn α, α ρ ( ( ) ) ( ) + ( ) mn max 296 / Vol. XXV, No. 3, July-September 23 ABCM

5 where ρ s the actual densty value for element, F k s the complance of load case k, α mn and α max are the mnmum and maxmum values for the move lmts and α s the move lmt offset factor. Ths local approxmaton of the objectve functon and constrants requres some frst order senstvtes wth respect to the desgn varables. The objectve functon n Eq. (7) s an explct functon of the desgn varable, so the senstvty s very easy to obtan: dv n ( 2 ) V. d n ρ ρ β ρ = + (9) The flexblty constrant s an mplct functon of the desgn varable, so the senstvty wth respect to the densty s more olved, but has a closed form. Dfferentatng the defnton of flexblty, T df df T du = u+ f (2) dρ dρ dρ where the senstvty of the dsplacements can be obtaned drectly from the equlbrum equaton, gves T df df T dk = 2 u u u. (2) dρ dρ dρ The last set of constrants n Eq. (8) are known as move lmts. They are added to keep the lnear approxmaton vald at the current pont. The move lmt strategy s crucal to the success of an SLP mplementaton and many researches have proposed dfferent strateges (Wujek and Renaud, 998a and 998b; Chen, 993). One of the smplest strateges s avodng the zgzag (Bazaraa and Shetty, 979), whch means that, f n the last two teratons the sgn of the change n one desgn varable changes, the move lmt of ths desgn varable s decreased by a factor of (-α ) and f there s no change n the sgn then the move lmt of ths varable s ncreased by a factor of (+α ). However ths scheme tends to be unstable n ths applcaton and was modfed to consder the last three teratons. After the evaluaton of the movng lmts, one of the proposed flters s appled separately, on the upper and on the lower movng lmts. Ths assures that the possble range of densty for each desgn varable s related to the surroundng elements. After the LP, elements denstes cannot dffer much from ts neghbors, avodng the checkerboard. As dscussed before, as the number of elements used n the averagng process ncreases, smpler s the topology. One drawback of the LP method, as ponted by Bruns and Tortorell, 2, s the loss of symmetry, due to the nature of the LP approxmaton. It was observed that the use of the flters proposed n ths work also allevate ths effect. Soluton of the Equlbrum Equatons Usng the strong form of the equlbrum equatons of the elastcty, ó + b =, (22) where ó s the Cauchy stress tensor and b s the body force vector, and consderng a lnear relaton between the stress tensor and the stran tensor t s possble to express Eq. (22) usng the materal propertes and the contnuum materal parameterzaton of Eq. (), wth p=, whch gves ( ρ ( )) EL u + b =. (24) Here the small dsplacement stran tensor s used T Lu ( ) = ( u + u ), (25) 2 so the results are vald only for small dsplacements (t s mportant to emphasze that both the flterng technque and the objectve functon can be used n non-lnear problems). The weak form of Eq. (22) s ( ) ρ ( ) dω= dω+ dγ, vlu ELuu b v tv (26) Ω Ω Γ where v s an arbtrary test functon. In ths work, the equlbrum problem s solved usng the fnte element method. The desgn doman s dvded n ne fnte elements, and t s assumed a local nterpolaton of the dsplacements and denstes. Usng the usual fnte element procedures used (Bathe, 996), the well known element stffness matrx s obtaned T ( ρ ) ρ K = B : E : B dω (27) Ω where a constant nterpolaton for the desgn varables s used for all elements n the mesh. Ths result s very mportant, because the densty s a smple scalng factor for the usual stffness matrx. The procedure to obtan the global stffness matrx s the subject of any fnte element book (Bathe, 996). The non conformng four nodes element (Kasper and Taylor, 2a and 2b), the 3D non conformng eght nodes element, and the GT9 trangular element wth drllng degrees of freedom (Yuqun and Yn, 994) are used n ths work. These are low order elements, so the checkerboard nstablty s not avoded. The use of these elements, however, mples n a better dsplacement soluton n the equlbrum problem, wth a small ncrease n the computatonal tme. The nonconformng formulaton allevates the parastc shear, so the fnal result s not corrupted by ths knd of fnte element error, especally n tny renforcements n bendng, so usual n topology optmzaton. The GT9 has a behavor between the 6 node trangular element and the constant stran trangle, wth a small ncrease n the computatonal tme when compared to the latter. Another advantage of the ncompatble elements s the stress