A 2D topology optimisation algorithm in NURBS framework with geometric constraints

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A 2D topology optimisation algorithm in NURBS

framework with geometric constraints

Giulio Costa, Marco Montemurro, Jérôme Pailhes

To cite this version:

Giulio Costa, Marco Montemurro, Jérôme Pailhes. A 2D topology optimisation algorithm in NURBS

framework with geometric constraints. International Journal of Mechanics and Materials in Design,

2017, 14 (4), pp.669-696. �hal-02354376�

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onstraints

GiulioCosta,Mar o Montemurro

∗

,Jér

ˆ

o

mePailhès

Arts etMétiers ParisTe h,I2M CNRSUMR 5295,F-33400 Talen e, Fran e.

Thisisapre-printofanarti lepublishedinInternationalJournalofMe hani sandMaterials

inDesign. Thenalauthenti atedversionis availableonlineat:

https://doi.org/10.1007/s10999-017-9396-z

Corresponding author:

Mar oMontemurro,PhD,

ArtsetMétiersParisTe h,

I2MCNRS UMR5295, F-33400Talen e,Fran e, tel: +33556845422, fax: +33540006964, e.mail: mar o.montemurroensam.eu, mar o.montemurrou-bordeaux.fr

∗

Correspondingauthor.

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In this paper, the Solid Isotropi Material with Penalisation(SIMP) method for

TopologyOptimisation(TO)of2DproblemsisreformulatedintheNon-Uniform

Ra-tionalBSpline (NURBS) framework. This hoi e impliesseveraladvantages,su has

thedenitionofanimpli itlterzoneandthepossibilityforthedesignertogeta

ge-ometri entityat theend oftheoptimisationpro ess. Therefore,importantfa ilities

areprovidedin CAD postpro essingphasesin orderto retrievea onsistentandwell

onne tednaltopology. Theee tofthemainNURBSparameters(degrees, ontrol

points,weightsandknot-ve tor omponents)onthenaloptimumtopologyis

investi-gated. Classi geometri onstraints,astheminimumandthemaximummembersize

havebeenintegratedandreformulateda ordingto theNURBSformalism.

Further-more,anew onstraintonthelo al urvatureradiushasbeendevelopedthankstothe

NURBS formalismand properties. The ee tiveness and the robustnessof the

pro-posed methodaretestedandproventhroughsomeben hmarkstakenfromliterature

andtheresultsare omparedwiththoseprovidedbythe lassi alSIMPapproa h.

Keywords:

TopologyOptimisation;NURBSsurfa es;FiniteElementMethod;CAD- ompatibility;Stru tural

Optimisation

1 Introdu tion

Inthelastthreede ades,TopologyOptimisation(TO)hasgainedanin reasingdegreeofinterestin

botha ademi andindustrialelds. TheaimofTOforstru turalappli ationsistodistributeone

ormorematerialphasesinapres ribeddomaininordertosatisfytherequirementsfortheproblem

at hand. Usually, the design problem is formulated as aConstrained Non-Linear Programming

Problem(CNLPP)whereinagiven ost(orobje tive)fun tionmustbeminimisedbymeeting,at

thesametime, the fullset of optimisation onstraints. Classi ally,rst TOmethods were based

ona FiniteElements (FE) des riptionof thegeometry [1℄. The basi idea onsists of dening a

ontinuous titiousdensity fun tion varyingbetweenzeroand oneonthe omputation domain.

Thedensityfun tionisevaluatedatthe entroidofea helementofapredenedmeshandprovides

informationabout thetopology: void"and solid"phasesareasso iated tothe lowerand upper

boundsofthedensityfun tion,i.e. zeroandone,respe tively. Meaninglessgray"elements(related

to intermediate valuesof the density fun tion) are allowedbut penalisedduring optimisationin

ordertoa hievea lear"solid-voiddesign. So,me hani alpropertiesofea helementare omputed

(and penalised) a ording to the lo al density value. Several interpolation s hemes have been

developed for evaluating me hani al properties, e.g. Solid Isotropi Material with Penalisation

(SIMP) or Rational Approximation of Material Properties (RAMP) [2℄. Sin e the pioneering

worksonTO,dierentstrategieswereproposed duringtheyearsin ordertoover ome lassi TO

drawba ks,su has he ker-boardee tandmeshdependen e. Proje tionmethodshavebeenused

in[3℄andtheirrobustnesshasbeeninvestigatedin[4℄. Thesealgorithms,basedonproje tionand

(4)

Density-suitablepostpro essingmust before astin order toobtainasmoothCAD- ompatible design. A

dierent TO method, originated from theexigen e ofpre isely ontrollingthe boundariesof the

optimisedstru ture,isreferredinliteratureasLevelSetMethod(LSM)[7℄. AFEmodelisutilised

onlytodes ribethephysi softheproblemathand,whilstthetopologyofthesystemisrepresented

throughasuitableLevelSetFun tion(LSF).Inthe3D ase,theLSFisas alarfun tion,dened

in

I ⊂ R

,whi hrepresentstheimageofthegeneri point(

P = {x, y, z, t}

)belongingtothedomain

D ⊂ R

4

. The sign of the LSF an be onventionallyasso iated to material orvoidzones while

thezerovaluerepresentstheboundaryoftheoptimisedstru ture[8℄. Furthermore,theevolution

of the boundary of the domain (i.e. the zero value of the LSF) is governed by the

Hamilton-Ja obiEquationandrepresentsitsfundamental solution. TheLSM is hara terised bytwomain

advantages:

a) theLSF givesa learimpli itgeometri representationoftheboundaryofthedomain;

b) unlikedensity-basedmethods,theLSFisnotae tedbygreynessee t.

A thoroughdis ussion onthe LSM forstru tural TOappli ations anbe foundin [9℄. TheLSF

anbe hosenamongdierentsetsoffun tions,a ordingtolo alorglobalsupport,dimensionof

thelo alsupport,mathemati alnatureofthefun tion[10℄. Often,RadialBasisFun tions(RBFs)

areutilisedbe auseoftheirversatilityandsimpli ity[11,12, 13℄.

Re enteortsarenalisedtollinsomegapsof urrentTOmethods. Inparti ular,themain

short omingof density-basedstrategiesis thetime onsumingpostpro essingphasene essaryto

rebuild theboundaries of theoptimum topologyof the stru ture starting from a FE pixelised"

domain (providing the required smoothness). The LSM represents a rst attempt to over ome

thisdi ultybyintrodu inggeometri entitiestoproperlydes ribethetopology. Inthepro edure

dis ussedin[14℄,theLSMis oupledtotheisogeometri analysisbymeansofNon-UniformRational

BSpline (NURBS) formalismfor bothLSF parametrisation and obje tive fun tion omputation.

However,theLSMisnotfreefromdrawba ks. Tipi ally,theLSMisbasedontheHamilton-Ja obi

equation,that needsparti ularattentionto besolved: itshould beproperlyreinitialisedand the

Courant-Friedri hs-Lewy onditionmustbemetto satisfyspe i requirementsonthetime step

related to the mesh dimension [7℄. Moreover,the LSM is ae ted bythe initial topology of the

domain (i.e. theinitial guess) and, roughlyspeaking, when onsidering the lassi alformulation

basedon theshapederivative, holes anmerge but annot nu leatein thestru ture [9℄. Su h a

sensitivity of thesolutionto the initial guess ofthe topology anbeeliminated by meansof the

topologi alderivativeanda titiousenergeti termin theobje tivefun tion [15℄. Unfortunately,

solutions exhibit a dependen e on the weight parameter of the energeti term in the obje tive

fun tion. Asfaras on ernsdensity-basedTOmethods,arstenhan ementwaspresentedin[16℄

by relating the titious density fun tion to a BSpline surfa efor 2D TOproblems. The main

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improvementof thepostpro essingphaserequiredtoget aCAD- ompatibleshape.

