Threshold Dynamics for Networks with Arbitrary Surface Tensions

We present and study a new algorithm for simulating the N-phase mean curvature motion for an arbitrary set of (isotropic) N(N-1)/2 surface tensions. The departure point is the threshold dynamics algorithm of Merriman, Bence, and Osher for the two-phase case. A new energetic interpretation of this algorithm allows us to extend it in a natural way to the case of N phases, for arbitrary surface tensions and arbitrary (isotropic) mobilities. For a large class of surface tensions, the algorithm is shown to be consistent in the sense that at every time step, it decreases an energy functional that is an approximation (in the sense of Gamma convergence) of the interfacial energy. A broad range of numerical tests shows good convergence properties. An important application is the motion of grain boundaries in polycrystalline materials: It is also established that the above-mentioned large class of surface tensions contains the Read-Shockley surface tensions, both in the two-dimensional and three-dimensional settings.(c) 2015 Wiley Periodicals, Inc.