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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�
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
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
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
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
whihdoesalsoapture 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 surfaeΓ
. TheWillmore energyhasaninterestinggeometrialinterpretation: itapturesthedeviationofasurfae
fromitsloalspheriity. Let
κ 1 ,κ 2
betheprinipalurvaturesandH = κ 1 +κ 2
,G = κ 1 κ 2
denote the respetive mean and Gaussian urvatures related byH 2 − 4G = (κ 1 − κ 2 ) 2
. The Gauss-Bonnet theorem ensure that the Gaus- sian urvature is a topologial invariant ieR
Γ G ds
is onstant for a losedsurfae. Therefore, up to an additive onstant, the Willmore energy writes
R
Γ (κ 1 − κ 2 ) 2 ds
. Asaresult,this energyvanisheswhenκ 1 = κ 2
and apturesthedeviationofasurfaefromasphere. Inordertotakethebendingenergyinto
aount,theforelinkedtotheWillmoreenergyhastobeomputed. Following
the virtual work priniple, this fore is related to the variation of the fun-
tional
J (Γ)
. SineΓ
is asurfae,theproblem is reastedasnding theshapederivativeof theWillmore funtional
J ′ (Γ)
. Twodierent methodshavebeendevelopedforthis problem in theliterature. While therstone is basedona
expliitrepresentationofthesurfaewithaparametrization[28℄,theseondone
isbasedonanimpliitrepresentationofthesurfaewithalevelsetfuntion[20℄.
Intherstmethod, thesurfaeisrepresentedbyaparametrization
X
and the Willmore funtional is denoted byJ ( X )
. In order to perform the shapederivative,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-BeltramioperatoronthesurfaeΓ
. Thismethodisnaturalsineitonlyinvolvesquantities 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 writtenJ (φ) = Z
{φ =0}
H(φ) 2 ds
. Thederivativewithrespetto
φ
istough tohandle sinethedomain ofintegration depends itself onφ
. To overome this diulty a volumi approximation ofJ (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
. Thisresult 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 ofφ
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
[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
. Leta
,b
betwovetorsofR 3
. Wedenotebya · b = P
i a i b i
theEulideansalarprodut,
| a |
theassoiatednormand[ a ⊗ b ] ij = a i b j
theouterprodut.Let
A
beamatrixofM 3 ( R )
. Wedenote byTr(A)
,det(A)
theusualtraeanddeterminant. We denote also by
Tr(Cof(A)) = 1 2 (Tr(A) 2 − Tr(A 2 ))
thetraeof 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 asmoothsalarfuntionφ
Γ = { x ∈ R 3 / φ(x) = 0 } .
Assumethat
|∇ φ | > 0
inaneighborhoodV
of{ φ = 0 }
. Theunitarynormalisthenwelldenedon
V
andgivenbyn (φ) = ∇ φ
|∇ φ | .
(3)Thedependeneon
φ
willbedroppedin thenextsetionsofthepaper. Wedenote by
U
the domainenlosed bythesurfaeΓ
. Wehoose theonventionthat
φ < 0
inU
andφ > 0
outsideU
so thatn
is theoutwardnormal. The
gradientoftherelation
| n | 2 = 1
writes[ ∇ n ] T n = 0, ([ ∇ n ] n ) · n = 0.
(4)1.2 Tangential operators
Itisnaturaltodenedierentialoperatorsonsurfaestostudythevariationof
salarandvetoreldson
Γ
. WhenΓ
isrepresentedexpliitlybyaparametriza- tion,these dierentialoperators arenaturallydened withthemetriinduedbytheparametrization[5,6,14℄. Intheaseofanimpliitrepresentationof
Γ
withalevelsetfuntion,thesalarandvetoreldsareextendedtoaneighbor-
hood
V
ofΓ
. ThedierentialoperatorsonsurfaesnamedtangentialoperatorsarethendenedastheprojetiononthetangentplaneofthelassialEulidean
dierentialoperators[18,1,26,9,15℄. Let
f
andv
besmoothsalarandvetor elds denedinV ⊂ 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
sinetheusualdierentialoperatorsareprojetedonthetangentplaneof
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)andmultiplytheresultbyn
gives(11).Let
f, g
be two smooth funtions andv
a smooth vetor eld. LetQ
asmoothopenset of
R 3
and∂Q
its boundary(whihis alosedsurfaeofR 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)Notethat thereis an extratermdepending onthe meanurvature
H
ifv
hasaomponentinthenormaldiretion. Weprovideanewsimpleproofofthis
formulain the Appendix1. Taking
v = ∇ Γ g
inthis formulaand using(5) weget
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 ]
, thesoalledextendedWirtingermap[15℄. Thistensor hastheeigenvetor
n
assoiatedto theeigen- value0
sine[ ∇ Γ n ] n = 0
. The remaining eigenvalues are the prinipal ur- vatures denoted byκ 1
andκ 2
. The mean and Gaussian urvatures are thendened 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 followingrelation
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
withparameterdomainU ⊂ R 2
. Wedenotebyg ij = X ,i · X ,j
theoeients of the rst fundamental form,g
its determinant andg ij
the oeientsof itsinverse. Thenormalisdenedby
n = X , 1 ∧ X , 2
| X , 1 ∧ X , 2 |
andtheseondfundamentalform by
h ij = X ,ij · n
. ThemeanurvatureisthengivenbyH = − 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
whereδ
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
whereΓ 0 = Γ
. Usingdierentialalulus
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 onthesurfae
Γ
. This operatorand the urvature involvedin (18) anbe omputedwith 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 ofafuntionφ
. WeonsiderthegeneralizedfuntionalI (φ) = Z
{φ =0 }
f [φ] ds,
(19)where
f [φ]
denotesafuntiondependingonthespatialderivativesofφ
. Forinstane, in thease of theWillmore funtional wehave
f [φ] = div ∇ φ
|∇ φ | 2
.
