A checkerboard-controlling filter for topology optimization based on stress distribution

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1 6 th World Congrsss o Structural and Multidisciplinary Optimization Rio d Janiro, 30 May - 03 Jun 2005, Brazil A chckrboard-controlling iltr or topology optimization basd on strss distribution Kazm Ghabrai, Sohil Mohammadi Dpartmnt o Civil Enginring, Faculty o Enginring, Univrsity o hran, hran, Iran kazm.ghabrai@gmail.com, smoham@ut.ac.ir Abstract: h chckrboard patchs ar known as on o major instabilitis in topology optimization algorithms that should b ovrcam in ordr to sur th manuacturability o th inal topology. In this papr th rlationship btwn strss/strain distribution and chckrboard ormation is invstigatd and with som ampls it is illustratd that smoothr strss/strain distributions rsult in topologis with wr chckrboards. Basd on this act, a chckrboard controlling iltr is introducd. his nw approach dos not rstricts th dsign domain and no dirct modiication is applid to dsign variabls or snsitivitis. In this approach th only rvision is smoothing strss/strain distributions. Dirnt numrical ampls ar solvd using this approach and all rsults wr r rom chckrboards. h rsults showd that th approach may also ovrcom th local minima problm which is anothr common instability in topology optimization problms. A dtaild comparison btwn th nw approach and a known chckrboard controlling iltr is mad. Kywords: topology optimization, chckrboard, strss distribution, controlling iltrs, SIMP mthod. Introduction Finding th bst topology o structural layout or opology Optimization is o grat importanc in dsign procss, and a grat amount o work has bn don in this ara during th last dcad. In structural topology optimization problms, matrial distribution mthods ar vry common du to thir prcis and simpl dinition o th problm and thir satisactory rsults. In ths mthods topology is rprsntd by matrial distribution in th dsign domain o th problm. By modiying matrial proprtis on can modl dirnt topologis in ths mthods and hnc thr is no nd or rmshing. Using Solid Isotropic Matrial with Pnalization or SIMP mthod is on o th matrial distribution mthods, which is rquntly usd in structural topology optimization. In this mthod, which is to b considrd in this papr, th lmnt rlativ dnsitis ar considrd as dsign variabls. Within ach lmnt, th matrial proprtis ar assumd constant and ar dscribd by th so-calld powr-low intrpolation [, 2, 3]. Dsign variabls (lmnt rlativ dnsitis) vary continuously btwn 0 (void) and (solid). h powr-law intrpolation pnalizs th intrmdiat dnsitis and prssurizs thm into boundary valus. h topology obtaind hr is rprsntd by a gry scal imag which consists o solid (black) lmnts, void (whit) lmnts and lmnts with intrmdiat dnsitis (gry lmnts). h SIMP mthod, lik th othr matrial distribution mthods, surs rom numrical instabilitis such as chckrboards, mshdpndncy and local minima [4]. In this papr w dal with th chckrboard problm which rrs to th ormation o aras o altrnating solid and void lmnts in a chckrboard-lik pattrn. It is dmonstratd by Diaz and Sigmund (995) that ths patchs may gnrat unralistic high stinss in displacmnt basd init lmnt analysis (s [5]) whil th rason is not quit known. In this study th rlationships btwn strss/strain distribution and chckrboard ormation ar invstigatd. his invstigation might b ound usul in inding th sourc o this instability. In th litratur thr ar som known approachs that can ovrcom th chckrboard problm [6]. hs approachs could b classiid into thr catgoris. h irst catgory, somtims rrrd to as Rstriction mthods, consists o thos approachs that rstrict th dsign spac by introducing additional constraints or ampl on primtr or dnsity slops (s [7], [8]). hr is anothr class o approachs in which dsign variabls or snsitivitis ar modiid in ordr to rmov th chckrboard pattrn. hs mthods ar usually rrrd to as Filtring approachs and ar mor widly usd with SIMP mthod [3]. Finally on may us spcial typs o init lmnts to ovrcom th instability. Highr ordr init lmnts as wll as non-conorming init lmnts could