Interfaces and Free Boundaries 1 , ( 1999 ) 107 123 Motion by intrinsic Laplacian of curvature

In this paper, we discuss numerical schemes to model the motion of curves and surfaces under the intrinsic Laplacian of curvature. This is an intrinsically dif"cult problem, due to the lack of a maximum principle and the delicate nature of computing an equation of motion which includes a fourth derivative term.We design and analyze a host of algorithms to try and follow motion under this #ow, and discuss the virtues and pitfalls of each. Synthesizing the results of these various algorithms, we provide a technique which is stable and handles very delicate motion in two and three dimensions. We apply this algorithm to problems of surface diffusion #ow, which is of value for problems in surface diffusion, metal re#ow in semiconductor manufacturing, sintering, and elastic membrane simulations. In addition, we provide examples of the extension of this technique to anisotropic diffusivity and surface energy which results in an anisotropic form of the equation of motion.

[1]  W. Mullins.  Theory of Thermal Grooving , 1957 .

[2]  J. Sethian.  AN ANALYSIS OF FLAME PROPAGATION , 1982 .

[3]  J. Sethian.  Curvature and the evolution of fronts , 1985 .

[4]  J. Sethian,et al.  Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations , 1988 .

[5]  James A. Sethian,et al.  Numerical Methods for Propagating Fronts , 1987 .

[6]  S. Osher,et al.  Algorithms Based on Hamilton-Jacobi Formulations , 1988 .

[7]  J. Sethian.  Numerical algorithms for propagating interfaces: Hamilton-Jacobi equations and conservation laws , 1990 .

[8]  Morton E. Gurtin,et al.  On the motion of a phase interface by surface diffusion , 1990 .

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

[10]  D. Chopp.  Flow under geodesic curvature , 1992 .

[11]  J. Sethian,et al.  Projection methods coupled to level set interface techniques , 1992 .

[12]  James A. Sethian,et al.  Flow under Curvature: Singularity Formation, Minimal Surfaces, and Geodesics , 1993, Exp. Math..

[13]  D. Chopp.  Computing Minimal Surfaces via Level Set Curvature Flow , 1993 .

[14]  Baba C. Vemuri,et al.  Front Propagation: A Framework for Topology Independent Shape Modeling and Recovery , 1994 .

[15]  Gal Alfred Vorst,et al.  Modelling and numerical simulation of viscous sintering , 1994 .

[16]  J. Taylor,et al.  Overview no. 113 surface motion by surface diffusion , 1994 .

[17]  J. Sethian,et al.  A Fast Level Set Method for Propagating Interfaces , 1995 .

[18]  J. Sethian,et al.  A level set approach to a unified model for etching, deposition, and lithography II: three-dimensional simulations , 1995 .

[19]  J. Sethian.  Level set methods : evolving interfaces in geometry, fluid mechanics, computer vision, and materials science , 1996 .

[20]  Charles M. Elliott,et al.  The Cahn–Hilliard equation with a concentration dependent mobility: motion by minus the Laplacian of the mean curvature , 1996, European Journal of Applied Mathematics.

[21]  Richard S. Falk,et al.  Space-Time Finite Element Methods for Surface Diffusion with Applications to the Theory of the Stability of Cylinders , 1996, SIAM J. Sci. Comput..

[22]  Scheffler,et al.  Ab initio calculations of energies and self-diffusion on flat and stepped surfaces of Al and their implications on crystal growth. , 1996, Physical review. B, Condensed matter.

[23]  J A Sethian,et al.  A fast marching level set method for monotonically advancing fronts. , 1996, Proceedings of the National Academy of Sciences of the United States of America.

[24]  Yoshikazu Giga,et al.  On pinching of curves moved by surface diffusion , 1997 .

[25]  L. M. Elliott,et al.  Existence Results for Di usive Surface Motion , 1997 .

[26]  George H. Gilmer,et al.  An atomistic simulator for thin film deposition in three dimensions , 1998 .

[27]  J. Escher,et al.  The surface diffusion flow for immersed hypersurfaces , 1998 .

[28]  James A. Sethian,et al.  The Fast Construction of Extension Velocities in Level Set Methods , 1999 .