Large-scale simulations of microstructural evolution in elastically stressed solids

We present a set of numerical methods for simulations of microstructural evolution in elastically stressed solids. We combine three powerful tools to achieve computational efficiency: the boundary integral method to provide excellent resolution, the fast multipole method to reduce computational cost, and the small-scale decomposition technique with a two-level time-stepping scheme to remove the stiffness from the time advance. Although we apply these methods to study the topic of our interest, the details of how they are implemented can be useful in many other situations. We extend the fast multipole method to calculate the anisotropic stress field in periodic two-dimensional domains and in doing so address issues associated with the conditional convergence of the summations. In addition, we introduce a new formula for the potential in periodically arranged two-dimensional cells in the absence of an applied field through a summation in physical space without using the Ewald sum. Furthermore, we implement a time-stepping scheme that enables us to speed up the calculation by an additional factor of 100 over a straightforward implementation of the small-scale decomposition technique. The computational complexity scales as the number of mesh points N, and thus we are able to employ N ∼ 500, 000 in a typical calculation.