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Higher-order Far-field Boundary Conditions for Crystalline Defects

Julian Braun, Christoph Ortner, Yangshuai Wang, Lei Zhang

TL;DR

The paper develops a principled, higher-order boundary-condition framework for crystalline defect simulations by exploiting a low-rank far-field expansion into discrete multipoles and continuum correctors. It introduces moment iterations and a continuous Green’s-function formulation to accelerate cell-size convergence, providing rigorous error bounds and truncation analyses for the multipole moments. Numerical experiments on tungsten point defects demonstrate accelerated decay of the far-field and improved geometry and energy errors as the domain size grows, validating the practical impact of higher-order boundary conditions in defect modeling. The work also clarifies the link between discrete and continuous multipole coefficients via force moments, enabling efficient, accurate implementations that extend to anisotropic and higher-order multipole scenarios.

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.

Higher-order Far-field Boundary Conditions for Crystalline Defects

TL;DR

The paper develops a principled, higher-order boundary-condition framework for crystalline defect simulations by exploiting a low-rank far-field expansion into discrete multipoles and continuum correctors. It introduces moment iterations and a continuous Green’s-function formulation to accelerate cell-size convergence, providing rigorous error bounds and truncation analyses for the multipole moments. Numerical experiments on tungsten point defects demonstrate accelerated decay of the far-field and improved geometry and energy errors as the domain size grows, validating the practical impact of higher-order boundary conditions in defect modeling. The work also clarifies the link between discrete and continuous multipole coefficients via force moments, enabling efficient, accurate implementations that extend to anisotropic and higher-order multipole scenarios.

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.
Paper Structure (20 sections, 6 theorems, 74 equations, 6 figures, 3 algorithms)

This paper contains 20 sections, 6 theorems, 74 equations, 6 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Background: Equilibrium of crystalline defects and its multipole expansions
  3. Equilibrium of crystalline defects
  4. Multipole expansion of equilibrium fields
  5. Moment Iterations and Continuous Computational Scheme
  6. Accelerated convergence of cell problems
  7. Multipole moment approximations
  8. The moment iteration
  9. Continuous coefficients of multipole expansion
  10. Numerical Experiments
  11. Model problems
  12. Convergence of cell problems
  13. Conclusion
  14. Proofs and Extensions
  15. Proofs of the results in Section \ref{['sec:moments']}
  16. ...and 5 more sections

Key Result

Theorem 2.1

Choose $p \geq 0, J \geq 0$ and suppose that $V \in C^{K}(\mathbb{R}^{d \times \mathcal{R}})$, such that $K \geq J+2+ \max \{0,\lfloor \frac{p-1}{d} \rfloor\}$. Let $g : \Lambda \to \mathbb{R}^{d \times \mathcal{R}}$ with compact support, and let $\bar{u} \in \mathcal{H}^{1}(\Lambda)$ such that Furthermore, let $\mathcal{S} \subset \Lambda$ be linearly independent with ${\rm span}_{\mathbb{Z}}\m

Figures (6)

  • Figure 2.1: 2D triangular lattice: Reference lattice $\Lambda$ (left); Defective lattice $\Lambda^{\rm def}$ with one self-interstitial atom inside $B_{R^{\rm def}}$ (right).
  • Figure 4.2: Defect cores for the five cases considered in this work, illustrated on the (001) plane, serving as benchmark problems for the numerical tests.
  • Figure 4.3: Decay of strains $|D\bar{u}_{i, R_{\rm dom}}(\ell)|$ for $i=0,1,2$ for all types of crystalline defect considered in this work.
  • Figure 4.4: Convergence of the relative moments error ${\rm ME}_{ki}$ defined by \ref{['eq:num:moments_err']} against domain size $R$. The blue and green lines illustrate the accelerate convergence of the dipole moment tensor ($k=1$) when improved boundary conditions are considered.
  • Figure 4.5: Convergence of geometry error $\|D\bar{u} - D\bar{u}_{i, R}\|_{\ell^2}$ for $i=0,1,2$ against domain size $R$.
  • ...and 1 more figures

Theorems & Definitions (11)

  • Theorem 2.1
  • Remark 2.1
  • Theorem 3.1
  • Lemma 3.1
  • Theorem 3.2
  • proof : Sketch of the proof
  • Corollary 3.1
  • Lemma 3.2
  • proof : Proof of Theorem \ref{['th:galerkin']}
  • proof : Proof of Lemma \ref{['th:galerkinfixedb']}
  • ...and 1 more