Phase field method for mean curvature flow with boundary constraints

Elie Bretin; Valerie Perrier

ESAIM: Mathematical Modelling and Numerical Analysis (2012)

  • Volume: 46, Issue: 6, page 1509-1526
  • ISSN: 0764-583X

Abstract

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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.

How to cite

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Bretin, Elie, and Perrier, Valerie. "Phase field method for mean curvature flow with boundary constraints." ESAIM: Mathematical Modelling and Numerical Analysis 46.6 (2012): 1509-1526. <http://eudml.org/doc/276375>.

@article{Bretin2012,
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.},
author = {Bretin, Elie, Perrier, Valerie},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Allen Cahn equation; mean curvature flow; boundary constraints; penalization technique; gamma-convergence; Fourier splitting method; numerical tests; convergence},
language = {eng},
month = {6},
number = {6},
pages = {1509-1526},
publisher = {EDP Sciences},
title = {Phase field method for mean curvature flow with boundary constraints},
url = {http://eudml.org/doc/276375},
volume = {46},
year = {2012},
}

TY - JOUR
AU - Bretin, Elie
AU - Perrier, Valerie
TI - Phase field method for mean curvature flow with boundary constraints
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2012/6//
PB - EDP Sciences
VL - 46
IS - 6
SP - 1509
EP - 1526
AB - 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.
LA - eng
KW - Allen Cahn equation; mean curvature flow; boundary constraints; penalization technique; gamma-convergence; Fourier splitting method; numerical tests; convergence
UR - http://eudml.org/doc/276375
ER -

References

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