An interpretation of the SIMP method based on topological sensitivity analysis

HAL Id: hal-00496925

https://hal.archives-ouvertes.fr/hal-00496925

Preprint submitted on 1 Jul 2010

HAL

is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire

destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

topological sensitivity analysis

Samuel Amstutz

To cite this version:

Samuel Amstutz. An interpretation of the SIMP method based on topological sensitivity analysis.

2010. �hal-00496925�

SAMUELAMSTUTZ

Abstrat. SIMPisaverypopularmethodthatonvertsatopologyoptimizationprobleminto

anonlinearprogrammingproblemdenedoveraonvexset.Itbasiallyinvolvesapenalization

funtion,sofaronstrutedinempirialways. ThispapergivesaninterpretationoftheSIMP

methodintermsoftopologialsensitivity,andthereforeprovidesargumentsastothehoieof

thepenalizationfuntion.Asimplealgorithmbasedontheseoneptsisproposedandillustrated

bynumerialexperiments.

1. Introdution

LetD ^{bea boundeddomainof}R^d,d= 2,3^{,withaLipshitzboundary}Γ^{madeoftwodisjoint}

parts ΓD ^and ΓN^{, with} ΓD ^{of nonzeromeasure and} ΓN ^{of lass} C^{1. We fous on the following}

minimizationproblem,withthenotationspeiedbelow:

(γ,u)∈E×Vmin J(γ, u) ⁽¹⁾

subjetto

Z

D

γ∇u.∇ηdx= Z

ΓN

ϕηds ∀η∈ V. ⁽²⁾

Forthetwounknownsγ ^andu^{thefeasiblesetsare}respetively

E:={γ:D→ {γ⁻, γ⁺}, γ ^measurable}, V :={u∈H¹(D), u|ΓD=0}.

Theonstantsγ⁺> γ⁻ >0^{, the}distribution ϕ∈H^−1/2(ΓN)^{,and thefuntional}J :E × V →R

aregivendata. Foronvenienewedenotebyj(γ) :=J(γ, uγ)^{thereduedost,orobjetive,with} uγ ^{thesolutionof (2) .} Subsequentlyγ ^anduγ ^{willbereferred toas thedensity(or} ondutivity) and the state, respetively. Note that, in many appliations, the weak phase approximates an

emptyregion,whihmeansthatγ⁻≪γ^+.

Duetothebang-bangnatureofthetargeteddensityγ^{,Problem(1) -(2)fallsintotheframework}

oftopologyoptimization. Anumberofmethodshavebeendevisedforitssolution,whihwebriey

reall. Arstlassofmethods,sometimesknownaslassialshapeoptimization[14 ,20 ℄,relieson

theontroloftheinterfaeΓγ ^whereγ^jumps. Sensitivityanalysisoftheobjetivewithrespetto thepositionofΓγ ^{leadstothenotionofshape}derivative. Algorithmsbasedontheshapederivative produeinpriniplesmoothvariationsofΓγ^{,inpartiular,thenumberofitsonnetedomponents}

annothange. This is a serious drawbakin many appliations. Animportant exeptionmust

neverthelessbementioned. Itonernslevelsetmethods[16 ,4 ,15℄,whereΓγ ^isrepresentedasthe zerolevel set of a smoothfuntion ψ ^{dened over} D^{. A} Hamilton-Jaobiequation is then often usedtomovetheinterfaeinthedesireddiretion. Withinthissetting,onnetedomponentsan

anelormerge,butanhardlybereated,atleastintwodimensions. Infat,thesemethodslak

anuleationmehanism. Topologialsensitivityanalysis[12 ,13,19 ℄hasbeenpreiselyintrodued

to evaluate the variation of the objetive when γ ^{is swithed within a small region. Therefore,}

Keywordsandphrases. topologyoptimization,SIMPmethod,topologialsensitivity,topologialderivative.

thetopologial sensitivitymay be advantageouslyombinedwith levelset methods[3 , 11, 7 ℄. A

basially dierent lass of methods onsists in relaxing the onstraintγ(x) ∈ {γ⁻, γ⁺}^{. This is}

lassially ahieved by invoking the homogenization theory [1 , 9℄. This latter approah benets

fromnietheoretialproperties,likedierentiabilityandexisteneofglobalminimizers. However,

eventually retrieving a feasible solution requires a rather heuristi penalization post-proessing.