Providing a areful des ription of the geometry is ru ial in TO not only to save time in

postpro essing but, mostly, to ensurethat the optimum shapeof the omponent(rebuilt at the

end of TO) ould meet the design onstraints. Nowadays, ommer ial software like OptiStru t

(a module of Altair HyperWorks pa kage)[17℄ orTos a(a module of Abaqus Pa kage)[18℄ are

widelyutilisedinindustrialeld. Sin etheymakeuseofadensity-basedmethod,suitabletoolsare

neededtorebuildtheboundaryoftheoptimumtopology. Althoughthesetoolsarequitefastand

e ient,usuallytherebuiltgeometrydoesnotsatisfypart(orthefullset)ofthedesign onstraints

onsideredfortheproblemathand.

Typi al onstraintsa ount forthe ontrol of geometri features on the onehand and

man-ufa turingrequirementson theother hand. Some onstraints an on ernbothaspe ts, likethe

onstraintsonthelo althi kness(referredinliteratureasminimumand maximummember size)

oftopologi al bran hesappearingduring optimisation. The minimummembersize anbetaken

into a ount in density-based TO algorithms through a simple riterion on the monotoni ity of

the titious density fun tion along

n

sear h dire tions: typi ally

n = 4

and

n = 13

for the 2D

and3D ase,respe tively[19℄. Ithasbeenproventhatthis onstraint onstitutesalsoanimpli it

lterthat anrepla e lassi aldistan e-based ltersin order to prevent the he ker-boardee t

in density-based TO methods. Further strategies to a ount for the minimum member size in

density-basedTO methods makeuseof proje tionand lteringte hniques, see[3℄and [20℄. Also

the dilated and eroded des ription of the topology outlined in [21℄ nally results in an impli it

ontrol on the minimum member size. An e ient way of handling the maximum membersize

onstraintispresentedin[5℄,whileexpli it onstraintsonthemembersdimensions anbedire tly

imposedin theframeworkoftheLSMbyintrodu ingthe`skeleton" on ept,see[22℄and[23℄.

Consideringallofthepreviousaspe ts,inthispaperanalternativedensity-basedTOmethod

for 2Dproblems is presented. Taking inspiration from the work of [16℄, the proposed method is

basedontheutilisationoftheNURBSformalism(inthe ontextoftheSIMPapproa h)torepresent

the titious density fun tion that be omes aNURBS surfa e(for 2D problems). Ea h point of

the NURBS ontrol net is then hara terised by three oordinates of whi h two are Cartesian

oordinates and the third oneis the density. Theproposed approa h, also alled NURBS-based

SIMP approa h [24℄, is hara terised by several original features whi h make it a general and

e ientmethodologyfordealingwith2Dtopologyoptimisationproblems. ThankstotheNURBS

formalism, the proposed methodology is fully ompatible with CAD software. The method is

extremely versatile and allthe lassi algeometri onstraints(minimum and maximummember

size,symmetries,et .) anbeeasilyreformulatedbymeansoftheNURBSsurfa es. Furthermore,

sin e the NURBS formalism gives an exa t representation of the boundary of the stru ture, a

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Someben hmarks,takenfrom literature[2,5℄,are onsideredtoprovethetheee tiveness of

theNURBS-basedSIMP method.

Thepaperisstru turedasfollows: rstly,thetheoreti alframeworkofbothNURBSsurfa es

andSIMPmethodispresented. Then,themathemati alformulationoftheproposedmethodology

isgiveninSe tion3. InSe tion4,themathemati alformulationofgeometri onstraintsisoutlined

togetherwith therelatedgradient omputation. Thenumeri alstrategyisthen brieydes ribed

in Se tion5. Se tion6is dedi atedto thedis ussion ofmeaningfulresultsand, nally,Se tion 7

endsthepaperwithsome on lusionsandperspe tives.

2 Theoreti al framework

In this Se tion, the fundamental on epts and notations of the NURBS theory and SIMP TO

methodarebrieyre alled.

2.1 The NURBS surfa e theory

A ordingto thenotationof[25℄, aNURBSsurfa eisdenedas:

S

(u, v) =

n

u

X

i=0

n

v

X

j=0

R

i,j

(u, v)

P

i,j

,

(1)

where

R

i,j

(u, v)

are thepie ewise rationalbasis fun tions,whi hare related to thestandard

NURBSblendingfun tions

N

i,p

(u)

and

N

j,q

(v)

bymeansoftherelationship

R

i,j

(u, v) =

N

i,p

(u)N

j,q

(v)w

i,j

P

n

u

k=0

P

n

v

l=0

N

k,p

(u)N

l,q

(v)w

k,l

.

(2)

In Eqs. (1) and (2), S

(u, v)

is a bivariate ve tor-valued pie ewise rational fun tion,

(u, v)

are

s alardimensionlessparametersbothdened in theinterval

[0, 1]

,while

p

and

q

aretheNURBS

degreesalong

u

and

v

dire tions,respe tively;

w

i,j

aretheweightsandP

i,j

= {x

i,j

, y

i,j

, z

i,j

}

the

Cartesian oordinatesof thegeneri ontrolpoint, with

i = 0, ..., n

u

and

j = 0, ..., n

v

. Thenetof

(n

u

+ 1) × (n

v

+ 1)

ontrolpoints onstitutestheso- alled ontrol net. Theblendingfun tions are

re ursivelydened bymeansoftheBernstein'spolynomials:

N

i,0

(u) =

(

1 if U

i

≤ u < U

i+1

,

0 otherwise,

(3)

N

i,p

(u) =

u − U

i

U

i+p

− U

i

N

i,p−1

(u) +

U

i+p+1

− u

U

i+p+1

− U

i+1

N

i+1,p−1

(u),

(4)

where

U

i

isthe

i

-th omponentofthefollowingnon-periodi non-uniformknotve tor:

U

= {0, . . . , 0

| {z }

p+1

, U

p+1

, . . . , U

m

u

−p−1

, 1, . . . , 1

| {z }

p+1

}.

(5)

(7)

Itisnoteworthythat thesizeoftheknotve toris

m

u

+ 1

,

m

u

= n

u

+ p + 1.

(6)

Analogously,the

N

j,q

(v)

aredenedontheknotve torV,whosesizeis

m

v

:

V

= {0, . . . , 0

| {z }

q+1

, V

q+1

, . . . , V

m

v

−q−1

, 1, . . . , 1

| {z }

q+1

},

(7)

m

v

= n

v

+ q + 1.

(8)

The knot ve tors U and V are two non-de reasing sequen es of real numbers that an be

interpretedastwodis rete olle tions of valuesofthe dimensionlessparameters

u

and

v

. As the

ontrol points,also theknotve tors omponentsform anet. Onebasi propertyof theblending

fun tionsisthelo alsupportproperty:

N

i,p

(u) = 0

if

u

isoutsidetheinterval

[U

i

, U

i+p+1

[

. Hen e,

itisevidentthat

R

i,j

(u, v) = 0

if

(u, v)

is outsidethere tangle

[U

i

, U

i+p+1

[×[V

j

, V

j+q+1

[

, i.e. the

lo al support asso iated to the ontrol point P

i,j

. The lo al support property is of paramount

importan e to understandall theadvantages of theNURBS formulationof theSIMP method in

the ontextofTO.ForadeeperinsightintheNURBStheory,thereaderisaddressedto[25℄.