b usd hr. Although highr ordr init lmnts do not compltly rsolv th chckrboard problm [5], using non-conorming init lmnts, as rportd by Jang t al. (2000), could ovrcom this instability compltly [9]. In this papr, it is shown that chckrboard ormation is inluncd by strss/strain distributions and smoothing strss/strain distributions in init lmnts may rmov chckrboard patchs compltly. Basd on this rsult, a simpl algorithm or supprssing chckrboard patchs is introducd. his approach dos not rstricts th dsign domain and dosn t nd spcial init lmnts. Bsids, no dirct modiication is applid to dsign variabls or snsitivitis. In this approach only modiication is smoothing strss/strain distributions and hnc it can b rgardd as a iltring approach. During vriication tsts, this approach did not show any kind o chckrboard instabilitis and bsids, som tsts showd that local minima problm is ovrcam as wll. An ovrviw o th papr is as ollows. In sction 2, through som ampls rlationships btwn chckrboard ormation and strss/strain distributions ar dmonstratd. In sction 3, w din our iltrd vrsion o th wll-known minimum complianc topology optimization problm and a simpl algorithm or smoothing strss/strain distribution is introducd. Som o th wll-known problms in structural topology optimization ar solvd using this nw approach and rsults ar rportd or vriication in sction 4. A comparison btwn th prsnt approach and Sigmund s snsitivity iltr [3] is mad in sction 5 and inally in sction 6, th indings ar summarizd and a conclusion is mad. 2. Inluncs o chckrboard ormation and strss distribution on ach othr In topology optimization problms, on may asily ind som rlationships btwn strss/strain distribution and ormation o chckrboard patchs. In th ollowing som ampls ar givn on this issu. As th irst primnt w invstigat th strss distribution o a chckrboard patch to ind th inlunc o ths patchs on strss distribution. o do this a sam problm (MBB problm, s ig 5-a) is solvd onc using normal SIMP mthod with no controlling

2 tchniqu and onc using a snsitivity iltr to ovrcom chckrboards. Optimum topologis ar illustratd in ig. -a and -b and strss distributions o ths topologis ar dpictd in ig -c and -d rspctivly. It should b notd that in ig -c and -d, absolut valus o strsss ar dpictd. It could b sn that in th lt solution, strsss luctuat considrably within chckrboard patchs (ig -c) whil strss distribution o th right solution is not oscillating (ig -d). h rason is that solid lmnts (black lmnts) hav highr stinss in contrast to void (whit) lmnts and so solid lmnts bar much mor strss than void lmnts. (a) (c) (d) Figur. opologis obtaind by normal SIMP mthod without any instability control (a) and by using snsitivity iltr strss distributions o optimum solutions (c), (d) At th nt stp, th inlunc o strss distribution on orming chckrboard patchs is invstigatd. It is known that utilizing highr ordr init lmnts which gnrat smoothr strss/stain solutions, rsults in topologis with wr or vn no chckrboards [5,0]. For urthr study w compar solutions o two dirnt typs o init lmnts namly Constant Strain riangls (CSs) and our-nod quadrilatral init lmnts. CS lmnts modl th strss (strain) distribution by a constant strss (strain) within ach lmnt whil within our-nod quadrilatral lmnts, strss and strain distributions ar assumd bilinar. Hnc strss and strain distributions obtaind by our-nod quadrilatral lmnts ar smoothr compard to CS rsults. h so-calld Short Cantilvr Bam or SCB problm (s Fig. 4-a) is solvd hr using SIMP mthod without any modiication with CS and our-nod quadrilatral init lmnts. Obtaind topologis ar illustratd in ig 6-a and 4-b rspctivly. It could b sn that th topology obtaind through using our-nod quadrilatral init lmnts has lss chckrboard patchs in contrast to th cas o CS init lmnts. hrough ths acts and rsults on may conclud two major conclusions that ar (a) strsss and strains luctuat trmly within chckrboard patchs and smoothr strss/strain approimations rsult in topologis with wr chckrboards. Basd on ths rsults an approach or controlling chckrboard ormation is introducd in th nt sction. 