Asimplied method, alledSolid IsotropiMaterial withPenalization(SIMP)[8 ,18, 10℄,is very

popularintheengineeringommunity. Itisbasedonthetwo followingpriniples. Firstly,theset

ofadmissiblevalues{γ⁻, γ⁺}^{issimplyhangedintoitsonvexhull}[γ⁻, γ⁺]^{,thusthewholetheory}

ofhomogenizationisnotneeded. Seondly,thepenalizationisdiretlyinludedintheoptimization

proess by a modiationof the stateequation. Typially, thedensity γ ^{in (2)is replaed by a}

power lawθ(γ) =γ^{p whih, forsome objetivefuntions,tends toenfore extremalvalues. This}

formulationhasprovenpartiularlysimpleandeientinmanyimportantases. However,ithas

nopropertheoretialjustiation,andthehoieoftheexponentp^isempirial.

The purpose of this paper is to give an insight into the SIMP method by interpreting the

assoiatedrstorderneessaryoptimalityonditionsin termsoftopologialsensitivity. Weshow

that speial penalizationfuntions θ ^{an berelatedto isotropi topologial}perturbations, whih providesalearmeaningtothesolutionsobtainedthroughthismodel. Thepowerlawpenalization

isretrievedinthelimitingaseγ⁻→0^.

The paper is organized as follows. Thenotion of topologial sensitivity is realled in Setion

2. The aforementioned relations between SIMP and the topologial sensitivity are exhibited in

Setion 3, omplemented by Appendix A. Those results are extended to the linear elastiity

settingin Setion 4. A gradient-likealgorithmbasedon theseoneptsis desribedin Setion 5.

Numerialexperimentsarereportedin Setions6and7.

2. Topologialsensitivity andoptimalityonditions

InthepaperweshallusethestandardnotationB(ˆx, ρ)^{fortheopenballofenter}ˆx^andradius ρ^,|A|^{fortheLebesguemeasure oftheset}A^,andχA^fortheharateristifuntionofA^{. Forany} γ∈ E ^{wedenethesets}

[γ=γ⁺] :={x∈D, γ(x) =γ⁺}, [γ=γ⁻] :={x∈D, γ(x) =γ⁻} D_γ⁺:= int[γ=γ⁺], D⁻_γ := int[γ=γ⁻].

Wedenethefuntions:D⁺_γ ∪D⁻_γ →R^by s(x) =

1 ^if x∈D⁻_γ,

−1 ^if x∈D⁺_γ.

Belowwedene a notion ofloal optimalityrelativeto apartiular lassof perturbations. Of

ourse,moregeneralperturbationsouldbeonsidered,butatoolargelassmayresultinthenon

existeneofloaloptima,as showninAppendixA.

Denition 2.1. We say that γ ∈ E ^{is a loal minimizer of} j ^{if, for every family} (x1, ..., xN) ∈ (D⁺_γ ∪D⁻_γ)^{N,thereexists}ρ >¯ 0^{suhthat,forall}(ρ1, ..., ρN)∈R^N+^,

i=1,...,Nmax ρi<ρ¯=⇒j γ+ (γ⁺−γ⁻)

N

X

i=1

s(xi)χ_B(xi,ρi)

!

≥j(γ).

