Boundary element based multiresolution shape optimisation in electrostatics

Contents lists available atScienceDirect

Journal

of

Computational

Physics

www.elsevier.com/locate/jcp

Boundary

element

based

multiresolution

shape

optimisation

in

electrostatics

Kosala Bandara

a

,

Fehmi Cirak

a

,∗

,

Günther Of

b

,

Olaf Steinbach

b

,

Jan Zapletal

c

a_{DepartmentofEngineering,UniversityofCambridge,TrumpingtonStreet,CambridgeCB21PZ,UnitedKingdom} b_{InstituteofComputationalMathematics,GrazUniversityofTechnology,Steyrergasse30,8010Graz,Austria}

c_{DepartmentofAppliedMathematics,VŠBTechnicalUniversityofOstrava,17. listopadu15/2172,70833Ostrava-Poruba,CzechRepublic}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received7February2014

Receivedinrevisedform4December2014 Accepted13May2015

Availableonline18May2015

Keywords:

Shapeoptimisation Shapederivative Boundaryelementmethod Subdivisionsurfaces Multiresolutionanalysis

We consider the shape optimisation of high-voltage devices subject to electrostatic field equations by combining fast boundary elements with multiresolution subdivision surfaces.Thegeometryofthedomainisdescribedwithsubdivisionsurfacesanddifferent resolutionsofthesamegeometryare usedforoptimisationandanalysis.Theprimaland adjointproblemsarediscretisedwiththe boundaryelementmethodusingasufficiently fine control mesh. For shape optimisation the geometry is updated starting from the coarsestcontrolmeshwithincreasinglyfinercontrolmeshes.Themultiresolutionapproach effectivelypreventstheappearanceofnon-physicalgeometryoscillationsintheoptimised shapes. Moreover, there is no need for mesh regeneration or smoothing during the optimisation due to the absence of a volume mesh. We present several numerical experimentsandoneindustrialapplicationtodemonstratetherobustnessandversatility ofthedevelopedapproach.

©2015TheAuthors.PublishedbyElsevierInc.ThisisanopenaccessarticleundertheCC BYlicense(http://creativecommons.org/licenses/by/4.0/).

1. Introduction

The shapeoptimisationofhigh-voltageelectricaldevices, suchasswitchgearortransformers,servesasthedriving ap-plicationforourwork.Thepreventionofelectricalbreakdownisoneofthekeyconsiderationsinthedesignofhigh-voltage devices[1,2].Inafirstapproximation,limitingtheelectricfieldstrengthoncriticalcomponentscanreduceadevice’s sus-ceptibility toelectricbreakdown.The electricfieldstrength isdetermined withtheelectrostatic field equations,which in absenceofspacechargesreducetotheLaplaceequationwithDirichletboundaryconditions[3].Byoptimisingtheshapeof criticalcomponentsthemaximumelectricfieldstrengthonthesurface,i.e.,thenormalflux,canoftenbeconsiderably re-duced.Thismaymakeitpossibletoshrinkthesizeofadeviceandinturnleadtocostsavings.Intheapproachintroduced inthispaperwesystematicallyoptimisethegeometryofadevice suchthatacostfunctionalconsistingofthe L2-normof theelectricfieldstrengthisminimised.

Theboundaryelementmethod(BEM)hasclearadvantageswhenappliedtoshapeoptimisationofhigh-voltagedevices, see [4–8] for an introduction to BEM. First of all, BEM relies only on a surface discretisation so that there is no need to maintain an analysis-suitablevolume discretisation during the shape optimisation process.Moreover, BEM is idealfor solving problemsinunboundeddomainsthat occurinelectrostaticfieldanalysis. Ingradient-basedshapeoptimisationthe

*

Correspondingauthor.

E-mailaddresses:[email protected](K. Bandara),[email protected](F. Cirak),[email protected](G. Of),[email protected](O. Steinbach), [email protected](J. Zapletal).

http://dx.doi.org/10.1016/j.jcp.2015.05.017

0021-9991/©2015TheAuthors.PublishedbyElsevierInc.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).

Fig. 1.Topology of the domainΩand its transformation to the domainΩt.

shape derivativeof thecostfunctional withrespecttogeometryperturbations is needed[9–11].Tothispurpose,we use the adjoint approach and solve the primary and the adjoint boundary value problems with BEM. The associated linear systemsofequationsaredenseandan accelerationtechnique, suchasthefastmultipolemethod[12,13],isnecessaryfor their efficientsolution. Forsome recentapplications offastBEM inshape optimisation andBernoulli-typefree-boundary problemswereferto[14–16].

The geometryparameterisation anditsinterplay withtheBEM surface discretisationplays a crucialrole in shape op-timisation.When theBEMsurface meshisusedforgeometryparameterisationitleads tonon-physicaloscillationsinthe optimised geometry, as already known in the finite element literature [17,18]. In addition, the BEM mesh may become severelydistortedafterafewoptimisationstepssothatauxiliarymeshsmoothingproceduresbecomenecessary.Toremedy bothdifficulties,geometriesinshapeoptimisationarecommonlyparameterisedwithb-splinesorrelatedtechniques,such asNURBSandsubdivisionsurfaces[17–21].Inthispaperwerepresentgeometrieswithsubdivisionsurfaces,whicharethe generalisationofsplinestoarbitraryconnectivitymeshes.Specifically,weusetheLoopschemebasedontriangularmeshes andquarticbox-splines[22].

In subdivisionschemes a limit surface isobtainedthrough the repeatedrefinement ofa coarse control mesh [23].In practice,thereareclosedformexpressionsforcomputingthelimitsurfaceforagivencontrolmesh[24,25].Thehierarchy ofcontrolmeshesunderlyingasubdivisionsurfacelendsitselfnaturallytomultiresolutionediting[26,27].Thecoarsecontrol meshvertexpositionsare modifiedtoperformlarge-scaleeditingandthefinecontrolmeshvertexpositionsaremodified to add localised changes. Inthe introduced multiresolutionshape optimisation approach we usea fine control meshfor BEM discretisation and coarser control meshes for geometry modification. More precisely, we start optimising with the coarsestcontrolmeshandprogresstooptimiseincreasinglyfinercontrolmeshes.Asournumericalexamplesdemonstrate, themultiresolutionoptimisationapproachdoesnotleadtonon-physicaloscillationsingeometry.Moreover,theoccurrence ofmeshpathologies,likeinvertedelements,isgreatlyreducedbecausethesupportsizeofthegeometrymodificationsand theelementsizesarewellcoordinated.

