Fast algorithms for evaluating the stress field of dislocation lines in anisotropic elastic media

TL;DR

This paper compares four fast algorithms, including a new kernel-independent method, for efficiently calculating dislocation stress fields in anisotropic elastic media, improving computational speed in dislocation dynamics simulations.

Contribution

The paper introduces and evaluates a new Lagrange FMM algorithm for stress field calculations in anisotropic media, enhancing efficiency and memory usage over existing methods.

Findings

Spherical FMM is more efficient than Taylor FMM in isotropic media.

Chebyshev FMM is easy to apply in anisotropic media but has high memory demands.

Lagrange FMM offers a memory-efficient, black-box solution with good scalability.

Abstract

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.