COMPUTATIONAL MECHANICS WCCM VI n conjuncton wth APCOM 4, Sept. 5-1, 24, Bejng, Chna 24 Tsnghua Unversty Press & Sprnger-Verlag Topology Optmzaton of Elastc Materal Mcrostructures wth Classc Models of Mcromechancs W.H. Zhang 1*, S.P. Sun 1, L. Wang 1, D. Wang 2 1 Insttute of Mechatronc Engneerng, Northwestern Polytechncal Unversty, X an, 7172, Shaanx, Chna 2 Insttute of Aeronautcs, Northwestern Polytechncal Unversty, X an, 7172, Shaanx,Chna e-mal: zhangwh@nwpu.edu.cn, shpngsun@163.com, lewang9@hotmal.com, wangdng6682@sna.com Abstract Nowadays, topology optmzaton consttutes a basc and powerful method for the desgn of materal mcrostructures. Recent advances show that ths method s manly based on the homogenzaton method that deals wth perodc unt cells usng the asymptotc expanson. Ths formulaton requres that the homogenzaton method be mplemented as an addtonal module n the context of a fnte element system to predct the effectve propertes of the mcrostructure. Theoretcally, the homogenzaton method s known to be vald only when the characterstc scale-sze of the concerned unt cell s small enough. In ths paper, topology optmzaton of the materal mcrostructure s carred out based on the classc model of mcromechancs wth Drchlet boundary condtons. The stffness of the mcrostructure s maxmzed wth the gven amount of materals. It s shown that ths approach s effcent to gve rse to satsfactory materal layouts n dfferent loadng condtons from the engneerng vewpont. Numercally, the advantage of ths approach les n that topology optmzaton of the mcrostructure can be performed n the same way as ts orgnal verson used for macrostructures wthout needs of addtonal software programmng. The key s to retan sutable boundary condtons n the fnte element modelng of the unt cell. Fnally, numercal tests are used to llustrate the valdty of the proposed method. Key words: topology optmzaton, materal desgn, mcrostructure, homogenzaton method INTRODUCTION Topology optmzaton s recognzed as an effcent tool n materals layout desgn. The mportance s based on the fact that the topology, shape, scale and volume fractons of the consttuents for a mcrostructure have a determnant nfluence upon ts effectve propertes. By topology optmzaton, structures can be, on one hand, effectvely optmzed n stffness, stablty and dynamc responses wth the gven amount of materals n the macro-scale; and on the other hand, materals can be optmally talored ether to attan prescrbed even extreme propertes or to provde nnovatve mcrostructures n the mcroscopc scale. For example, desgns of mcrostructures wth negatve Posson rato, zero thermal expanson coeffcents [1]. Topology optmzaton becomes thus a common technque to carry out smultaneous optmzaton of materals and structures. Hstorcally, the classcal model of mcromechancs (CMM) and the homogenzaton method (HM) consttute two basc approaches. The former s based on Drchlet or Neumann boundary condtons [2] whereas the latter s based on the asymptotc expanson wth perodc condtons of the unt cell [3]. Actually, desgnng mcrostructure s often carred out by ntegratng topology optmzaton wth the homogenzaton method. To ths end, a fnte element mplementaton has to be made for both evaluatons and senstvty analyss of macroscopc propertes. Conversely, the classcal model of mcromechancs s rather smple for the numercal predcton of effectve propertes of mcrostructures. But t seems to be not well exploted and combned wth topology optmzaton. 1 1
In ths paper, the classcal model of mcromechancs s used together wth topology optmzaton procedure to desgn the mcrostructure of materals. The procedure can be smply carred out as the same as for macrostructures wthout further mplementaton work. Moreover, after the nvolved mcrostructure s optmzed, attempts are made to perform topology optmzaton of macrostructures wth orthotropc materal propertes. CLASSICAL MODEL OF MICROMECHANICS WITH DIRICHLET BOUNDARY CONDITIONS v=1 u=1/2 u=1 v=1/2 (2) (3) Fg. 1 Three loadng cases wth unt deformaton boundary condtons Consder a 2-D unt cell shown n Fg. 1. The equvalent stffness matrx conssts of E1111 E1122 E = E 2211 E 2222 1212 E The equvalent macroscopc stffness matrx can be numercally predcted by usng the fnte element method. Under baxal and shear solctatons of ntal unt deformatons along the boundary, the deformaton energy and reacton forces along the boundary of the unt cell can be easly evaluated. Consequently, one can wrte (2) E 1111 = C, 2222 C (3) E =, E 1212 = C, 1122 = E 2211 = σ 22 E (2) Note that symbol C and the superscrpt corresponds to the deformaton energy and the load case, respectvely. Alternatvely, f the homogenzaton method s used, the equvalent macroscopc stffness matrx wll be evaluated as the ntegral over the unt cell E jkl 1 χ p = Y ( Ejkl Ejpq ) dy Y y kl q (3) Where Y and E jkl denote the volume of the unt cell and the rgdty coeffcents of the nvolved consttuents, respectvely. In ths formulaton, the unt cell has to be specfed wth perodc dsplacement boundary condtons and the loadng measurng the spatal varaton of materal propertes over the unt cell has to be defned when 2 2
