Multipole expansion of dislocation interactions: Application to discrete dislocations

The interaction between dislocations is long ranged and anisotropic, leading to severe computational challenges for large-scale simulations. Here we present a multipolar expansion for the interaction energy between dislocations in three dimensions, based on a straightforward expansion of some familiar energy expressions. An evaluation of the needed line integrals is conveniently performed by numerical integration over parametric representations of the dislocation loops. We show that the multipole expansion converges very quickly, leading to potential savings in computational time for evaluating dislocation interactions.