Thispaperisorganisedasfollows.InSection2we introducethe electrostaticshape optimisationproblemandthe re-quiredshape derivates.Wethen discussinSection 3thediscretisation ofthestate andadjointboundary valueproblems withthe BEM.Subsequently, in Section 4the multiresolutionsubdivision surfacesfor geometryparameterisation are ex-plained. The multiresolution optimisation algorithm is introduced in Section 5. Finally, in Section 6 we present several numericalexampleswithincreasingcomplexitytodemonstratetheefficiencyandrobustnessoftheproposedapproach. 2. Electrostaticshapeoptimisationproblem

TheelectrostaticfieldequationsinabsenceofspacechargesleadtoaDirichletboundaryvalue problemfortheLaplace equation

−

_u

=

_{0 in}

Ω,

u

=

0 on

Γ0,

u

=

1 on

Γ

f

,

(1)

where u isthe electric potential or voltage,

Ω

⊂

R

3 _{denotes a multiply connected,bounded Lipschitz domain withthe} boundary

Γ

:=

∂Ω

consistingofafreepart

Γf

andafixedpart

Γ0

.Inthispaper,weassumethatthepotentialson

Γf

and

Γ0

are constant.Thegeometryofthefreeboundary

Γf

istobedeterminedwithshapeoptimisation.Weassume thatthe topologyof

Ω

isasshowninFig. 1,i.e.,that

Γ0

and

Γf

aredisconnectedpartsoftheboundaryandthat

Γf

isinteriorto

Γ0

. Itis straightforwardtogeneraliseourapproachto othersituations. Moreover,itiswell knownthat theDirichletproblem

(1)admitsauniquesolutionu

∈

H1

(Ω)

_{.Forthepurposesoftheshapecalculus introducedbelow,however,we assumea} higherregularityofthesolutionuinthevicinityofthefreepartoftheboundary

Γ

f.

Inelectrostatic shape optimisationone mayseektominimisethe pointwisemaximumofthenormalflux onthe free partoftheboundary

Γf

.Thecorrespondingcostfunctionalreads

Jmax(Ω,u

)

:=

sup x∈Γf

∂

∂

un

(

x

)

(2)

withthenormalflux

∂

u

/∂

n

= ∇

u

,

n

andtheexteriorunitboundarynormaln.Recallthatthephysicalinterpretationofthe normalfluxistheelectricfield innormaldirection.Duetothenon-smoothnatureofthemax function in(2)we decided toconsideranalternativedifferentiablefunctionalinordertobeabletoexploitmethodsofthefirstordercalculus;namely

J

(Ω,

u

)

:=

1 2

∂

∂

un

−

Q

2 L2_(Γ f)

=

1 2

Γf

∂

u

∂

n

(

x

)

−

Q

2 dsx (3)

whereQ

≥

0 isaprescribedconstantexpectedvalue.Similarcostfunctionalsarealsoconsideredinthecontextof Bernoulli-typefree-boundaryproblems[16,28].

Duringtheiterativeshapeoptimisation(seeSection5)thederivativesofthecostfunctional(3)withrespecttodomain perturbationsareneeded.Tothisend,wemakeuseofshapecalculusmethodsasintroduced,e.g.,in[9,11].First,inorderto describe geometrychangesofthedomain

Ω

,wedefineafamilyofmappings Tt

:

Ω

→

R

3 astheperturbationofidentity, i.e., Tt

:=

I

+

tV withthepseudo-timeparametert

∈ [

0

,

τ

)

andsomevelocityfieldV.Thenewconfigurationofthedomain

Ω

atpseudo-timetisasillustratedinFig. 1givenby

Ω

t

:=

Tt

(Ω)

:= {

Tt

(

x

):

x

∈

Ω

}

.

Underreasonableregularityassumptionsonthevelocityfield V,sayV

∈ [

W1,∞

(Ω)

]

3_,andfor

τ

_{smallenough,T}

t canbe showntobeaone-to-onemappingof

Ω

onto

Ω

t preservingtheLipschitzregularity,see,e.g.,[10,11].Theshape optimisa-tionproblemisthus transformedtoasearch forasuitablevelocityfield V thatreducesthecostfunctional.Followingthe structuretheorembyHadamardandZolésio,see,e.g.,[9,Chapter9],oneonlyhastodefinethemappingontheboundary. Moreover,since

Γ0

isfixed,wecandefine V

|Γ

₀

:=

0,i.e.,

Γ0

,t

:=

Tt

(Γ0)

=

Γ0

.Theshapeoptimisationproblemissolvedby aniterativeprocedureandasuitablevelocityfield V hastobedeterminedineveryiterationstep.

Togetherwiththeoriginalboundaryvalueproblem(1)weconsidertheproblem

⎧

⎨

⎩

−

ut

=

0 in

Ω

t

,

ut

=

0 on

Γ0,

ut

=

1 on

Γf

,t (4)

inthetransformeddomain.Thevalueofthecostfunctional(3)inthisconfigurationchangesto

J

(Ω

t

,

ut

)

=

1 2

Γf,t

∂

ut

∂

nt

(

xt

)

−

Q

2 dsxt

.

To be ableto use firstorder minimisationtechniques, theshape derivative of thecost functional inthe directionofthe velocityfield V givenby

J

(Ω,

u

)(

V

)

:=

d dt J

(Ω

t

,

ut

)

t=0

=

lim t→0+ J

(Ω

t

,

ut

)

−

J

(Ω,

u

)

t

is needed. Underreasonable regularityassumptions (see [9,11]), theshape derivative canbe representedin theso-called Hadamard–Zolésioform

J

(Ω,

u

)(

V

)

=

Γf

g

(

x

)

V

(

x

),

n

(

x

)

dsx

,

where

V

(

x

),

n

(

x

)

denotesthescalarproductofthetwoinvolvedvectorsandgissomekernelfunctionwhichis indepen-dentofthevelocityfieldV.Notethatsetting

V

|

Γf

:= −

gn implies J

(Ω,

u

)(

V

)

= −

Γf

g2

(

x

)

dsx

≤

0 (5)

andthussuchavelocityfield V definesadescentdirection.ThisstructurewillbeexploitedinSection5foriterativeshape optimisation.