recovery, allowng a better descrpton of the stress feld when the stress s of nterest (especally n non-lnear problems). Results In the followng, some results obtaned wth the use of the proposed formulaton are shown. The non-conformng bdmensonal four node element and the non-conformng three dmensonal eght node are used. The trangular element consdered s the GT9. For all examples, the materal s assumed as sotropc and elastc, wth E=* 7 Pa and Posson's rato equal to.3. All the appled loads have a value of N. In all fgures, a gray scale s used to represent the densty feld, where whte means vod ( ρ =* -3 ) and black means base materal. ( ) ó= ρelu, (23) J. of the Braz. Soc. of Mech. Sc. & Eng. Copyrght 23 by ABCM July-September 23, Vol. XXV, No. 3 / 297

6 Complexty Control n the Topology Optmzaton of Contnuum Structures Influence of the Penalzaton As explaned before, nstead of penalzng the consttutve relaton, one can penalze the objectve functon. To show the nfluence of the penalty exponent n Eq. (6) t s used the tradtonal 8 x 5 short cantlever beam (See Bendøe, 995, for ths and other standard problems) dscretzed by 26 (6 x 36) fnte elements. The lmt complance s equvalent to 2% of the complance obtaned for % of volume. Four dfferent values of the penalzaton exponent are used:, /2, /6 and /8, and the volume fractons obtaned for those exponents are 62.3%, 62.95%, 64.53% and 64.6%, respectvely. When the exponent s equal to one, there s no penalzaton, so the result contans large areas wth ntermedate denstes. As the exponent s decreased, the relaton becomes non-coex, penalzng ntermedate denstes. Fgure 2 shows the topology obtaned wth these exponents. Smaller the exponent, fewer the gray areas. Ths s a good feature, but the noncoexty also makes the dscrete soluton non-unque. To avod ths dependency wth respect to the ntal densty dstrbuton, we used a contnuaton approach. Frst, the lnear problem s solved untl coergence. After that, the exponent n s decreased and the problem contnues untl the new coergence s reached. Ths approach needs more teratons, but t s effectve and we also observed that t s more stable than startng wth a noncoex problem. For the same problem of the Fg. 3, the same soluton for the random and homogeneous ntal densty dstrbuton s obtaned, as shown n Fg. 4. No assumpton on the ntal denstes dstrbuton was made, so t can be an nfeasble pont. Table 2 shows the volume fractons obtaned wth ths procedure. Two volume fractons are shown: the frst corresponds to the coerged coex soluton and the second to the coerged penalzed soluton. As n the prevous example, the second column n Fg. 4 s obtaned wth the use of the IWSF flter and the thrd wth the used of the AWSF flter. Table 2. Volume fractons obtaned wth the contnuaton approach, Fg. 4. Flter / Intal densty dstrbuton Volume fracton n = / n = 8(%) IWSF / Homogeneous / IWSF / Random / 6. AWSF / Homogeneous / AWSF / Random / 59.6 Table 3. Volume fractons obtaned for topologes n Fg 5. Fgure 2. Influence of the penalzaton n equaton Eq. (6). The penalzaton s ncreased from top left to bottom left, clockwse (n=, 2, 6 and 8). To show the effects of the non-coexty of the penalzed objectve functon, the prevous example s consdered agan, but wth a mesh of 3525 (75 x 47) fnte elements. The lmt complance s set to 25% of the complance obtaned for % of volume fracton and the penalzaton exponent s set to /8. The two flters proposed are used, wth a radus of.66* - m, (equvalent to select eght neghbors for each element). Two dfferent ntal densty dstrbutons are used, as shown n Fg. 3 (frst column), a homogeneous wth ρ =.5 and a random dstrbuton. The second column n Fg. 3 shows the results obtaned wth the use of the IWSF flter and the thrd wth the use of the AWSF flter. Flter / Intal densty dstrbuton Volume fracton for n=8 (%) IWSF / Homogeneous IWSF / Random AWSF / Homogeneous 25.3 AWSF / Random Fgure 4. Results obtaned wth the contnuaton approach. The frst column shows the ntal densty dstrbutons, the second row shows the results obtaned wth the IWSF flter and the thrd column shows the results obtaned wth the AWSF flter. Fgure 3. Non coexty of the objectve functon, Eq. (6), for exponent /8 and dfferent ntal densty dstrbutons. The fnal volume fractons are shown n Tab.. Table. Volume fractons obtaned for the second example, Fg. 3. Flter / Intal densty dstrbuton Volume Fracton (%) IWSF / Homogeneous 59,74 IWSF / Random 6,4 AWSF / Homogeneous 59,52 AWSF / Random 59,78 Fgure 5 shows the soluton of another common problem n the lterature, wth the use of the contnuaton method. The doman s a untary square and the lmt complance s set to 3% of the complance obtaned wth a volume fracton of %. In ths example, 8 (3 x 6) fnte elements are used and the radus of the flters s 2.6* -2 m (equvalent to 8 neghbors). The thrd row n Fg. 5 s obtaned wth the use of the AWSF flter and the fourth wth the use of the IWSF flter. Table 3 shows the volume fractons obtaned for ths example. 298 / Vol. XXV, No. 3, July-September 23 ABCM