2.2 The lassi SIMP method

TheSIMPmethodisbrieydis ussedherefortheminimum omplian e2Dproblemsubje ttoan

equality onstraintonthevolume[2℄. Letus onsiderare tangularreferen e domain

D ⊂ R

2

in

aCartesianorthogonalframe

O(x, y)

. Let

D

bedened as

D = {(x, y) ∈ R

2 |x ∈ [0, W ], y ∈ [0, H]},

(9)

where

W

and

H

aretworeferen elengthsofthedomain(that anvarydependingtothe onsidered

problem)along

x

and

y

axes, respe tively. Thegoalistond theoptimaldistributionof agiven

isotropi material on

D

by minimising the omplian e (i.e. thevirtual work of external applied

loads) withanimposed volume fra tion

f

ofthe designdomain. Thematerialdistribution (void

andmaterialzones)ae tsthestiness tensor

E

ijkl

(

x

)

,whi hisvariableoverthedomain

D

. Let

Ω ⊆ D

bethematerial domain. IntheSIMP approa h,

Ω

is determinedbymeans ofa titious

density fun tion

ρ(

x

) ∈ [0, 1]

dened over the whole design domain

D

. Su h a density eld is

relatedto thematerialdistribution:

ρ(

x

) = 0

meansabsen eof material,whilst

ρ(

x

) = 1

implies

ompletely dense base material. The relationship between the stiness tensor

E

ijkl

(x)

and the

densityeld

ρ(x)

is

E

ijkl

(ρ(

x

)) = ρ(

x

)

α

_E

0 ijkl

,

(10) where

E

0 ijkl

is thestiness tensor of the isotropi material and

α ≥ 3

asuitable parameterthat

(8)

eldand

l(

u

)

the omplian eofthestru ture,namely

l(

u

) =

Z

D

ρ(

x

)

α

_E

0 ijkl

ε

ij

(

u

)ε

kl

(

u

)dD.

(11)

InEq. (11),

ε

ij

(i, j = 1, 2, 3)

are the omponentsof thestraintensor. The FEM-dis retised

versionofEq. (11)is:

c = {U

FEM

}

T

[K] {U

FEM

} ,

(12)

where

[K]

istheglobalstinessmatrixofthestru turedened as

[K] =

N

e

X

e=1

ρ

α

e

[K

e

] .

(13)

InEq. (12),

{U

FEM

}

istheve torofthedegreesoffreedom(DOFs),also allednodalgeneralised

displa ements,ofthestru turerepresentingthesolutionoftheproblem:

[K] {U

FEM

} = {F} ,

(14)

wherein

{F}

istheve torofthenodalgeneralisedexternalfor es. InEq. (13),

ρ

e

isthe titious

density omputedatthe entroidofthegeneri element

e

,whilst

[K

e

]

isthenon-penalisedelement

stinessmatrixexpandedoverthefullsetofDOFsofthestru ture. Theproblemofdeterminingthe

optimumtopologywhi h minimisesthe omplian e subje tto a onstraintontheoverallvolume

anbestatedas follow:

min

ρ

e

c(ρ

e

),

subje tto:











[K]{U

F EM

} = {F},

V (ρ

e

)

V

tot

=

P

N

e

e=1

ρ

e

A

e

W H

= f,

ρ

min

≤ ρ

e

≤ 1, e = 1, ..., N

e

.

(15)

InEq. (15),

V

tot

istheoverallvolumeofthedenition domain

D

,

V (ρ

e

)

isthevolumeofthe

materialdomain

Ω

, while

f

isthexed volumefra tion;

A

e

istheareaofelement

e

onthe

x − y

plane and

ρ

min

representsthe lowerbound imposed to thedensity eld in order to preventany

singularityforthesolutionoftheequilibriumproblem.

ItmustbepointedoutthattheSIMPmethod anleadtonumeri alissues,e.g.thewell-known

he ker-boardee t,relatedtothela kofmutualdependen yamongthedesignvariables,i.e. the

pseudo-densities

ρ

e

denedatea helement entroid. Torepairtheseissues,adistan e-basedlter

isusuallyemployed[2℄. Forthesakeof ompleteness,thederivativeofthe omplian ewithrespe t

totheelements titiousdensities anbededu edbymeansof theadjoinsmethod [2℄:

∂c

∂ρ

e

= −αρ

α−1

(9)

c

e

= ρ

α

e

{U

F EM

}

T

[K

e

]{U

F EM

}.

(17)

Thus,thesensitivityanalysisforthe omplian e nallywrites:

∂c

∂ρ

e

= −

α

ρ

e

c

e

, e = 1, ..., N

e

.

(18)

Thesensitivityofthevolumefra tionis

1 V

tot

∂V

∂ρ

e

=

A

e

W H

, e = 1, ..., N

e

.

(19)

3 The NURBS-based SIMP method: mathemati al

formula-tion

In the framework of the proposed approa h, the pseudo-density eld hara terising the SIMP

methodisrelatedtoasuitableNURBS s alarfun tion:

ρ(u, v) =

n

u

X

i=0

n

v

X

j=0

R

i,j

(u, v)ρ

i,j

.

(20)

Theshapeof the NURBS is ae ted bythe valueof the titiousdensity at ea h ontrol point,

i.e.

ρ

i,j

,aswellasbythevalueoftheotherparametersinvolvedintothedenitionoftheNURBS

s alarfun tion,namely thedegreesof theblendingfun tion, i.e.

p

and

q

, the numberof ontrol

points

(n

u

+ 1) × (n

v

+ 1)

, the weights

w

i,j

and the value of the knot ve tors omponents, as

illustratedinEqs. (2)and (4). Thedimensionless parameters

u

and

v

anbearbitrarilydened.

For2D TOproblems, themostintuitive hoi e isto relatethem to theCartesian oordinates of

theglobalframeas:

(

u =

_W

x

,

v =

_H

y

.

(21)

InEq. (20),the ontrolpoints

ρ

i,j

andtheweights

w

i,j

arethedesignvariablesofthe

NURBS-basedSIMP method. Theyare olle tedin two olumn arrays

ξ

and

η

. Suitable boundariesare

imposedto satisfythedensityeldrequirementsfortheTOproblem:

ξ

_{= {ρ}

_0,0

, ..., ρ

n

u

,0

, ρ

0,1

, ..., ρ

n

u

,1

, ..., ρ

0,n

v

, ..., ρ

n

u

,n

v

},

ρ

i,j

∈ [ρ

min

, 1], ∀i = 0, ..., n

u

, ∀j = 0, ..., n

v

.

(22)

η

= {w

0,0

, ..., w

n

u

,0

, w

0,1

, ..., w

n

u

,1

, ..., w

0,n

v

, ..., w

n

u

,n

v

},

w

i,j

∈ [w

min

, w

max

], w

min

, w

max

∈ R, ∀i = 0, ..., n

u

, ∀j = 0, ..., n

v

.

(23)

Without lossof generality, in this work the two knotve torsU and V are onsidered uniformly

(10)

min

ξ,η

c(ρ(ξ, η)),

subje tto:











(

P

N

e

e=1

ρ

α

e

[K

e

]){U

F EM

} = [K]{U

F EM

} = {F},

V (ρ

e

)

V

tot

=

P

N

e

e=1

ρ

e

A

e

W H

= f,

{g(ξ, η)} ≤ {0},

ξ

k

∈ [ρ

min

, 1],

η

k

∈ [w

min

, w

max

],

∀k = 1, ..., (n

u

+ 1) × (n

v

+ 1).

(24)

In Eq. (24),

{g(ξ, η)}

is theve tor olle ting the onstraints of dierent nature (geometri ,

te hnologi alorphysi al),whilst

ρ

e

isthevalueofthepseudo-densityforthegeneri element,

ρ

e

= ρ(u

e

, v

e

) = ρ

x

e

W

,

y

e

H

,

(25)

where

(x

e

, y

e

)

aretheCartesian oordinatesoftheelement entroid.

Thenewformulationimpliestwonewvariablessets,dierentfromtheelementdensities,soa

sensitivityanalysisshould be arriedout. It anbeproven(seeAppendix A)thatthederivatives

ofboththe omplian eandthevolumewithrespe ttothe titiousdensityatea h ontrolpoint

anbeexpressedas

∂c

∂ρ

s,t

= −α

X

e∈I

s,t

c

e

ρ

e

R

s,t

(u

e

, v

e

),

(26)

1 V

tot

∂V

∂ρ

s,t

=

1 W H

X

e∈I

s,t

A

e

R

s,t

(u

e

, v

e

),

(27)

whilefortheweights

∂c

∂w

s,t

= −

α

w

s,t

X

e∈I

s,t

c

e

ρ

s,t

− ρ

e

ρ

e

R

s,t

(u

e

, v

e

),

(28)

1 V

tot

∂V

∂w

s,t

=

1 W Hw

s,t

X

e∈I

s,t

A

e

(ρ

s,t

− ρ

e

)R

s,t

(u

e

, v

e

).