Finally,inthecaseoftheconsideredcostfunctional(3)thekernelfunction gtakestheform

g

(

x

)

:= −

∂

p

∂

n

(

x

)

∂

u

∂

n

(

x

)

−

H

(

x

)

2

∂

u

∂

n

(

x

)

2

−

Q2

(6) accordingto[9,16,28].Here H istheadditivecurvature,uistheprimalsolutiongivenby (1),andp isthesolutionofthe adjointboundaryvalueproblem

⎧

⎪

⎨

⎪

⎩

−

p

=

0 in

Ω,

p

=

0 on

Γ

0

,

p

=

∂

u

∂

n

−

Q on

Γf.

(7)

Notice that the only difference between the primal andadjoint boundary value problems concerns the Dirichlet datum on

Γf

.ThenormalfluxoftheprimalsolutionservesastheDirichletdatumfortheadjointproblem.

3. Boundaryelementdiscretisation

Duringthe iterativeshape optimisationthe primal andtheadjoint boundaryvalue problems (1)and(7),respectively, have to be solved to compute the cost functional (3) andthe descent direction (5). The actual domain boundaries are described withmultiresolutionsubdivisionsurfacesaswillbe introducedinSection 4.Forallthecomputationswe apply the boundary element method to avoid remeshing or smoothing of the volume mesh in each step of the optimisation process.

Weapplyadirectboundaryintegralformulation,asweneedthenormalfluxes∂_∂p_nand ∂u

∂n fortheevaluationofthekernel functiong in(6).Fordetailsonboundaryintegralequations,thepropertiesoftheinvolvedboundaryintegraloperatorsand boundaryelementmethods,see,e.g.,[4–8].

Thesolutionoftheoriginalboundaryvalueproblem(1)isgivenbytherepresentationformula

u

(

x

)

=

Γ U∗

(

x

,

y

)

w

(

y

)

dsy

−

Γ

∂

∂

ny U∗

(

x

,

y

)

u

(

y

)

dsy for

x

∈

Ω,

(8)

where

Γ

=

∂Ω

=

Γf

∪

Γ0

istheboundaryofthedomain

Ω

andw

:=

∂_∂u_n denotesthenormalflux.Thefundamentalsolution U∗

(

x

,

y

)

oftheLaplacianinthreedimensionsreads

U∗

(

x

,

y

)

:=

1 4

π

1

|

x

−

y

|

.

Thelimitingcase

Ω

x

→

x

∈

Γ

providestheboundaryintegralequation

u

(

x

)

=

(

V w

)(

x

)

+

1

2u

(

x

)

−

(

K u

)(

x

)

for almost allx

∈

Γ,

(9)

whereV:H−1/2

(Γ )

→

_H1/2

(Γ )

_{denotesthesinglelayerboundaryintegraloperator}

(

V w

)(

x

)

:=

Γ

U∗

(

x

,

y

)

w

(

y

)

dsy forx

∈

Γ,

andK:H1/2

(Γ )

→

_H1/2

(Γ )

_{denotesthedoublelayerboundaryintegraloperator}

(

K u

)(

x

)

:=

Γ

∂

∂

ny U∗

(

x

,

y

)

u

(

y

)

dsy forx

∈

Γ.

Todeterminetheunknownnormalfluxesw

:=

∂_∂nu andq

:=

∂_∂np fortheboundaryvalueproblems(1)and(7),respectively, weusetheboundaryintegralequation(9)toobtain

(

V w

)(

x

)

=

1

2u

(

x

)

+

(

K u

)(

x

),

(10a)

(

V q

)(

x

)

=

1

2p

(

x

)

+

(

K p

)(

x

)

(10b)

foralmostallx

∈

Γ

.Fortheoriginalboundaryvalueproblem(1),theboundaryintegralequation(10a)canbesimplifiedto

(

V w

)(

x

)

=

0 forx

∈

Γ

0

,

1 forx

∈

Γ

f

duetothekernelpropertiesofthedoublelayerboundaryintegraloperator.Astheboundaryintegraloperatorsarebounded andthesinglelayerboundaryintegraloperatorV isH−1/2

(Γ )

_{-elliptic,allboundaryintegralequationsareuniquelysolvable.} Forthe boundaryelement discretisation we usetheGalerkin variationalformulationbased onpiecewise constant ap-proximations wh

:=

N

i=1 wi

ψ

i and qh

:=

N

i=1 qi

ψ

i

.

(11)

Therelatedconstantbasisfunctions

ψ

i

(

x

)

:=

_{1 forx}

∈

τ

i

,

0 elsewhere

aredefinedwithrespecttoadecompositionofthesurface

Γ

intoN planartriangularshaperegularboundaryelements

τ

i. Thus,wehavetosolvethefollowingtwosystemsoflinearequations

Vhw

=

Mhf

,

(12a)

Vhq

=

(

1

2Mh

+

Kh

)

h (12b)

todeterminethenormalfluxvectorst

∈

R

N _andq

∈

R

N _{withthecoefficientst}

iandqi.TheGalerkinmatricesaregivenby Vh

[

j

,

i

] :=

τj

τi U∗

(

x

,

y

)

dsydsx

,

Kh

[

j

,

i

] :=

τj

τi

∂

∂

ny U∗

(

x

,

y

)

dsydsx

,

Mh

[

j

,

i

] :=

τj

ψ

i

(

x

)

dsx

fori

,

j

=

1

,

. . . ,

N.Thecoefficientsofthetwovectors f andhontheright-hand sideof(12)aregivenby

fi

:=

0 for

τ

i

∈

Γ0,

1 for

τ

i

∈

Γf,

hi

:=

0 for

τ

i

∈

Γ0,

ti

−

Q for

τ

i

∈

Γf.

