Elie BRETIN and Valerie PERRIER - imag.fr · Elie BRETIN1 and Valerie PERRIER2 Abstract. This paper...

Mathematical Modelling and Numerical Analysis Will be set by the publisherModélisation Mathématique et Analyse Numérique

PHASE FIELD METHOD FOR MEAN CURVATURE FLOW WITH BOUNDARYCONSTRAINTS

Elie BRETIN1 and Valerie PERRIER2

Abstract. This paper is concerned with the numerical approximation of mean curvature flowt → Ω(t) satisfying an additional inclusion-exclusion constraint Ω1 ⊂ Ω(t) ⊂ Ω2. Classical phasefield model to approximate these evolving interfaces consists in solving the Allen-Cahn equationwith Dirichlet boundary conditions. In this work, we introduce a new phase field model, which canbe viewed as an Allen Cahn equation with a penalized double well potential. We first justify thismethod by a Γ-convergence result and then show some numerical comparisons of these two differentmodels.

1991 Mathematics Subject Classification. 49 Q, 35 B, 35 K.

April 2011.

IntroductionIn the last decades, a lot of work has been devoted to the motion of interfaces, and particularly to mo-

tion by mean curvature. Applications concern image processing (denoising, segmentation), material sciences(motion of grain boundaries in alloys, crystal growth), biology (modeling of vesicles and blood cells), imagedenoising, image segmentation and motion of grain boundaries.

Let us introduce the general setting of mean curvature flows. Let Ω(t) ⊂ Rd, 0 ≤ t ≤ T , denote theevolution by mean curvature of a smooth bounded domain Ω0 = Ω(0) : the outward normal velocity Vn ata point x ∈ ∂Ω(t) is given by

Vn = κ, (1)where κ denotes the mean curvature at x, with the convention that κ is negative if the set is convex. We willconsider only smooth motions, which are well-defined if T is sufficiently small [4]. Since singularities maydevelop in finite time, one may need to consider the evolution in the sense of viscosity solutions [5, 26].

The evolution of Ω(t) is closely related to the minimization of the following energy:

J(Ω) =∫∂Ω

1 dσ.

Indeed, (1) can be viewed as a L2-gradient flow of this energy.There is an important literature on numerical methods for the mean curvature flows. These can be roughly

classified into three categories: Parametric methods [6,7,22,23], Level set formulations [20,27,34–36] or Phase

Keywords and phrases: Allen Cahn equation, Mean curvature flow, Boundary constraints, penalization technique, Gamma-convergence, Fourier splitting method1 CMAP, Ecole Polytechnique, 91128 Palaiseau, France, [email protected] Laboratoire Jean Kuntzmann, Université de Grenoble and CNRS, UMR 5224, B.P. 53, 38041 Grenoble Cedex 9, France,[email protected]

© EDP Sciences, SMAI 1999

2 TITLE WILL BE SET BY THE PUBLISHER

field approaches [11, 18, 33, 38]. See for instance [24] for a complete review and comparison beetween thesethree differents strategies.

In this work, we focus on phase field method. Following [32, 33], the functional J can be approximatedby a Ginzburg–Landau functional:

Jε(u) =∫Rd

(ε

2 |∇u|2 + 1

εW (u)

)dx.

where ε > 0 is a small parameter, andW is a double well potential with wells located at 0 and 1 (for exampleW (s) = 1

2s2(1− s)2).

Modica and Mortola [32,33] have shown the Γ-convergence of Jε to cWJ in L1(Rd) (see also [9]), where

cW =∫ 1

0

√2W (s)ds. (2)

The corresponding Allen–Cahn equation [2], obtained as the L2-gradient flow of Jε, reads

∂u

∂t= ∆u− 1

ε2W ′(u). (3)

Existence, uniqueness, and a comparison principle have been established for this equation (see for examplechapters 14 and 15 in [4]). To this equation, one usually associates the profile

q = arg min∫

R

(12γ′2 +W (γ)

); γ ∈ H1

loc(R), γ(−∞) = 1, γ(+∞) = 0, γ(0) = 12

(4)

Remark 1. The profile q (when W is continuous) can also be obtained [1] as the global decreasing solutionof the following Cauchy problem

q′(s) = −√W (s), s ∈ R

q(0) = 12 ,

and satisfies ∫R

(12q′(s)2 +W (q(s))

)=∫ 1

0

√2W (s)ds.

Then, the motion Ω(t) can be approximated by

Ωε(t) =x ∈ Rd ; uε(x, t) ≥

12

,

where uε is the solution of the Allen Cahn equation (3) with the initial condition

uε(x, 0) = q(d(x,Ω(0))

ε

).

Here d(x,Ω) denotes the signed distance of a point x to the set Ω.The convergence of ∂Ωε(t) to ∂Ω(t) has been proved for smooth motions [10, 18] and in the general case

without fattening [5, 25,26]. The convergence rate has been proved to behave as O(ε2|log ε|2).

Various numerical methods have been used to solve the Allen Cahn equation, for instance, finite differencemethods [8,19,31], or finite element methods [29,30,44]. From the practical point of view, one usually solvesthis equation is in a box Q, with periodic boundary conditions, which allows solutions to be computed viaa semi-implicit Fourier-spectral method as in the paper [17].

TITLE WILL BE SET BY THE PUBLISHER 3

In this article, we investigate the approximation of interfaces evolving in a restricted area, which usuallyoccurs in several physical applications. More precisely, we will consider mean curvature flow t → Ω(t)evolving as the L2 gradient flow of the following energy:

JΩ1,Ω2(Ω) =∫

∂Ω 1 dσ if Ω1 ⊂ Ω ⊂ Ω2,

+∞ otherwise.

Here Ω1 and Ω2 are two given smooth subsets of Rd such that dist(∂Ω1, ∂Ω2) > 0. These evolving interfaceswill clearly satisfy the following constraint Ω1 ⊂ Ω(t) ⊂ Ω2.

