Efficient Flexible Boundary Conditions for Long Dislocations (Communications in Computational Physics, 2022)
We present a novel efficient implementation of the flexible boundary condition (FBC) method, initially proposed by Sinclair et al., for large single-periodicproblems. 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 almostlinear in the number of pad atoms. In addition, our implementation is supported byvarious other tools from numerical analysis, such as a residual-based transformationof the boundary conditions to accelerate the convergence. We assess the method fora comprehensive set of examples, relevant for predicting mechanical properties, suchas yield strength or ductility, including dislocation bow-out, dislocation-precipitate interaction, and dislocation cross-slip. The main result of our analysis is that the FBCmethod is robust, easy-to-use, and up to two orders of magnitude more efficient thanthe current state-of-the-art method for this class of problems, the periodic array ofdislocations (PAD) method, in terms of the required number of per-atom force computations when both methods give similar accuracy. This opens new prospects forlarge-scale atomistic simulations — without having to worry about spurious imageeffects that plague classical boundary conditions.