Algorithms for anisotropic mean curvature flow of networks

Threshold dynamics is a very efficient algorithm for moving an interface (e.g. a surface in 3D) by mean curvature motion. It was proposed by Merriman, Bence, and Osher in 1989, and also extended to networks of surfaces in the same paper. This dynamics arises as gradient flow for the sum of the areas of the surfaces in the network, and plays a prominent role in materials science applications where it describes the motion of grain boundaries in polycrystals (such as most metals) under heat treatment.

Further extension of the algorithm to weighted mean curvature flow of networks, where the surface tension of each interface in the network may be distinct (unequal) and may depend on the direction of the normal, is of great interest for applications, but has remained elusive. We describe how to extend threshold dynamics to unequal and anisotropic (normal dependent) surface tensions. Joint work with Matt Elsey and Felix Otto.