Duringshapeoptimisationweneedtheshapegradientatthemeshnodesxj,cf.Sections4,5.Tothisend,weapproximate thekernelfunction goftheshapegradient(6)with

gj

:= −

qj

wj

−

H

(

xj

)

2

wj

2

−

Q2

(13)

in thenodesxj.We estimate therequiredadditive curvature H withdiscrete differentialoperators givenin[29].Instead withthediscontinuousnormalfluxesin(11)wecomputetheshapegradients

gjwiththenodallyweightedaverages

wj

:=

i∈I(j)wiAi

i∈I(j)Ai and

qj

:=

i∈I(j)qiAi

i∈I(j)Ai

,

where the indexset I

(

j

)

contains all the element indices connected to node j and Ai denotes the surface area of the element

τ

i.Thiscanbeinterpretedasaquasi-interpolation,see,e.g.,[30].

As thefundamental solution U∗

(

x

,

y

)

isnon-local, theentriesofthe matricesVh and Kh arenon-zero ingeneral. As a result, both are fully populated.Therefore, accelerationtechniques,such as thefast multipolemethod [13], are crucial forthefastandefficientcomputation ofmatrixoperations.A descriptionofourimplementationoftheboundary element methodusingthefastmultipolemethodandadetailederroranalysisisgivenin[12].

4. Multiresolutionsubdivisionsurfaces

Next,we introducethe multiresolutionsubdivisionsurfacesfordescribing thegeometryofthecomputational domain. Subdivisionisapowerfulgeometricmodellingtechniqueforgeneratingsmoothsurfacesonarbitraryconnectivitymeshes. Thesmoothsurfaceisrecursivelygeneratedthroughrepeatedrefinementofaninitialcoarsecontrolmesh.Inthispaper,we limitourselvestothereviewoftheelementarypropertiesofsubdivisionsurfaces.Inparticular,weonlyconsidersubdivision schemesthatleadtocubicb-splinesintheunivariatesettingandquarticbox-splinesinthebivariatesettingwithtriangular elements.Formoredetailswereferto[23,25,20,31].

4.1. Subdivisionrefinement

Tobeginwith,weconsidertheunivariatesubdivisionrefinementofpolygons,asillustratedinFig. 2.Theinitialpolygon is referred toasthecontrolpolygonanddetermines the shapeofthe limitcurveresultingfromrepeatedsubdivision.It is instructivetothinkthat eachsubdivisionstepconsistsofarefinementandan averagingstep.Intherefinementstepeach segment ofthepolygonissubdividedintotwo segments,seeFig. 3(a).Subsequently,thevertexcoordinatesoftherefined polygonaredeterminedbyaveragingthecoarsevertexcoordinateswiththetwostencilsshowninFig. 3(b).Theevenvertex stencilappliestoverticesthatarealreadypresentinthecoarsepolygonandtheoddvertexstencilappliestoverticesthat areonlypresentintherefinedpolygon.Thenamingoddandevenismotivatedbytheconsecutivenumberingofverticesin a polygonwherenewlyinsertedverticesreceiveoddnumbers.AccordingtoFig. 3(b),foragivencoarsepolygonoflevel withvertexcoordinatesx_i arefinedpolygonoflevel

+

1 withvertexcoordinatesx_i+1iscomputedwith

Fig. 2.Subdivisionrefinementofagivencontrolpolygon(shownleft).Thethreepolygonstotherightaregeneratedbyrepeatedsubdivision.Noticethe increasingsmoothnessofthesubdividedpolygons.

Fig. 3.Univariate subdivision refinement.

Fig. 4.Bivariate Loop subdivision refinement.

x₂+_i1

=

1 8xi−1

+

3 4xi

+

1 8xi+1

,

(14a) x₂+_i+1₁

=

1 2xi

+

1 2xi+1

.

(14b)

Bydefinitionweassignthecoarsecontrolmeshthelevel

=

0.Withoutgoingintothedetails,wenotethattheweights in(14)havebeen chosen suchthat thelimit curve for

→ ∞

is auniformcubic b-spline.The weights canbe modified inorder tochangethe interpolationand/or smoothnesspropertiesof thecurve. Beingacubic b-splinethe limitcurve is C2-continuouswhichcanalsobereducedtoC0-continuousbymodifyingthestencilweights.Forthesakeofcompleteness wenotethatotherchoicesofstencilweightsmayalsoleadtosmoothcurves[23].

Itisconvenienttoexpressthesubdivisionequations(14)asamatrixvectormultiplication

x+1

=

S x

,

(15)

where x +1 and x are the vectors of all vertex coordinates of the coarse and refined polygons, respectively, and S is the subdivisionmatrix.The subdivisionmatrix S is abanded sparse matrixandits entries arethe stencil weightsgiven in Fig. 3(b). The dimensionsof S, forinstance, forthe 2D examplein Fig. 2are

(

4N

×

2N

)

, where N isthe number of verticesonlevel witheachhavingtwocoordinates.Althoughthedimensionsof S increasewithincreasing we denote allsubdivisionmatriceswithS sincethesamesubdivisionstencilsareusedineachstep.

Inthebivariatesurfacesetting,weusethesubdivisionschemeintroducedbyLoop[22].Thenotionofcontrolpolygonis nowreplacedwithcontrolmeshtoreflectthehigherdimension.TheLoopschemeisbasedontriangularmeshesandyields quarticbox-splines formesheswithregular vertices.Inthiscontext avertexisregular whenitis insidethedomain and incidenttosixedges,orisontheboundaryofthedomainandincident tofouredges. Thenumberofincidentedges toa vertexisusuallyreferredtoasthevalenceofthatvertex.ItisknownthattheLoopschemeyieldsaC1-continuoussurface onnon-regularverticesandaC2-continuoussurfaceeverywhereelse.

IntherefinementstepoftheLoopscheme,eachtriangleofthecontrolmeshissubdividedintofourtrianglesby intro-ducing newverticesattheedgemidpoints, asseen inFig. 4(a).Subsequently,thevertexcoordinates oftherefinedmesh

Fig. 5.Subdivisionrefinementofaconnector.Onthecoarsecontrolmeshonthelefttheedgesshowninredaretaggedascreaseedges.Thecontrolmesh inthecentreisgeneratedwithextendedLoopsubdivision.Thegeometryontherightisarenderingofthelimitsurface.(Forinterpretationofthecolors inthisfigure,thereaderisreferredtothewebversionofthisarticle.)

