Adaptive Quasicontinuum Methods and Simulations for Crystal Defects with a Theory based Unified a Posteriori Error Estimate

TL;DR

This work develops a unified residual-force based a posteriori error estimator for consistent quasicontinuum (QC) methods and leverages it to drive adaptive algorithms across sharp interfaced and blended QC schemes. By proving an upper bound on the true error via the residual and incorporating truncation and sampling strategies, the authors enable simultaneous refinement of the continuum mesh and reallocation of atomistic and blending regions for various crystalline defects. The approach is validated through adaptive simulations of micro-cracks, anti-plane screw dislocations, and anti-plane cracks, demonstrating optimal convergence rates, computational efficiency, and the practical viability of applying a single estimator across multiple QC couplings. The results suggest a robust framework for on-the-fly adaptivity in multi-scale crystal defect modeling, with potential extensions to three dimensions and more complex defects like grain boundaries. Overall, the methodology offers a scalable, method-agnostic path to reliable, efficient QC simulations in realistic material systems.

Abstract

Adaptive quasicontinuum (QC) methods are important methodologies in molecular mechanics for the simulations of materials with defects, intending to achieve the optimal balance of accuracy and efficiency on the fly. In this study, we propose a residual-force based a posteriori error estimate that is simple and is unified for consistent quasicontinuum methods, as opposed to the widely adopted residual-stress based a posteriori error estimates which are complicated and need to be derived for the particular QC method under consideration. The simple and unified formulation of the estimator, together with certain sampling techniques, leads to a highly efficient and adaptable implementation. We also prove in theory that the unified error estimator provides an upper bound of the true error. We develop adaptive algorithms based on this unified estimator and validate the algorithms by several representative quasicontinuum methods for various types of crystalline defects, in terms of convergence and efficiency. In particular, the adaptive simulations of the anti-plane crack, of which we possess little a priori knowledge, demonstrate the necessity and significance of the proposed a posteriori estimates and the adaptive strategies.

Figures (15)

• Figure 1: Illustration of three typical defects considered in this work.
• Figure 2: The convergence of the error, the CPU time of evaluating the estimators and solving the QC problem and the efficiency factors of the estimators for adaptive GRAC method for the micro-crack.
• Figure 3: The convergences of the error and the efficiency factors for both rigorous and approximated error estimators with respect to the number of degrees of freedom for the micro-crack.
• Figure 4: The CPU times in each steps (left) and the ratio between the radius of the atomistic region $R_{\textrm{a}}$ and the width of the blending region $R_{\rm b}$ (right) in the adaptive simulations for the micro-crack.
• Figure 5: Decay of $\mathcal{F}^{\textrm{a}}_{\ell}({\pmb 0})$ for the anti-plane screw dislocation.

Theorems & Definitions (3)

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