Formulation and Analysis of Blended Atomistic to Higher-Order Continuum Coupling Methods for Crystalline Defects

TL;DR

This work develops and analyzes blended atomistic-to-higher-order-continuum (a/c) coupling methods for crystalline defects in 1D. By replacing the classical second-order Cauchy-Born continuum with a higher-order continuum model, the authors establish rigorous a priori error results showing that energy-based blending (B-QHOCE) cannot improve overall accuracy due to coupling/interface errors, while force-based blending (B-QHOCF) can achieve higher accuracy comparable to the higher-order continuum model. The study integrates a detailed stress-error analysis, residual/interface considerations, and consistency bounds, and validates the theory with numerical experiments, including coarse-graining scenarios that reveal potential efficiency gains. The results provide fundamental insights into the limitations and benefits of higher-order continua in blended a/c schemes and outline future work on stability, sharp-interface formulations, and higher-dimensional extensions.

Abstract

Concurrent multiscale methods play an important role in modeling and simulating materials with defects, aiming to achieve the balance between accuracy and efficiency. Atomistic-to-continuum (a/c) coupling methods, a typical class of concurrent multiscale methods, link atomic-scale simulations with continuum mechanics. Existing a/c methods adopt the classic second-order Cauchy-Born approximation as the continuum mechanics model. In this work, we employ a higher-order Cauchy-Born model to study the potential accuracy improvement of the coupling scheme. In particular, we develop an energy-based blended atomistic to higher-order continuum method and present a rigorous a priori error analysis. We show that the overall accuracy of the energy-based blended method is not actually improved due the coupling interface error which is of lower order and may not be improved. On the contrast, higher order accuracy is achieved by the force-based blended atomistic to higher-order continuum method. Our theoretical results are demonstrated by a detailed numerical study.

Figures (4)

• Figure 1: Plot of the strain.
• Figure 2: Nodal error in strain.
• Figure 3: Convergence of relative error with respect to lattice spacing $\varepsilon$.
• Figure 4: coarsening error

Theorems & Definitions (4)

• proof
• proof
• proof
• proof : Proof of \ref{['a priori error of B-QHOCE']} in Theorem \ref{['a priori of bqhoce and bqhocf']}