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OVERVIEW APPLICATIONS INTERACTIVE APPLETS HISTORY OF THE METHODS/FLOW CHART PUBLICATIONS EDUCATIONAL MATERIAL ACKNOWLEDGEMENTS ABOUT THE AUTHOR/CV Copyright: 1996, 1999, 2006 J.A. Sethian 
Applications to Geometry


AbstractIn many physical problems, interfaces move with a speed that depends on the local curvature. Some common examples are flame propagation, crystal growth, and oilwater boundaries. We idealize the front as a closed, nonintersecting, initial hypersurface flowing along its gradient field with a speed that depends on the curvature. Because explicit solutions seldom exist, numerical approximations are often used. In this paper, we review some recent work on algorithms for attacking these problems. We show that algorithms based on direct parameterizations of the moving front face considerable difficulties. This is because such algorithms adhere to local properties of the solution, rather than the global structure. Conversely, the global properties of the motion can be captured by embedding the surface in a higherdimensional function. In this setting, the equations of motion can be solved using numerical techniques borrowed from hyperbolic conservation laws. We apply the algorithms to a variety of complicated shapes, showing corner formation and breaking and merging, and conclude with a study of a dumbbell in #R sup 3# moving under its mean curvature. We follow the collapsing dumbbell as the handle pinches off, a singularity develops, and the front breaks into two separate surfaces.
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AbstractWe study hypersurfaces moving under flow that depends on the mean curvature. The approach is based on a numerical technique that embeds the evolving hypersurface as the zero level set of a family of evolving surfaces. In this setting, the resulting partial differential equation for the motion of the level set function $\phi$ may be solved by using numerical techniques borrowed from hyperbolic conservation laws. This technique is used to analyze a collection of problems. First we analyze the singularity produced by a dumbbell collapsing under its mean curvature and show that a multiarmed dumbbell develops a separate, residual closed interface at the center after the singularity forms. The level set approach is then used to generate a minimal surface attached to a onedimensional wire frame in three space dimensions. The minimal surface technique is extended to construct a surface of any prescribed function of the curvature attached to a given bounding frame. Finally, the level set idea is used to study the flow of curves on 2manifolds under geodesic curvature dependent speed.
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AbstractThe Fast Marching Method is a numerical algorithm for solving the Eikonal equation on a rectangular orthogonal mesh in O(M \log M) steps, where M is the total number of grid points. In this paper we extend the Fast Marching Method to triangulated domains with the same computational complexity. As an application, we provide an optimal time algorithm for computing the geodesic distances and thereby extracting shortest paths on triangulated manifolds.
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AbstractThe collapse of a dumbbell moving under its mean curvature has attracted considerable attention. A new class of numerical algorithms have been developed recently that can follow hypersurfaces propagating with curvaturedependent speed in any number of space dimensions. The essential idea behind these algorithms is to view the propagating hypersurface as a particular level set of a higher dimensional function. The motion of this higherdimensional function is described by a scalar HamiltonJacobi equation with parabolic righthandside. This equation may be easily solved using techniques borrowed from the solution to hyperbolic conservation laws. We demonstrate this technique applied to the problem of a dumbbell collapsing under its own mean curvature. Our results show the breaking of the handle, the developing singularity, and the collapse of the two separate sections.
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AbstractIn this paper, we discuss numerical schemes to model the motion of curves and surfaces under the intrinsic Laplacian of curvature. This is an intrinsically difficult 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 flow, 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 flow, which is of value for problems in surface diffusion, metal reflow in semiconductor manufacturing, sintering, and elastic membrane simulations. In addition, we provide examples of the extension of this technique to anistotropic diffusivity and surface enery which results in an anisotropic form of the equation of motion.
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