On Equivalence between Optimality Criteria and Projected Gradient Methods with Application to Topology Optimization Problem

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1 Colloquium on Computr-Aidd Optimization of Mchanical ystm EUROMECH 442 Erlangn-urmbrg, 23 On Equivalnc btwn Optimality Critria and Projctd Gradint Mthods with Application to opology Optimization Problm ERGEY AAIEV Institut of Lightwight tructurs and Concptual Dsign Univrsity of tuttgart Pfaffnwaldring Grmany l.: mail: srgy.ananiv@po.uni-stuttgart.d Ky words: topology optimization, optimality condition, Lagrang multiplirs, Hstns multiplirs. Abstract. h papr dmonstrats th quivalnc btwn th optimality critria (OC) mthod, initially proposd by Bndsø & Kiuchi for topology optimization problm, and th projctd gradint mthod. h quivalnc is shown using Hstns dfinition of Lagrang multiplirs. Basd on this dvlopmnt, an altrnativ formulation of th Karush-Kuhn-ucr (KK) condition is suggstd. uch rformulation has som advantags, which will b also discussd in th papr. For vrification purposs th modifid algorithm is applid to th dsign of tnsion-only structurs.. OC mthod for minimization of strain nrgy opology optimization, bginning with th pionring wor of Bndsø & Kiuchi [], whr it was formulatd as an optimal matrial distribution problm, has won a broad accptanc in industry and scintific world. For a rcnt ovrviw of th succssful applications of this tchnology th papr of oto [2] can b consultd. Dspit ths dramatic dvlopmnts, th original optimality critria algorithm, which was usd for numrical solution, rmains without major changs. Evn in th rcnt wors in this fild (Bndsø & igmund [3], Maut [4]) it is still usd as a woring hors bcaus of its simplicity and stability. h intrsting thing in this OC algorithm that in spit of th fact that it is drivd from th Karush-Kuhn-ucr optimality condition it uss th huristic updat rul, without any mathmatical justification. his applis not only to th particular algorithm dvlopd by Bndsø & Kiuchi, but also to th gnralizd OC algorithm as prsntd by Vnayya [5] (s also Hafta [6] for ovrviw). o th author s nowldg, until now thr was no critical analysis of th qustion why th OC mthods wor. his papr tris to find an answr to this qustion. At first, th optimality critria for minimum complianc dsign will drivd. For simplicity rasons th isotropic matrial modl with pnalization (IMP) will b usd. h drivation follows th on prsntd in th Ph.D. hsis of igmund [7]. For linar-lastic structurs minimum of complianc mans minimization of strain nrgy, which is tan as objctiv function. o nsur that th rsulting structur has only uniaxial strss stat and to simplify its nginring intrprtation, som additional constraints ar introducd: (i) th total amount of matrial (V) to b distributd in th dsign domain is fixd (ii) th dnsity (ρ ) can chang only in crtain rang (bound constraints) and (iii) quilibrium

2 2 rgy Ananiv constraint. Aftr discrtization with finit lmnts th optimization problm can b formulatd as follows: min ρ,u = st.. ( ) ( ) K ρ u= p uk ρ u ρν = V < ρ < whr u vctor of nod s displacmnt ρ vctor of lmnt s dnsitis ρ dnsity in som lmnt K(ρ) stiffnss matrix of th whol structur K (ρ ) lmnt s stiffnss matrix, according to IMP concpt: K =(ρ ) p Elastic K p vctor of xtrnal loads V total amount of matrial to b distributd v lmnt s volum =.. squnc and th total numbr of lmnts (p pnalty paramtr) h KK optimality condition for this problm can b formally intrprtd (s Lunbrgr [8]) as a stationary point of th following Lagrangian function with rspct to dnsitis, displacmnts and multiplirs. A nonstrict charactr of such intrprtation can b illustratd by th fact that th KK thorm stats th xistnc of th Lagrang multiplirs only in th optimum, at th sam tim, in th Lagrangian, which is dfind by (2), th multiplirs do xist in ach dsign point. L = uku + λ ρν V + µ ( Ku p ) + α( ρ+ ) + β( ρ ) (2) = = = whr λ, µ, α, β ar Lagrangian multiplirs corrsponding to constraints on fixd amount of matrial, quilibrium constraint and bound s constraints, rspctivly. Using th standard procdur, which will b not rpatd hr (s igmund [7]), th optimality condition with rspct to th dnsity in ach lmnt loos as follows: () K u u + λ α + β = ρ = ρν V= ρ + = or ρ = (3) If bound constraints ar not activ, thn th corrsponding multiplirs (α, β ) ar qual to zro and th optimality condition simplifis to:

