Analysis of Boundary Conditions for Crystal Defect Atomistic 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.

Revised and corrected version extended from the published paper

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• (In-)Stability and Stabilisation of QNL-Type Atomistic-to-Continuum Coupling Methods
• QM/MM methods for crystalline defects. Part 2: Consistent energy and force-mixing
• Tucker tensor method for fast grid-based summation of long-range potentials on 3D lattices with defects
• QM/MM methods for crystalline defects. Part 1: Locality of the tight binding model
• Traction Boundary Conditions for Molecular Static Simulations
• Analysis of patch-test consistent atomistic-to-continuum coupling with higher-order finite elements
• Coupling Atomistic, Elasticity and Boundary Element Models
• Analysis of Blended Atomistic/Continuum Hybrid Methods
• Convergence Rates from Yukawa to Coulomb Interaction in the Thomas-Fermi-von Weizsäcker Model
• Development of an Optimization-Based Atomistic-to-Continuum Coupling Method
• Accuracy of computation of crystalline defects at finite temperature