Fig. 6.MultiresolutioneditingoftheconnectorgeometryintroducedinFig. 5.Thegeometryismodifiedbymovingtheedgeofoneoftheholesinthe verticaldirection.Onthefirstrowthemodificationisperformedonlevel =0 andonthesecondrowitisperformedonlevel =1.Noticetheeffectof themodificationlevelonthelimitsurface(lastcolumn).

arecomputedwiththestencilsshowninFig. 4(b).Similartounivariatesubdivisiontherearetwodifferenttypesofstencils. The even vertexstencilapplies tovertices that alreadyexisted onthe coarsemesh andtheodd vertexstencilapplies to newlyintroducedverticesontheedges.Noticethattheweightsinthestencilforevenverticesdependonthevalencev of thevertex.

In thepresentwork,we usethe extendedLoopsubdivisionschemeintroduced by Biermannetal.[32].In contrastto the original Loopscheme,the extended schemeallows themodelling ofnon-smoothandnon-manifoldfeatures, such as creasesorT-sections,whicharecrucialformanyengineeringapplications.Itisclearthatthelimitsurfacealongcreasesis only C0-continuous.Fig. 5showsamechanicalconnectorgeometrycontainingsharpfeaturesdescribedwiththeextended subdivisionsurfaces.Forimplementationdetailswereferto[31].

4.2. Multiresolutionsubdivisionsurfaces

Subdivision surfaces represent a limit surface with a nestedhierarchy of control meshes of increasing resolution. As known in computer graphics, this property lendsitself to efficient multiresolutionediting of surfaces [26,27]. The basic idea in multiresolution editingis to modify coarse mesh vertexcoordinates to perform large-scale changes (tothe limit surface) andtomodify finemeshcoordinatestoadd localisedchanges.Bywayofexample,thisis illustratedinFig. 6for the connectorgeometrypreviously introduced inFig. 5.First thecontrolmesh coordinatesx0 _{are modified withx}0

+

_d0_, where d0 can be thoughtas auser givenperturbation vector. In theconsidered example, d0 applies displacements only to thevertices placed on one ofthe hole edges. Subsequent computation ofthe limit surface (by repeatedsubdivision) leads toa geometrywith ratherlarge scale changes.Alternatively, the edgeof thehole can be displacedon level

=

1, i.e., x1

+

_d1

=

_{S x}0

+

_d1_{.Thisresultsinmorelocalisedchanges. Itcanbeshownthat theareaofinfluenceforeachvertex} extendsovertworingsofadjacenttriangles.

The multiresolutioneditingalgorithmsavailable incomputer graphicsallowustoalternately editcoarseandfine reso-lutions, see,e.g.,[26].Thisisachievedby a wavelet-likedecomposition ofthegeometryintoa lowresolution partanda collectionofwaveletcoefficientsexpressingperturbations fromthelowresolutionpart[26,27].Tocomputesucha decom-positionitisnecessarytodefine,inadditiontothesubdivisionrefinement,acoarseningoperation

x

=

Rx+1

.

(16)

The coarsening matrix R enablesthe computation of the coarse representation x corresponding to a given edited fine representationx +1_{.Incontrast,recallthatthesubdivisionmatrix S enablestocompute foragivencoarserepresentation}

Fig. 7.Coarsening ofthelimitcurve previouslyobtainedwith subdivisionrefinement,see Fig. 2.Toprow showsthecoarseningofthe unperturbed limitcurve andthe bottomrowthe coarseningofaperturbedlimit curve.For avertexwith index ithe perturbationisofthe formx˜i∞=xi∞+ n(xi)asin(fxi,e0),wherenisthenormaltothecurve,e0isthebasisvector(1,0,0)T,aistheamplitude,and f isthefrequency.

the corresponding fine representation, cf. (15). Different choices for the matrix R are possible. For instance, it can be determinedbytheleastsquaresfitting

x

=

argmin

y

x+1

−

S y

2

,

(17)

whichleadsto

ST_{S x}

=

_ST_x+1

.

₍₁₈₎

Bycomparisonwith(16)weobservethatthecoarseningmatrixhastobe R

=

(

ST_S

)

−1_ST_{.Noticethat R isthe} pseudoin-verseofS suchthat R S

=

I yieldstheidentitymatrix.

InFig. 7 the functioning ofthecoarseningmatrix R isillustrated. Wereconsider theone-dimensional subdivision re-finement examplepreviously introduced inFig. 2and investigatethe coarseningof two limit curves.Onthe top rowof

Fig. 7,thelimit curve previouslyobtainedvia subdivisionrefinementin Fig. 2issuccessivelycoarseneduntil theoriginal controlpolygonisrecovered.Onthebottomrowthecoarseningofaperturbedlimitpolygonisshown.Ascanbeseen,the coarseningoperationsuccessivelyremovesthehigh-frequencyoscillationsfromthegeometry.Theresultingcontrolpolygon representsalimitcurvewhichisavisuallyfaithfulsmoothrepresentationoftheperturbedoriginalcurve.

Inmultiresolutionanalysisofsurfacestherefinement andcoarseningmatrices S and R,respectively, areusedto con-structa wavelet-likedecompositionofthegeometry. Theresulting algorithmsallow theconcurrent editingofcoarse and fine levels.As will bediscussed inSection 5, inmultiresolution shapeoptimisation it issufficient tostart froma coarse controlmeshandsuccessivelyadddetailstoasurfacebyeditingincreasinglyfinersubdividedmeshes.Thiscanbeachieved withoutawavelet-likedecompositionofthesurface.

5. Multiresolutionshapeoptimisation

WearenowinapositiontodiscussthemultiresolutionoptimisationoftheelectrostaticproblemintroducedinSection2

bycombiningsubdivisionsurfaceswiththeshapegradientscomputedwithBEM.Thebasicideainourapproachistousea finemeshforcomputingtheshapegradientsandtouseacoarsermeshforgeometrymodification[20,33].Theunderlying subdivisionrepresentationensuresthatthesamelimitsurfaceisalwaysconsidered,independentlyoftheactualrefinement level.Moreover,itisstraightforwardtotransferanygeometryorfielddatabetweenthedifferentlevelsusingtheintroduced subdivisionrefinementandcoarseningoperations,see(15)and(16),respectively.