7 Fgure 5. Results obtaned wth the contnuaton approach. The frst column shows the geometry and boundary condtons for ths example. The second column shows the ntal densty dstrbuton, the thrd column the results obtaned wth the AWSF flter and the fourth column the results obtaned wth the use of the IWSF flter. Influence of the Flter The results shown above were obtaned wth flterng otherwse checkerboard would have been present. For the same problem and dscretzaton, t s possble to vary the radus of the flter to obtan a smpler topology or to obtan a more complex one. If one chooses a very small radus, checkerboard can appear, so there s a lower lmt on the value of the radus. For a coarse mesh, the flter can blur the result and many levels of penalzaton would be requred, so there s a mnmum resoluton to apply the flter, as observed before by Fonseca and Kkuch, 998. Ths s not a problem, because the fnte element soluton requres a mesh good enough to represent the problem and one can thnk n the fnte element soluton and n the optmzaton problem when dong the mesh. Fgure 6 shows the nfluence of the sze of the flter for a fxed mesh. As the radus of the flter s ncreased, thn renforcements are no longer present n the topology. Ths s very mportant, because the number of holes usually ncreases the cost of the part. Another mportant thng s that the fnte element soluton for these thn renforcements can be naccurate because they are usually one or two elements wde, especally when low order fnte elements are used n bendng. Therefore, wth the use of the flter, one can assure a mnmum mesh sze for some parts of the topology. Ths s not a strct control, but as shown t works and wth some experence the desgner can estmate the radus of the flter to ft the characterstcs of the desgn. For the short clamped beam, wth a mesh of 63 fnte elements, we changed the radus of the flter, to show ts nfluence n the complexty of the topology. Fgure 6 shows the topology obtaned for ncreasng values of the radus of the flter. The values correspond to a maxmum number of 4, 8 and 2 neghbors. It s clear that the larger the number of elements nsde the radus of the flter, smpler s the topology. patterns are shown n the frst column of the Fg. 7. The second column shows the solutons obtaned wthout flterng. It s clear that the mesh pattern changes the checkerboard, showng how strong the effects of the mesh qualty n the topology optmzaton are. When usng the flter, the same result s obtaned, ndependent of the mesh pattern (thrd column). The flter used n ths example s the IWSF and the radus s fxed for the three meshes (2* - m). It must be observed that the meshes n Fg. 7 have a dfferent number of fnte elements. The volume fractons obtaned for ths example are 63.36% for the frst pattern, 63.3% for the second pattern and 65.57% for the last pattern. Multple load cases The proposed formulaton can be used for multple load cases. As shown n Eq. (4), each load case represents one constrant. The computatonal cost assocated to each constrant s very low and t s possble to use as many load cases as needed. To exemplfy the soluton of multple load cases problems, t s consdered the two load cases short beam (Bendsøe et al., 995), dscretzed wth 64 fnte elements. The contnuaton method s used, wth fnal exponent equal to /8. Fgure 8 shows the results obtaned usng the IWSF flter and Fg. 9 shows the results obtaned usng the AWSF for the same value (8.8* -2 m) of the radus. Table 4 shows the volume fractons obtaned for ths example. Table 4. Volume fractons obtaned for the two-load cases example. Flter / Intal densty dstrbuton Volume fracton for n=8 (%) IWSF / Homogeneous IWSF / Random AWSF / Homogeneous AWSF / Random Fgure 7. Influence of the mesh pattern. The frst column shows the mesh patterns, the second column shows the results obtaned wthout flterng and the thrd column shows the results obtaned wth flterng. The proposed flter technque allevates the mesh nfluence. Fgure 6. Results obtaned for three dfferent