PEIECWISE CONSTANT LEVEL SET METHOD BASED FINITE ELEMENT ANALYSIS FOR STRUCTURAL TOPOLOGY OPTIMIZATION USING PHASE FIELD METHOD

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1 INTERNATIONAL JOURNAL OF OPTIMIZATION IN CIVIL ENGINEERING It. J. Optm. Cvl Eg., 05; 5(4): PEIECWISE CONSTANT LEVEL SET METHOD BASED FINITE ELEMENT ANALYSIS FOR STRUCTURAL TOPOLOGY OPTIMIZATION USING PHASE FIELD METHOD S. Shojaee *, A. Mohaghegh ad A. Haer Departmet of Cvl Egeerg, Shahd Bahoar Uversty of Kerma, Kerma, Ira ABSTRACT I ths paper the pecewse level set method s combed wth phase feld method to solve the shape ad topology optmzato problem. Frst, the optmzato problem s formed based o pecewse costat level set method the s updated usg the eergy term of phase feld equatos. The resultg dffuso equato whch updates the level set fucto ad optmzato problem s solved through fte elemet method. The proposed method ehaces the covergece rate ad soluto effcecy. Varous two-dmesoal examples are solved to verfy the performace of proposed method. Keywords: topology optmzato; level set method; phase feld method; pecewse costat level set method; fte elemet method. Receved: 5 May 05; Accepted: 8 July 05. INTRODUCTION The shape ad topology optmzato of structures s oe of the most mportat ssues varous egeerg applcatos. Sgfcat developmet shape ad topology optmzato methods s acheved by researchers recet decades [-5]. Level set method troduced by Osher ad Setha [6-8], has bee successfully used shape ad topology optmzato of structures. I level set method the er ad outer boudares are cosdered as desg varables ad the structural boudares are defed by zero level of level set fucto. By ths approach the doma boudares ca be easly combed or separated from each other. If oe uses the explct methods for soluto of Hamlto-Jacob equato, some restrctos wll arse, lke tme cosumg process of * Correspodg author: S. Shojaee, Departmet of Cvl Egeerg, Shahd Bahoar Uversty, Kerma, Ira E-mal address: saeed.shojaee@uk.ac.r (S. Shojaee)

2 390 S. Shojaee, A. Mohaghegh ad A. Haer talzato, satsfacto of Courat-Fredrchs-Lewy (CFL) codto ad depedece of fal topology to tal guess. Pecewse Costat Level Set (PCLS) method was frst troduced by Le et al. [9], for terface problems such as mage processg. I pecewse costat level set method ulke dscrete level set method, there s o eed to solve the Hamlto-Jacob equato, thus t s free of the CFL codto ad the retalzato scheme. Ths feature wll result sgfcat reducto of tme cost. I the PCLS method the terface s determed by forcg the value of the LSF at each mesh pot to be oe of the pecewse costat values. Therefore t eables ths method to create holes durg the evoluto of the LSF wthout usg topology dervatves. Recetly the PCLS method s used wth Lagrage multpler method costrat mmzato problem [0], however Lagrage multpler method the teratos umber for covergece s relatvely hgh. Shojaee ad Mohammada [] combed PCLS wth Merrma-Bece-Osher (MBO) scheme for topology optmzato problems whch resulted crease of covergece rate. I preset study the phase feld method s appled for crease of covergece rate ad reducg the cosumed tme of optmzato process. The phase feld method ad ts cocepts were developed by Cah ad Hllard [] ad Alle ad Cah [3]. Bourd ad Chambolle [4] proposed the dea of mplemetato of phase feld method structural optmzato. Takezawa et al. [5] adopted the phase feld method ad sestvty aalyss soluto of topology optmzato problems. I ths study the PCLS method s combed wth phase feld method to solve the topology optmzato problems. I detals, PCLS s used to form the optmzato problem whch s updated usg the eergy term of phase feld equatos. The resultg dffuso equato whch updates the level set fucto ad optmzato problem s solved through fte elemet method. The comparso of proposed method wth refereces cofrms ts covergece superorty ad soluto effcecy. Ths paper s orgazed as follows: Sec. PCLS bascs are brefly revewed. Sec. 3 deals wth PCLS optmzato platform. The cocepts of pahse feld are descrbed Sec. 4. I Sec. 5 PCLS ad phase feld are combed ad cosequetly the soluto of resultg dfferetal equato usg fte elemet method s descrbed Sec 6. The proposed method s verfed wth refrece methods Sec. 7 through solvg some D optmzato problems.. A PIECEWISE CONSTANT LEVEL SET METHOD I ths secto the pecewse costat level set (PCLS) method s brefly revewed []. Cosder parttog the doma to sub domas as follows: )( where s the uo of the boudares of the sub regos.,

