Division of Materials Science, Graduate School of Science and Engineering, Saitama University, Saitama, Japan

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1 CHECKERBOARD PROBLEM TO TOPOLOGY OPTIMIZATIO OF COTIUUM STRUCTURES Ju-ch Koga, Jro Koga ad Shu Homma Dvso of Materals Scece, Graduate School of Scece ad Egeerg, Satama Uversty, Satama, Japa Keywords: Structural optmzato; Flter; Partto of uty; SIMP; BESO; FEM Abstract: The area of topology optmzato of cotuum structures of whch s allowed to chage order to mprove the performace s ow domated by methods that employ the materal dstrbuto cocept. The typcal methods of the topology optmzato based o the structural optmzato of two phase compostes are the so-called varable desty oes, lke the SIMP (Sold Isotropc Materal wth Pealzato) ad the BESO (B-drectoal Evolutoal Structure Optmzato). The topology optmzato problem refers to the saddle-pot varato oe as well as the so-called Stokes flow problem of the compressve flud. The checkerboard patters ofte appear the results computed by the SIMP ad the BESO whch the Q -P 0 elemet s used for FEM (Fte Elemet Method), sce these patters are more favourable tha uform desty regos. Computatoal expermets of SIMP ad BESO have show that flterg of sestvty formato of the optmzato problem s a hghly effcet way that the checkerboard patters dsappeared ad to esure mesh-depedecy. SI ths paper, we dscuss the theoretcal bass for the flterg method of the SIMP ad the BESO ad as a result, the flterg method ca be uderstood by the theorem of partto of uty ad the covoluto operator of low-pass flter. Itroducto A commo structural optmzato mechacal structural desg feld s three categorzed to the sze, shape ad topology optmzato of a elastc structure gve certa boudary codtos. The sze optmzato s to determe oly the sze of the materals, ad the shape optmzato refers to the outer shape of materals. The purpose of topology optmzato s to fd the optmal layout of a structure cluded the holes wth a specfed rego. The oly kow quattes the problem are the appled loads, the possble support codtos, the volume of the structure to be costructed ad possbly some addtoal desg restrctos such as the locato ad sze of prescrbed holes or sold areas. I ths problem the physcal sze ad the shape, ad the coectvty of the structure are ukow. The area of the topology optmzato of cotuum structure s ow domated by methods that employ the materal dstrbuto cocept, lke the SIMP (Bedsøe ad Sgmud, 2003) ad BESO (Huag ad Xe, 200). The topology optmzato problem refers to the saddle-pot varato oe as well as the so-called Stokes flow oe of compressve flud (Brezz ad Fort, 99). The checkerboard patter ofte appear the results computed by the SIMP ad BESO whch the Q -P 0 elemet s used for FEM (Fte Elemet Method) as well as the Stokes problem, sce these patters are more favourable tha the uform desty regos (Daz ad Sgmud, 995, Jog ad Haber, 996, Sgmud ad Petersso, 998). Computatoal expermets of the SIMP ad BESO have show that the flterg of sestvty formato of the optmzato problem s a hghly effcet way that the checkerboard patter dsappeared. Sce the flter s usually used as low-pass flter for ose cleag the feld of the mage processg, ths type of flter was troduced to the problems of the topology optmzato by Sgmud (994). The fucto of the flter s explaed as the average of the desty by the desty of eghbourhood, or the covoluto operator of the low-pass flter (Sgmud ad Petersso, 998, Hassa ad Hsto, 999 ). However, the theoretcal bass for the flterg method s ot yet uderstood completely. I ths paper, we dscuss the theoretcal bass for the flterg method of SIMP ad BESO ad as the result, the flterg method ca be explaed by the theorem of partto of uty ad the covoluto. The paper s cossted of theory of FEM for elastcty equato of the sotropc materal ad of checkerboard problem (Sec.), the calculated procedure of SIMP ad BESO (Sec.2), the basc theory for the flterg (Sec.3), the computed Results of SIMP ad BESO (Sec.4), ad cocluso remarks.

