Phase field method for mean curvature flow with boundary constraints

This paper is concerned with the numerical approximation of mean curvature flow t → Ω(t) satisfying an additional inclusion-exclusion constraint Ω1 ⊂ Ω(t) ⊂ Ω2. Classical phase field model to approximate these evolving interfaces consists in solving the Allen-Cahn equation with Dirichlet boundary co...

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Bibliographic Details
Published in:ESAIM. Mathematical modelling and numerical analysis 2012-11, Vol.46 (6), p.1509-1526
Main Authors: Bretin, Elie, Perrier, Valerie
Format: Article
Language:English
Subjects:
35B
35K
49Q
Allen Cahn equation
Analysis of PDEs
Approximation
Boundaries
Boundary conditions
boundary constraints
Curvature
Differential geometry
Dirichlet problem
Exact sciences and technology
Fourier splitting method
gamma-convergence
Geometry
Grain boundaries
Mathematical analysis
Mathematical models
Mathematics
mean curvature flow
Numerical Analysis
Numerical analysis. Scientific computation
Numerical approximation
penalization technique
Sciences and techniques of general use
Studies
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Summary:This paper is concerned with the numerical approximation of mean curvature flow t → Ω(t) satisfying an additional inclusion-exclusion constraint Ω1 ⊂ Ω(t) ⊂ Ω2. Classical phase field model to approximate these evolving interfaces consists in solving the Allen-Cahn equation with Dirichlet boundary conditions. In this work, we introduce a new phase field model, which can be viewed as an Allen Cahn equation with a penalized double well potential. We first justify this method by a Γ-convergence result and then show some numerical comparisons of these two different models.
ISSN:0764-583X
1290-3841
DOI:10.1051/m2an/2012014