Analysis of Boundary Conditions for Crystal Defect Atomistic Simulations

TL;DR

This paper develops a rigorous framework to assess the accuracy of various artificial boundary conditions used in atomistic simulations of crystal defects, providing error estimates and numerical validation.

Contribution

It introduces a variational formulation and regularity estimates to rigorously compare different boundary conditions in defect simulations.

Findings

Error estimates for Dirichlet, periodic, linear elasticity, and nonlinear elasticity boundary conditions.

Numerical results confirm the sharpness of the theoretical error bounds.

Framework enables precise assessment of boundary condition accuracy in defect simulations.

Abstract

Numerical simulations of crystal defects are necessarily restricted to finite computational domains, supplying artificial boundary conditions that emulate the effect of embedding the defect in an effectively infinite crystalline environment. This work develops a rigorous framework within which the accuracy of different types of boundary conditions can be precisely assessed. We formulate the equilibration of crystal defects as variational problems in a discrete energy space and establish qualitatively sharp regularity estimates for minimisers. Using this foundation we then present rigorous error estimates for (i) a truncation method (Dirichlet boundary conditions), (ii) periodic boundary conditions, (iii) boundary conditions from linear elasticity, and (iv) boundary conditions from nonlinear elasticity. Numerical results confirm the sharpness of the analysis.