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An Approach to the Mesoscale Simulation of Grain Growth David Kinderlehrer1, Irene Livshits1, Florin Manolache1, Anthony D. Rollett2, and Shlomo Ta'asan1 1 Department of Mathematical Sciences and 2 Department of Materials Science and Engineering Carnegie Mellon University Pittsburgh, PA 15238-3890 Abstract The simulation of curvature driven growth in grain boundary systems is becoming an important tool in understanding the behavior of microstructure evolution and there is much distinguished work in this subject. Here we address the mesoscale simulation of large systems of grain boundaries subject to the Mullins equation of curvature driven growth with the Herring force balance equation imposed at triple junctions. We discuss several novel features of our approach which we anticipate will render it a flexible, scalable, and robust tool to aid in microstructural prediction. What is the result of the simulation? We discuss what such a simulation is capable of predicting, taking as a prototype the histogram of relative area population as it changes through the simulation. We do not use this data to seek the best distribution, like Hillert, Rayleigh, or lognormal. Instead we treat the set of distributions as the solution of an inverse problem for a time varying function and determine the equation they satisfy. This results in a coarse graining of the complex simulation to simpler system governed by a Fokker-Planck Equation. Even so, fundamental questions concerning the predictability of simulations of large metastable systems arise from these considerations. 1. Introduction In this paper we discuss our results, preliminary to date, about large scale simulation of twodimensional grain growth. Our simulations are termed mesoscale because the evolution is based on thermodynamical theories of Mullins and Herring applicable at the scale of grain boundaries and grains. Here we focus on normal grain growth. There are two parts to this enterprise. The first is simply to simulate accurately curvature driven grain growth complete with the resolution of the boundary condition imposed at triple junctions. The technique we employ is novel since we directly solve the differential equations governing the evolution of the grain boundaries and have no need to discretize the grains themselves. This data structure is one dimensional. Frost et al. have introduced a similar simulation algorithm for curvature driven grain growth based on a numerical estimate of curvature on discretized boundary segments, [9]. For related work on direct verification of curvature driven grain growth using a front tracking model, see Demirel et al. [7]. Simulations of grain growth based on Monte Carlo techniques have their origin in [3],[18]. What is the result of the simulation? The second part of this effort is directed toward this question. Our objective in undertaking this program is to estimate mobility during grain growth by designing a simulation that can be calibrated to agree with experiment, cf. Adams et al. [1],[2]. If we wish to find agreement betwee