Algorithms for Area Preserving Flows

Algorithms are proposed for computing certain area preserving geometric motions of curves, which are based on a new class of diffusion generated motion algorithms using signed distance functions. The convolution with the Gaussian kernel and construction of the distance function are alternated to generate the desired geometric flow in an unconditionally stable manner. The application of these area preserving flows to large scale simulation of coarsening are presented.

Motion by mean curvature has been extensively studied in mathematics literatures, like ‘Curvature Shortening Makes Convex Curves Circular’, ‘The Heat Equation Shrinking Convex Plane Curves’ and ‘Flow by Mean Curvature of Convex Surfaces into Spheres’. In this geometric flow case, each pointxon the curve moves with normal velocityuN x , in which x is the mean curvature. It is also called the Euclidean curve shortening flow.

In this paper, new schemes for area preserving flows based on a new algorithm which generates the desired interfacial motion by alternating construction of the signed distance function and convolution with a kernel are designed. In section 2, a brief description of previous diffusion generated motion based algorithm is given. In section 3, the new area preserving schemes are described. Applications are presented in section 4, and conclusions are made in section 5.

The algorithm proposed in this paper is motivated by the threshold dynamics idea proposed by Merriman, Bence and Osher, in the paper ‘Diffusion Generated Motion by Mean Curvature’, ‘Diffusion Generated Motion by Mean Curvature’, ‘Motion of Multiple Junctions: A Level Set Approach’. This algorithm is obtained by time-splitting the Allen-Cahn phase-field equation for motion by mean curvature.

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