3 On Equivalnc btwn Optimality Critria and Projctd Gradint Mthods... 3 B K u u = (4) λ ρ Up to this point th drivations wr mathmatically wll-foundd. h huristic bgins by th formulating an updat rul, which rads as: i i ρb i+ i i i i ρ = max{ ( ζ ) ρ,} if ( ρb < max{ ( ζ ) ρ,} ) (5) i i i i min{ ( + ζ ) ρ,} if ( ρb > min{ ( + ζ ) ρ, }) his updat rul provos som critical qustions, which confirm its huristic charactr: (i) if som dsign point is an optimal point of th problm (), thn th dsign variabls will not b changd by th multiplication with idntity (4), but if it is not th cas thn it is not clar why such multiplication will lad to th optimum all drivations wr carrid out undr assumption of stationarity of (2); (ii) th rol of th Lagrang multiplirs is also not clar and vry contradictory: on th on hand, th multiplir λ, which corrsponds to th fixd amount of matrial s constraint is calculatd at th ach optimization itration during th innr itration loop using wton s mthod (Bndsø & igmund [3], igmund [7]), on th othr hand, th multiplirs α, β, corrsponding to th bound constraints ar simply ignord, if ths constraints ar activ th dnsity is fixd at th bound valu; (iii) th rol of th mov limit paramtr ζ is also not clar: it is nown (from th practical xprinc) that it should b small, but why it is important is not xplaind in th classical wors ([], [3], [7]) h solution to ths contradictions is th subjct of th following sctions. 2. Rformulation of OC mthod using projctd gradint h problm formulatd in () blongs to th broad class of th optimization problms with stat constraints in th form of partial diffrntial quations (Gunzburgr [9]). h usual way to solv such problms is to us th primal mthods (Lunbrgr [8], Hafta [6]). h main fatur of ths mthods is that th stat constraint is nforcd at th ach optimization itration. In contrast to thm, th dual mthods or on-shot mthods (according to Gunzburgr [9]) do not forc th stat constraints to b fulfilld at th ach itration; thy try to solv th complt quation s systm of KK condition for primal variabls, for stat variabls and for Lagrang multiplirs simultanously. his is don usually using th wton s mthod. Which on is suitabl for xplaining th woring proprtis of optimality critria mthods can b rcognizd from th concrt form of KK quation s systm. h Lagrang multiplirs, corrsponding to th quilibrium constraint ar alrady xcludd from (3). his indicats that th intrprtation must b don using primal mthods. Enforcing th stat constraint mans that th displacmnt and dnsitis ar not mor indpndnt. An infinit small chang of th formr will caus an infinit small chang of th latr in such a way that th quilibrium constraint rmains fulfilld. his allows us to calculat thir snsitivitis.

4 4 rgy Ananiv u = = K Ku p K u ( ) ρ ρ ρ (6) h drivation of th objctiv function with rspct to dnsitis rads as: f,ρ u K = ( uku) = 2uK + u u ρ ρ ρ (7) ubstitution of (6) in (7) lads to th following xprssion for th gradint of th objctiv function: f,ρ K = u u (8) ρ which is ngativ. If w want to minimiz th function f w hav to go in dscnt dirction, which is th ngativ gradint of objctiv function. aing into account th local charactr of dpndnc btwn global stiffnss matrix and dnsity of som finit lmnt w find th following final xprssion for th dscnt dirction of th objctiv function: K f,ρ = u u (9) ρ his xprssion is almost quivalnt to th ons usd in th optimality critria (4). h only diffrnc is th absnc of th Lagrang multiplir λ. o undrstand its rol w will us its altrnativ dfinition, proposd by Hstns [] (s also Rocafllar [] for ovrviw). H has introducd th Lagrang multiplirs not at th optimal point of constraind optimization problm, but during projction of th objctiv function s gradint onto tangntial spac of activ constraints. h lmntary xampl shown in multiplir λ. Figur illustrats this altrnativ dfinition of th z minimiz: 2 2 f : x + y subjct to: h: x+ y = initial guss: x = [ ; 2] x h f f d= f + λ h x λ h h Figur tpst dscnt mthod for a constraind problm. y