(29)

InEqs. (26)-(29),

I

s,t

representsthelo alsupport relatedtothegeneri ontrolpoint[25℄. On e

thesensitivityanalysishasbeenprovided,asuitablegradient-basedalgorithm anbeutilisedasa

toolforthesolutionsear hofproblem(24).

TheSIMP approa hrevisitedin theNURBSmathemati alframeworkis hara terisedbythe

followingfeatures(whi h impliesjust asmanyadvantages):

1. Thenumberofdesignvariablesisunrelatedtothenumberofelements. Inthe lassi SIMP

approa h, ea h element introdu esa new design variable. In the NURBS framework, the

a ura yof thetopologydes riptionis hara terised solelyby thenumberof pointsof the

(11)

Thesizeofsu halterzoneisrelatedtothedimensionsofthelo alsupportoftheblending

fun tions,i.e. tothe omponents oftheknot ve torsaswell asto thedegreesof thebasis

fun tions. ItshouldberemarkedthatTOlters reateamutualdependen yareaamongthe

elementsdensities, i.e. thedesignvariables in standardSIMP formulations. Inthe ase of

NURBS,theinter-dependen eisautomati allyprovidedthankingtheNURBSlo alsupport,

withouttheneedofdeningalteronthemeshelementsdensities.

3. TheNURBSformalismallowstakingintoa ountawide onstraintsgamma,sin ea

math-emati ally well-dened des ription of the geometri al bounds of the optimum topology is

alwaysavailableduringtheiterationsoftheoptimisationpro ess. Nevertheless,lo al

infor-mation,su hasthelo alnormalandtangentve tors, anbeeasily dedu edfromstandard

NURBSformulae.

4 Geometri al onstraints: mathemati al formulation

InthisSe tion,two lassi algeometri onstraintsinTO,i.e. theminimumandmaximummembers

size,aswellasanew onstraintonthelo al urvatureradiuswillbeformulatedinthemathemati al

framework ofthe proposed NURBS-based SIMP approa h. Theywill onstitutethe omponents

ofthe onstraintve tor

{g(ξ, η)}

appearingin theCNLPP (24).

4.1 Minimum Member Size

The minimum member size onstraint is used in TO to provide a minimum admissible size of

stru tural elements. Here, the formulation proposed by [19℄ is onsidered. The intuitive idea

onsists of imposing the monotoni ityof the titiousdensity fun tion in a ir ular areahaving

adiameter equaltotheminimummembersize (

d

min

). The ir ularareais sket hedaroundea h

mesh elementand the monotoni ityis he ked every timealong four dire tions (

0 ◦

,

90 ◦

,

±45

◦

).

Mathemati allyspeaking,themonotoni ityofafun tion onaninterval

I

alongadire tion

γ

an

be he kedbymeansofthefollowingintegral:

M

γ

(f ) =

Z

I

|∇f · γ| − |

Z

I

∇f · γ|.

(30)

M

γ

(f )

isstri tlyequalto0if

f

ismonotoneandgraterthan0otherwise. Therefore,the onstraint

ontheminimummembersizeisformulatedasfollows:

g

d

min

=

N

e

X

e=1





X

γ

i

M

γ

i

(ρ)





θ

− σ ≤ 0,

(31)

where

N

e

isthe numberofmesh elements,

γ

i

the he kingdire tion (

i = 1, ..., 4

),

θ

apenalising

exponentand

σ

isusedtorelaxthe onstraintandtoprovidenumeri alstability. Of ourse,

M

γ

(12)

is themonotoni ityintegralandits evaluationdomain is the ir ular zonehavingdiameter

d

min

and entredatthe entroidofea helement. Theexpli itexpressionof

M

γ

i

(ρ)

is

M

γ

_i

(ρ) =

Z

d

min

/2

−d

min

/2

|∇ρ · γ

i

|ds −

Z

d

min

/2

−d

min

/2

∇ρ · γ

i

ds

.

(32)

In Eq. (32),

s

is a suitable abs issa along the urrent he king dire tion

γ

i

. In parti ular,

γ

1 = [1, 0]

t

,

γ

2 = [0, 1]

t

,

γ

3 = [

√

2/2,

√

2/2]

t

and

γ

4 = [

√

2/2, −

√

2/2]

t

. Inorder to formulate a

dis reteversionofEq. (32),letus onsideraregularmappedmeshofsquareelements. Then,

N

γ

i

is

thenumberofmeshelementsspanningthediameter

d

min

along

γ

i

dire tion. Itisstraightforward

toverify(see[19℄)thatEq. (32) hangesinto

M

γ

i

(ρ) =

N

_{γ i}

−1

X

j=1

|ρ

j+1

− ρ

j

| −

ρ

N

_{γ i}

− ρ

1 .

(33)

It isremarkedthat

j

in Eq. (33)is just amuteindex that sweepstheinterval

[0, d

min

]

, likethe

abs issa

s

in Eq. (32). Furthermore, asmoothapproximation ofthe absolute fun tion hasbeen

employedtoregularisethe onstraintevaluationforthegradientbasedalgorithm,namely

|z| ≈

p

z

2 _{+ ǫ}

2 _{− ǫ,}

(34)

with

ǫ = 0.01

. Thenalexpressionof

M

γ

i

(ρ)

tobeimplementedis

M

γ

i

(ρ) =

N

_{γ i}

−1

X

j=1

q

(ρ

j+1

− ρ

j

)

2 + ǫ

2 − ǫ

−

q

(ρ

N

_{γ i}

− ρ

1 )

2 + ǫ

2 + ǫ.

(35)

As far as the sensitivity analysis is on erned, the derivative of the minimum member size

onstraintwithrespe tto theNURBS ontrolpointswrites:

∂g

d

min

∂ρ

s,t

= θ

N

e

X

e=1





X

γ

i

M

γ

i

(ρ)





θ−1





X

γ

i

∂M

γ

i

(ρ)

∂ρ

s,t





.

(36)

Analogously,thederivativewithrespe tto theNURBS weightsis

∂g

d

min

∂w

s,t

= θ

N

e

X

e=1





X

γ

i

M

γ

i

(ρ)





θ−1





X

γ

i

∂M

γ

_i

(ρ)

∂w

s,t





.

(37)

Theonlydi ult onsists inevaluatingtheterms

∂M

γ

i

(ρ)

∂ρ

s,t

and

∂M

γ

i

(ρ)

∂w

s,t

. Thedetailed

om-putationis arriedoutinAppendix Bandthenalresultisprovidedhere:

∂M

γ

i

(ρ)

∂ρ

s,t

=

N

_{γ i}

−1

X

j=1

(ρ

j+1

− ρ

j

) (R

s,t

(u

j+1

, v

j+1

) − R

s,t

(u

j

, v

j

))

p

(ρ

j+1

− ρ

j

)

2 + ǫ

2 +

−

(ρ

N

γ i

− ρ

1 ) R

s,t

(u

N

γ i

, v

N

γ i

) − R

s,t

(u

1 , v

1 )

q

(ρ

N

_{γ i}

− ρ

1 )

2 + ǫ

2 ,

(38)

(13)

∂M

γ

i

(ρ)

∂w

s,t

=

ρ

s,t

w

s,t

∂M

γ

i

(ρ)

∂ρ

s,t

+

1 w

s,t

"

N

_{γ i}

−1

X

j=1

(ρ

j+1

− ρ

j

) (ρ

j

R

s,t

(u

j

, v

j

) − ρ

j+1

R

s,t

(u

j+1

, v

j+1

))

p

(ρ

j+1

− ρ

j

)

2 + ǫ

2 +

−

(ρ

N

γ i

− ρ

1 ) ρ

1 R

s,t

(u

1 , v

1 ) − ρ

N

γ i

R

s,t

(u

N

γ i

, v

N

γ i

)

q

(ρ

N

γ i

− ρ

1 )

2 _{+ ǫ}

2 #

.