Denition 2.2. Wesay that thefuntional j ^{admits a topologial derivative} g(ˆx)^{at thepoint} ˆ

x∈D⁺_γ ∪D⁻_γ ^{ifthefollowingasymptotiexpansionholdswhen}ρ→0^:

j(γ+s(ˆx)(γ⁺−γ⁻)χB(ˆx,ρ))−j(γ) =s(ˆx)(γ⁺−γ⁻)|B(ˆx, ρ)|gγ(ˆx) +o(|B(ˆx, ρ)|). ⁽³⁾

d 2 3

k⁻ 2

γ⁺+γ⁻

3 γ⁺+ 2γ⁻

k⁺ 2

γ⁺+γ⁻

3 2γ⁺+γ⁻

Table1. Expressionsofk^{± fortheondutivityproblem.}

Lemma 2.3. Supposethat γ ^{isa loal minimizer of} j ^{and that} j ^{admits atopologial derivative} gγ(x)^atallpointx∈D_γ⁺∪D_γ^{−. Then}

gγ(x)≥0 ∀x∈D_γ⁻,

gγ(x)≤0 ∀x∈D_γ⁺. ⁽⁴⁾

Proof. Wehoosea singleperturbationentered atapointxˆ^,forinstanexˆ∈D_γ^{−. By Denition}

2.2wehave

j(γ+ (γ⁺−γ⁻)χB(ˆx,ρ))−j(γ) = (γ⁺−γ⁻)|B(ˆx, ρ)|gγ(ˆx) +o(|B(ˆx, ρ)|).

InviewofDenition2.1thisquantityisnonnegativeassoonasρ^{issuientlysmall. Dividingby}

|B(ˆx, ρ)|^{andpassingtothelimitas}ρ→0^entailsgγ(ˆx)≥0^{. Likewise}gγ(ˆx)≤0^whenxˆ∈D_γ^+.

Tox ideas,weassumeheneforththat theobjetivefuntionalisoftheform

J(γ, u) = Z

ΓN

ψuds+ℓ Z

D

γdx, ⁽⁵⁾

withψ∈H^−1/2(ΓN)^{. Thesalar onstant}ℓ^{an beseen asaLagrange multiplierassoiatedwith}

avolumeonstraint. Thefollowingresultisprovenin[6℄.

Proposition 2.4. When J ^{is dened by (5) , the redued ost} j ^{admits a topologial derivative} gγ(x)^ateahpointx∈D_γ^{± givenby}

gγ(x) =k^±γ(x)∇uγ(x).∇vγ(x) +ℓ,

wherethe adjointstatevγ ∈ V ^solves Z

D

γ∇vγ.∇ηdx=− Z

ΓN

ψηds ∀η∈ V, ⁽⁶⁾

andtheexpressionof k^{± isreportedinTable1.}

3. Convexifiation andonnetion withthe SIMP method

Weintroduetheauxiliary problem

min

(γ,u)∈E×V˜

J(γ, u) ⁽⁷⁾

subjetto

Z

D

θ(γ)∇u.∇ηdx= Z

ΓN

ϕηds ∀η∈ V. ⁽⁸⁾

Comparedwith(1) -(2) ,γ ^{issoughtintheonvexset}

E˜:={γ:D→[γ⁻, γ⁺], γ ^measurable},

and the state equation is modied by the introdution of a smooth (at least C^{1) funtion} θ : (γ⁻⁻, γ⁺⁺) → (γ⁻⁻, γ⁺⁺)^{, with} 0 < γ⁻⁻ < γ⁻ < γ⁺ < γ^{++. We assume further that} θ ^is

inreasingandsatises

θ(γ⁻) =γ⁻, θ(γ⁺) =γ⁺.

Forthis problemwedenotebyjθ(γ) :=J(γ, uθ,γ)^{thereduedost,with}uθ,γ ^{thesolutionof (8).}

Proposition 3.1. The redued ost jθ ^{is Fréhet} dierentiable on L^∞(D,(γ⁻⁻, γ⁺⁺)) ^{with the}

derivativeinthe diretion δ∈L^∞(D)^givenby Djθ(γ)δ=

Z

D

gθ,γδdx, ⁽⁹⁾

with

gθ,γ :=θ^′(γ)∇uθ,γ.∇vθ,γ+ℓ,

andtheadjoint statevθ,γ ∈ V ^solutionof Z

D

θ(γ)∇vθ,γ.∇ηdx=− Z

ΓN

ψηds ∀η∈ V. ⁽¹⁰⁾

Proof. First,itstemsfromtheimpliitfuntiontheoremthatthemappingS:γ∈L^∞(D,(γ⁻⁻, γ⁺⁺))7→

uθ,γ ∈ V ^isFréhetdierentiable. Wewriteforsimpliityu˙θ,γ :=DS(uθ,γ)δ^{thederivativein the}

diretionδ∈L^∞(D)^{. Then}jθ^isdierentiablebyomposition,andthehain ruleyields

Djθ(γ)δ= Z

ΓN

ψu˙θ,γds+ℓ Z

D

δdx.