3 PEIECWISE CONSTANT LEVEL SET METHOD BASED FINITE ELEMENT ANALYSIS 39 A pece-wse costat fucto, : R, ca be defed o the ope ad bouded doma whch takes the followg values: Thus, for ay gve partto ( x), x,,,..., )( of the doma we just eed oe PCLS fucto whch takes the values of,,...,. A characterstc fucto, ( x), for each subdoma,, ca be defed as follows: ( j) )3( j, j ( k) (4) k, k Therefore, ( x) for x, ad ( x) 0 wherever Eq. () holds. To represet the dfferet propertes each sub doma, c, we defe a pecewse desty fucto as follow, Where ( ) c ( ) )5( K( ) ( )( )...( ) ( ) )6( A pecwse costat costrat s defed to avod vacuum or overlap betwee the dfferet phases: K( ) 0 )7( fally oe ca calculate the volume ad the permeter of dvdual subdomas the followg form: dx )8( dx, (9)

4 39 S. Shojaee, A. Mohaghegh ad A. Haer 3. PIECEWISE CONSTANT LEVEL SET FRAMEWORK FOR TOPOLOGY OPTIMIZATION PROBLEM Here a smple framework of PCLS for topology optmzato s revewed [6]. Wth respect to phase feld, PCLS s mplemeted two phases: 0 ad. The costat pecewse desty fucto s defed two phases as follows: ( ) ( ) c c )0( Where c ad c are prescrbed values respectvely for hollow ad sold parts. Now f oe cosders c 0 for hollow phase ad c for sold phase, the desty fucto wll have the followg relatoshp wth PCLS fucto. c 0 ( ) c )( A pecewse costat costrat should be defed to guaratee the covergece of level set fucto,, to a uque value: k ( ) 0, k( ) ( ) )( Ths dcates that every pot the desg doma must belog to oe ad oly oe phase ad there s o overlap ad vacuum betwee dfferet phases. I ths paper the optmzato objectve s to mmze the complace over the structural doma for geeral loadg codto ad varous boudary codtos. I other words the optmzato problem s defed as follows: m j(u, ) ( ) F(u)d d s. t H ( )dxv 0 H 0 K( ) 0 a(u,, ) l(, ) a(u,, ) ( ) E ( )d jkl l(, ) f. d p. d N kl )3( where s the structural doma ad ts boudary s represeted by. Also the

5 PEIECWISE CONSTANT LEVEL SET METHOD BASED FINITE ELEMENT ANALYSIS 393 lear elastc equlbrum equato, u deotes the dsplacemet feld, u 0 s the prescrbed dsplacemet o D, E s the elastcty tesor, jkl s the stra tesor. f ad p are body j force ad surface load respectvely. I the objectve fucto J(u), the frst term s the mea complace where fuctoal F (u) s the stra eergy desty ad s the materal desty rato. The secod term the objectve fucto s the regularzato term. s a o-egatve value to cotrol the effect of secod term. Ideed, ths term cotrols both the legth of terfaces ad the jump of, because the value of may ot be cotuous the PCLS. H defes the materal fracto for dfferet phases ad V 0 s the maxmum allowable volume of the desg doma. H s the pecewse costat costrat to guaratee that the level set fucto belogs to oly oe phase. If we use the augmeted Lagraga method to covert Eq. (3) to a ucostraed oe, the followg form s obtaed: L(, ) J( ) a(u,, ) l(, ) H H H d H d )4( where R ad L ( ) are Lagrage multplers ad, 0 are pealty parameters. Now, we eed to fd a saddle pot of the augmeted Lagraga fuctoal L. To fd the saddle pot of ths fucto where there s o body force, f, we have the followg equato as suggested by We ad Wag [7]: (u,,, ). d0 )5( (u,,, ) ( ) E jkl j (u) kl (u).( ) K ( ) (6) ( ) ( ) (c c ) (7) K( ) (8) K( ) K( ) (9) ( )d V0 (0) the steepest descet method ca be appled to satsfy Eq. (3) []:

