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Volume 32, Issue 3
Efficient Flexible Boundary Conditions for Long Dislocations

M. Hodapp

Commun. Comput. Phys., 32 (2022), pp. 671-714.

Published online: 2022-09

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  • 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.

  • AMS Subject Headings

65N55, 70C20, 65N80

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CiCP-32-671, author = {Hodapp , M.}, title = {Efficient Flexible Boundary Conditions for Long Dislocations}, journal = {Communications in Computational Physics}, year = {2022}, volume = {32}, number = {3}, pages = {671--714}, 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.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2021-0157}, url = {http://global-sci.org/intro/article_detail/cicp/21042.html} }
TY - JOUR T1 - Efficient Flexible Boundary Conditions for Long Dislocations AU - Hodapp , M. JO - Communications in Computational Physics VL - 3 SP - 671 EP - 714 PY - 2022 DA - 2022/09 SN - 32 DO - http://doi.org/10.4208/cicp.OA-2021-0157 UR - https://global-sci.org/intro/article_detail/cicp/21042.html KW - Atomistic/continuum coupling, flexible boundary conditions, local/global coupling, lattice Green functions, hierarchical matrices, dislocations. AB -

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.

M. Hodapp. (2022). Efficient Flexible Boundary Conditions for Long Dislocations. Communications in Computational Physics. 32 (3). 671-714. doi:10.4208/cicp.OA-2021-0157
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