Onusing theadjointequation(10)to rewritetherstintegralweobtain

Djθ(γ)δ=− Z

D

θ(γ)∇vθ,γ.∇u˙θ,γdx+ℓ Z

D

δdx. ⁽¹¹⁾

Nowdierentiating(8)yields

Z

D

[θ^′(γ)δ∇uθ,γ.∇η+θ(γ)∇u˙θ,γ.∇η]dx= 0 ∀η∈ V. ⁽¹²⁾

Combining(11)and(12)provides(9) .

Fortheoptimalityof jθ^{weusethestandard denitionrealledbelow. Inthesequelwesimply}

denotebyk.k ^theL^{∞ normon}D^.

Denition 3.2. Wesaythatγ∈E˜^{isaloal minimizerof}jθ^{ifthereexists} α >0 ^suhthat

∀˜γ∈E˜, kγ−γk ≤˜ α⇒jθ(γ)≤jθ(˜γ).

FromProposition3.1wederivethefollowingneessaryoptimalityonditions.

Corollary3.3. Supposethat γ ^{isaloalminimizer of}jθ^{. Then}

gθ,γ ≥0 ^{a.e. on} [γ=γ⁻], gθ,γ = 0 ^{a.e. on} [γ⁻ < γ < γ⁺], gθ,γ ≤0 ^{a.e. on} [γ=γ⁺].

(13)

Proof. By virtueoftheonvexityofE˜^{wehavetheoptimalityondition} Djθ(γ)(˜γ−γ)≥0 ∀˜γ∈E˜.

Consideranarbitrarypair(λ, δ)∈(γ⁻, γ⁺)×L^∞(D)^withδ≥0^a.e.,andset

˜

γ=γ+χ[γ≤λ]tδ.

Fort >0^{suientlysmallwehave}γ˜∈E˜^,hene Djθ(γ)(˜γ−γ) =t Z

[γ≤λ]

gθ,γδdx≥0.

Itfollowsthat

Z

[γ≤λ]

gθ,γδdx≥0 ∀δ∈L^∞(D), δ≥0,

andsubsequentlythat

gθ,γ≥0^a.e. [γ≤λ].

Similarlywendthat

gθ,γ≤0^a.e. [γ≥λ].

Usingthatλ∈(γ⁻, γ⁺)^{isarbitraryompletestheproof.}

FromomparisonofLemma2.3andCorollary3.3,itappearsthat,apartfromtheonvexiation

ofthefeasibleset,thetwooptimalitysystemsessentiallydierbytheexpressionofthesensitivities.

Then a natural question arises: is there a funtion θ ^{suh that those two} sensitivities oinide?

ThisisaddressedbythefollowingTheorem.

Theorem 3.4. Set

θ(γ) =γ²+γ⁺γ⁻

γ⁺+γ^{− if} d= 2,

θ(γ) =−γ³+ 3(γ⁺+γ⁻)γ²+ 2γ⁺γ⁻(γ⁺+γ⁻)

(2γ⁺+γ⁻)(γ⁺+ 2γ⁻) ^if d= 3,

(14)

andsupposethatγ∈ E ^and|D\D_γ⁻\D_γ⁺|= 0^{. Thenthereholds}θ(γ(x)) =γ(x)^andgθ,γ(x) =gγ(x)

foralmost all x∈D^. Consequently,jθ(γ) =j(γ)^{andthe onditions (4)and (13)are}equivalent.

