  In dislocation dynamics (DD) simulations, the most computationally intensive
step is the evaluation of the elastic interaction forces among dislocation
ensembles. Because the pair-wise interaction between dislocations is
long-range, this force calculation step can be significantly accelerated by the
fast multipole method (FMM). We implemented and compared four different methods
in isotropic and anisotropic elastic media: one based on the Taylor series
expansion (Taylor FMM), one based on the spherical harmonics expansion
(Spherical FMM), one kernel-independent method based on the Chebyshev
interpolation (Chebyshev FMM), and a new kernel-independent method that we call
the Lagrange FMM. The Taylor FMM is an existing method, used in ParaDiS, one of
the most popular DD simulation softwares. The Spherical FMM employs a more
compact multipole representation than the Taylor FMM does and is thus more
efficient. However, both the Taylor FMM and the Spherical FMM are difficult to
derive in anisotropic elastic media because the interaction force is complex
and has no closed analytical formula. The Chebyshev FMM requires only being
able to evaluate the interaction between dislocations and thus can be applied
easily in anisotropic elastic media. But it has a relatively large memory
footprint, which limits its usage. The Lagrange FMM was designed to be a
memory-efficient black-box method. Various numerical experiments are presented
to demonstrate the convergence and the scalability of the four methods.
