In this talk I will present a novel stable and efficient version 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 ($\mathcal{H}$-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, the implementation is supported by various other tools from numerical analysis, such as a residual-based transformation of the boundary conditions to accelerate the convergence.

The method is assessed 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. It will be shown 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.

REFERENCES

[1] Hodapp, M. (2021). Analysis of a Sinclair-type domain decomposition solver for atomistic/continuum coupling. SIAM Multiscale Modeling & Simulation, 19(4), 1499-1537
[2] Hodapp, M. (2022). Efficient flexible boundary conditions for long dislocations. Accepted for publication in Communications in Computational Physics, arXiv preprint arXiv:2105.08798

ACKNOWLEDGEMENTS

Financial support from the Fonds National Suisse (project 191680) is highly acknowledged.