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.
Highlights
Determining the geometry and energies of defects in crystalline solids is a key problem of computational materials science [47, Ch. 6]
To compute the equilibria we employ a robust preconditioned L-BFGS algorithm designed for large-scale atomistic optimisation problems [37]
We have introduced a flexible analytical framework to study the effect of embedding a defect in an infinite crystalline environment
Summary
Determining the geometry and energies of defects in crystalline solids is a key problem of computational materials science [47, Ch. 6]. Since practical schemes necessarily work in small computational domains (for example, “supercells”) they cannot explicitly resolve these fields but must employ artificial boundary conditions (periodic boundary conditions appear to be the most common). To assess the accuracy and in particular the cell size effects of such simulations, numerous formal results, numerical explorations, or results for linearised problems can be found in the literature; see for example [3,8,17,27] and references therein for a small representative sample. The novelty of the present work is that we rigorously establish explicit convergence rates in terms of computational cell size, taking into account the long-ranged elastic fields. Our framework encompasses both point defects and straight dislocation lines. Related results in a PDE context have recently been developed in [5]
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