Ω

Ω

Ω

1

2

c

Figure 1. Constrained Mean curvature flow

To our knowledge, the only phase field model known to approximate these evolving interfaces considersthe Allen Cahn equation in Ω2 \ Ω1 with Dirichlet boundary condition on ∂Ω1 and ∂Ω2 [37]. Yet, somelimitations appear in this model:

• The Dirichlet boundary condition prevent interfaces to reach boundaries ∂Ω1 and ∂Ω2. This can beseen as a consequence of thickness of the interface layer which is about O(ε ln(ε)). This highlightsthe fact that the convergence rate of this model can not be better than O(ε ln(ε)).

• From a numerical point of view, the resolution of the Allen Cahn equation with Dirichlet boundarycondition can be performed using a finite element method [28]. This choice appears more flexibleto describe complex geometries, despite the computational cost associated with the construction ofappropriate solution grid, especially in dimension 3. More recently, Fourier spectral discretizationshave been introduced to overcome the limit choice of finite element methods in irregular domain [16,43] and to produce hight-order accuracy schemes. These techniques are based on smooth immersedinterfaces and a penalization approach which can incorporate a large variety of boundary conditions.Note that these constructions used penalized differential operators which are ill-conditioned andshould raise some instability issues in their numerical integration.

To overcome these limitations, we introduce in this paper a new phase field model. The main idea willbe to consider the Allen-Cahn equation in the whole domain with a penalization technique on the doublewell potential W to take into account the boundary constraints. Note that our approach is quite differentfrom [16, 43]. Indeed, we are not looking for some penalization technique to solve the Dirichlet Allen Cahnequation but for the interface sharp-limit of a variational problem when ε goes to zero.We then expect to obtain a better convergence order than O(ε ln(ε)) for our phase field approximation.Moreover, this phase field model will be numerically solved by spectral method [17] as in [16, 43]. However,our penalization terms act on the double well potentialW , which can be explicitly integrated without addingnumerical instabilities.

The paper is organized as follows:

In section 2, we present in detail the two phase field model. In section 3, we justify our penalized approachby a Γ-convergence result. In section 4, we compare our method to the classical Finite Element model of [37],


through numerical illustrations. These simulations will investigate the numerical convergence rate of eachmodel.

1. Phase field model with boundary constraintsIn this section, we will introduce our Allen Cahn model for the approximation of mean curvature flow

t→ Ω(t) evolving as the L2 gradient flow of the following energy

JΩ1,Ω2(Ω) =∫

∂Ω 1 dσ if Ω1 ⊂ Ω ⊂ Ω2

+∞ otherwise

where Ω1 and Ω2 are two given smooth subsets of Rd satisfying dist(∂Ω1, ∂Ω2) > 0. We begin with thedescription of the classical approach.

1.1. Classical model with Dirichlet boundary conditionsThe classical strategy, see for instance [12,37], consists in introducing the function space

XΩ1,Ω2 =u ∈ H1(Ω2 \ Ω1) ; u|∂Ω1 = 1 , u|∂Ω2 = 0

,

and a penalized Ginzburg-Landau energy of the form

Jε,Ω1,Ω2(u) =∫

Ω2\Ω1

(ε2 |∇u|

2 + 1εW (u)

)dx if u ∈ XΩ1,Ω2

+∞ otherwise.

In such framework, Chambolle and Bourdin [12] have shown the Γ-convergence of Jε,Ω1,Ω2 to cWJΩ1,Ω2 inL1(Rd) (cW has been introduced in (2)). This approximation conduces to the following Allen-Cahn equation

ut = 4u− 1ε2W

′(u), on Ω2 \ Ω1

u|∂Ω1 = 1, u|∂Ω2 = 0u(0, x) = u0 ∈ XΩ1,Ω2 .

A more general Γ-convergence result for the Allen-Cahn equation with Dirichlet boundary conditions canbe found in [37].

1.2. Novel approach with a penalized double well potentialNow, we describe an alternative approach to force the boundary constraints, based on a penalized double

well potential. From W considered in section 1.1, we define two continuous and positive potentials W1 andW2 satisfying the following assumption :

(H1)W1(s) = W (s) for s ≥ 1/2W1(s) ≥ max(W (s), λ) for s ≤ 1/2

andW2(s) = W (s) for s ≤ 1/2W2(s) ≥ max(W (s), λ) for s ≥ 1/2,

.

for a given constant λ > 0.

For α > 1, ε > 0, and x ∈ Rd, we introduce also a penalized double well potential Wε,Ω1,Ω2,α defined by

Wε,Ω1,Ω2,α(s, x) = W1(s) q(dist(x,Ω1)

εα

)+W2(s) q

(dist(x,Ωc2)

εα

)+W (s)

(1− q

(dist(x,Ω1)

εα

)− q

(dist(x,Ωc2)

εα

)), (5)


0 0.2 0.4 0.6 0.8 1 1.2

0

0.02

0.04

0.06

0.08

0.1

0.12

WW

1W

2

Figure 2. Example of potentials W , W1 and W2

where dist(x,Ω1) and dist(x,Ωc2) are respectively the signed distance functions to the sets Ω1 and Ωc2,and q is the profile function associated to W defined in (4).

Our modified Ginzburg-Landau energy Jε,Ω1,Ω2,α reads

Jε,Ω1,Ω2,α(u) =∫Rd

[ε

2 |∇u|2 + 1

εWε,Ω1,Ω2,α(u, x)

]dx. (6)

We will prove in the next section that this energy Γ-converges to cWJΩ1,Ω2 . The associated Allen-Cahnequation reads in this context:

∂tu = 4u− 1ε2∂sWε,Ω1,Ω2,α(u, x) (7)

2. Approximation result of the penalized Ginzburg-Landau energyIn this section, we prove the convergence of the Ginzburg-Landau energy Jε,Ω1,Ω2,α introduced in (6), to

the following penalized perimeter

JΩ1,Ω2(u) =|Du|(Rd) if u = 1lΩ and Ω1 ⊂ Ω ⊂ Ω2

+∞ otherwise.