In thefollowing we briefly summarise ourmultiresolution optimisation algorithm.This descriptionshould be read in conjunctionwithFig. 8.Forboundaryelement discretisation,the computationallevel c isprescribedwith c

=

n, where nischosensufficientlyhightoensuretheaccuracyofthesolution,thegradientfield,andthefunctional.Theoptimisation level o is initialisedwith o

=

0 and successivelyincreaseduntil o

=

c.Forsimplicity, weassume that the discretised optimisationproblemissolvedusingthesteepestdescentalgorithmandnoconstraintsarepresent.

1. Initialisetheoptimisationlevelwith o

=

0,thecomputationallevelwith c

=

nandthediscretisedcostfunctionalwith J

(Ω

c

,

_uc

)

= ∞

_{.Here,nisusergivenandhastobelargeenoughsuch thattheaccuracyofthenumericalsolutionis}

sufficientforpracticalpurposes.

2. Repeatedlysubdividetheoptimisationmeshatlevel o untilthecomputationlevel cisreached.

3. Solvetheprimal boundaryelementsystem(12a)andtheadjointboundaryelement system(12b)usingthe computa-tionalmeshwith c

=

n.

4. Evaluatethecostfunctional J

(Ω

c

,

_uc

)

_{,i.e.,thediscretecounterpartto(3),andtheshapederivativekernel}

_gc i ateach vertexxiaccordingto(13).

Fig. 8.Multiresolution optimisation algorithm.

5. If the cost functional J

(Ω

c

,

_uc

)

_{is increasing in relation to the previous value, increment the optimisation level} o

←

(

o

+

1

)

.

6. Ifthe optimisationlevelexceedsthe maximaloptimisation level (e.g.thecomputationallevel,i.e., o

>

c), terminate theoptimisationprocedure.

7. Repeatedlycoarsenthecomputationalmeshandtheassociatedvertexshapederivativekernel

guntiltheoptimisation level o togetherwith

go isreached.

8. Perturbthevertexcoordinatesoftheoptimisationmeshaccordingto(5)with

xo

i

←

(

xio

−

α

gion

(

xio

))

where the indexi denotes vertex id, the kernelvalues

go

i are computedwith (13), and

α

≥

0 is a suitablychosen steplength parameter.Thenormaln iscomputeddirectlyfromxo

i usingsubdivisionlimitmasksforthesurface tan-gents[31].

9. GotoStep2.

In ouractual implementationwe use themethodof moving asymptotes(MMA)proposed by Svanberg [34,35]as im-plemented intheNLoptlibrary[36].ThischangesinparticularStep 8intheabovealgorithm.Usinga moresophisticated optimisationalgorithmthansteepestdescentsignificantlyreducesthenumberofoptimisationiterationsandallowstoapply boundconstraintsonthevertexcoordinatestopreventunwantedgeometries.Forfurtherdetails,suchastheconsideration ofvolumeorsurfaceconstraints,wereferto[33].

6. Examples

We presentthreeexamples todemonstrate thefunctioning oftheproposed BEM basedmultiresolutionshape optimi-sation approach.In allexamples, theDirichlet boundaryvalue problemfor theLaplace equation (1)is considered.Recall thatthedomainboundary

Γ

issplitintothetwodisjointparts

Γ0

and

Γ

f,withtheprescribedpotentialsu

=

0 on

Γ0

and u

=

1 on

Γf

.During theoptimisationonlytheshape oftheboundary

Γf

isupdated withmultiresolutionshape optimisa-tion described inSection 5. Theboundary

Γ0

andits meshresolution remains unchanged.As mentioned, thediscretised optimisationproblemissolvedwiththeMMAalgorithmasimplementedintheNLoptlibrary[36].TheinputtotheNLopt libraryconsistsofthecostfunctional J

(Ω

c

,

_uc

)

_{andforeachvertexontheoptimisationlevel}

o itspositionxio and gra-dient

go

i n

(

x o

i

)

.InallexamplesourobjectiveistominimisetheL2-normofthenormalflux(3).Asusualingradient-based optimisation, theaimisto achievea reduction incostfunctionalandnot tofindthe globalminimum ofthenon-convex optimisationproblem.

6.1. Boxinasphere

Asanintroductoryexampleweoptimisetheshapeofaboxplacedinsideasphere,seeFig. 9,withtheexpectednormal flux density Q in (3) set to 20. It can be shown that the optimal shape for the inner box is a sphere with half the diameteroftheoutersphere[37].Thebox,representingthetobeoptimisedboundary

Γ

f,isofsize0

.

16

×

0

.

2

×

0

.

24 and the outer sphere,representingthe fixedboundary

Γ0

, hasradius0

.

2.The coarse meshforthebox contains 48 elements whichincreasesto768 elementsinthetwicesubdividedfinemeshatlevel c

=

2.Duringthesubdivisionrefinement,the creases in the coarse mesh are maintained ascreases using the extended subdivision stencils mentioned in Section 4.1, see also[32].Withthe extendedsubdivision stencilsthe limit surface corresponding tothe coarsebox meshis abox of thesamegeometry.Notethat onthelimitsurface thecreasesareonlyC0-continuousandnotatleastC1-continuous.The resolutionoftheoutersphereremainsfixedwith3840 elements.Hence,themeshesfortheboundaryelementanalysisof thecubeandsphereconsistof768 and3840 elements,respectively.

Fig. 9.Boxinasphere.Initialandoptimisedgeometrieswithisocontoursofthenormalflux.Themeshesindicatetheoptimisationlevel o.Theisocontours

belongtothefinecomputationalmeshatlevel c=2.Thegeometriesshownin(b)and(c)representintermediateresultsand(d)representsthefinal

result.

Table 1

Boxinasphere.ReductionofthecostfunctionalcorrespondingtoFig. 9.

Initial o=0 o=1 o=2

J(Ωc_,uc_{) 1.54}·₁₀1 _1.68·₁₀0 _7.32·₁₀−2 _8.46·₁₀−3

Reduction 0.00% 89.06% 99.52% 99.95%

Fig. 10.Box in a sphere. Rendering of the limit surface of the final optimised shape.

Figure 9(a)showstheinitialcoarsegeometryyieldingacostfunctionalvalueof J

(Ω

c

,

_uc

)

=

₁₅

.

