A Posteriori Analysis and Adaptive Algorithms for Blended Type Atomistic-to-Continuum Coupling with Higher-Order Finite Elements

TL;DR

This paper develops a rigorous a posteriori error analysis and adaptive algorithms for blended atomistic-to-continuum coupling methods using higher-order finite elements, improving accuracy and efficiency in simulating material defects.

Contribution

It introduces a reliable a posteriori error estimator and adaptive mesh refinement for three blended a/c coupling methods with higher-order finite elements, enhancing simulation accuracy.

Findings

Achieved optimal convergence rates in numerical experiments.

Provided both upper and lower bounds for approximation error.

Demonstrated effectiveness of adaptive algorithms in defect simulations.

Abstract

The efficient and accurate simulation of material systems with defects using atomistic- to-continuum (a/c) coupling methods is a topic of considerable interest in the field of computational materials science. To achieve the desired balance between accuracy and computational efficiency, the use of a posteriori analysis and adaptive algorithms is critical. In this work, we present a rigorous a posteriori error analysis for three typical blended a/c coupling methods: the blended energy-based quasi-continuum (BQCE) method, the blended force-based quasi-continuum (BQCF) method, and the atomistic/continuum blending with ghost force correction (BGFC) method. We employ first and second-order finite element methods (and potentially higher-order methods) to discretize the Cauchy-Born model in the continuum region. The resulting error estimator provides both an upper bound on the true approximation error and a lower bound up to a theory-based truncation indicator, ensuring its reliability and efficiency. Moreover, we propose an a posteriori analysis for the energy error. We have designed and implemented a corresponding adaptive mesh refinement algorithm for two typical examples of crystalline defects. In both numerical experiments, we observe optimal convergence rates with respect to degrees of freedom when compared to a priori error estimates.