An extension of the fast-multipole Greengard-Rokhlin method was applied to the case of long-range interactions between parallel edge (at arbitrary orientations) and screw dislocations. By finding complex potentials from which the stress terms could be calculated, and by expanding those potentials in multi-pole series, it was possible to convert a computationally difficult O(N2) problem into a much faster O(N) approach. In order to obtain sufficient numerical accuracy, only a few terms were needed in the multi-pole expansions (4 for screws and 6 for edges) so that the interactions between millions of dislocations could be calculated within a few minutes. As an example, results were presented for the relaxed configurations of 16,384 edge dislocations of arbitrary orientation.

O(N) Algorithm for Dislocation Dynamics. H.Y.Wang, R.LeSar: Philosophical Magazine A, 1995, 71[1], 149-64