Remark 2. Given u ∈ L1(Rd), |Du|(Rd) is defined by

|Du|(Rd) = sup∫

Rdu div(g)dx ; g ∈ C1

c (Rd,Rd),

where C1c (Rd;Rd) is the set of C1 vector functions from Rd to Rd with compact support in Rd. If u ∈

W 1,1(Rd), |Du| coincides with the L1-norm of ∇u and if u = 1lΩ where Ω has a smooth boundary, |Du|coincides with the perimeter of Ω. Moreover, u→ |Du|(Rd) is lower semi-continuous in L1(Rd) topology.

We assume in this section that Ω1 and Ω2 are two given smooth subsets of Rd satisfying dist(∂Ω1, ∂Ω2) > 0,and that ε is sufficiently small such that

1− q(dist(x,Ω1)

εα

)− q

(dist(x,Ωc2)

εα

)> 1/2, (8)

for all x in Ω2 \ Ω1.We now state the main result for the modified Ginzburg-Landau energy Jε,Ω1,Ω2,α :


Theorem 1. Assume that W is a positive double-well potential with wells located at 0 and 1, continuouson R and such that W (s) = 0 if and only if s ∈ 0, 1. Assume also that W1 and W2 are two continuouspotentials satisfying assumption (H1). Then, for any α > 1, it holds

Γ− limε→0

Jε,Ω1,Ω2,α = cWJΩ1,Ω2 in L1(Rd).

Proof. We first prove the liminf inequality.i) Liminf inequality :Let (uε) converge to u in L1(Rd). As Jε,Ω1,Ω2,α ≥ 0, it is not restrictive to assume that the lim inf ofJε,Ω1,Ω2(uε) is finite. So we can extract a subsequence un = uεn such that

limn→+∞

Jεh,Ω1,Ω2,α(un) = lim infε→0

Jε,Ω1,Ω2,α(uε) ∈ R+.

Note that from remark (1) and assumption (8), it holds

q(dist(x,Ω1)

εα

)≥ 1/2 for x ∈ Ω1,

q(dist(x,Ωc2)

εα

)≥ 1/2 for x ∈ Ωc2,

1− q(dist(x,Ω1)

εα

)− q

(dist(x,Ωc2)

εα

)≥ 1/2 for x ∈ Ω2 \ Ω1,

1− q(dist(x,Ω1)

εα

)− q

(dist(x,Ωc2)

εα

)≥ 0 for x ∈ Rd,

for ε sufficiently small.This implies that ∫

Ω1

W1(un)dx ≤∫

Ω1

2q(dist(x,Ω1)

εαn

)W1(un)dx

≤ 2∫RdWεn,Ω1,Ω2,α(un, x)dx

≤ 2εhJεn,Ω1,Ω2,α(un).

In the same way :∫Rd\Ω2

W2(un)dx ≤ 2εnJεn,Ω1,Ω2,α(un) and∫

Ω2\Ω1

W (un)dx ≤ 2εnJεn,Ω1,Ω2,α(un).

At the limit n→∞, the Fatou’s Lemma and the continuity of W , W1 and W2 imply that∫

Ω1W1(u)dx = 0,∫

Rd\Ω2W2(u)dx = 0 and

∫Ω2\Ω1

W (u)dx = 0. Recall also that W , W1 and W2, vanish respectively ats = 0, 1, s = 1 and s = 0. This means that

u(x) ∈

1 a.e in Ω1

0 a.e in Rd \ Ω2

0, 1 a.e in Ω2 \ Ω1,

almost everywhere. Hence, we can represent u by 1lΩ for some Borel set Ω ∈ Rd satisfying Ω1 ⊂ Ω ⊂ Ω2.Using the Cauchy inequality, we can estimate

Jεn,Ω1,Ω2,α(un) ≥∫Rd

[εn|∇uh|2

2 + 1εnW (un)

]dx ( because W1 ≥W and W2 ≥W )

≥∫Rd

[εn|∇un|2

2 + 1εnW (un)

]dx ( where W (s) = min

W (s) ; sup

s∈[0,1]W (s)

)

≥∫Rd

√2W (un)|∇un|dx =

∫Rd|∇[φ(un)]|dx = |D[φ(un)]|(Rd),


where φ(s) =∫ s

0

√2W (t)dt. Since φ is a Lipschitz function (because W is bounded), φ(uε) converges in

L1(Rd) to φ(u). Using the lower semicontinuity of v → |Dv|(Rd), we obtain

limn→+∞

Jεn,Ω1,Ω2(un) ≥ lim infn→+∞

|Dφ(un)|(Rd) ≥ |Dφ(u)|(Rd).

The lim inf inequality is finally obtained remarking that φ(u) = φ(1lΩ) = cW 1lΩ = cWu.

Let us now prove the limsup inequality.ii) Limsup inequality :We first assume that u = 1lΩ for some bounded open set Ω satisfying Ω1 ⊂ Ω ⊂ Ω2 with smooth boundaries.We introduce the sequence

uε(x) = q

(dist(x,Ω)

ε

).

and two constants c1 and c2 defined by

c1 = sups∈[0,1]

W1(s)−W (s) , and c2 = sups∈[0,1]

W2(s)−W (s) .

Note that

Jε,Ω1,Ω2,α(uε) =∫Rd

[ε|∇uε|2

2 + 1εW (uε)

]dx+

∫Rd

1εq

(dist(x,Ω1)

εα

)(W1(uε)−W (uε)) dx

+∫Rd

1εq

(dist(x,Ωc2)

εα

)(W2(uε)−W (uε)) dx

Each of these 3 terms above is now analyzed.