2 . Theory of FEM for elastcty equato of a sotropc materal Let s cosder the 2-dmesoal elastcty problem. The s doma as show Fguree. f t Γ 0 Γ Fg. Dagramm of the exteral force f ad the boudary codtos Let the Γ0 be a part of Γ o whch we assume u = 0. We also assume the exstecee t of a dstrbuted force to f (eg. Gravty) ad o Γ of a tracto t that s decomposed to ormal part t ad a tagetal t t. The goverg equatos are gve as, σ ( u) = f u u σ ( u) = 2 µε ( u) + ςε ( u) δ kk where σ ( u) s stress, u dsplacemet, f (eg. Gravty). the body force, t, tt ad ε ( u) defed as The μad ς are Lame costats. The weak form of Eq.() s gve as, ε 2 µ ε ( u) : ε ( v) dx + ς dvu dv v dx = f v dx + t v ds + t vt ds Γ0 Γ = 0, σ = 0 = t, t t u u ( u) = ( + ). 2 x x o Γ o Γ. t Γ Γ 0 () (2) (3) Now, a( u, v) = 2 µ ε ( u) : ε ( v) dx, b( u, q) = dvu q dx, v V = (H ( )), q Q = L ( ) (4) Whe p = ς dv u Eq.(3) s rewrtte as, a a( u, v) + b( v, p) = ( f, v), v V, b( u, q) = ( p, q), ς q Q. (5) The exstece of the soluto of Eq. (5) s vald by the theorems of Lax-Mlgram ad the frst equalty of Kor as,

3 2 2 2 ε ( v ) dx C v, v V = (H ( )), 0. (6) The f-sup codto, however, s ot vald the FEM usg the Q -P 0 elemet. As a result, dv u = 0 at the corer pots of the macroelemet formed of four quadrlaterals. Whe dv u = 0, the materal s compressble. If the chage of volume of the materal lke a steel geerally occur, the stuato s ot practcal. Whe the u s the velocty feld, ad ε ( u) s the rate of stra, Eq. (5) s the expresso of Stokes problem for the low velocty of flud flow. Hece, checkerboard problem of the elastcty problem occurs for the same reaso as the Stokes problem why appearace of the checkerboard patter s gve the book by Gralt ad Ravart (986). 2. Calculato procedure of SIMP ad BESO We use the SIMP ad BESO methods ths paper. I each of methods, we fx the desg doma D cossted of both the materal rego ad the vod D\, ad the sgle equato obtaed by the vrtual work prcpal wth the Fuck s low s used. The equato s gve as, KU = F, (7) where U ad F are the global dsplacemet ad force vectors, respectvely. K s the global stffess matrx, u ad k = E( ρ ) k0 are elemet dsplacemet vectors ad stffess matrx, respectvely. Frst, the desg doma s assumed to be rectagular ad dscretsed by square fte elemets. I the SIMP codes (Sgmud, 200, Adreasse et al., 20), a topology optmzato problem based o the power-law s followed,.e. each elemet s assged a desty ρ that determes ts Youg s modulus E : E( ρ ) = E + ρ ( E E ) (8) p m 0 m where E o s the Youg s modulus of the materal, E m s a very small Youg s modulus assged to vod regos order to prevet the stffess matrx from becomg sgular, ad p s a pealzato factor (typcally p=3) troduced to esure black-ad-whte solutos. The obectve s to mmze complace ca be rewrtte as: T T m c( ρ) U KU E( ρ ) u k0u, x = = = V ( x) subect to : = f, V 0 : KU = F, :0 ρ ρ, < m < where K s the global stffess matrx, u ad k = E( ρ ) k0 are elemet dsplacemet vectors ad stffess matrx, respectvely, x the vector of desg varables, ρ m a vector of mmum relatve destes, the total umber of elemets the desg doma, p the pealzato power, V(x) ad V 0 are the materal volume ad desg volume, respectvely, ad f the prescrbed volume fracto. The optmzato problem Eq. () s solved by meas of a stadard optmalty crtera method as followed: η max(0, ρ m) f ρ B max(0, ρ m) ow η ρ = m(0, ρ + m) f ρ B m(, ρ + m) (0) η ρ B otherwse Where m s a postve move lmt, η(=/2) s a umercal dampg coeffcet, ad B s obtaed from the optmalty codto as: c V B = λ () ρ ρ (9)

4 Where λs the Lagraga multpler. The sestvtes of the obectve fucto c ad materal volume V wth respect to the elemet destes ρ are gve by: c = pρ u k u p T 0 V = (3) Equato (3) s based o the assumpto that each elemet has ut volume. I order to esure exstece of solutos to the topology optmzato problem ad to avod the formato of checkerboard patters, they troduced the mesh-depedecy flter as: (2) ρ c = H = H = c ρ (4) where H = rm dst(, ) dst(, ) r, =, 2,, { m }. (5) The operator dst(,) s defed as the dstace betwee the cetre of elemet ad the cetre of elemet. O the other had, the obectve of the BESO s rewrtte as: T T m c( ρ) = U KU = E( ρ ) u k0u, ρ 2 2 = V ( ρ) subect to : = f, V 0 : KU = F, = ρm or : ρ. (6) The ρ s dscrete that ρ = ρ or ad ths pot dfferet from the SIMP. The the sestvty umber s defed as, m T u ku f ρ = α = 2 0 f ρ = ρ m. (7) Now, by k α of the k-th terato ad k α of k-, α s defed as, where w s the same as k k = ( + ), α α α 2 = α = = w α w (8), (9) H of the SIMP. The optmzato crtera s very smple way that ρ at the elemet whch s smallest s chaged from to ρm ad ρ at the elemet whch α s largest D/ s chaged from ρm to. α

