Goal-Oriented Adaptive Mesh Refinement for the Quasicontinuum Approximation of a Frenkel-Kontorova Model

TL;DR

This paper introduces an adaptive mesh refinement method with an error estimator for the quasicontinuum approximation of a Frenkel-Kontorova model, enabling efficient and accurate computation of quantities of interest in crystalline solids.

Contribution

It presents a novel error estimator and adaptive refinement algorithm tailored for the quasicontinuum method applied to a Frenkel-Kontorova model, improving computational efficiency.

Findings

Effective error estimator for the quasicontinuum approximation.

Adaptive mesh refinement reduces computational cost.

Maintains desired accuracy in simulations.

Abstract

The quasicontinuum approximation is a method to reduce the atomistic degrees of freedom of a crystalline solid by piecewise linear interpolation from representative atoms that are nodes for a finite element triangulation. In regions of the crystal with a highly nonuniform deformation such as around defects, every atom must be a representative atom to obtain sufficient accuracy, but the mesh can be coarsened away from such regions to remove atomistic degrees of freedom while retaining sufficient accuracy. We present an error estimator and a related adaptive mesh refinement algorithm for the quasicontinuum approximation of a generalized Frenkel-Kontorova model that enables a quantity of interest to be efficiently computed to a predetermined accuracy.