3. Statmnt o th problm and introducing th nw approach o invstigat th inlunc o smoothing strss distribution on orming chckrboard patchs it is ncssary to rormulat th problm so that strss/strain modiications can adjust dsign variabls. h wll known complianc minimization is considrd which is dind as (s [6], [3]) min subjct to: : c() V() V 0 0< whr c() is th man complianc, is th vctor o dsign variabls, min is a vctor o minimum rlativ dnsitis (which ar considrd positiv to avoid singularitis), V() and V 0 ar matrial and dsign domain volums, rspctivly, and is a prscribd valu known as th volum raction. Man complianc, c(), can b ormulatd in trms o strss and strain vctors as: c Ω and i th dsign domain Ω is dividd into init lmnts, w hav: min εσdω (2) ()

3 c Ω εσ dω (3) In which Ω, ε, and σ rprsnt volum, strain vctor and strss vctor o th init lmnt, rspctivly. For simpliication a nw paramtr c is introducd as: Using Eq.(4), Eq.(3) can b rwrittn as ollows: c Ω εσ dω (4) c c (5) h strss vctor is a unction o dsign paramtrs. o show this, strss-strain rlationships rom th init lmnt mthod and th powr-law intrpolation rom SIMP mthod ar rcalld [, 6, 0] p 0 σ D ε, D D (6) Hr D 0 is th matrial matri and p is th pnalization powr. By introducing th paramtr σ 0 as 0 0 σ D ε (7) and using Eq.(6) and Eq.(7) in Eq.(4) rsults in: c εσ dω (8) p 0 Ω o mak us o th optimality critria tchniqu and th huristic updating schm [6, 3], it is also ncssary to rormulat th snsitivity o th objctiv unction in trms o strss and strain vctors which lads to: or simply: c c p 0 p d Ω c p c εσ Ω (9) ow i th strss and strain vctors wr modiid through a smoothing algorithm, th modiid complianc within ach lmnt can b ound as: ε ˆc ˆ εσ ˆ dω () Ω in whichˆ and ˆ ar smoothd strain and strss vctors rspctivly. Using Eq.() in Eq.(0), nw snsitivitis can b calculatd as ollows: σ c p ˆc In th nw statmnt o th problm, quations Eq.(4) and Eq.(9) ar rplacd by quations Eq.() and Eq.(2), rspctivly. 3.. Smoothing algorithm In th ollowing, a nw approach or smoothing strss/strain distribution is introducd. It should b notd that although this smoothing approach is not th only on which can liminat chckrboards. For ach nod o ach init lmnt, all connctd lmnts ar considrd. h nodal strss (strain) o ach spciid nod is thn calculatd through avraging strsss (strains) o that nod within ach connctd lmnt. As an ampl, a 33 msh o 4-nod quadrilatral init lmnts is considrd (ig 2). Within ach lmnt, nods ar numbrd in a countrclockwis ordr as dpictd in ig 2. h strss (strain) vctor at nod j, within lmnt k, is rprsntd by s k j. Using th dscribd algorithm, modiid strss (strain) vctors or thr sampl nods which ar markd in ig 2, ar calculatd as ollows: a a s3 s 3, ( ) ( ) c c s2 2 s2 + s s, s s + s + s + s s s s h g d h g In ordr to calculat th intgral in Eq.() or lmnts with constant or linar strss/strain distribution (lik CS and 4-nod quadrilatral lmnts), it is suicint to dtrmin th intgrand at th cntr o spciid lmnt multiplid by th lmnt s volum. For bilinar strss/strain distributions (also valid or constant strss/strain distributions), th strss (strain) vctor at th d (0) (2) (3)

4 cntr o ach lmnt can b computd by avraging nodal strsss (strains): i ( ) s s + s + s + s i i i i c In this papr, this approach is usd in combination with SIMP mthod. h approach is applid to smooth strsss and strains atr ach init lmnt analysis and th smoothd solutions ar thn usd in Eq.() and Eq.(2) to calculat adjustd snsitivitis. h low o this procdur is dpictd in ig 3. (4) Figur 2. An ampl o a msh o 9 our-nod quadrilatral lmnts Initializ FE analysis Smoothing strsss and strains Calculating objctiv unction and snsitivitis Updating dsign variabls (Optimality Critria) o Convrgnc? Stop Ys Figur 3. Algorithm o th nw approach 4. umrical ampls o vriy th prsnt approach iv ampls ar considrd. In th irst two ampls as wll as in th last two ons, uniorm msh o our-nod quadrilatral squar lmnts is usd whil an unstructurd msh o CS lmnts is usd in th third ampl. hrough th irst thr ampls it is dmonstratd that this approach could ovrcom th chckrboard problm and th ourth ampl illustrats that this approach could rliv th local minima instability. h last ampl dals with msh-dpndnc instability. his ampl shows that this approach could not ovrcom msh-dpndncy. In all o ths ampls th ollowing paramtrs ar adoptd: modulus o lasticity E.0, Poisson s ratio ν0.3 and applid orc F.0. All units ar assumd to b consistnt. h powr-law approach with th ponnt p3 is usd hr and th matrial volum is rstrictd to on hal o th dsign volum (0.5). 