Algorithms for Multiphase Motion with Applications to Materials Science.

The evolution of a collection of phases under multiphase motion by mean curvature has important applications, particularly in the modeling of microstructural development in polycrystalline materials. Models for the phenomena of grain growth and recrystallization require that the interface separating two phases moves with a normal velocity given by an affine function of the mean curvature of the interface. This work describes a class of algorithms for simulating motions of this type. These algorithms are called "distance function-based diffusion-generated motion" (DFDGM) algorithms. They are related to the threshold dynamics algorithm of Merriman, Bence, and Osher, and also share similarities with the level sets method of Zhao, et al. An idea of Almgren, Taylor, and Wang is applied to allow the DFDGM algorithms to be extended to motion by weighted mean curvature, the first distance function-based diffusion-generated motion-type algorithm to simulate this motion while correctly enforcing the natural boundary conditions at triple junctions. The DFDGM algorithms are demonstrated to be very accurate and very efficient in numerical tests. The efficiency of the algorithms allows for very large, well-resolved simulations to be performed. Large-scale simulations of isotropic grain growth and recrystallization in both two and three dimensions are presented and analyzed. The well-resolved simulations of isotropic grain growth are the largest simulations performed to date, containing over 130,000 grains initially in three dimensions. They are shown to agree well with theoretical predictions, experimental results, and prior simulations. The model of Srolovitz, et al. is used for the simulations of recrystallization. New analysis elucidating important model properties and making predictions for the type of microstructure resulting from parameter choices in the nucleation model is presented. It is shown that the DFDGM algorithm agrees with the analysis where other simulations have failed. This new analysis is also used to determine parameter choices which result in a final microstructure consisting of many very elongated grains, unlike any simulation results previously seen.Ph.D.Applied and Interdisciplinary MathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/86426/1/melsey_1.pd