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
1/60
Volume reconstruction from slices using phase fieldmethod
Elie Bretin (with F. Dayrens and S. Masnou)
Contrôle, Problème inverse et ApplicationsClermont-Ferrand
Sep. 2017
2/60
Motivation : Magnetic resonance Imaging
3/60
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
5/60
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
6/60
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 ]
7/60
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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1
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
ε = 0.0039062, δt = 1.5259e−05, N=256
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1
t = 0.0070038
ε = 0.0039062, δt = 1.5259e−05, N=256
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t = 0.020004
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
9/60
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),
10/60
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 !
11/60
Example of Willmore flow in dimension three
fig: Two smooth evolutions by Willmore flow ; a Clifford’s torus and aLawson-Kusner surface
12/60
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
13/60
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.
14/60
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|
15/60
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
16/60
Phase field method and Cahn Hilliard energy
Cahn Hilliard energy
Pε(u) =
∫Rd
(ε|∇u|2
2+
1ε
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
17/60
Cahn Hilliard energy:
Pε(u) =
∫ (ε|∇u|2
2+
1ε
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)
18/60
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)|
19/60
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)
20/60
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)
21/60
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
0
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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
23/60
Willmore Flow
Willmore Energy
W(Ω) =12
∫∂Ω
H2dHn−1,
L2 gradient flow
Vn = ∆SH + |A |2H −12
H3,
where |A |2 =∑κ2
i .
24/60
Classical approximation of Willmore energy
Phase field approximation
uε = q(dist(x,Ω)
ε
), with q′(s) = −
√2W(q(s)).
Remarks that
1ε
(ε∆uε −
1ε
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 −
1ε
W ′(uε))2
dx =⇒ε→0
cW12
∫∂Ω
H2dHn−1
25/60
De Giorgi conjecture : Γ-convergence of Wε ?
Definition ( Classical approximation of Willmore energy )
Wε(u) =12ε
∫Rd
(ε∆u −
1ε
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 !
26/60
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.
27/60
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 !
28/60
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
29/60
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.
30/60
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
42/60
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 !
43/60
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++
44/60
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
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0
0.1
0.2
0.3
0.4
0.5t = 6.1035e−05
Ω ∂ Ω
1
∂ Ω2
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−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
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−0.2
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0
0.1
0.2
0.3
0.4
0.5t = 0.033386
Ω ∂ Ω
1
∂ Ω2
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−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
47/60
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
48/60
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 ?
49/60
Numerical experiments
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0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
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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 = ∅.
51/60
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 !
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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
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Numerical experiments
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Perspectives : general inclusion-exclusion constraints
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Perspectives : multiphase reconstruction