4.. Structurd msh with quadrilatral init lmnts In th ollowing ampls a uniorm msh o our-nod quadrilatral squar init lmnts is usd. All lmnts ar qual in siz. h irst ampl is th wll-known short cantilvr bam (SCB) problm with a msh o lmnts. h lt sid o th bam is id and a downward concntratd orc is applid at th middl o th right sid. h dsign domain o this problm is shown in ig 4-a. h inal topologis obtaind by normal SIMP mthod and th nw approach ar illustratd in ig 4-b and 4-c rspctivly. h scond ampl is th MBB bam problm whos dsign domain is dpictd in ig 5-a. Hal o th dsign domain with symmtry conditions is modld hr (ig 5-b). A msh is usd or this modling. Figurs -a and 5-c illustrat th topologis obtaind using normal SIMP mthod and th nw approach rspctivly. It could b sn that in both cass using smoothd

5 strss/strain distribution rmovs th chckrboard pattrn. (a) (c) Figur 4. Short Cantilvr Bam problm - dsign domain (a) and optimum topologis obtaind by normal SIMP mthod and by th nw approach (c) (c) Figur 5. MMB Bam problm - dsign domain (a), hal dsign domain with symmtry conditions, and optimum solution obtaind by th nw approach (c) 4.2. Unstructurd msh with CS init lmnts his ampl is th sam SCB problm which was solvd in th irst ampl (ig 4-a) cpt th msh usd hr is an unstructurd msh o 3776 CS lmnts. Obtaind topologis ar shown in ig 6-a and 6-b. h topology obtaind by th nw approach in both cass is r rom chckrboards and qualitativly rmains unchangd (s ig 6-b and 4-c). his dmonstrats that unctioning o this nw approach is indpndnt o th typ o msh and lmnts. (a) Figur 6. Using unstructurd msh o CS lmnts - optimum topologis obtaind by normal SIMP mthod (a) and by th nw approach 4.3. Local minima and msh-dpndncy hrough th ollowing ampls w sk i th prsnt approach could rliv local minima and msh-dpndnc instabilitis as wll. h msh-dpndnc problm rrs to obtaining dirnt topologis or dirnt msh paramtrs whil th problm o obtaining dirnt solutions or th sam msh but dirnt algorithmic paramtrs rrrd to as local minima [4]. Again th MBB problm whos dsign domain is dpictd in ig 5-a, is considrd. o invstigat th local minima problm th nw approach was implmntd to dirnt initial conditions. Initial conditions ar dpictd in ig 7-a and 7-b and th obtaind topologis or ach on ar illustratd in ig 7-c and 7-d rspctivly. o qualitativ dirnc could b ound in ths solutions. In th nt ampl dirnt msh sizs with 6020 and 2040 quadrilatral lmnts ar considrd or th sam MMB problm. Obtaind solutions ar illustratd in ig 8-a and 8-b rspctivly. It could b sn that using inr mshs rsults in introduction o mor hols in th optimum solution. o dscrib th rason, Bndsø and Sigmund (2004) statd that th introduction o mor

6 hols, without changing th structural volum, will gnrally incras th icincy o a givn structur (s [6]). It is clar that th obtaind topologis ar msh-dpndnt and th nw approach could not ovrcom th msh-dpndncy. Howvr it should b notd hr that whilst msh-indpndnt solutions ar ncssary or obtaining practical simpl topologis, as corrctly notd by Rozvany (200), msh-dpndnc actually bcoms bnicial i w want to dmonstrat that topologis tnd to a known act solution []. (a) (c) (d) Figur 7. Local minima - topology optimization o MMB bam problm: dirnt initial dnsity distributions (a), and optimum solutions obtaind by th prsnt approach (c), (d) (a) Figur 8. Msh-dpndncy - optimum topologis o MMB problm using 6020 (a) and 2040 msh sizs 5. Comparison In this sction th prsnt approach is compard with Sigmund s snsitivity iltr which is a known chckrboard controlling tchniqu. In Sigmund s snsitivity iltr (SSF) th calculatd snsitivitis (Eq.