Volume reconstruction from slices using phase field...

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1/60 Volume reconstruction from slices using phase field method Elie Bretin (with F. Dayrens and S. Masnou) Contrôle, Problème inverse et Applications Clermont-Ferrand Sep. 2017

Transcript of Volume reconstruction from slices using phase field...

Page 1: Volume reconstruction from slices using phase field methodmath.univ-bpclermont.fr/~munch/Bretin17.pdf · 2017-09-27 · 1/60 Volume reconstruction from slices using phase field method

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Volume reconstruction from slices using phase fieldmethod

Elie Bretin (with F. Dayrens and S. Masnou)

Contrôle, Problème inverse et ApplicationsClermont-Ferrand

Sep. 2017

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Motivation : Magnetic resonance Imaging

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Surface reconstruction from slices

Find the set Ω∗ as a minimizer of

JΩ1,Ω2(Ω) =

J(Ω) if Ω1 ⊂ Ω ⊂ Ωc2

+∞ otherwise,

where J is a surface geometric energy as the perimeter or theWillmore energy.

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Other application

Regularization of discrete contours

Ω1

Ω2

Find the set Ω∗ such as

Ω∗ = arg minΩ1⊂Ω⊂Ωc

2

W(Ω), withW(Ω) =

∫∂Ω

H2dHn−1

where Ω1 and Ω2 are two given set such as Ω1 ⊂ Ωc2

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Perimeter and mean curvature flow

PerimeterP(Ω) =

∫∂Ω

1dHn−1

Shape derivative

P′(Ω)(θ) =

∫∂Ω

H θ.n dHn−1,

where n and H denote the normal and the mean curvature.L2 gradient flow of P ⇒ the normal velocity Vn satisfies

Vn = −H.

Γ0

Γt

V(x)n

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Some properties of mean curvature flow t → Ω(t)

Local existence for convex initial set. The set Ω(t) stay convex,converges to a point and becomes asymptotically spherical [Huisken1984]

In dimension 2, local existence for smooth closed curves. The setΩ(t) becomes convex in finite time, converges to a point and becomesasymptotically spherical [Gage and Hamilton 1986], [Grayson 1987]

In dimension n > 2 : singularities in finite time [Grayson 1989]

Inclusion principle [Ecker 2002]:

Ω1(0) ⊂ Ω2(0) then Ω1(t) ⊂ Ω2(t), ∀t ∈ [0,T ]

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Example of mean curvature flow in dimension two

t = 3.0518e−05

ε = 0.0039062, δt = 1.5259e−05, N=256

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t = 0.0020142

ε = 0.0039062, δt = 1.5259e−05, N=256

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t = 0.01001

ε = 0.0039062, δt = 1.5259e−05, N=256

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t = 0.027008

ε = 0.0039062, δt = 1.5259e−05, N=256

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t = 3.0518e−05

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t = 0.0070038

ε = 0.0039062, δt = 1.5259e−05, N=256

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t = 0.030014

ε = 0.0039062, δt = 1.5259e−05, N=256

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Example of mean curvature flow in dimension three

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Willmore Flow

Willmore Energy

W(Ω) =12

∫∂Ω

H2dHn−1,

L2 gradient flow

Vn = ∆SH + |A |2H −12

H3,

where |A |2 =∑κ2

i .

In dimension 2 :

Vn = ∆SH +12

H3,

In dimension 3 :

Vn = ∆SH +12

H(H2 − 4G),

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Existence and regularity of Willmore Flow,

Long time existence : single curve – [Dziuk Kuwert Schatzle-2002],

Long time existence : higher dimension (small energy) [KuwertSchatzle-2001]

But in general, singularities in finite time !

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Example of Willmore flow in dimension three

fig: Two smooth evolutions by Willmore flow ; a Clifford’s torus and aLawson-Kusner surface

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Mean curvature flowPerimeter

P(Ω) =

∫∂Ω

1dHn−1

L2 gradient flow of P ⇒ the normal velocity V satifies

V = H

where H denotes the mean curvature.Example of mean curvature flow

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A Parametric approach ( [Deckelnick,Dziuk,Elliott])

A parametric representation

Γ = X(s) = (x(s), y(s))s∈[0,2π]

Normal vector n(s) at X(s) :

n(X(s)) =Xs(s)⊥

|Xs(s)|=

(y′(s),−x′(s, t))√x′(s)2 + y′(s)2

Mean curvature at X(s)

κ(X(s)) =1

|Xs(s)|

(Xs(s)

|Xs(s)|

)s· n(X(s)) =

x′(s)y′′(s) − y′(s)x′′(s)

(x′(s)2 + y′(s)2)3/2

Mean curvature flow

Xt (s) = κ(X(s))n(X(s)) or equivalently Xt =1|Xs |

(Xs

|Xs |

)s.