6 394 S. Shojaee, A. Mohaghegh ad A. Haer d dt 0, 0 )( Thus, the optmzato problem s trasformed to a ordary dfferetal problem of tal value 0. The smplest approach for solvg the Eq. () s to use a explct scheme. However, here the phase feld method s combed wth PCLS method as descrbed secto 5. I proposed method the pecewse costat term s substtuted by eergy term phase feld method. Therefore the H costrat Eq. (3) s omtted ad there s o eed to update ad aymore. 4. PHASE FIELD METHOD I ths secto the cocepts of phase feld method are dscussed [5]. The phase feld fucto (x) s defed o whole aalyss doma to represet the phase of all local pots wth the doma. From the physcal aspect; the phase feld alters the mea phase of local pots. Cosder a system composed of two phases, ad The boudary of each phase s represeted lke a orm fucto whch s extrapolated amog dfferet values. Ths term s called the dffuse terface. The free eergy of Va Der Waals system s gve by [5]: F( ) ( f( )) dx )( where 0 s a coeffcet whch checks fluece of each term. The frst term of Eq. () depcts the teracto eergy of doma ma theory, ad the secod term depcts the double well potetal wth value of f( ) f( ) 0. Double well potetal mples the exstece of lower values of free eergy wth mmum respose to each phase. The tmedepedet evolutoary equatos are troduced the followg. The varato of feld phase fucto s assumed learly depedet to the drecto that the free eergy fucto s mmzed: F( ) M ( ) t )3( By replacg Eq. () Eq. (3): t M ( )( f ( )) )4( Eq. (4) s kow as Alle-Cah Equato [5].

7 PEIECWISE CONSTANT LEVEL SET METHOD BASED FINITE ELEMENT ANALYSIS Doma defed by phase feld fucto Here the optmzato doma, D, s dvded to two phases, 0 ad, ad the boudary betwee phases s deoted by whch s called dffuse terface rego. The doma (x D (x) ) s related to the optmal shape ad 0 (x D (x) 0) s related to D \. s represeted by a terpolato fucto betwee two phases. The doma of optmzer, D, cludes all acceptable shapes of. Where: D )5( ( ) )6( D \ ( ) (7) Therefore the phase feld doma fucto s defed as follows: 0 x 0 x 0 x 0 )8( (a) The ma doma (b) The doma represeted by phase feld fucto Fgure. Phase feld fucto doma [5] s the doma that the chages are appled o t durg optmzato process. A umber of partal dfferetal equatos defe ths doma. The D boudary of doma s dvded to two boudares, DD wth Drtchlet boudary codto ad DN wth Neuma boudary codtos. The sold phase of s depcted by martals of desty

8 396 S. Shojaee, A. Mohaghegh ad A. Haer c ad the hollow phase s depcted by the desty of c where for ( ): ( ) ( ) c c )9( The ma doma s preseted the form of a uo of ad. The locato of boudary s ukow uless whe t s located o the. Whe s small eough, the dffuse terface rego become extesvely th ad ths approxmates the represetato of. 5. COMBINATION OF PIECEWISE CONSTANT LEVEL SET METHOD WITH PHASE FIELD METHOD Now cosder the soluto of Eq. (3). I order to crease the covergece rate ad geerato of smooth ad th boudares, the eergy term of phase feld equato s combed wth Eq. (3). The obtaed equato s a partal secod order dfferetal equato from Alle-Cah equato [5]. Several methods are proposed to solve ths dfferetal equato. I ths paper, the fte elemet method s appled to solve ths equato, whch results accurate solutos ad better geerato of boudares. Now we ca defe the mmzato problem of stra eergy usg the eergy term stated phase feld: m j(u, ) ( ) F(u)d d s. t H ( )dxv 0 0 a(u,, ) l(, ) a(u,, ) ( ) E ( )d jkl l(, ) f. d p. d N kl )30( By addg ths term to the above equato H s omtted form specewse costat term whch results sgfcat crease of covergece rate. Here the Lagrage method s used to covert the costraed optmzato problem to a ucostraed oe. L(, ) J( ) a(u,, ) l(, ) H H )3(

9 PEIECWISE CONSTANT LEVEL SET METHOD BASED FINITE ELEMENT ANALYSIS 397 (u,,, ) ( ) E jkl j (u) kl (u) ( ) ( ) ( ) ( c c ) (3) (33) ( )d V0 (34) d 0, 0 dt (35) d ( ) Ejkl j kl dt (36) To update the level set fucto the followg explct form s used: d. t )37( dt fally by substtutg Eq. (35) Eq. (36) the followg equatos are obtaed: S t t S ( ) Ejklj kl o )38( 6. SOLUTION OF DIFFERENTIAL EQUATION USING FEM Usg FEM, the weak form of Eq. (38) ca be wrtte as follows: t T d d ( ) d t S d S ( ) Ejklj kl for o )39(

