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Shape derivative of the Willmore functional and applications to equilibrium shapes of vesicles

Thomas Milcent

To cite this version:

Thomas Milcent. Shape derivative of the Willmore functional and applications to equilibrium shapes

of vesicles. [Research Report] RR-7539, INRIA. 2011, pp.22. �inria-00565403�

(2)

a p p o r t

d e r e c h e r c h e

N 0 2 4 9 -6 3 9 9 IS R N IN R IA /R R -- 7 5 3 9 -- F R + E N G

Computational models and simulation

Shape derivative of the Willmore functional and applications to equilibrium shapes of vesicles

Thomas Milcent

N° 7539

Février 2011

(3)
(4)

Centre de recherche INRIA Bordeaux – Sud Ouest

Thomas Milent

Theme: Computationalmodelsand simulation

AppliedMathematis,Computation andSimulation

Équipe-ProjetMC2

Rapportdereherhe n° 7539Février201119pages

Abstrat: Thevesilesonstituteasimpliedbiologialmodeltodesribethe

mehanialbehaviorofredbloodells. Theequilibriumshapesofthesevesiles

are driven by the bending energy whih is given by the Willmore funtional.

We presenta method to ompute the shape derivativeof the Willmore fun-

tional in the levelset framework. Theequivalene withthe method originally

introduedbyWillmore,wherethesurfaeisrepresentedbyaparametrization,

is established. Finally,somenumerialsimulationsoftherelaxationofvesiles

towardstheirequilibriumshapesinthree dimensionsarepresented.

Key-words: Shape optimization, uid-struture interation, eulerian, level

set,nitedierenes

INRIABordeauxSud-Ouest,EquipeMC2

(5)

et appliations aux formes d'équilibre des

vésiules

Résumé: Lesvésiulesonstituentunmodèlebiologiquesimpliépourlade-

sriptionduomportementméaniquedesglobulesrouges. Lesformesd'équilibre

deesvésiulessontpilotéesparl'énergiedeexionqui estdonnéeparlafon-

tionnelledeWillmore. Nousprésentonsuneméthodepouralulerladérivéede

formedelafontionnelledeWillmoreavel'approhelevelset. Nousmontrons

l'équivalene ave la méthode initialement introduite par Willmore où la sur-

faeestreprésentéeàl'aided'uneparamétrisation. Enn,quelquessimulations

numériques de la relaxationde vésiules vers leur formes d'equilibre en trois

dimensionssontprésentées.

Mots-lés : Optimisation de formes, interation uide-struture, eulerien,

"levelset",dierenesnies

(6)

Introdution

Understandingtheblood rheologyisamajorissueforbiomedialappliations.

Wean ite, for example,thepreditionof vasularpathologies. Theblood is

mainlyomposedofredbloodells,whihareresponsibleofthenonNewtonian

behaviorof the ow. These ellshave omplexmembranes, whih are onsti-

tutedofphospholipidsandsprinkledoverwithproteins. Themodellingofboth

biologialandmehanialpropertiesofsuhellsisthendiulttohandle. The

vesilesinsuspensionareomposedofaphospholipidimembranewithelasti

properties. They onstitute asimplied biologial model to desribe the me-

hanialbehaviorofredbloodellsinvitro.

The strong interation between the elasti membrane and the uid leads

to a oupleduid-struture problem. A onvenientwayto taklethis kind of

problem is to use the immersed boundary method introdued by Peskin [22℄.

The main idea is to desribethe uid-struture medium with an uniqueon-

tinuousveloity. ThismethodinvolvesbothEulerianandLagrangianvariables

oupledthroughDiramass. Theelastiforesaretreatedasasouretermin

the uid equations. TheLagrangiantreatment ofthe elastiity leadsto some

drawbaksonthenumerialpointofviewasforinstanethetreatmentoflarge

deformationsandthelossofmass. Morereently,CottetandMaitrehaveintro-

dued in [7, 8℄afullEulerianformulationof theuid-strutureinteration for

membranes. Theinterfaeistrakedimpliitlywithalevelsetfuntion

φ

whih

doesalsoapture thehangeof areathrough

|∇ φ |

. Thisformulationallowsto overomethediulties inherenttotheimmersedboundarymethod.

Themembraneof the vesile responds mehaniallyonly to thehange of

areaandbending. Thevesileisonstituted ofaxedamountofmoleulesso

themembraneisnearlyinextensible. Asaresult,themainmodeofdeformation

ofthevesileis bending.

Forthisreason,wefousinthispaperonthederivationofthebendingfore

anditsappliation totheshapeequilibrium ofvesiles. Thebending energyis

generallygivenbytheWillmorefuntional [17,4,19℄

J (Γ) = Z

Γ

H (Γ) 2 ds.