5 3. The basc theory for the flter of SIMP ad BESO The f-sup codto s ot vald the mxed FEM used the Q -P 0 elemet that s wdely used the feld of egeerg (Brezz ad Fort, 99) ad the checkerboard patter appears (Daz ad Sgmud, 995, Jog et al. 996, Sgmud ad Petersso, 998). They also show that the dstrbuto parameter optmzato problem s the varato of saddle pot as well as the Stokes flow problem. The restrcto codto of the topology optmsato problem s w λ α ( 2 ρ ) = 0, (20) where w s the stra eergy ad α the Lagraga multpler. Petersso (999) suggested that the term α( 2 ρ ) comes from α ρ ( ρ ) dx whch s correct the sese that soluto to the problem wth pealty fucto wll approach the D correct oes, as oe creases the pealty actor α. The α s zero from ths dscusso ad Eq. (20) becomes w λ = 0. (2) I the eghbourhood of the saddle pot, the complace c s related to the stra eergy w, We obta the followg equato, c = w. (22) c λ = 0, (23) the, we product ρ to the both had sde of Eq. (22), we obta ρ ρ c = = ρ B. (24) ow λ Ths equato s the orgal optmzato crtera. We fd that Eq. (24) s the ecessary codto that gves the checkerboard patter of ρ. The ρ c of Eq. (4) s effectve way ot to occur ths patter, ad the materal desty ρ locally averaged ad chaged from the checkerboard patter to the uform desty rego. Frst, let s remember the theorem of a partto of uty as the followg proposto ( Follad, 999 ). If R s a topologcal space ad E R, a partto of uty o E s a collecto { h α } α Λ of fucto C( R, [0,]) such that, each x E has a eghbourhood o whch oly ftely may h 's α are ozero; h ( x) α Λ α = for x E. A partto of uty { h α } s subordate to a ope coveru of E f for α there exsts U U wth support ( hα ) U. I ths paper, h = H H =. (25)

6 Sce = s the cotuous fuctos havg a compact support, ad h = from the theorem of partto h H H = of uty ad ρ s a subset of ( C ) 0 R, the checkerboard patters are dsappeared. We ca also terpret the flterg defed by Eq.(9). The fucto of covoluto betwee a tempered dstrbuto ad a cotuous fucto s C 0 (D) whch s dese L 2 (D) from the formula of the Fourer trasform F [ F( ρ ) F( H )] where F s the Fourer trasform ad F - the verse trasform. = 3. Computed Results of SIMP ad BESO Fgure 2 shows the 2-dmesoal desg doma, the boudary codtos ad the exteral load for the catlever beam. I ths paper, all results are 2-dmesoal structures. The workg doma s a rectagle wth zero dsplacemet boudary codto o the left sde ad ut vertcal pot load at mddle of the rght sde. The doma sze s a rectagular mesh. Fgure 3 s the checkerboard patter of the catlever beam computed by SIMP. The rego of the checkerboard patter s relatvely wder tha that of uform desty regos, sce the patter appears the rego of hgh shear stress. Fgure 4 s the resultg shape after the flterg ad the pealzato SIMP. Fgure 5 shows the chage of the obectve complace vs. the terato umber of catlever beams as show Fgs.3 ad 4. The covergece s smooth ad quck. From the Fgure, the eergy of checkerboard patters s slghtly smaller tha the uform desty regos. The rego of the checkerboard patter s relatvely wder tha that of uform desty regos, sce the patter appear the rego of hgh shear stress. The uform desty regos occur due to the character that the fucto space C0( R ) s coected. Fg. 2 The desg doma, boudary codto, ad exteral load for the optmzato of a catlever beam Fg. 3 Checkerboard patters elemets; the flter radus r m = 0.2 wth pealzato wthout flterg, ν=0.22, p =3, f = 0400 complace terat