Proof. In orderto fulllthe assertions ofthe theoremweneedto onstruta smoothfuntionθ

suhthat

θ(γ⁻) =γ⁻, θ(γ⁺) =γ⁺, θ^′(γ⁻) =k⁻γ⁻, θ^′(γ⁺) =k⁺γ⁺. ⁽¹⁵⁾

Thesimplest waytoahievethisistoseeka ubipolynomialinterpolation. Thereforeweset

θ(γ) =a3γ³+a2γ²+a1γ+a0.

Thentheonditions (15)areequivalentto











a3(γ⁻)³+a2(γ⁻)²+a1γ⁻+a0=γ⁻, a3(γ⁺)³+a2(γ⁺)²+a1γ⁺+a0=γ⁺, 3a3(γ⁻)²+ 2a2γ⁻+a1=k⁻γ⁻, 3a3(γ⁺)²+ 2a2γ⁺+a1=k⁺γ⁺.

(16)

ByGausselimination,wendthat theabovesystemadmitsa uniquesolutiongivenby

a3=k⁺γ⁺+k⁻γ⁻−2 (γ⁺−γ⁻)² ,

a2=(1−k⁺γ⁺)(γ⁺+ 2γ⁻) + (1−k⁻γ⁻)(2γ⁺+γ⁻)

(γ⁺−γ⁻)² ,

a1= 1−(1−k⁺γ⁺)γ⁻(2γ⁺+γ⁻) + (1−k⁻γ⁻)γ⁺(γ⁺+ 2γ⁻)

(γ⁺−γ⁻)² ,

a0= γ⁺γ⁻ (γ⁺−γ⁻)²

(1−k⁺γ⁺)γ⁻+ (1−k⁻γ⁻)γ⁺ .

Nowusingtheexpressionsofk^+and k^{− fromTable1 resultsin (14) .}

0 0.2 0.4 0.6 0.8 1 0

0.2 0.4 0.6 0.8 1

Figure 1. Funtionθ^forγ⁺= 1^andγ⁻→0^{,fromtoptobottom: ondutivity}

3D,ondutivity2D,linearelastiity2D.

Remark 3.5. For γ⁺= 1^andγ⁻→0 ^wehave

θ(γ)∼γ^{2 if}d= 2, θ(γ)∼ −1

2γ³+3

2γ^{2 if}d= 3. ⁽¹⁷⁾

Thesefuntionsareplotted inFigure1.

Remark 3.6. Ofourse,Theorem3.4doesnotimplythatProblems(1)-(2)and(7) -(8)areequiva-

lentsinethereisnoguaranteethatthesolutions(globalorloal)to(7) -(8)belongtoE^{. However,}

forθ(γ) =γ^{p,in the}self-adjointaseandforadisreteversion,it isprovenin [17 ℄that solutions to(7)-(8)areneessarilyin E ^forp^{suientlyhigh.}

4. Generalizationin linearelastiity

Weonsidernowa linearelastiityproblemintwo spaedimensions. ThedomainD ^isdened

asbefore. Thestateisthedisplaementeld

u∈ V :={u∈(H¹(D))², u|ΓD=0},

andtheequilibrium equationreads

Z

D

γσ(u) :∇^sη= Z

ΓN

ϕ.ηds ∀η∈ V.

Here σ(u) ^{is the stress normalized to a unitary Young modulus,} ∇^sη := (∇η +∇^Tη)/2 ^{is the}

symmetrigradient(strain), andϕ∈(H^−1/2(ΓN))^{2 is apresribed load. Thestressisomputed}

bytheHookelaw

σ(u) =λ^tr∇^su+ 2µ∇^su,

where(λ, µ)^{aretheLaméoeients. Thefollowingresultistakenfrom[6 ℄.}

Proposition4.1. WhenJ ^{isdenedby (5) ,thereduedost}j ^{admitsatopologialderivative}g(x)

ateahpointx∈D^±_γ ^givenby gγ(x) = κ+ 1

2(κr^±+ 1)

2σ(uγ)(x) :∇^svγ(x) +(r^±−1)(κ−2)

κ+ 2r^±−1 ^trσ(uγ)(x)^tr∇^svγ(x)

+ℓ,

where

κ=λ+ 3µ

λ+µ , r⁺= γ⁻

γ⁺, r⁻ =γ⁺ γ⁻,

andtheadjoint statevγ ∈ V ^solves Z

D

γσ(vγ) :∇^sηdx=− Z

ΓN

ψ.ηds ∀η∈ V. ⁽¹⁸⁾

Heneforth weplaeourselvesin theplane stressase. Thenthe Laméoeientsarerelated

tothePoissonratioν ^by

λ= ν

1−ν², µ= 1 2(1 +ν),

whihentails

κ= 3−ν 1 +ν.