Here

H

is the meanurvature of the surfae

Γ

. TheWillmore energyhas

aninterestinggeometrialinterpretation: itapturesthedeviationofasurfae

fromitsloalspheriity. Let

κ 1 ,κ 2

betheprinipalurvaturesand

H = κ 1 +κ 2

,

G = κ 1 κ 2

denote the respetive mean and Gaussian urvatures related by

H 2 − 4G = (κ 1 − κ 2 ) 2

. The Gauss-Bonnet theorem ensure that the Gaus- sian urvature is a topologial invariant ie

R

Γ G ds

is onstant for a losed

surfae. Therefore, up to an additive onstant, the Willmore energy writes

R

Γ (κ 1 − κ 2 ) 2 ds

. Asaresult,this energyvanisheswhen

κ 1 = κ 2

and aptures

thedeviationofasurfaefromasphere. Inordertotakethebendingenergyinto

aount,theforelinkedtotheWillmoreenergyhastobeomputed. Following

the virtual work priniple, this fore is related to the variation of the fun-

tional

J (Γ)

. Sine

Γ

is asurfae,theproblem is reastedasnding theshape

derivativeof theWillmore funtional

J (Γ)

. Twodierent methodshavebeen

(7)

developedforthis problem in theliterature. While therstone is basedona

expliitrepresentationofthesurfaewithaparametrization[28℄,theseondone

isbasedonanimpliitrepresentationofthesurfaewithalevelsetfuntion[20℄.

Intherstmethod, thesurfaeisrepresentedbyaparametrization

X

and the Willmore funtional is denoted by

J ( X )

. In order to perform the shape

derivative,adeformationinthenormaldiretion isintrodued. Usingdieren-

tialalulus,

J

writes[28℄

J ( X )(δ) = − Z

Γ

2∆ Γ H + H (H 2 − 4G)

δ ds,

(1)

where

∆ Γ

istheLaplae-Beltramioperatoronthesurfae

Γ

. Thismethodis

naturalsineitonlyinvolvesquantities denedonthesurfae. This resultan

beusedin the frameworkof theimmersed boundarymethod sinethesurfae

andtermsappearingin(1)anbeomputedwithaparametrization.However,

when thesurfaeis desribedby alevel set funtion,aparametrizationis not

known.

Inthe seond method, the surfaeis representedby thezerolevelset ofa

funtion

φ

and the Willmore energy is written

J (φ) = Z

{φ =0}

H(φ) 2 ds

. The

derivativewithrespetto

φ

istough tohandle sinethedomain ofintegration depends itself on

φ

. To overome this diulty a volumi approximation of

J (φ)

isintroduedandtheresultisthefollowing[20℄

J (φ)(δ) = Z

{φ =0}

1

|∇ φ | div

− H (φ) 2 ∇ φ

|∇ φ | + 2

|∇ φ | P ∇φ ( ∇ ( |∇ φ | H (φ)))

δ ds,

(2)

where

P ∇φ

is theprojetorontheplanewhihis orthogonalto

∇ φ

. This

result an be used in an Eulerian framework sine it only involves quantities

dependingonthelevelset funtion

φ

.

The main goal of the paper is to investigate the equivalene of the shape

derivatives(1)and(2). Thisisnotstraightforwardsinethetworesultsusedif-

ferentrepresentationsofasurfaeanddierentdierentialoperators.Moreover,

theformula(2)dependsapriorion

φ

outsidethesurfae

{ φ = 0 }

through

|∇ φ |

,

while(1)dependsonbothurvatureandLaplae-Beltramioperator,whihare

independent of the parametrization. The equivalene of the approahes will

prove that

J (φ)

is in fat independent of the hoie of

φ

outside

{ φ = 0 }

.

Moreover,writing(2)undertheform(1)isalsorelevantfornumerialpurpose.

Indeed,weanfousseparatelyontheapproximationoftheurvatureandthe

Laplae-Beltramioperator.

Theseondgoalofthepaperistousetheformula(1)inthelevelsetframe-

worktostudytherelaxationofvesilestowardstheirequilibriumshapes. These

shapes an be determined by minimizing the bending energy with two on-

straints: xedarea(inextensiblemembrane) and xedvolume(inompressible

uid). Thenumerialmethods usedtosolvethisproblemin theliteraturerely

on the nite elements method on surfaes [3, 2℄ and the phase eld methods

(8)

[23, 24℄. It isworthmentioningtherelated works [12, 11, 13℄, whih fous on

theminimizationofthebendingenergywithoutonstraints. Ourapproahisto

onsideranEulerianmodelwhereanelastimembraneisimmersedinainom-

pressiblevisous uid. Taking thehydrodynamisinto aountis neessaryin

orderto beabletostudythedynamisofthesevesilesinaowinguid. The

volume of the vesile is naturallyonstraint with the inompressibility of the

uidwhiletheareaofthevesileisonstraintwithastielastioeient. We

obtainvarious equilibrium shapesin three dimensionsdepending onavolume

ratio,whihqualitativelyagreeswiththeliterature.

Thepaper isorganizedasfollow. Therstsetion isdevoted tothe intro-

dutionofthetoolsofdierentialgeometry. Theseond setiondealswiththe

shapederivativeoftheWillmore energyin thelevelset framework. Thethird

setionis devoted tonumerialsimulationsofequilibrium shapesof vesilesin

threedimensions. AsimpleproofoftheStokesformulaforsurfaesaswellasa

dierentshapederivativemethodfortheWillmorefuntionalwillbepresented

in theAppendix.

