Higher-Order Boundary Conditions for Atomistic Dislocation Simulations

TL;DR

This paper tackles slow convergence in atomistic dislocation simulations by developing higher-order boundary conditions built from a hierarchical continuum predictor (solved by a Galerkin spectral method) and discrete multipole corrections. The approach yields a low-rank representation of the dislocation far-field, enabling systematically improvable accuracy with controllable error through the radius $R$ and predictor order $p$, with rigorous convergence rates: geometry error scales as $R^{-1-p}\log^{p+1}(R)$ and energy error as $R^{-2-2p}\log^{2p+2}(R)$. A fully numerical framework combines predictor PDEs, moment iterations, and continuous multipole expansions (CMP), using domain truncation at $R_c$ and a rescaled, singularity-removing formulation solved via spectral Galerkin methods. Numerical experiments on screw and edge dislocations in tungsten demonstrate accelerated domain-size convergence and reduced cost, validating the theoretical error bounds and efficiency gains. The framework provides a foundation for faster, more accurate dislocation simulations and has potential extensions to more complex lattice models and defects.

Abstract

We present a higher-order boundary condition for atomistic simulations of dislocations that address the slow convergence of standard supercell methods. The method is based on a multipole expansion of the equilibrium displacement, combining continuum predictor solutions with discrete moment corrections. The continuum predictors are computed by solving a hierarchy of singular elliptic PDEs via a Galerkin spectral method, while moment coefficients are evaluated from force-moment identities with controlled approximation error. A key feature is the coupling between accurate continuum predictors and moment evaluations, enabling the construction of systematically improvable high-order boundary conditions. We thus design novel algorithms, and numerical results for screw and edge dislocations confirm the predicted convergence rates in geometry and energy norms, with reduced finite-size effects and moderate computational cost.

Figures (5)

• Figure 1: Single straight dislocations in BCC Tungsten considered in this work. Colored by Common Neighbor Analysis (CNA) in Ovito stukowski2009visualization.
• Figure 2: Decay of strains $|D \bar{w}_{i, R_{\rm dom}}(\ell)|$ as a function of the distance to the defect core $\lvert \ell \rvert$. Left: Antiplane screw dislocation. Right: (001)[100] Edge dislocation.
• Figure 3: Convergence of geometry error $\|D\bar{u} - D\bar{u}_{i, R}\|_{\ell^2}$ for $i=0,1$ against domain size $R$. Left: Antiplane screw dislocation; Right: (001)[100] edge dislocation.
• Figure 4: Convergence of energy error $|\mathcal{E}(\bar{u})-\mathcal{E}(\bar{u}_{i, R})|$ for $i=0,1$ against domain size $R$. Left: Antiplane screw dislocation; Right: (001)[100] edge dislocation.
• Figure 5: Computational efficiency of higher-order boundary conditions. Left: Antiplane screw dislocation; Right: (001)[100] edge dislocation.

Theorems & Definitions (12)

• Theorem 2.1
• Remark 2.2
• Theorem 3.1
• Theorem 3.2
• Lemma 4.1
• Lemma 4.2
• Remark 4.3
• Proof 1: Proof of Theorem \ref{['thm:dislocation']}
• Lemma B.1
• Proof 2: Proof of Lemma \ref{['th:galerkinfixedb']}