Convolution-Generated Motion and Generalized Huygens' Principles for Interface Motion

A physical interface can often be modeled as a surface that moves with a velocity determined by the local geometry. Accordingly, there is great interest in algorithms that generate such geometric interface motion. In this paper we unify and generalize two simple algorithms for constant and mean curvature based interface motion: the classical Huygens' principle and diffusion-generated motion. We show that the resulting generalization can be viewed both geometrically as a type of Huygens' principle and algebraically as a convolution-generated motion. Using the geometric-algebraic duality from the unification, we construct specific convolution-generated motion algorithms for a common class of anisotropic, curvature-dependent motionlaws. We validate these algorithms with numerical experiments and show that they can be implemented accurately and efficiently with adaptive resolution and fast Fourier transform techniques.

[1]  E. Müller,et al.  The Use of Classical Macroscopic Concepts in Surface Energy Problems , 1953 .

[2]  P. Pelcé.  Dynamics of curved fronts , 1988 .

[3]  J. Keller,et al.  Fast reaction, slow diffusion, and curve shortening , 1989 .

[4]  J. Keller,et al.  Reaction-diffusion processes and evolution to harmonic maps , 1989 .

[5]  G. Caginalp.  The Dynamics of a Conserved Phase Field System: Stefan-like, Hele-Shaw, and Cahn-Hilliard Models as Asymptotic Limits , 1990 .

[6]  Bruce J. MacLennan,et al.  Continuous Spatial Automata , 1990 .

[7]  L. Bronsard,et al.  Motion by mean curvature as the singular limit of Ginzburg-Landau dynamics , 1991 .

[8]  P. Souganidis,et al.  Phase Transitions and Generalized Motion by Mean Curvature , 1992 .

[9]  J. Rubinstein,et al.  Nonlocal reaction−diffusion equations and nucleation , 1992 .

[10]  J. Sethian,et al.  Crystal growth and dendritic solidification , 1992 .

[11]  J. Taylor,et al.  II—mean curvature and weighted mean curvature , 1992 .

[12]  L. Bronsard,et al.  On three-phase boundary motion and the singular limit of a vector-valued Ginzburg-Landau equation , 1993 .

[13]  L. Evans.  Convergence of an algorithm for mean curvature motion , 1993 .

[14]  D. Griffeath,et al.  Threshold growth dynamics , 1993, patt-sol/9303004.

[15]  S. Osher,et al.  Motion of multiple junctions: a level set approach , 1994 .

[16]  G. Barles,et al.  A Simple Proof of Convergence for an Approximation Scheme for Computing Motions by Mean Curvature , 1995 .

[17]  G. Beylkin.  On the Fast Fourier Transform of Functions with Singularities , 1995 .

[18]  Steven J. Ruuth.  An algorithm for generating motion by mean curvature , 1996 .

[19]  T. Chan,et al.  A Variational Level Set Approach to Multiphase Motion , 1996 .

[20]  Jin Yao,et al.  On the dynamics of multi-dimensional detonation , 1996, Journal of Fluid Mechanics.

[21]  L. Bronsard,et al.  Volume-preserving mean curvature flow as a limit of a nonlocal Ginzburg-Landau equation , 1997 .

[22]  S. Osher,et al.  THE WULFF SHAPE AS THE ASYMPTOTIC LIMIT OF A GROWING CRYSTALLINE INTERFACE , 1997 .

[23]  Steven J. Ruuth.  Eecient Algorithms for Diiusion-generated Motion by Mean Curvature , 1997 .

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

[25]  Steven J. Ruuth.  A Diffusion-Generated Approach to Multiphase Motion , 1998 .

[26]  Steven J. Ruuth,et al.  Convolution-Generated Motion as a Link between Cellular Automata and Continuum Pattern Dynamics , 1999 .