# Selim Esedoglu : Algorithms for anisotropic mean curvature flow of networks, with applications to materials science

Motion by mean curvature for a network of surfaces arises in many applications. An important example is the evolution of microstructure in a polycrystalline material under heat treatment. Most metals and ceramics are of this type: They consist of many small single-crystal pieces of differing orientation, called grains, that are stuck together. A famous model proposed by Mullins in the 60s describes the dynamics of the network of surfaces that separate neighboring grains from one another in such a material as gradient descent for a weighted sum of the (possibly anisotropic) areas of the surfaces. The resulting dynamics is motion by weighted mean curvature for the surfaces in the network, together with certain conditions that need to be satisfied at junctions along which three or more surfaces may intersect. Typically, many topological changes occur during the evolution, as grains shrink and disappear, pinch off, or junctions collide. A very elegant algorithm -- known as threshold dynamics -- for the motion by mean curvature of a surface was given by Merriman, Bence, and Osher: It generates the whole evolution simply by alternating two very simple operations: convolution with a Gaussian kernel, and thresholding. It also works for networks, provided that all surfaces in the network have isotropic surface energies with equal weights. Its correct extension to the more general setting of unequal weights and possibly anisotropic (normal dependent) surface energies remained elusive, despite keen interest in this setting from materials scientists. In joint work with Felix Otto, we give a variational formulation of the original threshold dynamics algorithm by identifying a Lyapunov functional for it. In turn, the variational formulation shows how to extend the algorithm correctly to the more general settings that are of interest for materials scientists (joint work with Felix Otto and Matt Elsey). Examples of how to use the new algorithms to investigate unsettled questions about grain size distribution and its evolution will also be given.

**Category**: Applied Math and Analysis**Duration**: 01:34:50**Date**: February 9, 2015 at 4:25 PM**Views**: 105-
**Tags:**seminar, Applied Math And Analysis Seminar

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