(39)

InEqs. (38)and(39),itisassumed

ρ

j

= ρ(u

j

, v

j

)

.

4.2 Maximum Member Size

Themaximummembersizeisusedin TOin ordertolimitthemaximumthi knessof topologi al

elements. Using the maximum and the minimum member size simultaneouslyis a smart hoi e

toobtainstru tures withuniform dimensions. This aspe t ouldbeparti ularly advantageousin

additivemanufa turingprodu tioninordertoavoidresidualstressesinthenalstru ture: infa t,

arelevantdieren einstru turalmemberssizeimpliesadieren eintheamountofexposedsurfa e

(thinnerelementswill hillfasterthanmassiveparts),thusleadingtoanon-uniformheatex hange.

Consequently,theo urringtemperaturegradientwill onstituteoneofthemostimportant auses

ofresidualstresses,see[26℄. Inthiswork,Guest'sformulation[5℄hasbeenrevisitedbymakinguse

ofNURBS. Thepro edure isnotsofarfrom thatproposedbyPoulsenfortheminimummember

size. However,the onstraintdoesnot on ernthemonotoni ityof the titiousdensityfun tion

butisexpli itlyimposedonthematerialphase: infa t,a he king ir ularregionisdrawnaround

ea h element entroid(whose diameter

d

max

isequaltothedesiredmaximummembersize). Let

Ω

e

be the ir ular region; thus, the following ondition must be met for ea h element in a 2D

stru ture:

X

i∈Ω

e

ˆ

ρ

i

A

i

≤

πd

2 max

4 (1 − ψ), ∀e.

(40)

InEq. (40),

i

is amuteindex to indi ate the elementsin the ir ularzone

Ω

e

(built around

element

e

),

ψ

isarelaxingparameter(

ψ = 0.05

),

A

i

istheareaofelement

i

and

ρ

ˆ

i

istheproje ted

titiousdensityfun tion. Inthiswork,su haproje tionisperformedthroughtherelation

ˆ

ρ

e

= ρ

α

e

,

(41)

where

α

isthesameparameterusedforthepenalisationoftheSIMP,refertoEq. (10). Of ourse,

itisnotpossibletoimposea onstraintforea hmeshelementandasuitableaggregationstrategy

(14)

a

e

bethelefttermofEq. (40),so

a

e

=

X

i∈Ω

e

ˆ

ρ

i

A

i

.

(42)

An e ient aggregation te hnique onsists of hoosing the maximum value of

a

among the

mesh elementsand makinguse ofit in theformulationof themaximummembersize onstraint.

However,inordertoinsertthemaximumoperatorinagradient-basedalgorithm,asuitablesmooth

approximationshould begiven:

a

max

=

N

e

X

e=1

a

χ

e

!

1 χ

,

(43)

wherein

χ

is atuning parameterwhose valueshould bebig enough. Therefore, the onstraint is

formulatedby ombiningEq. (40)withEq. (43):

a

max

=

N

e

X

e=1

X

i∈Ω

e

ˆ

ρ

i

A

i

!

χ

!

1 χ

≤

πd

2 max

4 (1 − ψ).

(44)

Then, Eq. (44) is arranged in order to be dimensionless and put in the form of astandard

inequality onstraintfortheCNLPP (24)asfollow

g

d

max

=

P

N

e

e=1

P

i∈Ω

e

ρ

ˆ

i

A

i

χ

1 χ

πd

2 max

4 (1 − ψ)

− 1 ≤ 0.

(45)

Itisremarkedthatthe

χ

parameterhasbeen hosen

χ = 10

atthebeginningoftheiterationsand

itisin reasedupto

35

bya ontinuationmethod.

Inthis ase,thesensitivityanalysiswithrespe t totheNURBS ontrolpointsandweightsis

straightforward(seeAppendixBformathemati alpassages):

∂g

d

max

∂ρ

s,t

= α(g

d

max

+ 1)

P

N

e

e=1

P

i∈Ω

e

ρ

α

i

A

i

χ−1

P

i∈Ω

e

ρ

α−1

i

R

s,t

(u

i

, v

i

)A

i

P

N

e

e=1

P

i∈Ω

e

ρ

ˆ

i

A

i

χ

,

(46)

∂g

d

max

∂w

s,t

=

ρ

s,t

w

s,t

∂g

d

max

∂ρ

s,t

+

−

α(g

d

max

+ 1)

w

s,t

P

N

e

e=1

P

i∈Ω

e

ρ

α

i

A

i

χ−1

P

i∈Ω

e

ρ

α

i

R

s,t

(u

i

, v

i

)A

i

P

N

e

e=1

P

i∈Ω

e

ρ

ˆ

i

A

i

χ

.

(47)

(15)

This kind of onstrainthas an interest for fun tional and manufa turing requirements. Ideally,

if theboundary of thestru ture is mathemati ally dened, thelo al radius of urvature anbe

evaluated and its minimum value anbe identied. Then, the minimum value of the urvature

radius an be onstrained to be superior to an admissible referen e value. In the framework of

lassi alSIMPapproa h,itisnotpossibletoformulatethiskindof onstraintssin etheboundary

ofthestru tureisnotdened(norinimpli itneitherinexpli itway). Conversely,inthe ontextof

theNURBSformulation,ades riptionoftheboundaryisavailableatea hiterationbyestablishing

a utting planefortheNURBS surfa erepresentingthe titiousdensityfun tion. Let

Ω ⊆ D

be

thematerialdomainand

ρ

cut

∈ [ρ

min

, 1]

thethreshold uttingvalueforthedensityeld. Inorder

tohaveapre isedes riptionof the ontour,it anbeassumedthat







(x, y) ∈ Ω, if ρ(x, y) > ρ

cut

,

(x, y) ∈ ∂Ω, if ρ(x, y) = ρ

cut

,

(x, y) ∈ D r Ω, if ρ(x, y) < ρ

cut

.

(48)

Foranimpli it2D urve,theexpressionofthe urvaturewrites [27℄

κ = −

∂ρ

∂y

−

∂ρ

∂x







∂

2 _ρ

∂x

2 ∂

2 _ρ

∂x∂y

∂

2 _ρ

∂x∂y

∂

2 _ρ

∂y

2 















∂ρ

∂y

−

∂ρ

_∂x











∂ρ

∂x

2 +

∂ρ

∂y

2 !

3

2 .

(49)

Similarly to the minimum member size onstraint, the derivatives an bearranged through the

NURBSnotation,sothe urvatureradius anbea hieved:

r = −

_{W H}

1 H

2 ∂ρ

∂u

2 + W

2 ∂ρ

∂v

2 !

3

2 ∂ρ

∂u

2 ∂

2 _ρ

∂v

2 − 2

∂ρ

∂u

∂ρ

∂v

∂

2 _ρ

∂u∂v

+

∂ρ

∂v

2 ∂

2 _ρ

∂u

2 .

(50)

Hen e,the onstraint an beformulatedas

min

∂Ω

|r(x, y)| ≥ r.

(51)

TheabsolutevalueisapproximatedbymeansofEq. (34),whilsttheminimumoperatorhasbeen

estimated through the Kreisselmeier-Steinhauserfun tion [28℄. Let

N

r

be the number of radius

evaluationonthe ontourof thestru ture. Eq. (51) hangesintothefollowingrelation:

g

r

= 1 +

1 rτ

ln

N

r

X

k=1

exp

−τ(

q

r

2 k

+ ǫ

2 − ǫ)

!