Toperformtheshapederivative,wedeformthesurfaeassumingthat
φ
satisesthetransportequation
φ t + u · ∇ φ = 0,
(20)where
u
is avetoreld. Wedenote thenΓ t = { x ∈ R 3 / φ(x, t) = 0 }
andΓ 0 = Γ
. Theshapederivativeisthengivenbyd d t
t =0 I (φ)
denotedby( I (φ)) t
.The major diulty to ompute
( I (φ)) t
relies in the fat that thedomain ofintegrationis 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
. Assumethatφ
isasmoothfuntionwith|∇ φ | > 0
inaneighbor-hoodof
{ φ = 0 }
. LetQ
beaanopensetofR 3
suhthat{ x ∈ R 3 /φ(x) ≤ 1 } ⊂ Q
,0 < ε < 1
andf [φ]
asmoothsalarfuntionwhihdependsonφ
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)
withrespettot
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 haveζ
φ ε
= 0
for| φ | > ε
beausethesupportof
ζ
isin[ − 1, 1]
. As{ x ∈ R 3 /φ(x) ≤ 1 } ⊂ Q
andε < 1
weget
ζ
φ ε
= 0
on∂Q
. Therefore, inserting( ∗∗ )
in( ∗ )
andusing thetransportequation(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,thetangentialomponentofu
has noinuenein theshapederivative.Thefollowingsetionisdevotedtotheomputation oftheshapederivative
oftheWillmorefuntionalwiththevolumiapproah. Weintroduethesalar
funtion
A : R −→ R
andthegeneralizedWillmore funtional 1J (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).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
. Thisproperty is used to develop the term involving
∇ Γ
and a Stokes formula forvolumesgives
( 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.
At this point we nd the result (2) when
ε
goes to0
with (21) andδ =
−|∇ φ | u · n
. This formula has thedrawbakto depend onφ
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
goesto0
,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 }
beauseitinvolvesonlyurvature and tangential operators. We propose in the Appendix 2 another
proofofthis resultbasedondiretomputationsonthesurfae. Intheparti-
ularase
A(r) = r 2
wendtheresult(18)obtainedwith aparametrizationof thesurfae. Thisresultallowustoonstrutmoreeientsnumerialshemesto approximate thebending fore. Indeed, weanfous separatelyon theap-
proximationoftheurvatureandtheLaplae-Beltrami operator.
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
isthevolumeandS
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
beadomain ontaininganinompressibleuid where thevesile isimmersed. The
motionof theinterfaeisaptured byalevelset funtion
φ
whih isadvetedbytheuidveloity. Apartoftheelastiity, thehangeofarea,isreordedin
|∇ φ |
andweintroduetheassoiatedregularizedenergyE e (φ) = Z
Q
E( |∇ φ | ) 1 ε ζ
φ ε
dx,
where
ζ
isaut-ofuntion usedto spreadtheinterfaenear{ φ = 0 }
. Weuseinoursimulationsthefollowingexpression
ζ(r) = 1 2 (1 + cos(πr))
on[ − 1, 1]
and
ζ(r) = 0
elsewhere. Theparameterε
isequalto1.5∆x
in thesimulations where∆x
isthegridsize.Theonstitutivelawr 7→ E ′ (r)
desribetheresponseof the membraneto thehange of area. Weuse in our simulations thelinear
law
E ′ (r) = λ(r − 1)
whereλ
is elastimodulus. Theassoiatedforeis givenby
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 lawA(r) = αr 2
whereα
is the bending modulus. Thepriniple ofvirtual works laims that
( E c (φ)) t = − Z
Q
F c (φ) · u dx
. Using theresult(26),weget
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
andµ 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|
tomeasurethe 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 with64
points ineahdiretion. Wehoose
R e = 0.01
toreproduethetypialregimeoftheow,
W e = 0.0001
toonstraintthehangeofareaandW c = 0.3
. Weimposezeroveloityfortheinitialandboundaryonditions. Weonsidertherelaxationof
twoinitialshapeswithdierentvolumeratio(28). Inthefollowingsimulations
werepresenttheevolutionofthezerolevelsetof
φ
andsliesofthepressureattheenterofthedomain.
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 =
0.35, b = 0.35, c = 0.1
. Weperforminitiallyaredistaningonthislevelsetfun- tion. Theassoiatedvolumeratioisτ = 0.6
. TheresultsarepresentedonFig1.Figure1: Evolutionoftheellipsoidattime
0, 0.5, 1, 1.5, 2
and4.5
.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))
withφ 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 isτ = 0.77
. We perform initially aredistaning on this levelset funtion. The resultsarepresentedonFig 2.Figure 2: Evolutionofthedumbbellat time
0, 0.05, 0.1, 0.15, 0.2
and2
.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
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 ofR 3
,f, g
smoothfuntionsandv , w
smoothvetor elds. Assumethatw
vanisheson∂Q
. Asaresult,aStokesformulaforvolumesandthedenitionoftangentialdivergene 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
goesto0
weobtainwiththelemma(21)Z
Γ
div Γ ( v ) ds = Z
Γ
H v · n ds.
Replae
v
byf 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
goesto0
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 orderto 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.
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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