1 Preliminaries

Thevetorsandthematrieswillbewrittenintheanonialorthonormalbasis

of

R 3

. Let

a

,

b

betwovetorsof

R 3

. Wedenoteby

a · b = P

i a i b i

theEulidean

salarprodut,

| a |

theassoiatednormand

[ a ⊗ b ] ij = a i b j

theouterprodut.

Let

A

beamatrixof

M 3 ( R )

. Wedenote by

Tr(A)

,

det(A)

theusualtraeand

determinant. We denote also by

Tr(Cof(A)) = 1 2 (Tr(A) 2 − Tr(A 2 ))

thetrae

of the omatrix. We denote by

,

div

and

the usual Eulidean gradient,

divergeneandLaplaeoperators,respetively.

1.1 Level set method

Throughoutthepaper,

Γ

denotesasmoothonneted,orientedandlosedsur-

faeof

R 3

. Thissurfaeisrepresentedimpliitlyasthezerolevelset[21,25℄of asmoothsalarfuntion

φ

Γ = { x ∈ R 3 / φ(x) = 0 } .

Assumethat

|∇ φ | > 0

inaneighborhood

V

of

{ φ = 0 }

. Theunitarynormal

isthenwelldenedon

V

andgivenby

n (φ) = ∇ φ

|∇ φ | .

(3)

Thedependeneon

φ

willbedroppedin thenextsetionsofthepaper. We

denote by

U

the domainenlosed bythesurfae

Γ

. Wehoose theonvention

that

φ < 0

in

U

and

φ > 0

outside

U

so that

n

is theoutwardnormal. The

gradientoftherelation

| n | 2 = 1

writes

[ ∇ n ] T n = 0, ([ ∇ n ] n ) · n = 0.

(4)

(9)

1.2 Tangential operators

Itisnaturaltodenedierentialoperatorsonsurfaestostudythevariationof

salarandvetoreldson

Γ

. When

Γ

isrepresentedexpliitlybyaparametriza- tion,these dierentialoperators arenaturallydened withthemetriindued

bytheparametrization[5,6,14℄. Intheaseofanimpliitrepresentationof

Γ

withalevelsetfuntion,thesalarandvetoreldsareextendedtoaneighbor-

hood

V

of

Γ

. Thedierentialoperatorsonsurfaesnamedtangentialoperators

arethendenedastheprojetiononthetangentplaneofthelassialEulidean

dierentialoperators[18,1,26,9,15℄. Let

f

and

v

besmoothsalarandvetor elds denedin

V ⊂ R 3

. Then,thetangentialoperatorsaredened by

Γ f = ∇ f − ( ∇ f · n ) n ,

Γ v = ∇ v − [ ∇ v ]( n ⊗ n ),

div Γ ( v ) = Tr( ∇ Γ v ) = div( v ) − ([ ∇ v ] n ) · n ,

∆ Γ f = div Γ ( ∇ Γ f ).

Thesedenitionsareindependentoftheextensionofthequantitiesoutside

thesurfae

Γ

sinetheusualdierentialoperatorsareprojetedonthetangent

planeof

Γ

. Thefollowingidentitieshold

Γ φ = 0,

(5)

Γ f · n = 0,

(6)

div Γ (f v ) = f div Γ ( v ) + ∇ Γ f · v ,

(7)

Γ f · ∇ g = ∇ f · ∇ Γ g = ∇ Γ f · ∇ Γ g,

(8)

[ ∇ n] n = ∇ Γ |∇ φ |

|∇ φ | .

(9)

Thesepropertiesaresimpleonsequenesofthedenitionsofthetangential

operatorsand(3). Wehavealsotherelations

([ ∇ ( ∇ Γ f )] n ) · n = −∇ f · ([ ∇ n ] n ),

(10)

([ ∇ ([ ∇ n] n)] n ) · n = − ([ ∇ n] n) · ([ ∇ n] n).

(11)

Taking gradientofequality(6) andmultiplyingby

n

leadsto (10). Taking thegradientof (4)andmultiplytheresultby

n

gives(11).

Let

f, g

be two smooth funtions and

v

a smooth vetor eld. Let

Q

a

smoothopenset of

R 3

and

∂Q

its boundary(whihis alosedsurfaeof

R 3

).

ForthelassialEulideanoperators,theStokesformulaforvolumeswrites

Z

Q

div( v )f dx = − Z

Q

v · ∇ f dx + Z

∂Q

f v · n ds.

(12)

For tangential operators, we have the following Stokes formula for losed

surfaes

Z

Γ

div Γ ( v )f ds = − Z

Γ

v · ∇ Γ f ds + Z

Γ

Hf v · n ds.

(13)

(10)

Notethat thereis an extratermdepending onthe meanurvature

H

if

v

hasaomponentinthenormaldiretion. Weprovideanewsimpleproofofthis

formulain the Appendix1. Taking

v = ∇ Γ g

inthis formulaand using(5) we

get

Z

Γ

f ∆ Γ g ds = Z

Γ

g∆ Γ f ds.

