Goal-oriented Atomistic-Continuum Adaptivity for the Quasicontinuum Approximation

TL;DR

This paper introduces a goal-oriented error estimator for the quasicontinuum method, enabling adaptive algorithms that efficiently compute quantities of interest with minimal atomistic degrees of freedom, demonstrated on dislocation models.

Contribution

It presents a novel a posteriori error estimator and an adaptive algorithm for atomistic-continuum modeling in the quasicontinuum approximation.

Findings

Effective adaptive algorithm reduces atomistic degrees of freedom.

Demonstrated success on dislocation array models.

Achieves desired accuracy with minimal computational effort.

Abstract

We give a goal-oriented a posteriori error estimator for the atomistic-continuum modeling error in the quasicontinuum method, and we use this estimator to design an adaptive algorithm to compute a quantity of interest to a given tolerance by using a nearly minimal number of atomistic degrees of freedom. We present computational results that demonstrate the effectiveness of our algorithm for a periodic array of dislocations described by a Frenkel-Kontorova type model.