≤ 0,

(52)

(16)

( ontrolpointsandweights) anbeexpressedasfollows:

∂g

r

∂ρ

s,t

= −

1 r

P

N

r

k=1

exp

−τ(

p

r

2 k

+ ǫ

2 − ǫ)

r

k

∂r

k

∂ρ

s,t

p

r

2 k

+ ǫ

2 P

N

r

k=1

exp

−τ(

p

r

2 k

+ ǫ

2 − ǫ)

,

(53)

∂g

r

∂w

s,t

= −

1 r

P

N

r

k=1

exp

−τ(

p

r

2 k

+ ǫ

2 − ǫ)

r

k

∂r

k

∂w

s,t

p

r

2 k

+ ǫ

2 P

N

r

k=1

exp

−τ(

p

r

2 k

+ ǫ

2 − ǫ)

,

(54)

wherethegradientofthegeneri urvatureradius anbeevaluatedthankingtoEq. (50). Details

areprovidedin AppendixB.

5 Numeri al Strategy

Asuitablenumeri alstrategyis des ribedwiththepre iseaimofsolvingtheCNLPP formulated

in Eq. (24). It is noteworthy that the proposed algorithm,ex luding the postpro essing phase,

has been developed in MATLAB: only the FEM analysis, whi h is needed for evaluating both

obje tiveand onstraint fun tions, has been arried out by means of a ommer ial FE ode (in

this aseANSYS).Thisisaveryimportantadvantagebe ausetheproposedmethodologyistested

and interfa ed with a widespreadand widely ustomisable FEM software. Usually, the Method

of Moving Asymptotes (MMA) is employed in TO [29℄. Instead of the MMA, in this work the

MATLAB lo al optimization toolbox and in parti ular the a tive-set algorithm of the fmin on

family[30℄ withnonlinear onstraintshasbeenutilised tosolveproblem(24).

Thea tive-setalgorithm(ACS)ispartofaspe ial lassofSequentialQuadrati Programming

(SQP)algorithmsfor onstrainedoptimisationproblemswhi h antoleratesomeiterativestepsout

ofthefeasibleregion. Thisfa tallowsforane ientexplorationofthefeasibledomain(espe ially

itsboundary)in CNLPPs. TheACSmethodprodu esasequen eofsub-problemsapproximating

theCNLPP athandbyexploitingtheinformationprovidedbythegradientandbyusingonlythe

violated onstraints,whi h onstitute,asamatteroffa t,theâa tive-setâ. Then,these

sub-problems are iteratively solved. Parti ularly, the ACS approximationis quadrati and the ACS

algorithm is a quasi-Newton method that makes use of the Broyden-Flet her-Goldfarb-Shanno

(BFGS)formula to approximate the Hessian in order to save omputational time, see [31℄. The

robustnessandthee ien yofSQPmethodsinsolvingCNLPPshavebeenlargelytestedandthey

onstituteawell-established lassofgradient-basedalgorithms[31℄. Moreover,SQPalgorithms an

be lassiedasglobally onvergentalgorithms,inthesensethat,providedafeasiblestartingpoint,

the onsequentsolution foundby thealgorithm will alwaysrespe t the optimality onditions for

onstrained optimisation, i.e. the so- alled Karush-Kuhn-Tu ker(KKT) onditions. Of ourse,

(17)

Furthermore, the problem nature depends on the hosen obje tive and onstraint fun tions, so

nothing anbeapriori statedaboutthe onvexityoftheCNLPP athand whenthese quantities

are omputed numeri allythroughaFEM analysis. Adeeperdis ussiononthe onvexity ofTO

problems, uniqueness of solution and dependen e on initial data an be found in [2℄ and it is

out of the s opes of this paper. A syntheti s heme of the proposed numeri al strategy and its

appli ation to TOproblems is shown in Fig. 1: inherentdetails on erning ea h blo karegiven

inthefollowing.

Figure1: NURBS-based SIMPalgorithm - syntheti s heme

•

Prepro essing The design domain together with geometry, mesh, loads and boundary

onditionsfor the stru tural problem at hand are established. Meanwhile, the NURBS is

(18)

Regions(NDRs)orsymmetry onditions anbesetat thisstage.

•

Optimisation blo k. The generi CNLPP of omplian e minimisation subje t to

m

inequality onstraints anbestatedintheform

min

ξ,η

c(ξ, η),

subje tto:

G

i

(ξ, η) ≤ 0, i = 1, ..., m,

(55)

wherethe FEequilibrium stateequationhasbeennegle tedandallinequality onstraints,

in ludingbounds onthe designvariables, are expressedin the ompa tform

G

(ξ, η) ≤ 0

.

Classi SQP methods hange the onstrained optimisation problem into an un onstrained

minimisationproblemthroughasuitableLagrangianfun tion

L(ξ, η, Λ)

denedas

L(ξ, η, Λ) = c(ξ, η) + Λ

T

G

_{(ξ, η),}

(56)

where

Λ

isa olumnve tor onstitutedof

m

Lagrangemultipliers,whose omponentsmust

fullthefollowing onditions

λ

i

= 0, if G

i

(ξ, η) ≤ 0

λ

i

> 0, if G

i

(ξ, η) > 0

∀ i = 1, ..., m.

(57)

Adire tsolutionofproblem(55)intermsofbothdesignvariablesandLagrangemultipliers

isnotpossible. Therefore,asequen eofquadrati approximationsof theinitialproblem is

generatedandea h problemissolvedin aniterativeloop.

Before pro eeding with the algorithm des ription, it should be remarked that a suitable

initialisationisneeded. Inparti ular,aninitialguessforthedesignvariables(NURBS

on-trolpointsandweights)isrequiredforstartingthesolutionsear h. A ordingtothenotation

ofSe tion 3, let

ξ

0

and

η

0

betheinitialve torsof ontrol pointsandweightsrespe tively:

suitablelower

lb

andupper

ub

bounds mustbedened

_lb

ξ

≤ ξ

0 ≤ ub

ξ

,

lb

_η

_{≤ η}

0 ≤ ub

η

.

(58)

Moreover, someoptionsfor the ACS fmin on algorithm are set in this phase: the

onver-gen etoleran e ontheobje tive fun tion andon thedesign variablesmust be established.

Moreover,athresholdonthemaximumnumberofiterationsisalsointrodu ed asafurther

stop riterion. The optiononthegradientprovision(sensitivity analysisformulaeforboth

obje tivefun tion and non-linear onstraints) is a tivated here. Let

(ξ

k

, η

k

)

be thevalues

(19)

NURBSrepresentingthe titiousdensityisevaluatedatthe entroidsoftheelementsand

thisinformationistransferredfromMATLABtoANSYSinasuitableformat(whi h anbe

denedbytheuser). TheSIMP riterion(13)isusedtoa hievethepenalisationof

me han-i alpropertiesandtoperformtheFEManalysis. Then,therequiredme hani alquantities

areregisteredforea helementand transferredto MATLAB.Now,itis possibletoupdate

the obje tivefun tion (the omplian e

c

k

= c(ξ

k

, η

k

)

) and onstraints(

G

k

= G(ξ

k

, η

k

)

).

Furthermore,thankstothesensitivityanalysis(refertoSe tion3andSe tion4),the

deriva-tives anbeeasily omputed(

∇c

k

, (∇G

i

)

k

, i = 1, ..., m

).

Se ondly, the aforementioned quantities are used, together with the previous evaluation

ofLagrangemultipliers,toapproximatetheHessianmatrixa ordingtotheBFGSformula

(referto[30℄and [31℄). TheapproximatedHessianmatrix

H

k

= H

BF GS

(ξ

k

, η

k

, c

k

, (G

i

)

k

,

∇c

k

, (∇G

i

)

k

, (λ

i

)

k−1

), i = 1, ..., m,

isemployed to statethe a tive-setquadrati

program-ming(QP) approximatedproblem,namely

min

d

Q

k

(d) = min

d

1

2 d

T

_H

k

d

+ ∇c

T

_k

d

,

subje tto:

A

_k

d

_{≤ b}

_k

_,

(59)

whose design variables are the sear h dire tion omponents olle ted in the ve tor

d

. In

Eq. (59),

A

k

is the oe ient matrix, whilst

b

k

is a ve tor of onstants, both given by

the linearization employed by ACS algorithm for the optimisation onstraints. The QP

problem (59) an be solved taking advantage of one of the several methods in literature,

see [31℄. Hen e, the urrent sear h dire tion (

d

k

) is evaluated, together with the urrent

valueofLagrangemultipliers (

(λ

i

)

k

). Thelast operation onsists ofndingasuitablestep

length(

s

k

)along thesear h dire tion

d

k

. Foradeeperinsightinto thematter, thereader

isaddressedto[31℄. Finally,variables anbeupdatedasfollows:

ξ

k+1

η

k+1

=

ξ

k

η

k

+ s

k

d

k

,

k = k + 1.