10 398 S. Shojaee, A. Mohaghegh ad A. Haer where: ( x) (x) H ( D) wth o )40( Usg FEM Eq. (39) ca be rewrtte as follows: t t A A A o T T N NdA N BNdA S da )4( where N s the shape fucto ad cotrols the stablty of results. The approprate selecto of may result sutable shapes. Four-ode square elemets of sze are used. 7. NUMERICAL EXAMPLES I ths secto some mportat ssues about mplemetato of the proposed PCLS method wth phase feld method are dscussed. These mplemetatos are developed order to mprove the performace of the proposed method. The fte elemet aalyss s based o ersatz materal scheme [], whch flls the vod areas wth oe weak materal. All umercal examples have the followg data: Youg s modulus of real materal s assumed 3 ad ersatz materal 0. Ths also meas c ad c.00. Posso rato for two materals s assumed 0.3 ad the thckess s t. Fgure : A catlever beam 7. The catlever beam Fg. shows the desg doma of a catlever beam. The boudary of the left sde s fxed, ad a vertcal cocetrated force F N s loaded at the bottom of ts rght free sde. The sze of the desg doma s wth a squared mesh of sze ad the volume

11 PEIECWISE CONSTANT LEVEL SET METHOD BASED FINITE ELEMENT ANALYSIS 399 fracto s 50%. Ths example s solved by PCLS-PFM scheme. For ths method the sze of tme step s 46 ad the other parameters are cosdered as e 5, 0.0, 700. The dffcult part s to fd these parameters that ca be chose after testg dfferet values for these parameters. Therefore, dfferet values may lead to dfferet optmal topologes. The evoluto process of the optmal topology ad the bary level set surface are show Fg. 3. Fg. 4 shows the covergece rate of the objectve fucto ad the volume rato for the short catlever beam. It ca be see that the complace of the optmal soluto s cosderably better tha that of the tal desg ad the complace coverges a fast ad stable way because of the preset phase feld method (PFM). The pecewse costat costrat s satsfed each terato ad the PCLS takes the value 0 or everywhere the desg doma. Therefore, there s o overlap ad vacuum betwee the subdomas of dfferet phases. I the covetoal PCLS ths costrat ca be appled wth the pealzato method or augmeted Lagraga method. The pealty method s more stable ad ca be used easly, but to satsfy the costrat exactly, oe has to set t very small. However, ths may cause the stablty of the umercal process. By usg the augmeted Lagraga method, we do ot eed to use a small value for pealty parameter but the terato process s hgh at covergece. Thus, by couplg the PCLS ad the PFM scheme, we ca release ths costrat. It should be oted there s o eed to use the pealty method or the augmeted Lagraga method for ths costrat. (a) Ital desg (b) Step 0

12 400 S. Shojaee, A. Mohaghegh ad A. Haer (c) Step 5 (d) Step 3 (e) Fal desg Fgure 3. The evoluto process of optmal desg ad the level set fucto wth Phase feld method

13 PEIECWISE CONSTANT LEVEL SET METHOD BASED FINITE ELEMENT ANALYSIS 40 Fgure 4. The evoluto process of the complace ad the volume rato The evoluto process of topology optmzato s llustrated Fg. 3. Fg. 4 depcts the hstory of covergece rate ad the volume rato for catlever beam problem. The covergece rate ad stablty of stra eergy verfes the effcecy of proposed method. Moreover, couplg phase feld method wth FEM results smoother ad ther boudares comparso wth other methods. The proposed method coverges wth fewer teratos comparg to AOS-MBO ad MOS-MBO methods. 7. The catlever beam Fg. 5 shows the desg doma of a catlever beam. The boudary of the left sde s fxed, ad a vertcal cocetrated force F N s loaded the mddle of ts rght free sde. The sze of desg doma s wth a squared mesh of sze ad the volume fracto s 50%. Ths example s solved by PCLS-PFM scheme. Fgure 5. A catlever beam

14 40 S. Shojaee, A. Mohaghegh ad A. Haer (a) Ital desg (b) Step 4 (c) Step 8

15 PEIECWISE CONSTANT LEVEL SET METHOD BASED FINITE ELEMENT ANALYSIS 403 (d) Step (e) Fal desg Fgure 6. The evoluto process of optmal desg ad the level set fucto wth Phase feld method Fgure 7. The evoluto process of the complace ad the volume rato 7.3 The Messerschmtt Bölkow Blom beam I the ext example, we cosder Messerschmtt Bölkow Blom (MBB) beam. The desg

