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Phase field method for mean curvature flow with boundary constraints

Published online by Cambridge University Press:  13 June 2012

Elie Bretin  and
Valerie Perrier
Elie Bretin
Affiliation:
CMAP, École Polytechnique, 91128 Palaiseau, France. [email protected]
Valerie Perrier
Affiliation:
Laboratoire Jean Kuntzmann, Université de Grenoble and CNRS, UMR 5224, B.P. 53, 38041 Grenoble Cedex 9, France; [email protected]
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Abstract

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.

Keywords

Allen Cahn equationmean curvature flowboundary constraintspenalization techniquegamma-convergenceFourier splitting method
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 46 , Issue 6 , November 2012 , pp. 1509 - 1526
Copyright
© EDP Sciences, SMAI, 2012

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