Simulation of mean curvature flows on surfaces of revolution (Q2313437)

This article describes a numeric simulation for the normalized mean curvature flow of a closed surface of revolution. At first, the author constructs a finite element discretization of the flow equation for arbitrary triangulated surfaces. The actual simulation is performed on a triangulation derived from an icosahedral subdivision of the sphere. If this is done naively, the resulting schemes will be instable, even in case of triangulated spheres. This problem is overcome by an equidistant redistribution of triangle vertices along certain ``characteristic polylines'' in each iteration. Examples demonstrate stability of the modified scheme and agree with known theoretical results on smooth limit shapes. Flow simulation for dumbbell shapes shows dependence on the profile curve: The author observes convergence to a sphere as well as the development of a singularity.