_{38.Firstweselectthis}

coarsegeometryasoptimisationlevel,i.e., o

=

0,andobtaintheoptimisedgeometryshowninFig. 9(b).Afterconsecutively selecting o

=

1 and o

=

2 andoptimisingweobtainthefinaloptimisedgeometryshowninFig. 9(d).AsshowninTable 1, everysubdivisionofthecontrol meshleadstoadecreaseofthecost.Thefinal shapeoftheinitialboxisnearly asphere ofdiameter0

.

215 andthecostfunctionalvalue is J

(Ω

c

,

_uc

)

=

₈

.

₄₆

·

₁₀−3_{,whichrepresentsa reductionof99}

.

_95%.The

maximumnormalfluxintheinitialgeometryis Jmax(Ωc

,

_uc

)

=

₈₁

.

_{49 and reducesto Jmax(Ω}c

,

_uc

)

=

₂₁

.

_{43 inthefinal}

geometry.Astobeexpected, theoptimisationleadstoageometrywithnearlyuniformdistributionofnormalfluxasseen inFig. 9(d).Sincetheextended subdivisionschemewasusedinthiscase, themarkedcreaseswerenot smoothedout by thesubdivisionitself,butratherbytheshapeoptimisationprocedure.InFigure 10 arenderingofthelimitsurface ofthe finaloptimisedgeometryisshown.

Inordertodemonstratetherobustnessandbenefitsoftheproposedmultiresolutionoptimisationapproachwecompute theboxinasphereexampleusingafixedsingleoptimisationlevel.Allboundaryelementcomputationsareperformedwith c

=

2.First,thefixedoptimisationlevelischosenwith o

=

1,that istheoncesubdividedboxmeshservesastheinitial

Fig. 11.Boxinasphere.Singleleveloptimisationforfixedlevels oand c.Thewireframemeshindicatestheoptimisationmesh,theisocontoursonthe

analysismeshindicatethenormalflux.

Fig. 12.L-shapeddomain.PlanviewoftheinitialgeometrywiththeouterpolygonrepresentingtheboundaryΓ0andtheinnerpolygonrepresentingthe boundaryΓftobeoptimised.Theout-of-planelengthofeachpartis0.2 and0.1,respectively.

geometry.TheobtainedgeometryisshowninFig. 11(a)andthecorrespondingreductioninthecostfunctional J

(Ω

c

,

_uc

)

is83

.

0%.Inasecondindependentcomputationthefixedoptimisationlevelischosenwith o

=

2 andthetwicesubdivided boxmeshservesastheinitialgeometry.TheresultinggeometryisshowninFig. 11(b)andthecorrespondingreductionin thecostfunctional J

(Ω

c

,

_uc

)

_is80

.

_{0%.Itisworthemphasisingthatforbothcomputationsthestartinggeometryisabox,}

each witha differentmeshresolution.The advantagesofmultilevel optimisationareevident fromthecomparisonof the single leveloptimisationresultsinFig. 11withmultileveloptimisationresultsinFig. 9.Moreover,noticethenon-physical oscillationsinFig. 11(b)whicharereminiscentofproblemsreportedinthefiniteelementcontext[17,18].

6.2. L-shapeddomain

As asecondexampleweconsidertheoptimisationofanL-shaped domainplacedinside alargerL-shapeddomain,see

Fig. 12.The geometrycontainsconvexandnon-convexcornerswhich usuallyrepresentchallengesforgeometryupdating during optimisation.TheinnerL-shapedboundaryistheboundary

Γf

tobeoptimisedandtheouterL-shapedboundaryis the fixedboundary

Γ0

.The numberofelements on

Γf

increasesfrominitially28 to448 inthetwice subdividedmeshat level c

=

2. Theresolution ofthe boundary

Γ0

remains fixed with7168 elements.During thesubdivisionrefinement of the innerL-shaped boundary

Γ

f thecreasesare notretainedasvisibleinFig. 13(a).Aspreviously mentioned,inorderto maintainthecreasesitisnecessarytousespecialsubdivisionstencils[32,31].

Figure 13(a)showstheinitialgeometrywiththemaximumnormalflux Jmax(Ωc

,

_uc

)

=

₆₂

.

_{04 andthecostfunctional} J

(Ω

c

,

_u c

)

=

₁

.

_{32 (withtheexpectedvalueQsetto30).Noticetherelativelylow valuefor Jmax(Ω} c

,

_uc

)

_{resultingfrom}

thesmoothnessofthecomputationalgeometryatlevel c

=

2.Theellipsoidalgeometryresultingfromthemultiresolution shape optimisation isshownin Fig. 13(d). Ascan be deducedfrom theisocontourplots onthe optimised boundary, the normalfluxisnearlyuniformlydistributed.Moreover,duringoptimisation Jmax(Ω c

,

_uc

)

_reducesby48

.

_{69% tothevalueof}

31

.

83.Thecostfunctional J

(Ω

c

,

_uc

)

_reducesto8

.

₆₄

·

₁₀−3₍₉₉

.

_{35% reduction).Thecosthistoryissummarised inTable 2.} 6.3. Gasinsulatedswitchgear

In thisexample we apply theproposed shapeoptimisation approach tothe design ofan electrode in agas insulated switchgearcomponent, seeFig. 14.Suchdevicesarewidelyusedasdisconnectorsinhigh-voltagepowertransmission.The objective of shape optimisation is to reduce the propensity of the componentfor electricbreakdown with the ultimate aim to enable more compact component geometries. This can be achievedby modifying the electrode geometries such

Fig. 13.L-shapeddomain.Initialandoptimisedgeometrieswithisocontoursofthenormalflux.Themeshesindicatetheoptimisationlevel oandthe

isocontoursbelongtothefinecomputationalmeshatlevel c=2.Thegeometriesshownin(b)and(c)representintermediateresultsand(d)represents

thefinalresult.

Table 2

L-shapeddomain.ReductionofthecostfunctionalcorrespondingtoFig. 13.