5 On Equivalnc btwn Optimality Critria and Projctd Gradint Mthods... 5 o solv this problm using th stpst dscnt mthod (without lin-sarch) a projctd gradint has to b built. h trm projctd mans that an infinitly small stp in this dirction will not violat th activ constraint. h trm gradint mans that th rduction of th objctiv function happns in quicst possibl way. hs rquirmnts ar fulfilld by th following vctor: d= f + λ h () whr λ is a solution of th orthogonality quation: f h + = λ = h h ( f λ h) h () It is clar that an infinitly small stp along vctor d will not violat th activ constraint. In th spcial cas, whr constraint is linar, a stp of any lngth (paramtr γ) will not violat it: i+ i x = x+γ d (2) h fact that vctor d rducs th objctiv function in th quicst possibl way (and in this sns can b undrstood as gradint) will b provd in th nxt sction. Rcalling th original problm () thr is a simpl arithmtic intrprtation of paramtr λ in (). If all componnts of th constraint s gradint ar qual to on (for xampl, structurd grid with lmnts of qual volum), thn paramtr λ is qual to th man valu of th objctiv function s gradint. Figur 2 illustrats this. f,ρ λ = f,ρ d = f +λh,ρ,ρ man Figur 2 Paramtr λ as a man valu of th objctiv function s gradint. h OC formula (5) dos actually th sam job, but in an implicit way (in th innr itration loop, s igmund [7]). It itrats until th valu of Lagrang multiplir will b qual to th wightd man of th strain nrgy (in th gnral cas of unstructurd grid, s Bndsø & igmund [3]). Aftrwards ach dnsity will b multiplid with th scald drivation of objctiv function. If som lmnt has a highr lvl of strain nrgy than th man valu, its dnsity will incras (B is highr than.), if lowr, thn its dnsity dcrass (B is lowr than.). It is rally a mattr of tast what to do: on can add a small positiv numbr to th dsign variabl or multiply it with a numbr highr than on. If in som dsign point mor than on constraint is activ, th xprssion () gnralizs to: d = f + λ h (3)

6 6 rgy Ananiv whr =.. squnc and total numbr of activ constraints h gradint of activ constraint λ corrsponding Lagrang (Hstns) multiplir h rsulting vctor has to b orthogonal to th ach of th activ constraint s gradint. his rsults in xactly quations, which allow calculating th uniqu valu of Hstns multiplirs. At this point it is intrsting to not a full analogy to th Galrin basd Finit Elmnt Mthod (Bath [2]). h cofficints of shap functions (nod s displacmnts) ar dtrmind in this mthod also using orthogonality condition btwn rsiduum of th partial diffrntial quation, which is bing solvd and ach of th shap functions. h innr product in this cas is dfind as a volum intgral ovr th finit lmnt. h final form of quation s systm, rprsnting orthogonality condition rads: f + λ h h =... λ = ( HH ) ( fh) f + λ h hs = whr H x dimnsional matrix of th activ constraint s gradints (H=[h,ρ ]) λ dimnsional vctor of Hstns multiplirs (4) For th cas of simpl bound constraints th orthogonality condition (4) lads to th following rsult: th componnt of th projctd gradint d, corrsponding to an activ constraint, must b zro. Only such a vctor can b orthogonal to th gradint of th bound constraint. h following xprssions illustrat this statmnt. h f = f,... f, f, f,..., f,ρ,ρ,ρ,ρ,ρ - + n [ ] : ρ bound = h =,...,,,,... d h = ( f ) ( f ) ( f )... + λ + λ + λ +... = λ = f,ρ,ρ,ρ,ρ - + d = (5) If w do a prturbation of dsign variabls according to (2), than th dnsity, whr a bound constraint is activ, will not chang. And this is xactly what updat rul (5) dos. h projctd gradint mthod allows also to undrstand th importanc of th mov limit ζ in (5). h snsitivity analysis linarizs th objctiv function in an infinitly small ara around th currnt dsign point. If w us this snsitivity for a finit prturbation of dsign variabls, w hav to p it small, hoping at th sam tim that th linarizd function is smooth nough.