1) Estimation of the first term :

I1ε =

∫Rd

[ε|∇uε|2

2 + 1εW (uε)

]dx.

By co-area formula, we estimate

I1ε = 1

ε

∫Rd

[q′(d(x,Ω)/ε)2

2 +W (q(d(x,Ω)/ε))]dx

= 1ε

∫Rg(s)

[q′(s/ε)2

2 +W (q(s/ε))]ds

=∫Rg(εt)

[q′(t)2

2 +W (q(t))]dt

where g(s) = |D1ld≤s|(Rd).By the smoothness of ∂Ω, g(εt) converges to |D1ldist(x,Ω)≤0|(Rd) as ε → 0; moreover, by definition of theprofile q, uε converges to 1lΩ and

lim supε→0

I1ε ≤ |D1lΩ|(Rd)

∫ +∞

−∞

[12 |q′(s)|2 +W (q(s))ds

].

According to remark (1), it follows that∫ +∞

−∞

[12 |q′(s)|2 +W (q(s))

]ds =

∫ 1

0

√2W (s)ds = cW ,


which implies that

lim supε→0

I1ε ≤ cW |D1lΩ|(Rd).

2) Estimation of the second term :

I2ε =

∫Rd

1εq

(dist(x,Ω1)

εα

)(W1(uε)−W (uε)) dx.

The function dist(x,Ω) is negative on Ω1, thus uε(x) ≥ 12 on Ω1 and W1(uε(x)) = W (uε(x)) for all x ∈ Ω1.

This means that

I2ε =

∫Rd\Ω1

1εq

(dist(x,Ω1)

εα

)(W1(uε)−W (uε)) dx ≤ c1

∫Rd\Ω1

1εq

(dist(x,Ω1)

εα

)dx,

where c1 = sups∈[0,1] W1(s)−W (s).Using co-area formula, we estimate∫

Rd\Ω1

1εq

(dist(x,Ω1)

εα

)dx =

∫ ∞0

1εg1(s)q

( sεα

)ds = εα−1

∫ ∞0

g1(εαs)q(s)ds,

where g1(s) = |D1ldist(x,Ω1)≤s|(Rd).By the smoothness of Ω1, g(εαt) converges to |D1ldist(x,Ω1)≤0|(Rd) as ε→ 0. We then deduce that

lim supε→0

I2ε = 0,

as α > 1 and∫∞

0 q(s)ds is bounded.

3) Estimation of the last term :

I3ε =

∫Rd

1εq

(dist(x,Ωc2)

εα

)(W2(uε)−W (uε)) dx.

This estimation is similar to the second one. The function dist(x,Ω) is positive on Rd \ Ω2, this meansuε(x) ≤ 1

2 on Rd \ Ω2 and W2(uε(x)) = W (uε(x)) for all x ∈ Rd \ Ω2. Then, we have

I3ε =

∫Ω2

1εq

(dist(x,Ωc2)

εα

)(W2(uε)−W (uε)) dx ≤ c2

∫Ω2

1εq

(dist(x,Ωc2)

εα

)dx,

and using co-area formula, it holds∫Ω2

1εq

(dist(x,Ωc2)

εα

)dx =

∫ ∞0

1εg2(s)q

( sεα

)ds = εα−1

∫ ∞0

g2(εαs)q(s)ds,

where g2(s) = |D1ldist(x,Ω2)≤−s|(Rd). We deduce as before that

lim supε→0

I3ε = 0.

Finally, we conclude thatlim supε→0

Jε,Ω1,Ω2,α(uε) ≤ cW |D1lΩ|(Rd).


Remark 3. This theorem is still true in the limit case α→∞, where Jε,Ω1,Ω2,α=∞(u) reads

Jε,Ω1,Ω2,∞(u) =∫

Ω1

[ε|∇u|2

2 + 1εW1(u)

]dx+

∫Ω2\Ω1

[ε|∇u|2

2 + 1εW (u)

]dx

+∫Rd\Ω2

[ε|∇u|2

2 + 1εW2(u)

].

2.1. Some remarks about sharp interface limit of the gradient flowWe have just established a connection between the penalized perimeter JΩ1,Ω2 and its smooth approxima-

tion Jε,Ω1,Ω2,α. In section 3, we will introduce a numerical scheme associated to the penalized Allen Cahnequation (7) to approximate the mean curvature flow with obstacle defined as the gradient flow of JΩ1,Ω2 .

A first remark concerns the geometric evolution of a mean curvature flow with obstacle Ω(t) ⊂ Rd,0 ≤ t ≤ T , from a smooth domain Ω0 (Ω1 ⊂ Ω0 ⊂ Ω2). The outward normal velocity at point x ∈ ∂Ω(t)satisfies for regular interfaces

Vn(x) =

κ(x) if x ∈ Ω2 \ Ω1

maxκ(x), 0 if x ∈ ∂Ω1

minκ(x), 0 if x ∈ ∂Ω2,

where κ denotes the mean curvature at the interface. Some recent results about existence, unicity and C1,1

regularity of such flow have been established in [3] for smooth obstacles. Notice also that in this case, Vnis discontinuous on ∂Ω(t), then the classical viscosity theory [21] does no apply. The existence of motion ingeneral sense (when singularities appear) is still an open issue to our knowledge.

A second remark concerns the gradient flow of Jε,Ω1,Ω2,α defined by

ut = 4u− 1ε2∂sWε,Ω1,Ω2,α(u, x).

Since the potential (s, x)→Wε,Ω1,Ω2,α(s, x) is smooth, the proof of existence, unicity and comparison prin-ciple can be easily derived from the original Allen Cahn equation properties.

Finally, the global convergence result of the gradient flow of Jε,Ω1,Ω2,α to JΩ1,Ω2 is a difficult problem forwhich classical approaches can not apply directly.