evaluatng the ntermedate dsplacement feld kl χ p by the fnte element method. For 2D and 3D unt cells, evaluatons of kl χ p wll be carred out under 3 and 6 loadng cases, respectvely wth kl = 11,22,12 or kl = 11,22,33,12,13, 23. From (3), t s clearly seen that each materal consttuent n the unt cell has ts proper contrbuton to the equvalent propertes. In fact, the frst part of the ntegral n (3) corresponds to a volume fracton average whereas the second one can be regarded as a correcton term dependng upon the shape, confguraton and poston of consttuents nsde the unt cell. TOPOLOGY OPTIMIZATION PROCEDURE OF THE MICROSTRUCTURE As llustrated below, both above methods agree well n precson for the numercal predcton of the stffness matrx. Snce prescrbed dsplacement boundary condtons are appled, the stffest mcrostructure,.e, the maxmzaton of dagonal terms of the stffness matrx s the same as the maxmzaton of deformaton energy n 3 loadng cases. Mathematcally, when the unt cell s meshed wth fnte elements whose pseudo denstes X are used as desgn varables, the optmzaton problem s expressed as Max C ( k ) n ( X ) V ( X ) = xv V = 1 TVa ( X ) P < δ x 1 = 1, n (4) where V(X) s the total materal volume lmted by ts upper bound V over the unt cell. A small value of δ =1-5 s used to avod the sngularty of the elementary stffness matrx durng optmzaton. TV a denotes the total varaton (TV) control whose role s, lke the permeter control, to regularze the pattern of the materal layout. The role of upper bound P s twofold. On one hand, the avodance of checkerboards can be ensured wth small values of P, on the other hand, ts relaxaton wll elmnate ntermedate values between and 1 for densty varables. Here, we adopt TV 2 whose expresson s M 2 TV2 = lk ( x x j ) (5) k = 1 lk denotes the edge length of the kth nterface between two adjacent fnte elements and j. Jumps of materal densty varatons at all adjacent elements are controlled by the upper bound P. Detals about ths control can be found n [4]. To solve problem (4), the deformaton energy of the unt cell assocated wth prescrbed dsplacements shown n Fg. 1 wll be frstly evaluated at current values of densty varables. Thereafter, senstvty analyss of the deformaton energy s evaluated n such a way that prescrbed dsplacements are regarded as equvalent prescrbed loads. From ths vewpont, the maxmzaton problem (4) becomes a mnmzaton one and the senstvty of the deformaton energy wll be a scalng of the concerned element deformaton energy under equvalent prescrbed loads. Namely, C x ( k ) = p x C ( k ) (6) where p s the exponent used as the penalty of the power law for the element stffness matrx (see Ref. [4]). NUMERICAL TESTS In ths secton, nvestgatons wll be focused on the followng aspects: 3 3
1) Comparson between the homogenzaton method and the classc model of mcromechancs. 2) Topology optmzaton of mcrostructures 3) Numercal computng of macrostructure flexblty wth and wthout homogenzaton 4) Topology optmzaton of macrostructures consstng of mcrostructures wth orthotropc propertes 1. Numercal predctons of the stffness matrx of the mcrostructure A set of unt cells studed prevously n [5] wth dfferent scale szes and elastc constants of consttuents s gven n Fg. 2. Both CMM and HM methods are appled, respectvely to evaluate the macroscopc stffness matrx. Note that snce boundary condtons used here are dfferent from those n [5], our results wll be dfferent from those obtaned n [5] for the stffness matrx. Fg. 2 Illustraton of mcrostructures of unt cells Assume that the mcrostructure s made of two sotropc materals ndcated by red and blue colors, respectvely. The volume fracton s 5%. Case 1: materal propertes are E1 = 1, E2 = 1; ν 1 =ν 2 =.3 Stffness matrces obtaned wth the classc model (2) and the homogenzaton method (3) are as follows, respectvely. 3.745972 1.3716327 3.745972 1.183233 3.745974 1.371632 3.745974 1.183233 (7) (CMM) (HM) Case 2: materal propertes are E 1 = 1, E 2 = 1 ; ν 1 =ν 2 =.3 Stffness matrces obtaned wth the classc model (2) and the homogenzaton method (3) are as follows, respectvely. 2.423664.713 2.423664.87422 (CMM) 2.43366.713 2.4366.8874 (HM) From above results, t turns out that the scale-sze of the unt cell has no nfluence upon the results wth the current mcromechancs model and that both methods produce almost the same stffness matrx. Therefore, both methods can be used n the numercal predcton of the equvalent stffness matrx. 