(14)

1.3 Curvature

Intuitively, the urvature represents the amount by whih a surfae deviates

from beingaplane. Let usintroduethetensor

[ ∇ Γ n ]

, thesoalledextended

Wirtingermap[15℄. Thistensor hastheeigenvetor

n

assoiatedto theeigen- value

0

sine

[ ∇ Γ n ] n = 0

. The remaining eigenvalues are the prinipal ur- vatures denoted by

κ 1

and

κ 2

. The mean and Gaussian urvatures are then

dened by

H = κ 1 + κ 2 = Tr([ ∇ Γ n ]) = div Γ ( n ), G = κ 1 κ 2 = Tr(Cof([ ∇ Γ n ])).

(15)

Sine

n

isextendedby(3), weobtainwith(4)

H = Tr([ ∇ n ]) = div( n ), G = Tr(Cof([ ∇ n ])).

(16)

Theidentity

Tr([ ∇ n ] 2 ) = κ 2 1 + κ 2 2 = H 2 − 2G

and (16)provethe following

relation

div([ ∇ n] n) = ∇ H · n + H 2 − 2G.

(17)

2 Shape derivative of the Willmore funtional

The aim of this setion is to ompute the shape derivative of the Willmore

funtional

J (Γ) = Z

Γ

H (Γ) 2 ds.

The original method of Willmore where the surfae is represented by a

parametrizationispresentedinsetion2.1. Thesetion2.2dealswiththeshape

derivative of ageneral funtional in the level set framework with thevolumi

approah. Inthesetion2.3,thismethodisappliedfortheWillmorefuntional

andtheequivaleneofthelevelsetandtheparametriapproahesisproved.

2.1 Shape optimization with a parametrization

Inthemethoddesribedin[28℄,thesurfaeisparametrizedloallyby

X : U −→

R 3

withparameterdomain

U ⊂ R 2

. Wedenoteby

g ij = X ,i · X ,j

theoeients of the rst fundamental form,

g

its determinant and

g ij

the oeientsof its

inverse. Thenormalisdenedby

n = X , 1 ∧ X , 2

| X , 1 ∧ X , 2 |

andtheseondfundamental

(11)

form by

h ij = X ,ij · n

. Themeanurvatureisthengivenby

H = − X

i,j

g ij h ij

.

TheWillmoreenergyisthen

J ( X ) = Z

U

(H (u 1 , u 2 )) 2 p

g(u 1 , u 2 ) du 1 du 2 .

Inordertoperformtheshapederivative,adeformationinthenormaldire-

tionis introdued withtheassoiateparametrization

X (t) = X + tδ n

where

δ

is asalar funtion. This hoie is justied bythe fat that adeformationin

the tangentialdiretion has no inuene on theshapederivative. The result-

ingsequene ofperturbedsurfaes isdenoted by

(Γ t ) t> 0

where

Γ 0 = Γ

. Using

dierentialalulus

J

writes[28℄

J ( X )(δ) := d dt t =0

J ( X + tδ n ) = − Z

Γ

2∆ Γ H + H (H 2 − 4G)

δ ds,

(18)

where

∆ Γ H = 1

√ g X

i,j

( √ gg ij H ,j ) ,i

isthe LaplaeBeltramioperator onthe

surfae

Γ

. This operatorand the urvature involvedin (18) anbe omputed

with a parametrization of the surfae. However, this formula annot be used

diretlyinalevelset frameworksineaparametrizationisnotknown.

2.2 Shape optimization in the level set framework

In an Eulerian framework, the surfae

Γ

is representedimpliitly as the zero levelset ofafuntion

φ

. Weonsiderthegeneralizedfuntional

I (φ) = Z

{φ =0 }

f [φ] ds,

(19)

where

f [φ]

denotesafuntiondependingonthespatialderivativesof

φ

. For

instane, in thease of theWillmore funtional wehave

f [φ] = div ∇ φ

|∇ φ | 2

.

Toperformtheshapederivative,wedeformthesurfaeassumingthat

φ

satises

thetransportequation

φ t + u · ∇ φ = 0,

(20)

where

u

is avetoreld. Wedenote then

Γ t = { x ∈ R 3 / φ(x, t) = 0 }

and

Γ 0 = Γ

. Theshapederivativeisthengivenby

d d t

t =0 I (φ)

denotedby

( I (φ)) t

.

The major diulty to ompute

( I (φ)) t

relies in the fat that thedomain of

integrationis asurfaewhihdependsontime. Tooveromethis diulty,an

ideaintroduedin [8℄ isto approximatethesurfaeintegralon axed domain

of

R 3

withthefollowinglemma.

Lemma: Letbe

ζ

asmoothpositiveut-ofuntionwithsupportin

[ − 1, 1]

and

R

R ζ = 1

. Assumethat

φ

isasmoothfuntionwith

|∇ φ | > 0

inaneighbor-

hoodof

{ φ = 0 }

. Let

Q

beaanopensetof

R 3

suhthat

{ x ∈ R 3 /φ(x) ≤ 1 } ⊂ Q

,

(12)

0 < ε < 1

and

f [φ]

asmoothsalarfuntionwhihdependson

φ

anditsderiva-

tives. Then

I ε (φ) :=

Z

Q

f [φ] |∇ φ | 1 ε ζ

φ ε

dx −→

ε−→ 0

Z

{φ =0}

f [φ] ds.