A first way would be to used the recent theory of Serfaty [42] which insures the Γ-convergence of thegradient flow associated to a family of energy. In the case of Allen Cahn equation, the assumptions neededin this theory are relied to the De Giorgi conjecture about the Gamma-convergence of

Fε(u) = ε

∫Rd

(∆u− 1

ε2W ′(u)

)2dx

to Willmore energyF (Ω) =

∫∂Ωκ2dσ(x).

This conjecture has been recently established [39–41], but only in the case of C2 interfaces, which does notcorrespond to the regularity of mean curvature flow with obstacle.

Other approaches [10,18] should be more suitable but need a deeper analysis. Indeed, the proofs are basedon a comparison principle which is still satisfied by our modified Allen Cahn equation, and an explicit con-struction of a subsolution and supsolution. Unfortunately, such last constructions also need a C2 regularityof the interface.


3. Algorithms and numerical simulationsWe now compare numerically the two phase field models described in section 1. The first and classical

model is integrated by a semi-implicit finite element method whereas our penalized Allen Cahn equation issolved by the semi-implicit Fourier spectral algorithm. In particular, we will observe that both approachesgive similar solutions but, the convergence rate of the phase field approximation appears to behave as aboutO(ε ln(ε)) for the Dirichlet model and as O(ε2 ln(ε)2) (when α is sufficiently large) for our penalized versionof Allen Cahn equation.

3.1. A semi-implicit finite element method for the Allen Cahn equation with Dirichletboundary conditions

Let us give more precision about the classic semi-implicit finite element method used for the equation

ut(x, t) = ∆u(x, t)− 1ε2W ′(u)(x, t), on Ω2 \ Ω1 × [0, T ], (9)

where u|∂Ω1 = 1, u|∂Ω2 = 0 and W (s) = 12s

2(1− s)2.

Note that when the initial condition u0 is chosen of the form u0 = q (dist(Ω0, x)/ε) with Ω0 satisfying theconstraint Ω1 ⊂ Ω0 ⊂ Ω2, then we expect that the set Ωε(t) defined by

Ωε(t) = Ω1 ∪ x ∈ Ω2 \ Ω1 ; u(x, t) ≥ 1/2 ,

should be a good approximation to the constrained mean curvature flow t→ Ω(t).

Let us introduce a triangulation mesh Th on the set Ω2 \Ω1 and the discretization time step δt. Then, weconsider the approximation spaces Xh,0 and Xh defined by

Xh =v ∈ H1(Ω2 \ Ω1) ∪ C0(Ω2 \ Ω1) ; v|K∈Th ∈ Pk(K), v|Ω1 = 1 and v|Ω2 = 0

Xh,0 =

v ∈ H1

0 (Ω2 \ Ω1) ∪ C1(Ω2 \ Ω1) ; v|K∈Th ∈ P2(K)

where Pk denotes the polynomial space of degree k. We take k = 2 in the future numerical illustrations.Then, the solution u(x, tn) at time tn = nδt is approximated by Uh,n, defined for n > 1 as the solution onXh of∫

Ω2\Ω1

Uh,nϕ dx+ δt

∫Ω2\Ω1

∇Uh,n∇ϕ dx =∫

Ω2\Ω1

(Uh,n−1 − δt

ε2W ′(Uh,n−1)

)ϕ dx, ∀ϕ ∈ Xh,0,

and for n = 0 byUh,0 = arg min

v∈Xh‖v − u0‖L2(Ω2\Ω1).

This algorithm is known to be stable under the condition

δt ≤ cW ε2,

where cW =[supt∈[0,1] W ′′(s)

]−1. More results about stability and convergence of finite element methods

for the resolution of the Allen Cahn equation can be found in [13,28–30,44].

3.2. A time-splitting Fourier spectral method for the penalized Allen-Cahn equationWe now consider the second model

ut(x, t) = 4u(x, t)− 1ε2∂sWε,Ω1,Ω2,α(u(x, t), x), on Q× [0, T ], (10)


with periodic boundary conditions on a given box Q, chosen sufficiently large to contain Ω2. In futurenumerical tests, we use α = 2, W (s) = 1

2s2(1− s)2 and the potentials W1,W2 are defined by

W1(s) =

12s

2(1− s)2 if s ≥ 12

10(s− 0.5)4 + 1/32 otherwiseand W2(s) =

12s

2(1− s)2 if s ≤ 12

10(s− 0.5)4 + 1/32 otherwise ,

which clearly satisfy the assumption (H1) (see figure (2)).The initial condition u0 satisfies u0 = q (dist(Ω0, x)/ε) and we will show that the set

Ωε(t) = x ∈ Q ; u(x, t) ≥ 1/2 ,

will be a good approximation of Ω(t) as ε tends to zero.

Our numerical scheme for solving equation (10) is based on a splitting method between the diffusion andreaction terms. We take advantage of the periodicity of u by integrating exactly the diffusion term in theFourier space. More precisely, the solution u(x, tn) at time tn = t0 + nδt is approximated by its truncatedFourier series :

unP (x) =∑|p|∞=P

cnpe2iπp·x.

Here |p|∞ = max1≤i≤d |pi| and P represents the number of Fourier modes in each direction.

The step n of our algorithm writes :

• un+1/2P (x) =

∑cn+1/2p e2iπp·x, with c

n+1/2p = cnp e

−4π2δt |p|2 .

• un+1P = u

n+1/2P − δt

ε2 ∂sWε,Ω2,Ω1,α(un+1/2P , x).

In practice, the first step is performed via a Fast Fourier Transform, with a computational costO(P d ln(P )).In the simplest case of traditional Allen Cahn equation, the corresponding numerical scheme turns out to bestable under the condition

δt ≤ cW ε2,

and the convergence of this splitting approach has been established in [15]. In practice, the parameter ε hasto be chosen great than the spatial discretization step δx = 1/P .

