A Numerical Scheme for Generalized Peierls-Nabarro Model of Dislocations Based on the Fast Multipole Method and Iterative Grid Redistribution (Communications in Computational Physics, 2020)
Dislocations are line defects in crystalline materials. The Peierls-Nabarromodels are hybrid models that incorporate atomic structure of dislocation core intocontinuum framework. In this paper, we present a numerical method for a generalizedPeierls-Nabarro model for curved dislocations, based on the fast multipole methodand the iterative grid redistribution. The fast multipole method enables the calculationof the long-range elastic interaction within operations that scale linearly with thetotal number of grid points. The iterative grid redistribution places more mesh nodesin the regions around the dislocations than in the rest of the domain, thus increasesthe accuracy and efficiency. This numerical scheme improves the available numericalmethods in the literature in which the long-range elastic interactions are calculateddirectly from summations in the physical domains; and is more flexible to handleproblems with general boundary conditions compared with the previous FFT basedmethod which applies only under periodic boundary conditions. Numerical examplesusing this method on the core structures of dislocations in Al and Cu and in epitaxialthin films are presented.