Weassumethat ν= 1/3^,thusκ= 2^{,andthetopologialderivativeadmitsthesimplerexpression} gγ(x) =k^±γ(x)σ(uγ)(x) :∇^svγ(x) +ℓ ∀x∈D^±_γ,

with

k⁻= 3

2γ⁺+γ⁻, k⁺= 3 γ⁺+ 2γ⁻.

Obviously, Lemma 2.3, Proposition 3.1 and Corollary 3.3 straightforwardly extend to the linear

elastiityaseforthegradient

gθ,γ=θ^′(γ)σ(uθ,γ) :∇^svθ,γ+ℓ.

Byarguingasin Theorem3.4,weobtain thefollowingresult.

Theorem 4.2. Set

θ(γ) =2γ³+ 3γ⁺γ⁻γ+ 2γ⁺γ⁻(γ⁺+γ⁻)

(γ⁺+ 2γ⁻)(2γ⁺+γ⁻) , ⁽¹⁹⁾

andsuppose thatγ∈ E ^and|D\D⁻_γ \D⁺_γ|= 0^{. Then} jθ(γ) =j(γ)^,gθ,γ =gγ ^{andthe onditions}

(4)and (13)areequivalent.

Remark 4.3. For γ⁺= 1^andγ⁻→0 ^wehave

θ(γ)∼γ³. ⁽²⁰⁾

Thispowerlawpenalizationisthemostfrequentlyused withintheSIMPmodel.

5. Optimizationalgorithm

Thenumerialissueisthentosolvetheauxiliary problem(7) -(8) ,orequivalently

min

γ∈E˜

jθ(γ).

Two lasses of methods are of ommon usage in topology optimization in the ontext of SIMP,

for whih we refer to [10 ℄. On one hand the so-alled optimality riteria methods are eient

in some ases but theyare quite heuristi. On theother hand onvexapproximations methods,

liketheMethodofMovingAsymptotes(MMA),onsistiniterativelysolvingsimplersubproblems

onstrutedsoastoaountforapproximationsoftheobjetiveandpossibleonstraints,withthe

propertyofbeingseparableintheelements. Herewesimplyuseaprojetedgradientalgorithm.

Algorithm 1. (1) Initialization: hooseβ >0^,α∈(0,1)^,γ0∈E˜^.

(2) Loop whilekγn+1−γnk/kγnk ≤β^:

γn+1= max(γ⁻,min(γ⁺, γn−tn∇jθ(γn))),

where tn=t⁰_nα^{m and}m^{is thesmallestintegersuhthat} jθ(γn+1)< jθ(γn).

Intheomputationswehavealwaysusedα= 0.5^,γ0≡(γ⁻+γ⁺)/2^,andt⁰_n=kγnk/k∇jθ(γn)k^.

For thedisretization of the stateequation weuse nite elements with pieewiselinearshape

funtionsona triangularmesh. WedenotebyVh ^{theniteelementspaeandby}{e1, ..., eN} ^the

niteelementbasis. Thedensityγ^isalsorepresentedonthis basis,as

γ=

N

X

i=1

γiei,

and thevetor (γ1, ..., γN) ^{is the designvariable. The equivalent density within eah elementis}

omputedbyapplyingalinearinterpolationoperatorS∈ L(Vh,Th)^,whereTh^{isthesetoffuntions}

denedonD ^{whihareonstantper element. Thenthedisretestate}uθ,γ ∈ Vh ^isomputedby

Z

D

θ(Sγ)∇uθ,γ.∇ηdx= Z

ΓN

ϕηds ∀η∈ Vh,

andthedisreteostisdened by

jθ(γ) = Z

ΓN

ψuθ,γds+ℓ Z

D

Sγdx.