3.3. Simulations and numerical convergenceWe compare in this part numerical solutions obtained with these two algorithms. For each test, we took

ε = 2−8 and δt = ε2. The P2 finite element algorithm was implemented in Freefem++. The mesh Th used inthese simulations are plotted in figure (3). The penalization method was implemented in MATLAB wherewe have taken P = 28.

We first plot two situations on figures (4) and (5). The functions uε are plotted only on the admissibleset Ω2 \ Ω1 for the FE Dirichlet method and on all the set Q for the pseudo-spectral penalization versionwith α = 2. We note that the solutions obtained by both methods are very similar.

In order to estimate the convergence rate of both models, we consider the case where Ω1 and Ω2 are twocircles of radii equal to R1 = 0.3 and R2 = 0.4. The situation is thus very simple when the initial set Ω0is also a circle with radius R0 satisfying R1 < R0 < R2. Indeed, the penalized mean curvature motion Ω(t)evolves as a circle, with radius satisfying

R(t) = max(√

R20 − 2t, R1

),


Figure 3. Mesh Th generated by Freefem++, and used in simulations plotted on figures(4) and (5). In all cases, ∂Ω1 and ∂Ω2 are respectively identified as the green and the yellowboundaries.

t = 1.5259e−05

−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

0.5

t = 0.022781

−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

0.5

t = 0.033386

−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

0.5

t = 0.055603

−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

0.5

Figure 4. Numerical solutions obtained at different times t0 = 0, t1 = 0.022, t2 = 0.033and t = 0.055. The first line corresponds to the FE Dirichlet method and the second line tothe penalization pseudo-spectral approach.

t = 1.5259e−05

−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

0.5

t = 0.0055695

−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

0.5

t = 0.0083466

−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

0.5

t = 0.016678

−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

0.5

Figure 5. Numerical solutions obtained at different times t0 = 0, t1 = 0.0055, t2 = 0.0083and t3 = 0.016. The first line corresponds to the FE Dirichlet method and the second lineto the penalization pseudo-spectral approach.


that decreases until R(t) = R1.

The solutions of the two different models are computed for different values of ε with P = 28, δt = 1/P 2

and R0 = 0.35. In both cases, the set Ωε(t) appears as a circle of radius Rε(t). We then estimate thenumerical error between Rε(t) and R(t). The results obtained for the first method are plotted on figure (6):the first figure corresponds to the evolution t→ Rε(t) for 4 different values of ε and the second figure showsthe error

ε→ supt∈[0,T ]

|R(t)−Rε(t)|,

in logarithmic scale, compared to curves of slope√ε, ε and ε ln(ε) . It clearly appears an error of O(ε ln(ε)).

The same test is done for the penalization algorithm with α = 2: the results are plotted on figure (7) andwe now clearly observed a convergence rate of O(ε2 ln(ε2)).

Remark 4 (About violation of the constraint Ω1 ⊂ Ω ⊂ Ω2).We remark that with our penalized Allen Cahn approach, the constraint Ω1 ⊂ Ω ⊂ Ω2 is always violated.Indeed, figure (7) left shows that for a given ε, the radii of the circle Ω(t) converge in time to a value lowerthan the radius of Ω1. In fact, this error on the constraint appears only with an order of O(ε2 ln(ε2)) as εtends to zero. Moreover, the approximation of mean curvature flow by Allen Cahn equation is also of orderO(ε2 ln(ε2)) [10, 18]. This means that the precision on the constraint Ω1 ⊂ Ω ⊂ Ω2 do not affect the initialprecision of the approximation of phase field method.

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04

0.29

0.3

0.31

0.32

0.33

0.34

0.35

0.36

0.37

t

R(t

)

exactε= 1/Nε= 2/Nε= 3/Nε= 4/N

−5.6 −5.4 −5.2 −5 −4.8 −4.6 −4.4 −4.2 −4 −3.8−4.6

−4.4

−4.2

−4

−3.8

−3.6

−3.4

−3.2

−3

ε

Err

eur(

ε) =

|R(∞

) −

Rε (∞

)|

erreur(ε)ε ln(ε)εε1/2

Figure 6. Dirichlet algorithm : numerical error |Rε(t)−R(t)| ; Left: t→ Rε(t) for differentvalues of ε; Right: ε → supt∈[0,T ] |R(t)−Rε(t)| in logarithmic scale compared to curvesof slope

√ε (red), ε (green) and ε ln(ε) (blue).

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04

0.29

0.3

0.31

0.32

0.33

0.34

0.35

0.36

t

R(t

)

exactε= 1/Nε= 2/Nε= 3/Nε= 4/N

−5.6 −5.4 −5.2 −5 −4.8 −4.6 −4.4 −4.2 −4 −3.8−7.5

−7

−6.5

−6

−5.5

−5

−4.5

ε

Err

eur(

ε) =

|R(∞

) −

Rε (∞

)|

erreur(ε)

ε2 ln2(ε)

ε2

ε

Figure 7. Penalization algorithm with α = 2: numerical error |Rε(t) − R(t)| ; Left:t → Rε(t) for different values of ε; Right: ε → supt∈[0,T ] |R(t)−Rε(t)| in logarithmicscale compared to curves of slope ε (red), ε2 (green) and ε2 ln(ε)2 (blue).


In the following last test, we numerically analyze the impact of the coefficient α on our penalized AllenCahn equation. As above, we consider the case where Ω1 and Ω2 are two circles of radii equal to R1 = 0.3and R2 = 0.4. We consider as initial set Ω0 = Ω1. The exact penalized mean curvature motion Ω(t) is thenequal to Ω1 for all t > 0.