(21)

Aproof ofthis lemma involvestheo-areaformula andis givenin [8℄. We

nowusethislemmato giveasimpleproofoftheshapederivative

( I (φ)) t

.

Thederivativeof

I ε (φ)

withrespetto

t

givesthree terms

( I ε (φ)) t =

Z

Q

(f [φ]) t |∇ φ | 1 ε ζ

φ ε

dx + Z

Q

f [φ] 1 ε ζ

φ ε

∇ φ

|∇ φ | · ∇ φ t dx +

Z

Q

f [φ] |∇ φ | 1 ε 2 ζ

φ ε

φ t dx. ( ∗ )

TheStokesformulaforvolumes(12)gives

Z

Q

f [φ] 1 ε ζ

φ ε

∇ φ

|∇ φ | · ∇ φ t dx = − Z

Q

div

f [φ] ∇ φ

|∇ φ | 1

ε ζ φ

ε

φ t dx

− Z

Q

f [φ] ∇ φ

|∇ φ | · ∇ φ 1 ε 2 ζ

φ ε

φ t dx, ( ∗∗ )

sine the term on

∂Q

vanishes. Indeed, we have

ζ

φ ε

= 0

for

| φ | > ε

beausethesupportof

ζ

isin

[ − 1, 1]

. As

{ x ∈ R 3 /φ(x) ≤ 1 } ⊂ Q

and

ε < 1

we

get

ζ

φ ε

= 0

on

∂Q

. Therefore, inserting

( ∗∗ )

in

( ∗ )

andusing thetransport

equation(20)leadsto

( I ε (φ)) t = Z

Q

(f [φ]) t + div (f [φ] n ) u · n

|∇ φ | 1 ε ζ

φ ε

dx.

(22)

Notethat

u · n

appearsin (29). Hene,thetangentialomponentof

u

has noinuenein theshapederivative.

Thefollowingsetionisdevotedtotheomputation oftheshapederivative

oftheWillmorefuntionalwiththevolumiapproah. Weintroduethesalar

funtion

A : R −→ R

andthegeneralizedWillmore funtional 1

J (φ) = Z

{φ =0}

A(H (φ)) ds.

(23)

2.3 Shape derivative with a volumi approximation

Weonsiderthevolumiapproximation(21)oftheWillmore funtional(23)

J ε (φ) = Z

Q

A(H(φ)) |∇ φ | 1 ε ζ

φ ε

dx.

1

Thehoie

A ( r ) = ( r − c 0 ) 2

iswidely usedinthe appliations (

c 0

isthe spontaneous urvature).

(13)

Weapply thegeneralvolumi approximationformula(22)with

f = A(H )

toobtain

( J ε (φ)) t = Z

Q

A (H)H t + div (A(H ) n ) u · n

|∇ φ | 1 ε ζ

φ ε

dx.

Asthemeanurvaturedependsonthevariationofthenormal,westartby

omputingthederivativeofthenormal. Wegetfrom (20)and(9)

n t = ∇ φ

|∇ φ |

t

= ∇ Γ φ t

|∇ φ | = −∇ Γ (u · n) − ([ ∇ n] n)u · n.

(24)

Asbeforetherewillbenoontributionon

∂Q

intheStokesformulaforvol-

umes sine

ζ

φ ε

= 0

on

∂Q

.

Weobtainwiththederivativeofthenormal(24)

( J ε (φ)) t = Z

Q

A (H) div ∇ Γ φ t

|∇ φ |

+ div (A(H ) n ) u · n

|∇ φ | 1 ε ζ

φ ε

dx.

(25)

Thersttermdenoted by

( J 1 ,ε (φ)) t

. TheStokesformulaforvolumes(12)

andthesymmetryrelation(8)areusedtoobtain

( J 1 ,ε (φ)) t = − Z

Q

Γ φ t

|∇ φ | · ∇

A (H ) |∇ φ | 1 ε ζ

φ ε

dx

= − Z

Q

∇ φ t

|∇ φ | · ∇ Γ

A (H ) |∇ φ | 1 ε ζ

φ ε

dx.

Using(5),wehavetherelation

Γ 1

ε ζ φ

ε

= 1 ε 2 ζ

φ ε

Γ φ = 0

. This

property is used to develop the term involving

Γ

and a Stokes formula for

volumesgives

( J 1 ,ε (φ)) t = − Z

Q

∇ φ t

|∇ φ | · ∇ Γ (A (H ) |∇ φ | ) 1 ε ζ

φ ε

dx

= Z

Q

div 1

|∇ φ | ∇ Γ (A (H ) |∇ φ | )) 1 ε ζ

φ ε

φ t dx.

Withtherelation(6),weget

Γ (A (H ) |∇ φ | ) · ∇ 1

ε ζ φ

ε

= 1 ε 2 ζ

φ ε

|∇ φ |∇ Γ (A (H ) |∇ φ | ) · n = 0.

