Error Estimation and Atomistic-Continuum Adaptivity for the Quasicontinuum Approximation of a Frenkel-Kontorova Model

TL;DR

This paper introduces a goal-oriented error estimator and adaptive algorithm for atomistic-continuum modeling in the quasicontinuum method, improving accuracy and efficiency in simulating crystallographic defects.

Contribution

It develops a novel a posteriori error estimator and an adaptive algorithm for atomistic-continuum partitioning in the quasicontinuum method, tailored for defect modeling.

Findings

Efficient error estimation for quasicontinuum simulations

Adaptive algorithm achieves nearly optimal atomistic-continuum partitioning

Numerical experiments validate the method's accuracy and efficiency

Abstract

We propose and analyze a goal-oriented a posteriori error estimator for the atomistic-continuum modeling error in the quasicontinuum method. Based on this error estimator, we develop an algorithm which adaptively determines the atomistic and continuum regions to compute a quantity of interest to within a given tolerance. We apply the algorithm to the computation of the structure of a crystallographic defect described by a Frenkel-Kontorova model and present the results of numerical experiments. The numerical results show that our method gives an efficient estimate of the error and a nearly optimal atomistic-continuum modeling strategy.