Multiphase mean curvature flows approximation : the case of non harmonically additive mobilities

ABSTRACT. This paper concerns the robust approximation of multi-phase mean curvature flow by phase fields even when the phase mobility are highly contrasted. Recent work suggested that harmonically additive mobilities could be incorporated in the metric of the associated gradient flow. We generalize this approach to arbitrary mobilities, by splitting them as a sum of a harmonically additive mobilities. We establish the consistency of the resulting method, by analyzing the sharp interface limit of the flow : a formal expansion of the phase field shows that the method is of order 2. Finally, we present some numerical experiments in dimensions 2 and 3 that illustrate the interest of our method, in particular in the modeling of flows in which some of the phases have 0 mobility.

[1]  K. Takasao.  Convergence of Landau-Lifshitz equation to multi-phase Brakke's mean curvature flow. , 2018, 1804.08805.

[2]  Paul C. Fife,et al.  Dynamics of Layered Interfaces Arising from Phase Boundaries , 1988 .

[3]  É. Oudet,et al.  Phase-field modelling and computing for a large number of phases , 2019, ESAIM: Mathematical Modelling and Numerical Analysis.

[4]  Elie Bretin,et al.  Multiphase mean curvature flows with high mobility contrasts: A phase-field approach, with applications to nanowires , 2017, J. Comput. Phys..

[5]  Harald Garcke,et al.  A MultiPhase Field Concept: Numerical Simulations of Moving Phase Boundaries and Multiple Junctions , 1999, SIAM J. Appl. Math..

[6]  Jeffrey W. Bullard,et al.  Computational and mathematical models of microstructural evolution , 1998 .

[7]  Steven J. Ruuth.  Efficient Algorithms for Diffusion-Generated Motion by Mean Curvature , 1998 .

[8]  P. Loreti,et al.  Propagation of fronts in a nonlinear fourth order equation , 2000, European Journal of Applied Mathematics.

[9]  Michelle Schatzman,et al.  Geometrical evolution of developed interfaces , 1995, Emerging applications in free boundary problems.

[10]  Elie Bretin,et al.  A modified phase field approximation for mean curvature flow with conservation of the volume , 2009, 0904.0098.

[11]  Xinfu Chen,et al.  Generation and propagation of interfaces for reaction-diffusion equations , 1992 .

[12]  L. Ambrosio,et al.  Geometric evolution problems, distance function and viscosity solutions , 1997 .

[13]  A. Chambolle,et al.  Consistency result for a non monotone scheme for anisotropic mean curvature flow , 2010, 1005.4794.

[14]  S. Baldo.  Minimal interface criterion for phase transitions in mixtures of Cahn-Hilliard fluids , 1990 .

[15]  Robert L. Pego,et al.  Front migration in the nonlinear Cahn-Hilliard equation , 1989, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[16]  A. Karma,et al.  Phase-field model of vapor-liquid-solid nanowire growth , 2017, 1710.09460.

[17]  Maurizio Paolini,et al.  Quasi-optimal error estimates for the mean curvature flow with a forcing term , 1995, Differential and Integral Equations.

[18]  Édouard Oudet,et al.  Approximation of Partitions of Least Perimeter by Γ-Convergence: Around Kelvin’s Conjecture , 2011, Exp. Math..

[19]  J. Cahn,et al.  A microscopic theory for antiphase boundary motion and its application to antiphase domain coasening , 1979 .

[20]  P. Bassanini,et al.  Elliptic Partial Differential Equations of Second Order , 1997 .

[21]  Harald Garcke,et al.  A phase-field approach for wetting phenomena of multiphase droplets on solid surfaces. , 2014, Langmuir : the ACS journal of surfaces and colloids.

[22]  S. Masnou,et al.  A new phase field model for inhomogeneous minimal partitions, and applications to droplets dynamics , 2017 .

[23]  John W. Cahn,et al.  Critical point wetting , 1977 .

[24]  Jie Shen,et al.  Second-order Convex Splitting Schemes for Gradient Flows with Ehrlich-Schwoebel Type Energy: Application to Thin Film Epitaxy , 2012, SIAM J. Numer. Anal..

[25]  G. Bellettini,et al.  Lecture Notes on Mean Curvature Flow: Barriers and Singular Perturbations , 2014 .

[26]  Antonin Chambolle,et al.  Computing the Equilibrium Configuration of Epitaxially Strained Crystalline Films , 2002, SIAM J. Appl. Math..

[27]  Harald Garcke,et al.  On anisotropic order parameter models for multi-phase system and their sharp interface limits , 1998 .

[28]  W. Mullins.  Two‐Dimensional Motion of Idealized Grain Boundaries , 1956 .

[29]  Chert,et al.  Applications of semi-implicit Fourier-spectral method to phase field equations , 2004 .