7 Fg. 5 Obectve fucto (complace) vs. Fg elemets; the flter radus rm=.3 terato, -;wthout flterg(fg.3), wth pealzato ad flterg, ν= 0.22, p = 3, f = ---;wth flterg (Fg.4) Fgure ad 7 show the results of 40 20, elemets for the same catlever beam, respectatvely. The result of elemet s the same topology as that of elemet ad s slght dfferet from that of From fgures, we fd that ths flterg method has ot mesh-depedecy. Fg elemets; the flter radus r m =.5 wth pealzato, ν=0.22, p = 3, f = 0400 Fg elemets; the flter radus r m =.5 wth pealzato, ν=0.22, p = 3, f = 0400 Fgure 8 s the checkerboard patter of the short catlever beam. ad the pealzato. Fgure 9 s the resultg shape after the flterg Fg. 8 Checkerboard patters elemets; the flter radus r m = 0.2 wth pealzato wthout flterg, p =3, f = 025 Fg elemets; the flter radus rm=.5 wth pealzato ad flterg, p = 3, f = 0.25

8 Fg elemets; the flter radus r m =.5 wth pealzato, p = 3, f =0.25 Fg elemets; the flter radus r m =.5 wth pealzato, p = 3, f =025 Fgures 0 ad show the results of ad elemets for the same short catlever beam computed by the BESO. Fgure 2 shows that the complaces of ad elemets vs. terato. From fgure, we fd that ths flterg method s ot mesh-depedecy. complace terat Fg. 2 Obectve fucto (complace) vs. terato of short catlever beam BESO, -;80 40 elemets (Fg. 9) ---; elemets (Fg. ) Cocluso remarks

9 We vestgate the topology optmzato whch the obect fucto s the total complace by the SIMP ad BESO. Sce ths problem s the saddle pot varato, ad the Q -P 0 elemet used for FEM s ot vald for the f-sup codto, the checkerboard patter ofte appears. I order to avod ths patter, the flter of sestvty s used. We dscuss the theoretcal bass for the flterg method of SIMP ad BESO. As the result, the flterg method ca be explaed from the theorem of partto of uty. Nome creature c, c(ρ) = complace [J m -3 ] E = Youg modules [N m -2 ] F = force [N m -3 ] f = volume rato [-] H = dstace defed by Eq.(5) [-] K = total stffess matrx [N m -3 ] k = stffess matrx of elemet [N m -3 ], = umber of elemet, drecto of R [-] p = pealty expoet [-] r m = radus of flter [m] U = dsplacemet vector [m] u = dsplacemet [m] V(ρ) = volume of materal [ m 3 ] V 0 = volume of desg doma [ m 3 ] w = stra eergy [J m -3 ] Greek-letters ε = stra [-] λ = Lagraga multpler [-] ρ = desty [-] Lterature cted Adreasse, E., Clause, A., Scheveels, M., Lazarov, B.S., Sgmud, O., Effcet Topology Optmzato Matlab Usg 88 Les of Code, Struct. Multdsc. Optm., 43, -6 (20). Bedsøe, M.P., O.Sgmud, Topology Optmzato Theory, Methods ad Applcatos 2d ed., Sprger, New York Berl Hedelberg (2003). Brezz, F., Fort, M., Mxed ad Hybrd Fte Elemet Methods, Sprger, New York Berl Hedelberg (99). Daz, A., Sgmud, O., Checkerboard Patters Layout Optmzato, Struct. Optm., 0, (995) Follad, G.B., Real aalyss, Moder Techques ad Ther Applcatos 2 d ed., Wley, New York (999) Gralt, V., Ravart, P.A., Fte Methods for Naver-Stokes Equatos, Theory ad Algorthms, Sprger, New York Berl Hedelberg (986). Huag, X., Xe, Y.M., Evolutoary Topology Optmzato of Cotuum Structures Methods ad Applcato, Weley (200) Hassa, B., Hto, E., Homogezato ad Structural Topology Optmzato, Theory, Practce ad Software, Sprger, Berl Hedelberg New York (999). Jog, S.H., Haber, R.B., Stablty of Fte Elemet Models for dstrbuted-parameter Optmzato ad Topology desg, Comput. Methods Appl. Mech. Egrg., 30, (996). Petersso, J., Some Covergece Results Permeter-cotrolled Topology Optmzato, Compt. Methods Appl. Mech. Egrg., 7, (999) Sgmud, O., Desg of Materal Structures Usg Topology Optmzato, Ph.D. thess, Tech. Uv. of Demark (994). Sgmud, O., A99 Le Topology Optmzato Code Wrtte Matlab, Struct. Multdsc.Optm., 2, (200). Sgmud, O., Petersso, J., Numercal Istabltes Topology Optmzato: A Survey o Procedures Dealg wth Checkerboards, Mesh-depedeces ad Local Mma, Struct. Optm., 4, (998).

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