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The level set method ( [Osher,Sethian])

An implicit representation of the interface

Γ =x ; ϕ(x, t) = 0

Normal vector n and curvature κ :

n(ϕ) =∇ϕ

|∇ϕ|and κ(ϕ) = div

(∇ϕ

|∇ϕ|

)

Mean curvature flow

∂tϕ = κ(ϕ)|∇ϕ| = div(∇ϕ

|∇ϕ|

)|∇ϕ|

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The level set method

A Hamilton-Jacobi equation

∂tϕ = div(∇ϕ

|∇ϕ|

)|∇ϕ| = ∆ϕ −

〈∇2ϕ∇ϕ,∇ϕ〉

|∇φ|2

Weak solution in sense of viscosity [Evan,Spruck][Chen,Giga,Goto])

Numerical approach : fast marching method (for transport equation)where the velocity κ(ϕ) is estimated explicitly

Stability problems as for the explicit parametric approach

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Phase field method and Cahn Hilliard energy

Cahn Hilliard energy

Pε(u) =

∫Rd

(ε|∇u|2

2+

W(u)

)dx,

where W(s) = 12s2(1 − s)2 is a double well potential,

and ε is a small parameter.Approximation in sense of Γ-convergence

(1) ∀uε → u, lim infε→0

Pε(uε) ≥ P(u)

(2) ∀u ∃uε → u such that lim supε→0

Pε(uε) ≤ P(u)

Γ-convergence in L1(Rd) [Modica-Mortola77]

Γ − limε→0

Pε = cwP

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Cahn Hilliard energy:

Pε(u) =

∫ (ε|∇u|2

2+

W(u)

)dx

Allen Cahn equation as L2 gradient flow of Pε

∂tuε = ∆uε −1ε2

W ′(uε)

the solution uε is of the form

uε(x, t) = q(dist(x,Ωε(t))

ε

)+O(ε2)

where q is a profile function.

The normal velocity of Ωε(t) satisfies[Mottoni-Schatzmann89] [Chen92] [Bellettini-Paolini95]

Vε = H + O(ε2)

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Resolution of Allen Cahn equation

Resolution of the Allen Cahn equation in Q = [0, 1]n with periodicboundary condition :ut (x, t) = ∆u(x, t) − 1

ε2 W ′(u(x, t)), for (x, t) ∈ Q × [0,T ],

u(x, 0) = u0 ∈ [0, 1]

Classical semi-implicit scheme :

un+1 − un = δt

(∆un+1 −

1ε2

W ′(un)

)Stability constraint : δt ≤ M−1ε2 where M = sups∈[0,1]|W

′′(s)|

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A convex-concave decomposition

Cahn Hilliard energy

Pε(u) = JC(u) + JN(u)

Semi-implicit scheme is equivalent to minimize implicitly

Pε,un (u) = JC(u) + JN(un) + J′N(un)(u − un)

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J(x) = J

c(x) + J

N(x)

Jx

0

(x)= Jc(x) + J

N(x

0) +J’

N(x

0)*(x − x

0)

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An unconditionally stable scheme

Semi-implicit minimization

Pε(u) =

∫Rd

[ε|∇u|2

2+α

ε

u2

2

]dx +

∫Rd

[1ε

(W(u) −α

2u2)

]dx

An Pε-unconditionally stable scheme

un+1 − un = δt

(∆un+1 −

α

ε2un+1 −

1ε2

(W ′(un) − αun)

)as soon as α ≥ M.

Optimization process : define un+1 as the minimum of Pε,un :

un+1 = −1ε2

(∆ −

α

ε2I)−1

(W ′(un) − αun)

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Validation of this numerical method

Initial set : a circle of radius R0

The motion by mean curvature remains a circle of radius

R(t) =√

R20 − 2t ,

Extinction time : text = 12R2

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fig: ∂Ωε(t) at different times t

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εE

rreu

r(ε)

= |t

ext −

t ext

ε|

erreur(ε)

ε2 ln2(ε)

ε2

ε

fig: Error ε → |text − t εext |

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Some simulations

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Willmore Flow

Willmore Energy

W(Ω) =12

∫∂Ω

H2dHn−1,

L2 gradient flow

Vn = ∆SH + |A |2H −12

H3,

where |A |2 =∑κ2

i .