Arguingas inProposition3.1, wendthatthederivativeofthedisreteostis

Djθ(γ)δ= Z

D

[θ^′(Sγ)∇uθ,γ.∇vθ,γSδ+ℓSδ]dx,

withthedisreteadjointstatevθ,γ ^solutionof

Z

D

θ(Sγ)∇vθ,γ.∇ηdx=− Z

ΓN

ψηds ∀η∈ Vh.

DenotingbyS^{⋆ theadjointoperatorof}S ^{weobtain thegradient}

∇jθ(γ) =S^⋆(θ^′(Sγ)∇uθ,γ.∇vθ,γ+ℓ).

Remark 5.1. Pieewise linear nite elements are seldom used in topology optimization beause

theyareknown to produeinstabilities in theform ofhekerboard patterns[2 , 10 ℄. However in

all the numerial tests performed this phenomenon has not been enountered. Probably this is

duetothefatthatthedesignvariableisheredenedatthenodes,whereasitisusuallyattahed

tothe elements. Hene, thanksto thepossibility to uselow ordertriangular nite elements, the

presentalgorithmprovestobeofremarkablysimpleimplementationandappropriateforarbitrary

domains.

6. Numerial examples inondutivity

Weusetheparameters γ⁺ = 1^, γ⁻ = 10^{−5 and}β = 10^{−2. Intheexampleunder} onsideration the domain D ^{is a square of size} 1.5^{, with the boundary onditions indiated on Figure 2. In}

theobjetivefuntional wetakeψ=ϕ^{, henetheproblemis} self-adjoint, i.e., vθ,γ =−uθ,γ^{. We}

usea mesh ontainingN = 29041 ^{nodes. Figures 3 and4 show theresultsoftwo} omputations, orrespondingto the Lagrangemultipliers ℓ = 1 ^andℓ = 10^{. The} implementation is doneunder Matlab. TheamountsofCPU timeused inthedierentomputations,performedona standard

∂_nu=−1

u= 0

Figure 2. Computationaldomainandboundaryonditions.

Figure3. Optimaldensityforℓ= 1^{(ase1,left),and}ℓ= 10^{(ase 2,right).}

0 10 20 30 40 50 60

1.5 1.6 1.7 1.8 1.9 2 2.1 2.2 2.3 2.4 2.5

number of iterations

cost function

0 5 10 15 20 25 30 35

5 6 7 8 9 10 11 12 13

number of iterations

cost function

Figure 4. Convergenehistoryforase1(left),andase2(right).

7. Numerialexamples inelastiity

For strutural optimization problems, in order to redue the risk to fall in loal minima and

saveomputertime,weuseasequeneofiterativelyrenedmeshes,likein[7℄. Theoptimizationis

rstperformedonaoarsemesh. Afteronvergene,themeshisrened,thedensityγ^isprojeted

ontothenewmesh,andtheoptimizationis ontinued. Thisproedureisrepeateduptothenal

desiredmesh. Wepresentthreeexamples.

7.1. Cantilever. Hereandinthesubsequentsetion7.2weuseγ⁺= 1^,γ⁻= 10^−5andβ= 10^−3.

Again we plae ourselves in the self-adjoint ase (ψ = ϕ^{), whih orresponds to the standard}

omplianeminimization problem. Thedomain D ^{isa retangleofsize} 2×1 ^{(seeFigure5,left).}

TheLagrange multiplier is hosen as ℓ= 100^{. The suessive meshesonsist of 431,1661, 6521}

and25841nodes. Theobtained distributionofmaterialisdepitedonFigure5,right.

7.2. Mast. Thedomain D ^{isshownonFigure6,left,where thevertialandhorizontalbranhes}

areretanglesofsizes 2×4^and 4×2^, respetively. TheLagrange multiplieris hosenasℓ= 50^.