Initial o=0 o=1 o=2

J(Ωc_,uc_{) 1.32}·₁₀1 _5.56·10−1 _3.17·10−2 _8.64·10−3

Reduction 0.00% 57.85% 97.60% 99.35%

Fig. 14.Gasinsulatedswitchgear.Initialgeometryandgeometricoptimisationconstraints.(Forinterpretationofthecolorsinthisfigure,thereaderis referredtothewebversionofthisarticle.)

that the maximum normal flux Jmax(Ωc

,

_uc

)

_{is minimised. We attempt to achieve this by minimising the cost}

func-tional J

(Ω

c

,

_u c

)

_.

InFigure 14 the gasinsulatedswitchgear componentisshownwiththeelectrode intheformofa primitivecylinder. The cylinderrepresents theelectrode geometry

Γ

f to be optimised. The initialcoarse mesh ofthe cylindercontains 264 elements. The creases on the cylinder are not tagged.Therefore, the geometrybecomes smootherwhile it is refined by subdivision.Asadesignconstraint,theinnersurfaceofthecylinderisrequiredtohaveaconstantradiusforaboltpassing through it. Geometric bounds on the positions of vertices lying on the inner surface are applied to prevent any radial movementthatwouldviolatethisdesignrequirement,seeFig. 14(b).

The once subdivision refined mesh with 1056 elements is chosen asthe computational level, i.e., c

=

1. As can be seeninFig. 15,theendsofthecylinderbecomesmootherbecausetheusual(vs.extended)subdivisionstencilsareapplied throughout the mesh.In thisexample, we consider the geometry at level o

=

0 for optimisation. In the initial design,

Fig. 15.Gas insulated switchgear. Isocontours of the normal flux for the initial cylindrical electrode design.

Fig. 16.Gas insulated switchgear. Isocontours of the normal flux for the optimised electrode design.

Fig. 17.Gas insulated switchgear. Comparison of multiresolution shape optimised design with manually optimised electrode created by ABB engineers.

in the optimised shape shown in Fig. 16, corresponding to a reduction of 17

.

94%. However, the reduction in the cost function J

(Ω

c

,

_uc

)

_{ismuchhigherwith38}

.

_{24%.InFigure 17we alsoshowthecomponentwiththeelectrodegeometry}

as currentlymanufactured by ABB. This electrode geometryhas beenobtainedover the years by combiningengineering intuitionwithsimplecalculationsandtesting.Thesimilaritiesbetweenthemethodicallyshapeoptimisedandtheelectrode geometryinproductionarestriking.Noticeinparticularthesaddleshapeatthetwoendsoftheoriginal cylinder.Ithelps tolowerthelargenormalfluxatthesharpcreaseattheboundaryoftheinnerhole.

7. Summaryandconclusions

We haveintroduced anovel multiresolutionshapeoptimisation approachforelectrostatic problemsby combiningfast boundary element methods with subdivision surfaces. For the considered electrostatic problems only a surface mesh is requiredfortheboundaryelementdiscretisation.Thisisparticularlyappealingforthree-dimensionalproblemsbecauseof the difficultiesingeneratingandupdating volumemeshes. Itwascriticalforthesuccessfulcomputation ofthe presented optimisationexamplesthattherewerenodomainmeshes.

Forthedescriptionofthedomainboundarieswe usedthesubdivisionsurfaces.Theinherenthierarchyofthe subdivi-sionsurfacesallowstoconsiderthesamesurface atdifferentresolutionsandtotakeadvantageofmultiresolutionediting techniques.Starting fromthecoarsest controlmesh increasinglyfiner meshesareused forgeometryupdating andafine controlmeshisusedfortheboundaryelementdiscretisation.Asdemonstratedwiththecomputedexamples,thiseffectively inhibitsthenon-physical geometryoscillationsthat mayoccur inshape optimisation.Moreover,anypathologicalelement distortions onthe computationalmesh arepractically avoided. Asa result, thereisno needto regenerateorsmooth the surfacemeshduringtheoptimisation.

Theproposedmultiresolutionoptimisationapproachhassomesimilaritiestomultigridshapeoptimisation,see[38]and referencestherein.Inbothcasesthegeometryisrepresentedthroughahierarchyofsuccessivelyrefinedmeshesandboth exhibitbetterglobalisationproperties.Differentfrommultigridshapeoptimisation,inthepresentapproachthehierarchyof meshesisonlyusedtoimprovethestabilityoftheoptimisationapproach.Theprimaryaimisnottoacceleratetheiteration process.However,theintroducedrefinementandcoarseningoperators,thatistheprolongationandrestrictionoperatorsin multigridterminology,appeartobealsoidealformultigridoptimisationandacceleratingtheoptimisationprocess.

Inthepresentedapproachsubdivisionsurfaceswereusedonlyforgeometryparametrisation.Itis,however,possibleto usesubdivision surfaces,orthe underlying b-splinebasis functions, forboth analysisandgeometry description.Thishas alreadybeensuccessfullydemonstratedinisogeometric analysisofsolidsandshellswiththe finiteelementmethod,see, e.g.,[39,25].Recently, theisogeometricanalysisofsolidswiththe boundaryelementmethodhasalsobeenexplored [40]. Inclosing,wenotethattheintroducedshapeoptimisationtechniqueiseasilyextendibletootherboundaryvalueproblems withknownfundamentalsolutions, likeelasticity,StokesandHelmholtz.Moreover,duetothesimilarities betweenshape optimisationandinverseproblems,seee.g.[41,42],thepresenttechniqueisalsopromisingforinverseboundaryproblems.

Acknowledgements

WegratefullyacknowledgethesupportprovidedbytheEUcommissionthroughtheFP7MarieCurieIAPPprojectCASOPT (PIAP-GA-2008-230224). K.B. and F.C. thank for the additional support provided by EPSRC through #EP/G008531/1. J.Z. thanks forthesupport provided bythe European Regional DevelopmentFund inthe IT4InnovationsCentre ofExcellence project(CZ.1.05/1.1.00/02.0070)andbytheprojectSPOMECH–CreatingaMultidisciplinaryR&DTeamforReliableSolution ofMechanicalProblems,reg.no.CZ.1.07/2.3.00/20.0070withintheOperationalProgramme‘EducationforCompetitiveness’ fundedby theStructural FundsoftheEuropean Unionandthestate budgetofthe CzechRepublic. Specialthanks to An-dreasBlaszczykfromtheABBCorporateResearchCenterSwitzerlandforfruitfuldiscussionsandforprovidingtheindustrial applications.

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