(9)) ar rplacd by In which Ĥ is a wight actor writtn as c c Ĥ ˆ H Ĥ ma { 0, r dist(, )}, (, ), {,..., }. Whr dist(,) is dind as th distanc btwn cntr o lmnt and cntr o lmnt and r is a paramtr that dins th radius o th iltrd ara (s [3]). It could b sn that in SSF, snsitivity o ach lmnt is modiid with rspct to snsitivitis o nighboring lmnts and th iltr works similar to iltrs usd in imag procssing. I within ach lmnt strsss and strains assumd constant, modiid snsitivitis obtaind by th nw approach and SSF will b somhow similar to ach othr as thy both normaliz lmnt snsitivity with rspct to snsitivitis within nighboring lmnts. For mor planation w rwrit Eq.(5) using Eq.(0). c ˆ H ( oting that p is constant or all lmnts and using Eq.(2) in Eq.(7) w hav or simply p ˆc p Hˆ c Hˆ Ĥpc ) (5) (6) (7) (8)

7 ĉ As strss and strain vctors within ach lmnt ar assumd constant, th lmnt complianc c (s Eq.(4)) could b ound through ollowing c εσv (20) whr V dnots volum o lmnt. Finally using Eq.(20) in Eq.(9) th ollowing quation is drivd or SSF Ĥ εσ V ĉ A similar quation could b ound or th nw approach. h lmnt modiid complianc ĉ (s Eq.()) can b writtn as ˆc ˆ εσ ˆ V (22) Using th smoothing algorithm dscribd in sction 3, smoothd strss and strain vctors at th cntr o lmnt can b dscribd by ˆε F Ĥc Ĥ Ĥ F ε σ F, ˆσ F Ŵ In Eq.(23) Ŵ is a wight actor which can b calculatd using smoothing algorithm and F dnots lmnts connctd to lmnt. Using Eq.(23) in Eq.(22) th ollowing quation could b drivd or th nw approach ĉ ε F F 2 F Ŵ σ It should b notd that th Eq.(24) which is drivd or th nw approach is basd on th assumption o constant strss and strain stats within lmnts. his assumption is not valid or many cass (such as our-nod quadrilatral or highr ordr init lmnts) and in ths cass this quation would b mor complicatd. In this comparison, howvr, w us Eq.(24) bcaus o its simplicity and its similarity with Eq.(2). It is clar that both approachs ind iltrd lmnt complianc (and subsquntly lmnt snsitivity) through avraging strsss and strains within nighboring lmnts. Howvr whil th nw approach normalizs strss and strain vctors sparatly, in SSF thir scalar product is normalizd. h othr obvious dirnc is about th inlunc o volums o nighboring lmnts in ach tchniqu. In Eq.(2) (SSF) th iltrd paramtrs o ach lmnt ar dirctly inluncd by volums o nighboring lmnts whil in th nw approach th iltrd lmnt complianc (and subsquntly lmnt snsitivity) within ach lmnt only dpnds on volum o that lmnt itsl. In uniorm structurd mshs lik thos usd in ampls, 2, 4 and 5, this atur is not important as all lmnts ar o th sam siz. Howvr in using SSF with non-uniorm or unstructurd mshs on should choos th iltr s radius carully bcaus i sizs o som lmnts ar gratr than th chosn iltr s radius, liminating chckrboards is not assurd. In ths cass larg lmnt sizs, vn i thy ar smallr than th radius o iltring ara, act iltrd snsitivitis o nighboring lmnts considrably (s Eq.(2)). A noticabl advantag o SSF in contrast to th nw approach is that in SSF, on could obtain dirnt topologis by changing th radius o iltring ara. In this tchniqu numbr o hols in th inal topology could b controlld by adjusting th iltring radius. As a gnral rul, gratr iltring radii rsult in simplr topologis with wr hols. his atur could b usd to obtain sam topologis rom dirnt msh sizs. In th othr words SSF tchniqu could ovrcom th msh-dpndncy problm. 6. Conclusions In this study it has bn shown that th ormation o chckrboard patchs in topology optimization dpnds on strss distribution. Filtrd vrsion o minimum complianc topology optimization problm has bn drivd and a simpl algorithm or smoothing strss and strain distributions has bn introducd. his algorithm was usd in combination with th so-calld SIMP mthod and th approach was vriid by numrical ampls. Vriication ampls hav shown that this tchniqu may rliv th local minima problm as wll. Also it was dmonstratd that unctioning o this approach is indpndnt o th typ o msh and lmnts. at th nd o th papr th approach was compard with Sigmund s snsitivity iltr. V (9) (2) (23) (24)

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