We consider the problem of microstructural evolution in binary alloys in two dimensions. The microstructure consists of arbitrarily shaped precipitates embedded in a matrix. Both the precipitates and the matrix are taken to be elastically anisotropic, with different elastic constants. The interfacial energy at the precipitate-matrix interfaces is also taken to be anisotropic. This is an extension of the inhomogeneous isotrpic problem considered by H.-J. Jou et al. (1997, J. Comput. Phys. 131, 109). Evolution occurs via diffusion among the precipitates such that the total (elastic plus interfacial) energy decreases; this is accounted for by a modified Gibbs-Thomson boundary condition at the interfaces. The coupled diffusion and elasticity equations are reformulated using boundary integrals. An efficient preconditioner for the elasticity problem is developed based on a small scale analysis of the equations. The solution to the coupled elasticity-diffusion problem is implemented in parallel. Precipitate evolution is tracked by special non-stiff time stepping algorithms that guarantee agreement between physical and numerical equilibria. Results show that small elastic inhomogeneities in cubic systems can have a strong effect on precipitate evolution. For example, in systems where the elastic constants of the precipitates are smaller than those of the matrix, the particles move toward each other, where the rate of approach depends on the degree of inhomogeneity. Anisotropic surface energy can either enhance or reduce this effect, depending on the relative orientations of the anisotropies. Simulations of the evolution of multiple precipitates indicate that the elastic constants and surface energy control precipitate morphology and strongly influence nearest neighbor interactions. However, for the parameter ranges considered, the overall evolution of systems with large numbers of precipitates is primarily driven by the overall reduction in surface energy. Finally, we consider a problem related to the microstructure of fully orthotropic geological materials.
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It is a pleasure to thank I. Schmidt and P. Voorhees for stimulating and helpful discussions. The authors also acknowledge the support of the Minnesota Supercomputer Institute and the MCS division at Argonne National Laboratory for use of their computational facilities. In addition, P.H.L. was partially supported by the National Science Foundation Grant CMS-9503393. J.S.L. was partially supported by National Science Foundation Grants and the Sloan Foundation. Q.N. was partially supported by the Accelerated Strategic Computing Initiative Center (DOE) and Materials Research Center (NSF) at the University of Chicago. Finally, Q.N. thanks the Institute for Mathematics and Its Applications for its hospitality.