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Classical approximation of Willmore energy

Phase field approximation

uε = q(dist(x,Ω)

ε

), with q′(s) = −

√2W(q(s)).

Remarks that

(ε∆uε −

W ′(uε))2

= (∆dist(x,Ω))2 1ε

q′(dist(x,Ω)

ε

)2

→ H(x)2cWδ∂Ω

Then, at least for smooth set Ω, we have

Wε(uε) =12ε

∫Rd

(ε∆uε −

W ′(uε))2

dx =⇒ε→0

cW12

∫∂Ω

H2dHn−1

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De Giorgi conjecture : Γ-convergence of Wε ?

Definition ( Classical approximation of Willmore energy )

Wε(u) =12ε

∫Rd

(ε∆u −

W ′(u)

)2

dx

Gamma-convergence of Wε to cWW ?Ok in the case of C2 Set adding a perimeter term [Röger,Schätzle2006],[Nagase,Tonegawa 2007],

Γ − limε→0

(Wε + Pε) = cW (W + P)

ButW is not lower semi-continuous !

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A relaxation of Willmore energy

Semi-continuous envelope ofW for L1-topology of set

W(Ω) = inflim infW(Ωh), ∂Ωh ∈ C2, Ωh → Ω in L1(Ω).

Characterization of finite relaxed Willmore energy in dimension 2 :[Bellettini,Maso and Paolini 1993],[Bellettini,Mugnai,2004]

ifW(E) < +∞, then a non oriented tangent must exist everywhereon the boundary of E.

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Γ-convergence ofWε to cWW ?

Existence of Allen Cahn solutions [Dang Fife Peletier 92],[KowalczykPacard 2012]

∆uε −1ε2

W ′(uε) = 0,

such as uε → χE withW(E) = +∞

Example of Allen Cahn solutions

Find another relaxation (see varifold) but requirement of aclassification of all Allen Cahn solutions !

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Other approximation of Willmore Energy

Mugnai’s approximation in dimension n = 2

We have H2 = |A |2 in dimension 2

When uε = q(

dist(x,Ω)ε

), then

ε∇2uε −1ε

W ′(u)∇u|∇u|

⊗∇u|∇u|

= A q′(dist(x,Ω)

ε

)Willmore energy approximation

WMε (u) =

12ε

∫ ∣∣∣∣∣ε∇2u −1ε

W ′(u)∇u|∇u|

⊗∇u|∇u|

∣∣∣∣∣2 dx

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Other approximation of Willmore Energy

Willmore energy approximation

WMε (u) =

12ε

∫ ∣∣∣∣∣ε∇2u −1ε

W ′(u)∇u|∇u|

⊗∇u|∇u|

∣∣∣∣∣2 dx

Control of the mean curvature of the isolevel surfaces

|∇u|∣∣∣∣∣div∇u|∇u|

∣∣∣∣∣ ≤ 1ε

∣∣∣∣∣ε∇2uε −1ε

W ′(u)∇u|∇u|

⊗∇u|∇u|

∣∣∣∣∣Γ-convergence of WM

ε + Pε to cW (W + P) [Mugnai 2010],[Bellettini,Mugnai 2010] in dimension 2.

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Approximating the Willmore flow with the classicalapproach

Willmore energy

Wε(u) =12ε

∫ (ε∆u −

W ′(u)

ε

)2

dx

Its L2 gradient flow

∂tu = −∆

(∆u −

1ε2

W ′(u)

)+

1ε2

W ′′(u)

(∆u −

1ε2

W ′(u)

),

or ε2∂tu = ∆µ − 1ε2 W ′′(u)µ

µ = W ′(u) − ε2∆u.

Well-posedness and existence at fixed parameter ε : [ColliLaurencot-2011-2012] with volume and area constraints

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Asymptotic expansion in smooth case

Formal asymptotic expansion in smooth case [Loreti March-2000]uε(x, t) ' q

(d(x,Ωε(t))

ε

)+ ε2

(A2 − 1

2H2)η1

(d(x,Ωε(t))

ε

)µε(x, t) ' −εHq′

(d(x,Ωε(t))

ε

)+ ε2H2η2

(d(x,Ωε(t))

ε

) ,

where η′′1 (s) −W ′′(q(s))η1(s) = sq′(s),

η′′2 (s) −W ′′(q(s))η2(s) = q′′(s)

Formal convergence

V ε = ∆SH + |A |2H −12

H3 + O(ε)

the velocity limit depends on the second term of order 2 in theasymptotic expansion of uε and µε !