Weusethispropertytodevelopthedivergeneterm. Sine

φ t = −|∇ φ | u · n

withthetransportequation(20),weget

( J ε (φ)) t = Z

Q

div

A(H) n − 1

|∇ φ | ∇ Γ (A (H ) |∇ φ | ))

u · n |∇ φ | 1 ε ζ

φ ε

dx.

(14)

At this point we nd the result (2) when

ε

goes to

0

with (21) and

δ =

−|∇ φ | u · n

. This formula has thedrawbakto depend on

φ

outside

{ φ = 0 }

.

Wegofurthertoobtainamoreintrinsiresultdependingonlyonurvatureand

tangentialoperators.Following(9)weget

div 1

|∇ φ | ∇ Γ (A (H ) |∇ φ | )

= div(A (H )[ ∇ n] n) + div( ∇ Γ (A (H))).

Forthersttermweuse(17)

div(A (H )[ ∇ n] n) = A (H ) ( ∇ H · n + H 2 − 2G) + ∇ (A (H)) · ([ ∇ n] n).

Fortheseondtermweuse(10)

div( ∇ Γ (A (H ))) = div Γ ( ∇ Γ (A (H ))) + ([ ∇ ( ∇ Γ (A (H )))] n ) · n

= ∆ Γ (A (H)) − ∇ (A (H)) · ([ ∇ n] n).

Combiningthese expressionsgives

div 1

|∇ φ | ∇ Γ (A (H ) |∇ φ | )

= ∆ Γ (A (H)) + A (H) ( ∇ H · n + H 2 − 2G).

Aordingto(16),wehavetherelation

div(A(H ) n ) = A(H )H + A (H) ∇ H · n ,

andusingthetransportequation(20),wenally get

( J ε (φ)) t = Z

Q

A(H )H − ∆ Γ (A (H )) − A (H )(H 2 − 2G)

u · n |∇ φ | 1 ε ζ

φ ε

dx.

(26)

When

ε

goesto

0

,weobtainwiththelemma(21)

( J (φ)) t = Z

{φ =0}

A(H)H − ∆ Γ (A (H)) − A (H )(H 2 − 2G)

u · n ds.

(27)

Thisresultdoesnotdependson

φ

outside

{ φ = 0 }

beauseitinvolvesonly

urvature and tangential operators. We propose in the Appendix 2 another

proofofthis resultbasedondiretomputationsonthesurfae. Intheparti-

ularase

A(r) = r 2

wendtheresult(18)obtainedwith aparametrizationof thesurfae. Thisresultallowustoonstrutmoreeientsnumerialshemes

to approximate thebending fore. Indeed, weanfous separatelyon theap-

proximationoftheurvatureandtheLaplae-Beltrami operator.

(15)

3 Numerial simulations of equilibrium shapes

Vesilesarelosedmembranesoflipidssuspendedin anaqueous solution. The

membraneanbeonsideredasatwo-dimensionaluidsinethereisnostress

response to shear deformation. Themembrane isalso nearlyinextensible and

thus the equilibrium shapes of vesiles is driven by the bending energy. An

importantingredientisthevolumeratiogivenby

τ = V

4 3 π 4 S π

3 2

,

(28)

where

V

isthevolumeand

S

theareaofthevesile. Thisparametermea-

surestheratiobetweenthevolumeofthevesile andthevolumeofthesphere

withthesamearea.

Our approah is to onsider an Eulerian uid-struture model where an

elastimembraneisimmersed inaninompressiblevisousuid. Twoenergies

willbeintroduedtotakethehangeofareaandthebendingintoaount. The

volume of the vesile is naturallyonstraint with the inompressibility of the

uidwhiletheareaofthevesileisonstraintwithasti elastioeient.

3.1 Eulerian model

WeonsidertheEulerianuid-struturemodelintroduedin [7,8℄. Let

Q

bea

domain ontaininganinompressibleuid where thevesile isimmersed. The

motionof theinterfaeisaptured byalevelset funtion

φ

whih isadveted

bytheuidveloity. Apartoftheelastiity, thehangeofarea,isreordedin

|∇ φ |

andweintroduetheassoiatedregularizedenergy

E e (φ) = Z

Q

E( |∇ φ | ) 1 ε ζ

φ ε

dx,

where

ζ

isaut-ofuntion usedto spreadtheinterfaenear

{ φ = 0 }

. We

useinoursimulationsthefollowingexpression

ζ(r) = 1 2 (1 + cos(πr))

on

[ − 1, 1]

and

ζ(r) = 0

elsewhere. Theparameter

ε

isequalto

1.5∆x

in thesimulations where

∆x

isthegridsize.Theonstitutivelaw

r 7→ E (r)

desribetheresponse

of the membraneto thehange of area. Weuse in our simulations thelinear

law

E (r) = λ(r − 1)

where

λ

is elastimodulus. Theassoiatedforeis given

by

F e (φ) =

Γ (E ( |∇ φ | )) − E ( |∇ φ | )H (φ) n (φ)

|∇ φ | 1 ε ζ

φ ε

.