7 On Equivalnc btwn Optimality Critria and Projctd Gradint Mthods Comparison with th gnralizd OC mthod In this sction it will b shown that th gnralizd OC mthod dvlopd by Vnayya [5] is much closr to th projctd gradint mthod than th fix-point updat rul (5). Vnayya calculats Lagrangian multiplirs in a way, which is almost quivalnt to th orthogonality condition (4). For simplicity rasons w will show this similarity assuming thr dimnsional dsign spac and two activ constraints in th currnt dsign point. h optimality condition rads as: h h2 f λ + λ2 = x x x h h2 f λ + λ2 = y y y h h2 f λ + λ2 = z z z E λ = f (6) his is th original form of th KK condition. In Eq.(7) in [5] ach quation is scald by corrsponding componnt of th objctiv function s gradint (Eq.() in [5]): Eλ = (7) Vnayya obtains th rsolving systm of quations by multiplying th scald KK condition with th wightd original matrix. (s Eq.(2) in [5]) whr A is a diagonal wighting matrix (Eq.(3) in [5]): h xpandd form of th lft sid of quation (8) rads EAEλ= EA (8) f x x f A = y y (9) f z z EAE= ( ) ( ) ( ) xh x yh y zh z xh xh2x yh yh2y zh zh2z x f y f z f x f y f z f xh xh2x yh yh2y zh zh2z ( xh2) x ( yh2) y ( zh2) z x f y f z f x f y f z f (2) h xpandd form of th right sid of quation (8) rads

8 8 rgy Ananiv h h h x + y+ z x y z E A= h2 h2 h 2 x + y+ z x y z (2) A clos loo at xprssions (8), (2) and (2) allows us to rwrit thm using vctor-matrix format, similar to (4), which rads: - (( )( )) = ( ) Hx H f λ Hx (22) h xprssion (22) dos not allow a clar intrprtation in th sns of orthogonality condition (4) yt. o simplify th intrprtation w can multiply th original form (6) with a wighting matrix (9). In this cas w gt th xprssion, which stats th orthogonality condition btwn scald projctd gradint and th gradints of activ constraints. (( ) ) ( ) EAEλ= EA f HxH λ = H x f or x f + xλ H H = ( ) (23) As on can s from th last quation, multiplying by th dsign vctor x is actually not ncssary for th dfinition of Lagrangian multiplirs. Without it, xprssion (23) would b quivalnt to th orthogonality condition (4). As alrady mntiond, aftr calculating th multiplirs, th dsign variabls can b changd in two quivalnt ways (dpnding on on s tast): usually usd in mathmatical programming (with small positiv γ): i+ i ρ = ρ + γ f,ρ + λ h,ρ (24) usually usd in optimality critria mthods (with small positiv γ): h γ i+ i,ρ ρ = ρ λ (25) f,ρ h scond basic lmnt of th OC algorithm according to [5] is th scaling of nw dsign vctor in ordr to fulfill th violatd constraints. Its ncssity bcoms clar if w rmmbr that w do a finit stp in tangntial dirction of th constraints. h only xcption whr scaling is not ncssary would b linar constraints. 4. Last squars stimats or ral multiplirs? h xprssion (4) is actually wll nown in th optimization thory (s Hafta [6]), but it is undrstood in a diffrnt way as last-squar stimat of Lagrang multiplirs. For xampl,

9 On Equivalnc btwn Optimality Critria and Projctd Gradint Mthods... 9 Fltchr has proposd in [3] th following augmntd Lagrangian to solv constraind optimization problms: F L = f + λ h + h h (26) 2µ whr th multiplirs λ minimiz th squar norm of th vctor, which w call projctd gradint. Indd, th xprssion (4) can b obtaind as a solution of th following auxiliary problm: F min ( f + λh) ( f + λh) = ( f + λh) H= λ 2 λ ( ) ( ) λ = HH H f (27) Formally, it is not wrong to undrstand th Hstns multiplirs as last-squar stimats of Lagrang multiplirs. h classical KK thorm introducs th multiplirs only in th optimum, so any xprssion for thir calculation in th non-optimal point can b only an stimation. In contrast to that, th main ida of this papr is that th Hstns multiplirs from (4) ar not stimats, but th ral multiplirs. And only in spcial cas if rsulting projctd gradint is zro vctor, thy obtain th maning of classical Lagrang multiplirs. uch intrprtation obviously rquirs rformulation of Karush-Kuhn-ucr condition, which will b don in th following sction. 5. Altrnativ formulation of KK condition h projctd gradint rsulting from (4) can b dirctly usd for an altrnativ dfinition of th optimal point of optimization problm with constraints. his condition tas th following simpl form: d = (28) his rformulation has th following advantags: (i) full analogy with unconstraind optimization (gradint is a zro vctor); (ii) multiplirs xist not only in optimum, but in ach dsign point. hy also hav a clar gomtric intrprtation (s Figur ); (iii) th projctd gradint can b dirctly usd in som stpst dscnt li optimization algorithms. It was shown that ths algorithms ar quivalnt to th optimality critria mthods. It is important to prov that thr is no othr vctor, which rducs th objctiv function without violation of constraints. It can b provn by contradiction, assuming that thr is som vctor s, which is linar indpndnt on d. uch a vctor can b always constructd by th following linar combination: s = α d + β r (29) whr th vctors d and r ar orthogonal, i.. d r=, or