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An implicit spectral scheme using a fixed point iteration

Phase field system ∂tu = ∆µ − 1ε2 W ′′(u)µ

µ = 1ε2 W ′(u) −∆u.

Implicit discretization in timeun+1 = δt

[∆µn+1 − 1

ε2 W ′′(un+1)µn+1]

+ un

µn+1 = 1ε2 W ′(un+1) −∆un+1,

Computed with a Fixed-point iteration

φ

(un+1

µn+1

)=

(Id + δt ∆

2)−1

(Id δt ∆−∆ Id

) (un −

δtε2 W ′′(un+1)µn+1

1ε2 W ′(un+1)

)Convergence of the fixed point iteration if

δt ≤ C minε4, δ2

xε2.

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A convex-concave splitting ofWε

1εWε(u) =

12

∫Ω

(∆u)2dx −∫

Ω

1ε2

∆uW ′(u)dx +

∫Ω

12ε4

W ′(u)2dx.

=12

∫Ω

(∆u)2dx + Jα,β(u)

∫Ω

1ε2

∆uW ′(u)dx +

∫Ω

12ε4

W ′(u)2dx − Jα,β(u)

withJα,β(u) =

α

2

∫Ω|∇u|2dx +

β

2

∫Ω

u2dx.

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An unconditionally stable scheme

A new semi-implicit scheme

(un+1 − un)/δt = −∆2un+1 + α∆un+1 − βun+1

−∆

[αun +

1ε2

W ′(un)

]+

[βun −

W ′′(un)

ε2

(∆un −

1ε2

W ′(un)

)]Unconditionally stability as soon as

α >2ε2

sups|W ′′(s)| and β >

1ε4

sups

|W ′′(s)|2 + |W (3)(s)|

√2W(s)

.

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Validation of this numerical method

Willmore flow of an initial circle with radius equals to R0 :

R(t) =(R4

0 + 2t)1/4

.

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Validation of this numerical method

fig: Two smooth evolutions by Willmore flow ; a Clifford’s torus and aLawson-Kusner surface

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Union of two disjoint circles

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Comparison phase field // parametric [Dziuk-2008]

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Other experiments

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Inclusion-exclusion boundary constraints

Minimization of

JΩ1,Ω2(Ω) =

J(Ω) if Ω1 ⊂ Ω ⊂ Ω2

+∞ otherwise,

for two given smooth sets Ω1 and Ω2 with dist(∂Ω1, ∂Ω2) > 0.

Ω1Ω(t)Ω2

c

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Perimeter and Dirichlet boundary conditions :

Dirichlet Cahn Hilliard energy approximation

Pε,Ω1,Ω2(u) =

Ω2\Ω1

(ε2 |∇u|2 + 1

ε W(u))dx if u ∈ XΩ1,Ω2

+∞ otherwise.

withXΩ1,Ω2 =

u ∈ H1(Ω2 \ Ω1) ; u|∂Ω1 = 1 , u|∂Ω2 = 0

,

Γ-convergence [Chambolle Bourdin] of Pε,Ω1,Ω2 to cW Pε,Ω1,Ω2 in theL1(Rd) topology :

Order of convergence ? But dist(∂Ω, ∂Ω1) > ε and dist(∂Ω, ∂Ω2) > ε !

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Dirichlet boundary conditions :

Allen Cahn equation with boundary Dirichlet conditionsut = 4u − 1

ε2 W ′(u), on Ω2 \ Ω1

u|∂Ω1 = 1, u|∂Ω2 = 0

u(0, x) = u0 ∈ XΩ1,Ω2 .

Numerical scheme : Implicit Euler scheme in time and finite elementsdiscretization in space.

A numerical experiment with Freefem++

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An other penalized Cahn Hilliard energy

Penalized Cahn Hilliard Energy

Pε,u1,ε ,u2,ε (u) =

∫ε|∇u|2 + 1

ε W ′(u)dx if u1,ε ≤ u ≤ 1 − u2,ε

+∞ otherwise,,

where u1,ε and u2,ε are defined by

u1,ε = q(dist(x,Ω1)

ε

)and u2,ε = q

(dist(x,Ω2)

ε

)Γ-convergence of Pε,u1,ε ,u2,ε to cW PΩ1,Ω2 ?Yes, slightly adaptation of Modica-Mortola proof !

