Efficient flexible boundary conditions for long dislocations

TL;DR

This paper introduces an efficient hierarchical matrix-based implementation of the flexible boundary condition (FBC) method, significantly reducing computational complexity and enabling large-scale atomistic simulations of dislocations with improved accuracy and robustness.

Contribution

The authors develop a hierarchical matrix approach for FBC, making it up to two orders of magnitude more efficient than existing methods for simulating dislocations.

Findings

FBC method is robust and easy-to-use.

Achieves up to 100x efficiency over the PAD method.

Enables large-scale atomistic simulations without spurious image effects.

Abstract

We present a novel efficient implementation of the flexible boundary condition (FBC) method, initially proposed by Sinclair et al., for large single-periodic problems. Efficiency is primarily achieved by constructing a hierarchical matrix ($\mathscr{H}$-matrix) representation of the periodic Green matrix, reducing the complexity for updating the boundary conditions of the atomistic problem from quadratic to almost linear in the number of pad atoms. In addition, our implementation is supported by various other tools from numerical analysis, such as a residual-based transformation of the boundary conditions to accelerate the convergence. We assess the method for a comprehensive set of examples, relevant for predicting mechanical properties, such as yield strength or ductility, including dislocation bow-out, dislocation-precipitate interaction, and dislocation cross-slip. The main result of our analysis is that the FBC method is robust, easy-to-use, and up to two orders of magnitude more efficient than the current state-of-the-art method for this class of problems, the periodic array of dislocations (PAD) method, in terms of the required number of per-atom force computations when both methods give similar accuracy. This opens new prospects for large-scale atomistic simulations - without having to worry about spurious image effects that plague classical boundary conditions.