Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-28T09:00:52.906Z Has data issue: false hasContentIssue false
  • English
  • Français

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]
Get access

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

G. Alberti, Variational models for phase transitions, an approach via γ-convergence, in Calculus of variations and partial differential equations (Pisa, 1996). Springer, Berlin (2000) 95–114.
Allen, S.M. and Cahn, J.W., A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metall. 27 (1979) 1085–1095. Google Scholar
L. Almeida, A. Chambolle and M. Novaga, Mean curvature flow with obstacle. Technical Report Preprint (2011).
L. Ambrosio, Geometric evolution problems, distance function and viscosity solutions, in Calculus of variations and partial differential equations (Pisa, 1996). Springer, Berlin (2000) 5–93.
G. Barles, Solutions de viscosité des équations de Hamilton-Jacobi, in Mathématiques & Applications (Berlin) [Mathematics & Applications]. Springer-Verlag, Paris 17 (1994).
Barrett, J.W., Garcke, H. and Nürnberg, R., On the parametric finite element approximation of evolving hypersurfaces in r3. J. Comput. Phys. 227 (2008) 4281–4307. Google Scholar
Barrett, J.W., Garcke, H. and Nürnberg, R., A variational formulation of anisotropic geometric evolution equations in higher dimensions. Numer. Math. 109 (2008) 1–44. Google Scholar
Bates, P.W., Brown, S. and Han, J.L., Numerical analysis for a nonlocal Allen-Cahn equation. Int. J. Numer. Anal. Model. 6 (2009) 33–49. Google Scholar
Bellettini, G., Variational approximation of functionals with curvatures and related properties. J. Convex Anal. 4 (1997) 91–108. Google Scholar
Bellettini, G. and Paolini, M., Quasi-optimal error estimates for the mean curvature flow with a forcing term. Differ. Integral Equ. 8 (1995) 735–752. Google Scholar
Bellettini, G. and Paolini, M., Anisotropic motion by mean curvature in the context of Finsler geometry. Hokkaido Math. J. 25 (1996) 537–566. Google Scholar
Bourdin, B. and Chambolle, A., Design-dependent loads in topology optimization. ESAIM : COCV 9 (2003) 19–48. Google Scholar
M. Brassel, Instabilité de Forme en Croissance Cristalline. Ph.D. thesis, University Joseph Fourier, Grenoble (2008).
Brassel, M. and Bretin, E., A modified phase field approximation for mean curvature flow with conservation of the volume. Math. Meth. Appl. Sci. 34 (2011) 1157–1180. Google Scholar
E. Bretin, Méthode de champ de phase et mouvement par courbure moyenne. Ph.D. thesis, Institut National Polytechnique de Grenoble (2009).
Bueno-Orovio, A., Pérez-García, V.M. and Fenton, F.H., Spectral methods for partial differential equations in irregular domains : The spectral smoothed boundary method. SIAM J. Sci. Comput. 28 (2006) 886–900. Google Scholar
Chen, X., Generation and propagation of interfaces for reaction-diffusion equations. J. Differ. Equ. 96 (1992) 116–141. Google Scholar
Chen, L.Q. and Shen, J., Applications of semi-implicit Fourier-spectral method to phase field equations. Comput. Phys. Commun. 108 (1998) 147–158. Google Scholar
Chen, Y.G., Giga, Y. and Goto, S., Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations. Proc. Jpn Acad. Ser. A 65 (1989) 207–210. Google Scholar
Chen, X.F., Elliott, C.M., Gardiner, A. and Zhao, J.J., Convergence of numerical solutions to the Allen-Cahn equation. Appl. Anal. 69 (1998) 47–56. Google Scholar
Crandall, M.G., Ishii, H. and Lions, P.-L., User’s guide to viscosity solutions of second order partial differential equations. Bull Amer. Math. Soc. 27 (1992) 1–68. Google Scholar
Deckelnick, K. and Dziuk, G., Discrete anisotropic curvature flow of graphs. ESAIM : M2AN 33 (1999) 1203–1222. Google Scholar
Deckelnick, K., Dziuk, G. and Elliott, C.M., Computation of geometric partial differential equations and mean curvature flow. Acta Numer. 14 (2005) 139–232. Google Scholar
Evans, L.C. and Spruck, J., Motion of level sets by mean curvature I. J. Differ. Geom. 33 (1991) 635–681. Google Scholar
Evans, L.C., Soner, H.M. and Souganidis, P.E., Phase transitions and generalized motion by mean curvature. Commun. Pure Appl. Math. 45 (1992) 1097–1123. Google Scholar
Feng, X. and Prohl, A., Numerical analysis of the Allen-Cahn equation and approximation for mean curvature flows. Numer. Math. 94 (2003) 33–65. Google Scholar
Feng, X. and Prohl, A., Analysis of a fully discrete finite element method for the phase field model and approximation of its sharp interface limits. Math. Comput. 73 (2004) 541–567. Google Scholar
Feng, X. and Wu, H.-J., A posteriori error estimates and an adaptive finite element method for the Allen-Cahn equation and the mean curvature flow. J. Sci. Comput. 24 (2005) 121–146. Google Scholar
Li, Y., Lee, H.G., Jeong, D. and Kim, J., An unconditionally stable hybrid numerical method for solving the Allen-Cahn equation. Comput. Math. Appl. 60 (2010) 1591–1606. Google Scholar
Modica, L. and Mortola, S., Il limite nella Γ-convergenza di una famiglia di funzionali ellittici. Boll. Un. Mat. Ital. A 14 (1977) 526–529. Google Scholar
Modica, L. and Mortola, S., Un esempio di Γ − -convergenza. Boll. Un. Mat. Ital. B 14 (1977) 285–299. Google Scholar
S. Osher and R. Fedkiw, Level Set Methods and Dynamic Implicit Surfaces, Springer-Verlag, New York. Appl. Math. Sci. (2002).
S. Osher and N. Paragios, Geometric Level Set Methods in Imaging, Vision and Graphics. Springer-Verlag, New York (2003).
Osher, S. and Sethian, J.A., Fronts propagating with curvature-dependent speed : algorithms based on Hamilton-Jacobi formulations. J. Comput. Phys. 79 (1988) 12–49. Google Scholar
Owen, N.C., Rubinstein, J. and Sternberg, P., Minimizers and gradient flows for singularly perturbed bi-stable potentials with a dirichlet condition. Proc. R. Soc. London 429 (1990) 505–532. Google Scholar
M. Paolini, An efficient algorithm for computing anisotropic evolution by mean curvature, in Curvature flows and related topics, edited by Levico, 1994. Gakuto Int. Ser. Math. Sci. Appl. 5 (1995) 199–213.
Röger, M. and Schätzle, R., On a modified conjecture of De Giorgi. Math. Z. 254 (2006) 675–714. Google Scholar
Schätzle, R., Lower semicontinuity of the Willmore functional for currents. J. Differ. Geom. 81 (2009) 437–456. Google Scholar
Schätzle, R., The Willmore boundary problem. Calc. Var. Partial Differ. Equ. 37 (2010) 275–302. Google Scholar
Serfaty, S., Gamma-convergence of gradient flows on hilbert and metric spaces and applications. Disc. Cont. Dyn. Systems 31 (2011) 1427–1451. Google Scholar
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 (2011). Submitted.
Zhang, J. and Du, Q., Numerical studies of discrete approximations to the Allen-Cahn equation in the sharp interface limit. SIAM J. Sci. Comput. 31 (2009) 3042–3063. Google Scholar