10 rgy Ananiv f + λ h r = (3) At th sam tim, vctor s must not violat any of activ constraints. his mans it must b orthogonal to th ach constraint s gradints, i.. s h = or ( ) αd+ βr h = αd h + βr h = (3) but according to (4) w hav d h = from which follows that r h =. ubstituting th last rsult in (3) on obtains that th vctor r is orthogonal to th objctiv function s gradint. f r = (32) his orthogonality proofs that thr is no othr vctor than d, which rducs th objctiv function without violation of constraints. hrfor, if this vctor is a zro vctor, thn th currnt dsign point is an optimal point of constraind optimization problm. 6. Dsign of tnsion-only structurs o vrify th dvlopd mthod it will b applid to on nginring problm: topology dsign of tnsion-only structurs. Its practical application is concptual dsign of cabl supportd bridgs. h problm was alrady considrd by Miss in [4], whr th optimality critria mthod was usd. Hr w prsnt a mor rigorous way (using numrical intgration) to supprss th comprssiv strss stat. Rcalling again th original problm () and taing into account th IMP-typ dpndnc btwn Young s modulus and dnsity of matrial in som lmnt, it follows from (9) that th drivativ of th strain nrgy, stord in th lmnt, is proportional to itslf: K p p f = u u = u K u = u B CBu dt J dv (33),ρ ρ ρ ρ V h intgration of th stiffnss matrix is usually don numrically, i.. th intgral in (33) is rplacd by Gauss quadratur. For xampl, for a 2D-bilinar lmnt th four-point intgration schm is xact (s Bath [2]). s.577, r dω= Ω 2 2 ubcbudt J ubcbudt J (34) -.577, At th sam tim, th stord lastic nrgy can b calculatd not only through strains, but also through strsss:

12 2 rgy Ananiv charactr and simplifis th undrstanding of constraind optimization thory. his can b probably considrd as th main rsult of th prsntd wor. Acnowldgmnts h financial support to th author from th Grman Acadmic Exchang rvic (DAAD) is gratfully acnowldgd. h author is also gratful to th anonymous rviwrs for th critical commnts and suggstions. Rfrncs [] Bndsø M.P., Kiuchi Gnrating optimal topologis in structural dsign using a homognization mthod. Computr Mthods in Applid Mchanics and Enginring, 7: [2] oto C.A. 22. Applications of tructural opology Optimization in th Automotiv Industry: Past, Prsnt and Futur. In: Mang H.A., Rammrstorfr F.G., Ebrhardstinr J. Fifth World Congrss on Computational Mchanics, Vinna, Austria. [3] Bndsø M.P., igmund O. 23. opology Optimization. hory, Mthods and Applications. pringr, Brlin. [4] Maut K Optimization of opology and Form of slndr structurs. Ph.D. hsis. Univrsity of tuttgart. (in Grman) [5] Vnayya V.B Gnralizd optimality critria mthod. In: Kamat M. (d.). tructural optimization: status and promis. Amrican Institut of Aronautics and Astronautics. [6] Hafta R Elmnts of tructural Optimization. Kluwr Acadmic Publishrs, Dordrcht. [7] igmund O Dsign of matrials and structurs using topology optimization. Ph.D. hsis, U Dnmar. [8] Lunbrgr D Linar and nonlinar programming. Addison-Wsly, cond dition. [9] Gunzburgr M.D. 23. Prspctivs in Flow Control and Optimization. IAM, Philadlphia. [] Hstns M.R Optimization thory. h finit dimnsional cas. John Wily & ons, w Yor. [] Rocafllar R Lagrang multiplirs and optimality. IAM Rviw, 35(2): [2] Bath K.-J Finit lmnt procdurs. Englwood Cliffs, J: Prntic Hall. [3] Fltchr R An xact pnalty function for nonlinar programming with inqualitis. Mathmatical Programming, 5: [4] Miss K. 22. w dvlopmnts in nginring application of structural optimization. In: chißl P. (d.). 4 th Intrnational Ph.D. ymposium. pringr-vdi, Düssldorf.

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