(60)

Thus, the onvergen e riteria an be he ked. The algorithm stops either if the

maxi-mumnumberof iterationsis rea hedorifthepredi ted hangeofoneamongthe following

quantities islessthanasuitablethresholdvalue(

10 −6

):

theobje tivefun tion

thegradientnormoftheLagrangefun tion

(20)

onstraintsaredimensionless.

•

Postpro essing. The resultof theoptimisationis a titiousdensitydistribution on the

referen edomainrepresentedthroughaNURBSs alarfun tion. Theoutstandingadvantage

provided by the NURBS formulation stands onthe possibilityto export a CAD

ompati-ble entity in order to rebuild in astraightforward way the boundaryof the optimised 2D

stru ture. However,itis ne essarytoadapt theresulttotheNURBS formalism: in fa t,a

standardNURBS surfa eisimpli itlydened bymeansofrelationssimilar toEq. (1).

In-deed,ea hphysi al oordinateisfun tionofthedimensionlessparameters

u

and

v

. Thus,a

fulldes riptionofaNURBSsurfa eisobtainedthroughrelationslike

x = x(u, v)

,

y = y(u, v)

and

z = z(u, v)

, where the respe tive oordinates of ontrol pointsappear. Normally, the

dimensionlessparametersare notrelatedto thephysi al oordinates

x

,

y

and

z

. Standard

formatles forNURBSdataex hange(.igs)require ontrolpoints oordinatesinthethree

physi al dire tions. However, in this dis ussion, wemakeuse of aNURBS s alarfun tion

ρ (u, v)

whereinthephysi al oordinates

x

and

y

arerelatedto

u

and

v

throughEqs. (21).

Hen e,itisnotne essaryto introdu e

x

and

y

oordinatesof ontrol pointsduringthe

op-timisationbuttheymustbeprovidedin thepostpro essingphaseinordertosetupthe.igs

le. Now,the

z

- oordinateofea h ontrol pointissimplythevalueofthe titiousdensity

forthe onsidered ontrol point (i.e.

ρ

i,j

, i = 1, ..., n

u

, j = 1, ..., n

v

) provided by theTO

optimisation;

x

and

y

oordinatesof ontrol pointsshouldbe hosenin su hawaythatthe

following onditionsaremet:

(

x =

P

n

u

i=0

P

n

v

j=0

R

i,j

(u, v)P

x

i,j

,

y =

P

n

u

i=0

P

n

v

j=0

R

i,j

(u, v)P

y

i,j

.

(61)

Thesolutionofproblem(61)isknowninliteratureandreferredasGreville'sabs issae[32℄,

denedthroughtheknotve torsanddegrees,namely











P

x

i,j

=

W

p

P

p

k=0

U

i+k

, ∀j = 1, ..., n

v

,

P

y

i,j

=

H

q

P

q

k=0

V

j+k

, ∀i = 1, ..., n

u

.

(62)

On ethe omputationisdone,alltheNURBSinformationare olle tedintoan.igsleand

theNURBS anbe imported in aCAD software. Beforepro eeding, athresholdvalue for

thedensityfun tion is omputedin MATLAB in su h awaythatoptimisation onstraints

aremet: asuitableplane pla edat thisdensityvalue anbe ut bymeansof theNURBS

surfa ein the CAD software in order to retrievethe nal 2D boundaries of theoptimised

stru ture. Finally, a further .igs le is reated with the nal 2D geometry and it an be

transferredtotheFEM softwarefor he kingoperations(i.e. inorderto verifythevalueof

bothobje tiveand onstraintfun tionsontherebuiltstru ture). Asummarizings hemeof

(21)

6 Results and dis ussion

Some meaningful results are presented in this Se tion in order to provethe ee tiveness of the

proposedalgorithm. Thetopologiesofea hoptimised geometryisshownhereonthenalrebuilt

stru ture after postpro essing phase. The CAD ompatibility of NURBS is fully exploited, so

uselesselementshavebeeneasily uto: therefore,theobje tivefun tionand onstraintsare

eval-uatedonthetruestru ture"insteadofonthemeshedreferen edomain(whereinvoid"elements

stillholdontogetherwiththematerial"elements),that ismeaninglessfromanengineering

view-point. Forea hanalysedtest ase,resultsarealso omparedwiththoseprovidedbythe ommer ial

softwareOptiStru t[17℄ byusing thesamemesh ofthereferen edomain. Asstatedin Se tion3,

twoset of designvariables, namely the NURBS ontrol points and the weights, olle tedin the

arrays

ξ

and

η

respe tively(refertoEqs. (22)and(23)),tunethetopologyofthe2Ddomain. For

ea h ase,thelowerandupperbounds arexedasfollows:

·

on erning the titious density at ea h ontrol point, standard bounds are hosen, i.e.

lb

ξ

i,j

= 10

−3

and

ub

ξ

i,j

= 1

,

∀i = 0, ..., n

u

and

∀j = 0, ..., n

v

;

·

weights enjoy greater freedom and it has been hosen

lb

η

i,j

= 1/2

and

ub

η

i,j

= 10

,

∀i =

0, ..., n

u

and

∀j = 0, ..., n

v

.

The rstpartof this se tionis dedi ated to a omparison betweentheresultsobtained with

NURBSandBSpline basisfun tions,respe tively,onastandardben hmark. Then,moregeneral

examplesareprovidedbyimposingaspe i materialphase,namelyvoid"(

ρ = 0

)ormaterial"

(

ρ = 1

)in somepres ribedNon-DesignRegions(NDRs)of the omputationaldomain. Moreover,

anappli ationwithasymmetry onstraintisshown. Finally,theee tsofthegeometri onstraints

dis ussedinSe tion 4areinvestigated.

6.1 ComparisonbetweenBspline and NURBSsurfa es for topology

op-timisation

Theproblemof the omplian eminimisationwith animposed volume fra tionis onsideredhere

(22)

Figure 3: Cantilever plate problem -

W = 320 mm

,

H = 200 mm

,Thi kness

t = 2 mm

,

YoungModulus

E = 72000M P a

,PoissonModulus

ν = 0.33

,Load

P = 1000 N

.

ofthe domainresultingfrom the utilisationofBspline andNURBS surfa es,respe tively. When

usingtheBsplinesurfa etheTOproblem anbestatedas

min

ξ

c(ρ(ξ)),

subje tto:











[K]{U

F EM

} = {F},

V (ρ

e

)

V

tot

= 0.4,

lb

_ξ

_{≤ ξ ≤ ub}

_ξ

_,

(63)

thus allthe weights getthe samevalue and an be ex luded from thedesign spa e. Conversely,

whenusingthemoregeneralNURBS formalism,theTOproblemis

min

ξ,η

c(ρ(ξ, η)),

subje tto:











[K]{U

F EM

} = {F},

V (ρ

e

)

V

tot

= 0.4,

lb

_ξ

_{≤ ξ ≤ ub}

_ξ

_,

lb

_η

_{≤ η ≤ ub}

_η

_.

(64)

Bothproblems(63)and(64)havebeensolvedthroughthepro eduredes ribedinSe tion5for

threevaluesofsurfa edegrees, i.e.

p, q = 2, 3, 4

,and threedierentvaluesoftheoverallnumber

of ontrol points, i.e.

