An efficient algorithm for accurately simulating curvature flow for large networks of curves in two dimensions and surfaces in three dimensions on uniform grids is proposed. This motion arises in the technologically important problem of simulating grain boundary motion in polycrystalline materials. In this formulation grains are zero level sets of signed distance functions. Curvature motion is achieved by first diffusing the signed distance functions followed by a reinitialization step. A technique is devised to allow a single signed distance function to represent many grains allowing the simulation of a very large number (hundreds of thousands) of grains using modest computational hardware.

An accurate and efficient algorithm, closely related to the level set method, is presented for the simulation of Mullins' model of grain growth with arbitrarily prescribed surface energies. The implicit representation of interfaces allows for seamless transitions through topological changes. Well-resolved large-scale simulations are presented, beginning with over 650,000 grains in two dimensions and 64,000 grains in three dimensions. The evolution of the misorientation distribution function (MDF) is computed, starting from random and fiber crystallographic textures with Read-Shockley surface energies. Prior work had established that with random texture the MDF shows little change as the grain network coarsened whereas with fiber texture the MDF concentrates near zero misorientation. The lack of concentration about zero of the MDF in the random texture case has not been satisfactorily explained previously since this concentration would decrease the energy of the system. In this study, very large-scale simulations confirm these previous studies. However, computations with a larger cutoff for the Read-Shockley energies and an affine surface energy show a greater tendency for the MDF to concentrate near small misorientations. This suggests that the reason the previous studies had observed little change in the MDF is kinetic in nature.In addition, patterns of similarly oriented grains are observed to form as the MDF concentrates.

We study extensions of Merriman, Bence, and Osher’s threshold dynamics scheme to weighted mean curvature flow, which arises as gradient descent for anisotropic (normal dependent) surface energies. In particular, we investigate, in both two and three dimensions, those anisotropies for which the convolution kernel in the scheme can be chosen to be positive and/or to possess a positive Fourier transform. We provide a complete, geometric characterization of such anisotropies. This has implications for the unconditional stability and, in the two-phase setting, the monotonicity, of the scheme. We also revisit previous constructions of convolution kernels from a variational perspective, and propose a new one. The variational perspective differentiates between the normal dependent mobility and surface tension factors (both of which contribute to the normal speed) that results from a given convolution kernel. This more granular understanding is particularly useful in the multiphase setting, where junctions are present.

Diffusion-generated motion is used to perform a very large-scale simulation of normal grain growth in three dimensions with high accuracy. The method is based on the diffusion of signed distance functions and shares similarities with level-set methods. The Herring-angle condition at junctions and topological transitions are naturally captured with this formulation. This approach offers significant advantages over existing numerical methods and allows for accurate computations on scales not previously possible. A fully resolved simulation of normal grain growth, initially containing over 130 000 grains in three dimensions, is presented and analysed. It is shown that the average grain radius grows as the square root of time and the grain-size distribution is self-similar. Good agreement with other theoretical predictions, experimental results and simulation results via other techniques is also demonstrated.