Higher-order Far-field Boundary Conditions for Crystalline Defects

TL;DR

This paper introduces a new numerical scheme leveraging low-rank far-field expansions to improve boundary conditions in atomistic simulations of crystalline defects, reducing domain size effects.

Contribution

It presents a novel method that systematically improves boundary conditions using asymptotic far-field expansions, with rigorous error estimates and empirical validation.

Findings

The method accelerates defect simulations by reducing domain size effects.

Rigorous error bounds are established for the proposed scheme.

Numerical tests demonstrate convergence and robustness of the approach.

Abstract

Crystalline materials exhibit long-range elastic fields due to the presence of defects, leading to significant domain size effects in atomistic simulations. A rigorous far-field expansion of these long-range fields identifies low-rank structure in the form of a sum of discrete multipole terms and continuum correctors. We propose a novel numerical scheme that exploits this low-rank structure to accelerate material defect simulations by minimizing the domain size effects. Our approach iteratively improves the boundary condition, systematically following the asymptotic expansion of the far field. We provide both rigorous error estimates for the method and a range of empirical numerical tests, to assess it's convergence and robustness.