Computing dislocation stress fields in anisotropic elastic media using fast multipole expansions

The calculation of stress fields due to dislocations and hence the forces they exert on each other is the most time consuming step in dislocation dynamics (DD) simulations. The fast multipole method (FMM) can reduce the computational cost at each simulation step from O(N-2) to O(N) for an ensemble of N dislocation segments. However, FMM has not yet been applied to three-dimensional DD simulations which take into account anisotropic elasticity. We demonstrate a systematic procedure to establish this capability by first obtaining the derivatives of the elastic Green's function to arbitrary order for a medium of general anisotropy. We then compute the stress field of a dislocation loop using multipole expansions based on these derivatives, and analyze the dependence of numerical errors on the expansion order. This method can be implemented in large scale DD simulations when the consideration of elastic anisotropy is necessary, for example the technologically important cases of iron and ferritic steels at high temperatures.