2.Topology optmzaton of mcrostructures Based on three boundary condtons gven n Fg. 1, the mcrostructure wll be desgned wth a two-phase of vod and sotropc materal. To do ths, relatons n (2) wll be used to evaluate stffness terms, optmzaton wll be carred out by means of (4) and senstvty analyss wll be performed by means of (6). In these tests, volume fractons of 3% and 5% wll be taken nto account for the sold materal n the unt cell, respectvely. As plotted n Fg. 3, results of materal layouts are gven n columns 1 and 3. To have a global vew about the dstrbuton of mcrostructures, columns 2 and 4 provde a perodc repetton of the related unt cell. From the engneerng vewpont, the above mcrostructures possess ndeed the maxmum stffness n tracton and shear. (8) 4 4
Volume fracton 3% Volume fracton 5% Fg. 3 Topology optmzaton of mcrostructures Now, consder a combned load case of b-objectve optmzaton wth horzontal tracton ( u =. 1 ) and shear boundary condtons ( u = v =.1). Gven a volume fracton of 5%, the desgn soluton of the mcrostructure s shown n Fg. 4. One can see that the TV control s effectve to regularze the desgn pattern. Fg. 4 Topology optmzaton of mcrostructures In ths case, equvalent stffness matrces obtaned are as follows 1.91269 1.572636 1.91269 1.341544 ( 9 ) 3.Flexblty analyss and topology optmzaton of the macrostructure As s known, topology optmzaton wth stffness desgn n the macroscopc level depends strongly upon the accuracy of flexblty analyss. To nvestgate the nfluence of the mcrostructure, consder now a cantlevered beam wth perforated holes dstrbuted perodcally over the doman. As shown n Fg. 5, the beam s loaded by a 5 5
vertcal force on the mddle pont of the rght sde. Suppose that Young s modulus of the beam materal s E 1 =26.6667. Perforaton 4 8, E 1 =26.6667 (2) E 1 /E 2 =26.6667/1.e-3 (3) Perforaton 8 16, E 1 =26.6667 (4) Homogenzaton Fg. 5 Flexblty analyss In Fg. 5, plot corresponds to the dsplacement result when the structure has a perforaton of 4 8 and s drectly dscretzed and evaluated by the fnte element method. Plot (2) corresponds to the dsplacement result for whch vods are all flled wth softenng materals (E 2 =1.e-3). Plot (3) corresponds to the result wth a perforaton of 8 16. Plot (4) corresponds to the result for whch an equvalent sold sheet of orthotropc materal propertes s used wth the followng homogenzed stffness matrx. 12.96861 3.21648 17.51743 2.751778 It s seen that wth an ncrease of perforatons, the dsplacement response has the tendency of convergng to the homogenzed result. Now, consder the topology optmzaton of mnmzng the complance of the cantlevered beam n the macroscopc level. By means of the above equvalent stffness matrx, the power law wth exponent p=4 s used to penalze the above stffness matrx n the fnte element formulaton. A macroscopc volume fracton of 6% s used. The desgn soluton s shown n Fg. 5. Fg.5 Topology optmzaton of the macrostructure 6 6
CONCLUSIONS In ths paper, the classc model of mcromechancs and homogenzaton method are nvestgated. It s shown that both methods have a good coherence for the numercal predcton of effectve propertes of the mcrostructure f boundary condtons are sutably used. Ths coherence provdes an alternatve approach of ntegratng the classcal model wth the topology optmzaton methodology. Under dsplacement boundary condtons, deformaton energes assocated wth dagonal terms of the equvalent stffness matrx can be maxmzed to fnd the optmal layout of the stffest mcrostructures. Wth the avalable materals, topology optmzaton of mcrostructures can be performed n the same way as for structures of macro-scale. Furthermore, the last example provdes an dea of jonng mcrostructure optmzaton wth macrostructure optmzaton. It s shown that topology optmzaton of the macrostructure can be carred out based on the desgn result of mcrostructures. Ths concept wll promote the methodology of smultaneous desgn of materals and structures. Acknowledgements The support of the Natonal Natural Scence Foundaton of Chna (Grant No. 137283) s gratefully acknowledged. REFERENCES [1] O. Sgmund, S. Torquato, Desgn of materals wth extreme thermal expanson usng a three-phase to[ology optmzatuon method, J. Mech. Phys. Solds., 45, (1997), 137-167. [2] Z. Hashn, Analyss of composte materals, ASME Journal of Appled Mechancs, 5, (1983), 481-55. [3] A. Bensoussan, J.L. Lons, G. Papancolaou, Asymptotc analyss for perodc structures, North Holland, Amsterdam (1978). [4] W.H. Zhang, P. Duysnx, Dual approach usng a varant permeter constrant and effcent sub-teraton scheme for topology optmzaton, Computers & Structures, 81(22/23), 2173-2181. [5] S. Pecullan. L.V. Gbansky, S. Torquato, Scale effects on the elastc behavor of perodc and herarchcal two-dmensonal compostes, J. Mech. Phys. Solds., 47, (1999), 159-1542. 7 7