Figure (8) shows experiment results obtained with α = 1. We clearly observe an error on the localizationwhich does not decrease as ε tends to zero. This is consistent with our theoretical analysis for which α isassumed to be strictly greater than 1. Figure (9) presents the numerical errors for different values of alpha.We observe that the convergence rate is not affected by the value of α, when α is chosen sufficiently far than1: α = 1.5, 1.6, 1.75, 2, 50. In practice, we could have chosen α = +∞, but this corresponds to use adiscontinuous potential (s, x)→Wε,Ω1,Ω2,+∞(s, x) in variable x.

Figure 8. Penalization algorithm with α = 1: numerical error |Rε(t) − R(t)| ; Left: t →Rε(t) for different values of ε; Right: ε → supt∈[0,T ] |R(t)−Rε(t)| in logarithmic scale,compared to curves of slope ε (red), ε2 (green) and ε2 ln(ε)2 (blue).

Figure 9. Penalization algorithm, numerical error ε → supt∈[0,T ] |R(t)−Rε(t)| in loga-rithmic scale, for different values of α: α = 1.5, 1.6, 1.75, 2, 50 .

Moreover, our approach allows us to simulate very easily and efficiently three dimensional experiments,whatever the geometry of the sets Ω1 and Ω2. See for instance figure (10) where the numerical solution ofthe Allen Cahn equation is plotted for different times t.

3.4. Possible extension of the methodAnother advantage of our penalization approach is that it can be easily extended for more general sit-

uations of evolving interfaces. For example, we have recently considered a mean curvature flow with anadditional forcing term g and a conservation of the volume in [14]. Then, the model of [14] can be modified


Figure 10. Minimal surface estimation : solution of the Allen Cahn equation at differenttimes t. The domain Ω1 is the union of the two red tori and Ω2 is the box Q with N = 27,ε = 1/N and δt = ε2

to take into account additional inclusion-exclusion constraints by simply using potential WΩ1,Ω2 instead ofW in phase field equation. In this case, this leads to the following perturbed Allen-Cahn equation

ut = ∆u− 1ε2F (u),

with

F (u) = W ′Ω1,Ω2(u)− εg

√2WΩ1,Ω2(u)−

∫QW ′Ω1,Ω2

(u)− εg√

2WΩ1,Ω2(u)dx∫Q

√2WΩ1,Ω2(u)dx

√2WΩ1,Ω2(u).

Two simulations obtained from this model are plotted in figure (11). We can observe that both constraints(conservation of the volume and inclusion-exclusion set) are well respected.

Figure 11. Two numerical experiments with a forcing term (gravity force), a volume con-servation and an exclusion constraint : the domain Ω1 is empty and Ω2 is plotted in red ineach picture. We used N = 27, ε = 1/N and δt = ε2.

4. ConclusionThis paper has presented a new phase field model for the approximation of mean curvature flow with

inclusion-exclusion constraints. The classical method in such a situation is to solve the Allen-Cahn equation


with Dirichlet boundary conditions. Since this method appears to be not optimal while its convergence isobserved with a rate about O(ε ln(ε)) only, we have introduced a new approach based on a penalized doublewell potential. This method was firstly motivated by a Γ-convergence result and secondly numerical testssuggesting that its convergence rate is about O(ε2 ln(ε)2). The proof of the numerical precision of the schemeis still an open problem and will be the subject of a forthcoming paper. Another advantage of our methodlies in its simplicity to be implemented, since it is only based on Fourier Transforms. This simplicity allowsone to consider 3D Geometries, and to elaborate new strategies in more general situations, such as meancurvature flow with forcing term and conservation of volume.

References[1] G. Alberti. Variational models for phase transitions, an approach via γ-convergence. In Calculus of variations and partial

differential equations (Pisa, 1996), pages 95–114. Springer, Berlin, 2000.[2] S. M. Allen and J. W. Cahn. A microscopic theory for antiphase boundary motion and its application to antiphase domain

coarsening. Acta Metall., 27:1085–1095, 1979.[3] L. Almeida, A. Chambolle, and M. Novaga. Mean curvature flow with obstacle. Technical Report Preprint, 20011.[4] L. Ambrosio. Geometric evolution problems, distance function and viscosity solutions. In Calculus of variations and partial

differential equations (Pisa, 1996), pages 5–93. Springer, Berlin, 2000.[5] G. Barles. Solutions de viscosité des équations de Hamilton-Jacobi, volume 17 of Mathématiques & Applications (Berlin)

[Mathematics & Applications]. Springer-Verlag, Paris, 1994.[6] J. W. Barrett, H. Garcke, and R. Nürnberg. On the parametric finite element approximation of evolving hypersurfaces in

r3. J. Comput. Phys., 227:4281–4307, April 2008.[7] J. W. Barrett, H. Garcke, and R. Nürnberg. A variational formulation of anisotropic geometric evolution equations in

higher dimensions. Numer. Math., 109:1–44, February 2008.[8] P. Bates, S. Brown, and J. Han. Numerical analysis for a nonlocal allen-cahn equation. Int. J. Numer. Anal. Model.,

6:33–49, 2009.[9] G. Bellettini. Variational approximation of functionals with curvatures and related properties. J. Convex Anal., 4(1):91–108,

1997.[10] G. Bellettini and M. Paolini. Quasi-optimal error estimates for the mean curvature flow with a forcing term. Differential

Integral Equations, 8(4):735–752, 1995.[11] G. Bellettini and M. Paolini. Anisotropic motion by mean curvature in the context of Finsler geometry. Hokkaido Math.

J., 25:537–566, 1996.[12] B. Bourdin and A. Chambolle. Design-dependent loads in topology optimization. ESAIM: Control, Optimisation and

Calculus of Variations, 9:19–48, 2003.[13] M. Brassel. Instabilité de Forme en Croissance Cristalline. PhD thesis, University Joseph Fourier, Grenoble, 2008.[14] M. Brassel and E. Bretin. A modified phase field approximation for mean curvature flow with conservation of the volume.