(n

u

+ 1) × (n

v

+ 1) = 16 × 10, 32 × 20, 48 × 30

. The FE model of the

re tangulardomainisdis retisedbymeansofAnsysSHELL181elements,i.e. shell elementswith

4nodes and 6DOFs per node[33℄. Aftera preliminary he kon the onvergen e ofthe results,

thesizeofthemapped meshofthere tangulardomainhasbeen hosenequalto

80 × 50

elements.

Theequality onstraintissplitintwoinequality onstraintsby onsideringatoleran eof

0.005

on

thevalueof

f

. Then,thevolumefra tion onstraintwillbemetif

0.395 <

V (ρ

e

)

V

tot

< 0.405

. Results

areprovidedin termsof omplian e

c

andvolumefra tion

V /V

tot

ofthenaloptimumtopologies

(23)

x [mm]

y [mm]

0

50

100

150

200

250

300

0

50

100

150

200

(a)

(n

u

+ 1) × (n

v

+ 1) = 16 ×

10

,

c = 426.31 N mm

,

V /V

tot

=

0.4003

.

x [mm]

y [mm]

0

50

100

150

200

250

300

0

50

100

150

200

(b)

(n

u

+ 1) × (n

v

+ 1) = 32 ×

20

,

c = 400.63 N mm

,

V /V

tot

=

0.4134

.

x [mm]

y [mm]

0

50

100

150

200

250

300

0

50

100

150

200

( )

(n

u

+ 1) × (n

v

+ 1) = 48 ×

30

,

c = 403.45 N mm

,

V /V

tot

=

0.4020

.

Figure4: BSplineresults for

p, q = 2

x [mm]

y [mm]

0

50

100

150

200

250

300

0

50

100

150

200

(a)

(n

u

+ 1) × (n

v

+ 1) = 16 ×

10

,

c = 413.96 N mm

,

V /V

tot

=

0.4042

.

x [mm]

y [mm]

0

50

100

150

200

250

300

0

50

100

150

200

(b)

(n

u

+ 1) × (n

v

+ 1) = 32 ×

20

,

c = 403.49 N mm

,

V /V

tot

=

0.4044

.

x [mm]

y [mm]

0

50

100

150

200

250

300

0

50

100

150

200

( )

(n

u

+ 1) × (n

v

+ 1) = 48 ×

30

,

c = 394.81 N mm

,

V /V

tot

=

0.4036

.

Figure 5: NURBSresults for

p, q = 2

x [mm]

y [mm]

0

50

100

150

200

250

300

0

50

100

150

200

(a)

(n

u

+ 1) × (n

v

+ 1) = 16 ×

10

,

c = 432.29 N mm

,

V /V

tot

=

0.4011

.

x [mm]

y [mm]

0

50

100

150

200

250

300

0

50

100

150

200

(b)

(n

u

+ 1) × (n

v

+ 1) = 32 ×

20

,

c = 408.37 N mm

,

V /V

tot

=

0.3994

.

x [mm]

y [mm]

0

50

100

150

200

250

300

0

50

100

150

200

( )

(n

u

+ 1) × (n

v

+ 1) = 48 ×

30

,

c = 402.39 N mm

,

V /V

tot

=

0.4025

.

p, q = 3

Furthermore,numeri al results on erning thetrue omplian e asfun tion of thenumber of

ontrol points are syntheti ally plotted in Fig. 10. It is interesting to ompare this omplian e

withthe omplian e al ulatedonthereferen edomainattheendofthesolutionphasebeforethe

uttingoperation,i.e. the omplian eofthewholere tangulardomain onstitutedbyallelements

withtherespe tivepseudo-densityvalue(itisreferredasproje ted omplian e",seeFig. 11).

Thefollowingremarksarisefromtheanalysisofthenumeri alresults:

a) TopologiesobtainedthroughaNURBS-basedrepresentationofthe titiousdensityfun tion

(24)

x [mm]

y [mm]

0

50

100

150

200

250

300

0

50

100

150

200

(a)

(n

u

+ 1) × (n

v

+ 1) = 16 ×

10

,

c = 416.96 N mm

,

V /V

tot

=

0.4048

.

x [mm]

y [mm]

0

50

100

150

200

250

300

0

50

100

150

200

(b)

(n

u

+ 1) × (n

v

+ 1) = 32 ×

20

,

c = 404.07 N mm

,

V /V

tot

=

0.4050

.

x [mm]

y [mm]

0

50

100

150

200

250

300

0

50

100

150

200

( )

(n

u

+ 1) × (n

v

+ 1) = 48 ×

30

,

c = 394.45 N mm

,

V /V

tot

=

0.4047

.

p, q = 3

x [mm]

y [mm]

0

50

100

150

200

250

300

0

50

100

150

200

(a)

(n

u

+ 1) × (n

v

+ 1) = 16 ×

10

,

c = 446.71 N mm

,

V /V

tot

=

0.4020

.

x [mm]

y [mm]

0

50

100

150

200

250

300

0

50

100

150

200

(b)

(n

u

+ 1) × (n

v

+ 1) = 32 ×

20

,

c = 407.31 N mm

,

V /V

tot

=

0.4032

.

x [mm]

y [mm]

0

50

100

150

200

250

300

0

50

100

150

200

( )

(n

u

+ 1) × (n

v

+ 1) = 48 ×

30

,

c = 401.69 N mm

,

V /V

tot

=

0.4009

.

p, q = 4

x [mm]

y [mm]

0

50

100

150

200

250

300

0

50

100

150

200

(a)

(n

u

+ 1) × (n

v

+ 1) = 16 ×

10

,

c = 433.01 N mm

,

V /V

tot

=

0.4039

.

x [mm]

y [mm]

0

50

100

150

200

250

300

0

50

100

150

200

(b)

(n

u

+ 1) × (n

v

+ 1) = 32 ×

20

,

c = 409.09 N mm

,

V /V

tot

=

0.4044

.

x [mm]

y [mm]

0

50

100

150

200

250

300

0

50

100

150

200

( )

(n

u

+ 1) × (n

v

+ 1) = 48 ×

30

,

c = 403.12 N mm

,

V /V

tot

=

0.4047

.

p, q = 4

takes lower values of the proje ted omplian e when ompared to those resulting from a

BSpline-basedrepresentation. Thisfa tjusties theutilisationofthemoregeneralNURBS

surfa esformalism. However,whenlookingat thetrue omplian e(Fig. 10),NURBShave

still signi antly better performan es than the respe tiveBSplines onlywhen the number

of ontrol points is kept small. If the number of ontrol pointsin reases, even ifNURBS

topologiesarestillsmootherthanBSplinetopologies,thede reaseoftheobje tivefun tion

disappearsand,sometimes,aBSplinesolution ouldbebetterthanaNURBSsolution(refer

to the asesof Fig. 8b and Fig. 9b). Thus, the utilisation ofNURBS insteadof BSpline

(25)

0

500 1000

1500

0.112

0.114

0.116

0.118

0.12

0.122

0.124

0.126

0.128 (n

u

+1)x(n

v

+1)

c/c

ref

Bspline, p,q=2

NURBS, p,q=2

Bspline, p,q=3

NURBS, p,q=3

Bspline p,q=4

NURBS, p,q=4

Figure 10: Cantilever plate problem- True omplian e trends.

0

500 1000

1500

0.12

0.13

0.14

0.15

0.16

0.17

0.18

0.19

0.2

0.21

0.22 (n

u

+1)x(n

v

+1)

c/c

ref

Bspline, p,q=2

NURBS, p,q=2

Bspline, p,q=3

NURBS, p,q=3

Bspline p,q=4

NURBS, p,q=4

Figure11: Cantileverplate problem- Proje ted omplian e trends.

b) Takinginspirationfrom[16℄,thebehaviourofthesolutionshasbeeninvestigatedbyvarying

boththenumberof ontrolpoints

(n

u

+ 1) × (n

v

+ 1)

and thedegreesofthesurfa e

(p, q)

.

Theseparametersae t thedimension ofthelo al support of theblendingfun tions. The