Order of convergence ? but ∂Ω can now touch ∂Ω1 and ∂Ω2 !

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Numerical experiments

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5t = 6.1035e−05

Ω ∂ Ω

1

∂ Ω2

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5t = 0.022278

Ω ∂ Ω

1

∂ Ω2

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5t = 0.033386

Ω ∂ Ω

1

∂ Ω2

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5t = 0.055603

Ω ∂ Ω

1

∂ Ω2

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Numerical experiment : example of minimal surface

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Case of Willmore energy

Example of application : regularization of discrete contours

Ω1

Ω2

Find the set Ω∗ such as

Ω∗ = arg minΩ1⊂Ω⊂Ωc

2

W(Ω), withW(Ω) =

∫∂Ω

H2dHn−1

where Ω1 and Ω2 are two given set such as Ω1 ⊂ Ωc2

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Phase field versus

Cahn Hilliard Energy

Wε,u1,ε ,u2,ε (u) =

12ε

∫ (ε∆u − W ′(u)

ε

)2dx if u1,ε ≤ u ≤ 1 − u2,ε

+∞ otherwise,,

where u1,ε and u2,ε are defined by

u1,ε = q(dist(x,Ω1)

ε

)and u2,ε = q

(dist(x,Ω2)

ε

)Γ-convergence ofWε,Ω1,Ω2 to cWWΩ1,Ω2 ?

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Numerical experiments

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

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Case of thin constraints

Find the set Ω∗ as a minimizer of

JΩ1,Ω2(Ω) =

J(Ω) if Ω1 ⊂ Ω ⊂ Ωc2

+∞ otherwise,

with Ω1 = Ω2 = ∅.

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About semi-continuity of PΩ1,Ω2 in L1-topology

Note that PΩ1,Ω2 is not lowersemi-continuous

PΩ1,Ω2(Ω) =

P(Ω) if Ω1 ⊂ Ω ⊂ Ωc2

+∞ otherwise

hΩΩ

Relaxation of the penalized perimeter

PΩ1,Ω2(Ω) = inflim inf PΩ1,Ω2(Ωh), ∂Ωh ∈ C2, Ωh → Ω in L1(Ω).

Identification of PΩ1,Ω2 ?

PΩ1,Ω2(Ω) = P(Ω) + 2Hn−1(Ω0 ∩ Ω1) + 2Hn−1(Ω1 ∩ Ω2)

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About penalized Cahn Hilliard energy

Penalized Cahn Hilliard Energy

Pε,u1,ε ,u2,ε (u) =

∫ε|∇u|2 + 1

ε W ′(u)dx if u1,ε ≤ u ≤ 1 − u2,ε

+∞ otherwise,,

where u1,ε and u2,ε are defined by

u1,ε = q(dist(x,Ω1)

ε

)and u2,ε = q

(dist(x,Ω2)

ε

)Γ-convergence of Pε,u1,ε ,u2,ε to cW P∗Ω2,Ω2

where

P∗Ω1,Ω2(Ω) = P(Ω) +Hn−1(Ω0 ∩ Ω1) +Hn−1(Ω1 ∩ Ω2)

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With thickened constraints

Use some thickened constraints

εα

Γ-convergence of Pε,ue1,ε ,u

e2,ε

to cW PΩ2,Ω2 !

−0.5 0 0.5−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

−0.5 0 0.5−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

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3D numerical experiments

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Case of Willmore energy

Note thatWΩ1,Ω2 is not lower semi-continuous

WΩ1,Ω2(Ω) =

W(Ω) if Ω1 ⊂ Ω ⊂ Ωc2

+∞ otherwise

Relaxation of the penalized Willmore energy

WΩ1,Ω2(Ω) = inflim infWΩ1,Ω2(Ωh), ∂Ωh ∈ C2, Ωh → Ω in L1(Ω).

Identification ofWΩ1,Ω2 ?

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Phase field approximation

Phase field approximation

Wε,u1,ε ,u2,ε (u) =

Wε(u) if u1,ε ≤ u ≤ 1 − u2,ε

+∞ otherwise,

with ui,ε = q(

dist(x,Ωi)ε

).

Γ-convergence ofWε,u1,ε ,u2,ε to cWWΩ1,Ω2 ?

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Numerical experiments

−0.5 0 0.5−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

−0.5 0 0.5−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

−0.5 0 0.5−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

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Numerical experiments

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Perspectives : general inclusion-exclusion constraints

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Perspectives : multiphase reconstruction