Rapport de recherche arXiv:0904.0098v1, LJK, 2009. submitted.[15] E. Bretin. Méthode de champ de phase et mouvement par courbure moyenne. PhD thesis, Institut national polytechnique

de Grenoble, 2009.[16] A. Bueno-Orovio, V. M. Pérez-García, and F. H. Fenton. Spectral methods for partial differential equations in irregular

domains: The spectral smoothed boundary method. SIAM J. Sci. Comput., 28:886–900, March 2006.[17] L. Chen and J. Shen. Applications of semi-implicit Fourier-spectral method to phase field equations. Computer Physics

Communications, 108:147–158, 1998.[18] X. Chen. Generation and propagation of interfaces for reaction-diffusion equations. J. Differential Equations, 96(1):116–

141, 1992.[19] X. Chen, C. Elliott, A. Gardiner, and J. Zhao. Convergence of numerical solutions to the allen-cahn equation. Appl. Anal.,

69:47–56, 1998.[20] Y. G. Chen, Y. Giga, and S. Goto. Uniqueness and existence of viscosity solutions of generalized mean curvature flow

equations. Proc. Japan Acad. Ser. A Math. Sci., 65(7):207–210, 1989.[21] M. G. Crandall, H. Ishii, and P.-L. Lions. User’s guide to viscosity solutions of second order partial differential equations.

Bulletin of the American Mathematical Society, 27(1):1–68, 1992.[22] K. Deckelnick and G. Dziuk. Discrete anisotropic curvature flow of graphs. M2AN Math. Model. Numer. Anal., 33:1203–

1222, 1999.[23] K. Deckelnick, G. Dziuk, and C. M. Elliott. Computation of geometric partial differential equations and mean curvature

flow. Acta Numer., 14:139–232, 2005.[24] K. Deckelnick, G. Dziuk, and C. M. Elliott. Computation of geometric partial differential equations and mean curvature

flow. Acta Numer., 14:139–232, 2005.[25] L. Evans, H. Soner, and P. Souganidis. Phase transitions and generalized motion by mean curvature. Comm. Pure Appl.

Math., 45:1097–1123, 1992.[26] L. C. Evans, H. M. Soner, and P. E. Souganidis. Phase transitions and generalized motion by mean curvature. Comm.

Pure Appl. Math., 45(9):1097–1123, 1992.[27] L. C. Evans and J. Spruck. Motion of level sets by mean curvature. I. J. Differential Geom., 33(3):635–681, 1991.


[28] X. Feng and A. Prohl. Numerical analysis of the Allen-Cahn equation and approximation for mean curvature flows.Numerische Mathematik, 94:33–65, 2003.

[29] X. Feng and A. Prohl. Analysis of a fully discrete finite element method for the phase field model and approximation ofits sharp interface limits. Math. Comput, pages 541–567, 2004.

[30] X. Feng and H.-J. Wu. A posteriori error estimates and an adaptive finite element method for the allen–cahn equation andthe mean curvature flow. J. Sci. Comput., 24:121–146, August 2005.

[31] Y. Li, H. G. Lee, D. Jeong, and J. Kim. An unconditionally stable hybrid numerical method for solving the allen–cahnequation. Computers Mathematics with Applications, 60(6):1591–1606, 2010.

[32] L. Modica and S. Mortola. Il limite nella Γ-convergenza di una famiglia di funzionali ellittici. Boll. Un. Mat. Ital. A (5),14(3):526–529, 1977.

[33] L. Modica and S. Mortola. Un esempio di Γ−-convergenza. Boll. Un. Mat. Ital. B (5), 14(1):285–299, 1977.[34] S. Osher and R. Fedkiw. Level Set Methods and Dynamic Implicit Surfaces. Springer-Verlag New York, Applied Mathe-

matical Sciences, 2002.[35] S. Osher and N. Paragios. Geometric Level Set Methods in Imaging, Vision and Graphics. Springer-Verlag, New York,

2003.[36] S. Osher and J. A. Sethian. Fronts propagating with curvature-dependent speed: algorithms based on hamilton-jacobi

formulations. J. Comput. Phys., 79:12–49, 1988.[37] N. C. Owen, J. Rubinstein, and P. Sternberg. Minimizers and gradient flows for singularly perturbed bi-stable potentials

with a dirichlet condition. Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences,429(1877):pp. 505–532, 1990.

[38] M. Paolini. An efficient algorithm for computing anisotropic evolution by mean curvature. In Curvature flows and relatedtopics (Levico, 1994), volume 5 of GAKUTO Internat. Ser. Math. Sci. Appl., pages 199–213. Gakkotosho, Tokyo, 1995.

[39] M. Röger and R. Schätzle. On a modified conjecture of De Giorgi. Math. Z., 254(4):675–714, 2006.[40] R. Schätzle. Lower semicontinuity of the Willmore functional for currents. J. Differential Geom., 81(2):437–456, 2009.[41] R. Schätzle. The Willmore boundary problem. Calc. Var. Partial Differential Equations, 37(3-4):275–302, 2010.[42] S. Serfaty. Gamma-convergence of gradient flows on hilbert and metric spaces and applications. Disc. Cont. Dyn. Systems,,

31(4):1427–1451, 2011.[43] H.-C. Y. Yu, H.-Y. Chen, and K. Thornton. Extended smoothed boundary method for solving partial differential equations

with general boundary conditions on complex boundaries. Technical Report arXiv:1107.5341v1, 20011. submitted.[44] J. Zhang and Q. Du. Numerical studies of discrete approximations to the allen-cahn equation in the sharp interface limit.

SIAM J. Sci. Comput., 31:3042–3063, July 2009.

Elie BRETIN and Valerie PERRIER - imag.fr · Elie BRETIN1 and Valerie PERRIER2 Abstract. This paper...

Documents

Transcript of Elie BRETIN and Valerie PERRIER - imag.fr · Elie BRETIN1 and Valerie PERRIER2 Abstract. This paper...