flter raduses. The radus s ncreased from left to rght. The proposed flters can also mnmze the nfluence of the mesh pattern on the topology. It s well known that the way the fnte elements are dsposed n the mesh can nfluence the result. Ths s known as numercal ansotropy. Most of the papers dealng wth topology optmzaton use a regular quadrlateral mesh, so the mesh qualty s not consdered. To show the nfluence of the proposed flterng technque, the short clamped beam s used. The lmt complance s set to 2% of the complance obtaned for % volume fracton and dfferent mesh patterns are used. The mesh Fgure 8. Short clamped beam wth two load cases. The frst column shows the ntal densty dstrbutons and the second column shows the results obtaned wth the IWSF flter. J. of the Braz. Soc. of Mech. Sc. & Eng. Copyrght 23 by ABCM July-September 23, Vol. XXV, No. 3 / 299

8 Complexty Control n the Topology Optmzaton of Contnuum Structures The results obtaned are very close to the results obtaned n references where a permeter constrant s used (Fernandes, Guedes and Rodrges, 999). It s very nterestng verfyng that smpler topologes, obtaned wth flterng, have smaller permeter than the complex ones, as predcted by the permeter control theory. Although dfferent, these approaches lead to smlar results n terms of complexty. Fgure 9. Short clamped beam wth two load cases. The frst column shows the ntal densty dstrbutons and the second column shows the results obtaned wth the AWSF flter. Three-dmensonal Results The proposed formulaton can be appled to three-dmensonal problems, wth no modfcatons. To llustrate three-dmensonal results, t s used as desgn doman a untary cube, clamped n one face. In the frst example, a pont load s appled n the bottom of the opposte face. The result s shown n Fg.. Due to the symmetry of the problem, only half doman s dscretzed wth 4 elements. The AWSF flter s used wth a radus equvalent to take nto account each edge neghbor of each element. The lmt complance s set to 2.6* -5 Nm. Fgure. Three dmensonal short clamped beam. Just half of the result s shown due to the symmetry of the problem. Fgure llustrates the result obtaned for the same geometry, but wth the load appled n the center of the opposte face. The same mesh s used and the lmt complance s 5.5* -6 Nm (4% of the complance for % volume fracton). Both results where obtaned wth the contnuaton method, wth a penalzaton exponent of /8. The results where exported to the ANSYS format, usng a threshold value of.9 (no element wth densty less than.9 s consdered). The complance evaluated by ANSYS dffers less than % for all examples. Fgure. Three dmensonal short clamped beam. Just half result s shown due to the symmetry of the problem. Conclusons A new spatal flter was proposed, together wth a penalzed verson of the tradtonal formulaton for volume mnmzaton. The spatal flter (n two dfferent formulatons) can prevent the checkerboard nstablty and also controls the complexty (number of holes) of the topology. It s shown that usng ths flter, mesh nfluence can be allevated. Ths s mportant, due to the mnmum requrements of mesh qualty to ensure a good fnte element approxmaton for the response of the structure. Low order elements have a poor behavor n bendng, so t s advsable to use mproved formulatons, lke some nonconformng and subntegrated fnte elements n structural optmzaton, especally when desgnng low fracton or very flexble structures. The objectve functon used n ths work allows reducng gray areas, ncreasng the cost of the ntermedate denstes, whch are expensve to buld. A contnuaton method was used to further penalze the ntermedate denstes, when t s coenent. Some results wth multple load cases and some 3D results were shown ndcatng that ths formulaton s very general and can be used wth any knd of mesh or fnte element. Lke other formulatons based on mathematcal programmng methods, other types of constrants, lke dsplacement constrants, can be added wth lttle effort. As shown n some examples the use of non-regular meshes can nduce very strong nfluence of the fnte element soluton on the fnal topology. In these cases, the use of mesh ndependent flters can allevate ths nfluence and also control the complexty of the topology. It s mportant to notce that the ntroducton of the flter restrcts the possble solutons, because the set of admssble solutons s modfed by the flter. Therefore, t s pontless to compare the results obtaned wth dfferent flterng technques, because they may not belong to the same set of