Totakethebendingenergyintoaount,weintroduetheregularizedenergy

E c (φ) = Z

Q

A(H(φ)) |∇ φ | 1 ε ζ

φ ε

dx,

where

A

is a onstitutivelawfor the bending energy. Weuse in oursim- ulations the Helfrih law

A(r) = αr 2

where

α

is the bending modulus. The

priniple ofvirtual works laims that

( E c (φ)) t = − Z

Q

F c (φ) · u dx

. Using the

result(26),weget

(16)

F c (φ) = α

2∆ Γ (H(φ)) + H (φ)(H (φ) 2 − 4G(φ))

|∇ φ | 1 ε ζ

φ ε

n (φ).

ThemodelisthengivenbytheinompressibleNavier-Stokesequationswith

thebendingandelastiforesasasouretermoupledwithatransportequation

forthelevelset. Let

L

,

U

,

ρ ref

and

µ ref

representtherespetiveharateristi length,veloity,densityandvisositysales. Thedimensionlessequationswrites

R e ( u t + ( u · ∇ ) u ) − ∆ u + ∇ p = W 1

e

F e (φ) + W 1

c

F c (φ), φ t + u · ∇ φ = 0,

div( u ) = 0,

where

R e = LU ρ ref

µ ref

, W e = µ ref U

λ , W c = µ ref U L 2 α ,

are respetivelythe Reynolds, Weissenberg and bending numbers. In this

model,theuidvisosityandthedensityarethesameinthetwouids. More-

overwe havenegleted themassof the membrane. These equationsare om-

pletedwithappropriateinitialandboundaryonditions.

ThetransportequationisdisretizedwithfthorderWENOshemeinspae

[16℄ andexpliitEuler shemein time. A projetionmethod isused todeou-

pled thepressureandtheveloityin timein theNavier-Stokesequations. The

resolution ofthePoisson equationforthepressureisperformedwith FFT.An

impliitshemein time isused forthe diusionto avoidarestritivestability

ondition on the time step for low Reynolds numbers. The others terms are

disretizedinspaewithlassialenteredshemesofordertwo. Sinethelevel

set funtion isused throughitsgradientto omputethestrething, wedonot

performtheredistaning. Instead,weusetherenormalization

φ

|∇φ|

tomeasure

the distane to interfae. Thus,

|∇ φ | 1 ε ζ

φ ε

is replaed by

1 ε ζ

φ

|∇φ|ε

. This

approahwasprovedin[8℄tobeeientfromthepointofviewofbothvolume

onservationandinterfaeforealulations.

3.2 Numerial results

The domain

Q = [0, 1] 3

is disretized on a Cartesianmesh with

64

points in

eahdiretion. Wehoose

R e = 0.01

toreproduethetypialregimeoftheow

,

W e = 0.0001

toonstraintthehangeofareaand

W c = 0.3

. Weimposezero

veloityfortheinitialandboundaryonditions. Weonsidertherelaxationof

twoinitialshapeswithdierentvolumeratio(28). Inthefollowingsimulations

werepresenttheevolutionofthezerolevelsetof

φ

andsliesofthepressureat

theenterofthedomain.

Inthe rst simulation, we hoose an ellipsoid representedby the levelset

funtion

φ(x, y, z) = x−0 a . 5 2

+ y−0 b . 5 2

+ z−0 c . 5 2

with theparameters

a =

(17)

0.35, b = 0.35, c = 0.1

. Weperforminitiallyaredistaningonthislevelsetfun- tion. Theassoiatedvolumeratiois

τ = 0.6

. TheresultsarepresentedonFig1.

Figure1: Evolutionoftheellipsoidattime

0, 0.5, 1, 1.5, 2

and

4.5

.

Intheseond simulation,wehoosea"dumbbell" representedbythe level

set funtion

φ(x, y, z) = min(φ 1 (x, y, z), φ 2 (x, y, z), φ 3 (x, y, z))

with

φ 1 (x, y, z) = p

x 2 + y 2 + (z + α) 2 ) − r, φ 2 (x, y, z) = p

x 2 + y 2 + (z − α) 2 ) − r, φ 3 (x, y, z) = max( | z | − α, p

x 2 + y 2 − w).

and the parameters

α = 0.22, w = 0.13, r = 0.18

. The volume ratio is

τ = 0.77

. We perform initially aredistaning on this levelset funtion. The resultsarepresentedonFig 2.

(18)

Figure 2: Evolutionofthedumbbellat time

0, 0.05, 0.1, 0.15, 0.2

and

2

.

Notethattheinitialshapesarelosetotheoptimalshapesinordertoavoid

loalminimums. Theseshapesqualitativelyagreewiththeresultspresentedin

theliterature[3,2,23,24℄.

4 Conlusion and forthoming works

In this paper, we haveintrodued a method to omputethe shapederivative

of theWillmorefuntional inthe levelset framework. This approahrelies on

(19)

theapproximationofsurfaeintegralson axeddomain and allowto usethe

standarddierentialalulusin

R 3

. Wehaveprovedthatthis methodleadsto a geometrial result involvingthe urvature and the Laplae-Beltrami opera-

tor. TheequivalenewiththemethodoriginallyintroduedbyWillmore,where

the surfae is represented by a parametrization, is established. The level set

bendingfore assoiated totheWillmore energyis thenaddedon anEulerian

uid-struturemodelwhereanelastimembraneisimmersedinavisousuid.