16 404 S. Shojaee, A. Mohaghegh ad A. Haer doma ad the boudary codto of ths type of structure are represeted Fg. 8. I ths example the desg doma s dscretzed wth 30 0 squared elemets of sze. The volume fracto of the sold materal s 50%. To fd the optmal topology of ths example, we apply the PCLS-PFM Scheme. Fg. 9 dsplays the evoluto of a optmal topology of the MBB beam. Fgure 8. A MBB beam (b) Step 3 (c) Step 6

17 PEIECWISE CONSTANT LEVEL SET METHOD BASED FINITE ELEMENT ANALYSIS 405 (d) Step 7 (e) Fal desg Fgure 9. The evoluto process of optmal desg ad the level set fucto wth Phase feld method Fgure 0. The evoluto process of the complace ad the volume rato

18 406 S. Shojaee, A. Mohaghegh ad A. Haer 8. CONCLUSIONS I ths study the mprovemet of pecewse costat level set method s vestgated. The pecewse costat costrat s used to prevet the overlap or vacuum betwee two phases. The pealzato method or Lagrage multpler method ca be used to apply the pecewse costat costrat. The pealzato method s a stable method ad feasble to use, however for accurate satsfacto of ths costrat a small pealty coeffcet should be cosdered. Ths wll lead to stablty of umercal soluto process. I augmeted Lagrage method there s o eed to use a small value, but the umber of teratos wll crease. I the proposed method by omttg the pecewse costat costrat ad addto of eergy term based o phase feld the covergece rate s creased. Moreover er ad outer boudares are formed smoother ad ther. The resultg dffuso equato s solved through FEM whch results accurate solutos ad better shapes. REFERENCES. Allare G, et al. Shape optmzato by the homogezato method, Numer Math 997; 76(): Allare G, Koh R. Optmal bouds o the effectve behavor of a mxture of two well-ordered elastc materals, Quarf Appl Math 993; 5(4): Bedsøe MP. Optmal shape desg as a materal dstrbuto problem, Struct Optm 989; (4): Bedsøe MP, Kkuch N. Geeratg optmal topologes structural desg usg a homogezato method, Comput Meth Appl Mech Eg 988; 7(): Suzuk K, Kkuch N. A homogezato method for shape ad topology optmzato, Comput Meth Appl Mech Eg 99; 93(3): Setha JA. Level Set Methods ad Fast Marchg Methods: Evolvg Iterfaces Computatoal Geometry, Flud Mechacs, Computer Vso, ad Materals Scece, Cambrdge Uversty Press, Vol. 3, Osher S, Setha JA. Frots propagatg wth curvature-depedet speed: algorthms based o Hamlto-Jacob formulatos, J Comput Phys 988; 79(): Osher S, Fedkw R. Level Set Methods ad Dyamc Implct Surfaces, Sprger Scece & Busess Meda, Vol. 53, Le J, Lysaker M, Ta XC. A varat of the level set method ad applcatos to mage segmetato, Math Comput 006; 75(55): Luo Z, et al. Desg of pezoelectrc actuators usg a multphase level set method of pecewse costats, J Comput Phys 009; 8(7): Shojaee S, Mohammada M. Pecewse costat level set method for structural topology optmzato wth MBO type of projecto, Struct Multdscp Optm 0; 44(4): Cah JW, Hllard JE. Free eergy of a o uform system. I. Iterfacal free eergy, J Chem Phys 958; 8()

19 PEIECWISE CONSTANT LEVEL SET METHOD BASED FINITE ELEMENT ANALYSIS Alle SM, Cah JW. A mcroscopc theory for atphase boudary moto ad ts applcato to atphase doma coarseg, Acta Metallurgca 979; 7(6): Bourd B, Chambolle A. The phase-feld method optmal desg, I IUTAM Symposum o Topologcal Desg Optmzato of Structures, Maches ad Materals, Sprger, Takezawa A, Nshwak S, Ktamura M. Shape ad topology optmzato based o the phase feld method ad sestvty aalyss, J Comput Phys 00; 9(7): Mohamada M, Shojaee S. Bary level set method for structural topology optmzato wth MBO type of projecto, It J Numer Meth Eg 0; 89(5): We P, Wag MY. Pecewse costat level set method for structural topology optmzato, It J Numer Meth Eg 009; 78(4):