admssble solutons. Features to be compared are the mesh ndependence, checkerboard control and complexty control. Acknowledgements The authors gratefully acknowledge the support of ths research by CAPES and the computatonal facltes of CESUP-UFRGS. References Ambroso, L., Buttazzo, G., 993, An optmal desgn Problem wth Permeter Penalzaton, Calc. Var.,, pp Bathe, K., 996, Fnte Element Procedures, Prentce-Hall. Bazaraa, M. S., Shetty, C. M., 979, Nonlnear Programmng - Theory and Algorthms, John Wley & Sons, New York. Beckers, M., 997, Optmsaton de Structures en Varables Dscretes, Uerste de lege, Collecton des Publcatons de la faculté des Scence Applquées, No. 8. Bendsøe, M. P., 995, Optmzaton of Structural Topology, Shape and Materal, Sprnger-Verlag, New York. Bendsøe, M. P., Díaz, A. R., Lpton, R., Taylor, J., E., 995, Optmal Desgn of Materal Propertes and Materal Dstrbuton for Multple Loadng Condtons, Internatonal Journal of Numercal Methods n Engneerng, Vol. 38, pp Bendsøe, M. P., Kkuch N., 988, Generatng Optmal Topologes n Structural Desgn Usng Homogenzaton Method, Computer Methods n Appled Mechancs and Engneerng, Vol. 7, No. 2, pp / Vol. XXV, No. 3, July-September 23 ABCM

9 Bendsøe, M. P., Sgmund, O., 999, Materal Interpolaton Schemes n Topology Optmzaton, Archve of Appled Mechancs, Vol. 69, pp Bourdn, B., Flters n topology optmzaton, In. J. Numer. Meth. Engng, 2: 5: Bruns, T. E., Tortorell D. A., Topology optmzaton of non-lnear elastc structures and complant mechansms, Comput, Methods Appl. Mech. Engrg., 9 (2) Cardoso, E. L., Fonseca, J., S., O., 999c, Spatal Gradent Control n the Structural Topology Optmzaton, Frst ASMO UK/ISSMO Conference on Engneerng Desgn Optmzaton, Ilkley, West Yorkshre, UK. Chen, T. 993, Calculaton of the Move Lmts for The Sequental Lnear Programmng Method, Internatonal Journal of Numercal Methods n Engneerng, Vol. 36, pp Cheng, K.T, 98, On Non-Smoothness n Optmal Desgn of Sold, Elastc Plates, Int. J. Solds Structures, Vol. 7,pp Díaz, A., Sgmund, O., 995, Checkerboard Patterns n Layout Optmzaton, Structural Optmzaton, Vol., No., pp Fernandes, P., Guedes, J., M., Rodrgues, H., 999, Topology Optmzaton of Three-Dmensonal Lnear Elastc Structures wth a Constrant on Permeter, Computers & Structures, Vol. 73, No. 6, pp Fonseca, J. S. O., Kkuch, N., 998, Dgtal Imagng Flterng n Topology Optmzaton, Computatonal Mechancs New Trends and Applcatons, CIMNE, Span. Hanson, R. J., Hrbert, K., L., 98, A sparse Lnear Programmng Subprogram, Report SAND8-297, Sanda Natonal Laboratores. Hassan, B., Hnton, E., 998a, A Revew of Homogenzaton and Topology I - Homogenzaton Theory for Meda Wth Perodc Structure, Computer and Structures, Vol. 69, pp Hassan, B., Hnton, E., 998b, A Revew of Homogenzaton and Topology II- Analtycal and Numercal Soluton of Homogenzaton Equatons, Computer and Structures, Vol. 69, pp Jog, C., Haber, R. B., 996, Stablty of Fnte Element Models for Dstrbuted-Parameter Optmzaton and Topology Desgn, Computer Methods n Appled Mechancs and Engneerng, Vol. 3, pp Kasper, E. P., Taylor R. L., 2a, A mxed-enhanced stran method Part I: Geometrcally lnear problems, Computers & Structures, 75, pp Kasper, E. P., Taylor R. L., 2b, A mxed-enhanced stran method Part II: Geometrcally nonlnear problems, Computers & Structures, 75, pp Nordson, F. I., 983, Optmal Desgn of Plates wth a Constrant on the Slope of the Thckness Functon, Int. J. Solds Structures, Vol. 9, pp Park, Y. K., 995, Extensons of Optmal Layout Desgn Usng the Homogenzaton Method, PhD theses, Uersty of Mchgan. Petersson, J., Sgmund, O., 998, Slope constraned Topology Optmzaton, Internatonal Journal of Numercal Methods n Engneerng, Vol. 4, pp Sgmund, O. 997, On the desgn of complant mechansms usng topology optmzaton, Mechancs of Structures and Machnes, 25(4), pp , 997. Swan, C. C., Kosaka, I., 997, Vogt-Reuss Topology Optmzaton for Structures wth Lnear Elastc Materal Behavor, Internatonal Journal of Numercal Methods n Engneerng, Vol. 4, No., pp Wujek, B. A., Renaud, J., E., 998a, New Adaptatve Move-Lmt Management Strategy for Approxmate Optmzaton, Part, AIAA Journal, Vol. 36, No., pp Wujek, B. A., Renaud, J., E., 998b, New Adaptatve Move-Lmt Management Strategy for Approxmate Optmzaton, Part 2, AIAA Journal, Vol. 36, No., pp Yuqun, L., Yn, X., 994, Generalzed Conformng Trangular Membrane Element wth Rgd Rotatonal Freedoms, Fnte Elements n Analyss and Desgn, Vol. 7, pp J. of the Braz. Soc. of Mech. Sc. & Eng. Copyrght 23 by ABCM July-September 23, Vol. XXV, No. 3 / 3