ThedisretizationonaxedCartesianmesh ofthelevelsetfuntion allowsto

overomethediulties inherent toaLagrangiantrakingof theinterfae,as

forinstane thetreatmentoflargedeformations. Wehavepresentedsomenu-

merialsimulationsrelatedtotherelaxationofvesilestowardtheirequilibrium

shapesinthreedimensions. Thevariousshapesobtainedareingoodagreement

with theresultspresentedintheliterature. Thesenumerialresultsarearst

stepto understandthe behaviorof vesilesin thebloodow. Aongoingwork

onernthesimulationofdynamis ofvesilesinashearow.

5 Appendix 1: Stokes formula for surfaes

Theaim ofthisrstAppendix isto presentanewproofoftheStokesformula

forsurfaesbasedonthevolumiapproximationoffuntionals.

Let

Q

beanopenset of

R 3

,

f, g

smoothfuntionsand

v , w

smoothvetor elds. Assumethat

w

vanisheson

∂Q

. Asaresult,aStokesformulaforvolumes

andthedenitionoftangentialdivergene give

Z

Q

div Γ ( w ) dx = Z

Q

div( w ) − [ ∇ w ] : n ⊗ n dx = Z

Q

div( n ⊗ n ) · w dx.

Taking

w = v |∇ φ | 1 ε ζ

φ ε

(whihvanisheson

∂Q

)andusingtheproperties

(7)and(16),weobtain

Z

Q

div Γ ( v ) |∇ φ | 1 ε ζ

φ ε

dx = − Z

Q ∇ Γ

|∇ φ | 1 ε ζ

φ ε

· v dx +

Z

Q

H n + [ ∇ n] n

· v |∇ φ | 1 ε ζ

φ ε

dx

Using(5)and(9),wehave

Γ

|∇ φ | 1 ε ζ

φ ε

= ∇ Γ ( |∇ φ | ) 1 ε ζ

φ ε

= [ ∇ n] n |∇ φ | 1 ε ζ

φ ε

Therefore

Z

Q

div Γ ( v ) |∇ φ | 1 ε ζ

φ ε

dx =

Z

Q

H v · n |∇ φ | 1 ε ζ

φ ε

dx.

When

ε

goesto

0

weobtainwiththelemma(21)

(20)

Z

Γ

div Γ ( v ) ds = Z

Γ

H v · n ds.

Replae

v

by

f v

giveswith(7)

Z

Γ

f div Γ ( v ) ds = − Z

Γ

Γ f · v ds + Z

Γ

f H v · n ds.

6 Appendix 2: Anotherproofofthe shapederiva-

tive of the Willmore funtional

This seond Appendix is devoted to a dierent proof of the shape derivative

oftheWillmorefuntional. Themajordierenewiththevolumiapproahis

that theomputationsareperformeddiretlyonthesurfae

{ φ = 0 }

.

Thestartingpointisthevolumishapederivative(22)

( I ε (φ)) t = Z

Q

(f [φ]) t + div (f [φ] n ) u · n

|∇ φ | 1 ε ζ

φ ε

dx.

When

ε

goesto

0

wegetwiththelemma(21)

( I (φ)) t =

Z

{φ =0}

(f [φ]) t + div (f [φ] n ) u · n ds.

(29)

We obtainthe general shape derivativeformula for surfaeintegral whih

anbefoundin[1,18℄. WeonsidertheWillmorefuntional

J (φ) = Z

{φ =0 }

A(H (φ)) ds.

Weapply(29)with

f = A(H)

to obtain

( J (φ)) t =

Z

{φ =0 }

A (H )H t + div(A(H) n ) u · n ds.

(30)

Wehoose thedenition

H = div Γ ( n )

of themeanurvature (15)in order

to introdue thetangentialoperators. Then aStokesformulafor surfaes(14)

whihinvolvestangentialoperatorswill beusedtogetthenalresult.

Wehavewiththedenitionoftangentialoperatorsand(4)

H t = (div( n )) t − (([ ∇ n] n) · n ) t ,

= div( n t ) − ([ ∇ n t ] n ) · n − ([ ∇ n ] n t ) · n − ([ ∇ n ] n ) · n t ,

= div Γ ( n t ) − ([ ∇ n] n) · n t .

Inserting(24)in theprevious expressionand developingthetangentialdi-

vergenewith(7)gives

H t = − ∆ Γ (u · n) − div Γ ([ ∇ n] n) u · n + ([ ∇ n] n) · ([ ∇ n] n) u · n.

(21)

Usingtherelations(17)and(11)fortheseondtermgives

H t = − ∆ Γ (u · n) − ( ∇ H · n + H 2 − 2G) u · n.

Wehavealsowith(16)therelation

div(A(H ) n ) = A(H )H + A (H) ∇ H · n .

UsingtheStokesformulaforsurfaes(14)wenally obtain

( J (φ)) t = Z

{φ =0 }

A(H)H − ∆ Γ (A (H)) − A (H )(H 2 − 2G)

u · n ds.

(31)

Weobtain thesame shape derivative(27)obtained with thevolumi app-

proah.

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(23)

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