Development of an Optimization-Based Atomistic-to-Continuum Coupling Method

TL;DR

This paper extends an optimization-based atomistic-to-continuum coupling method from 1D to multi-dimensional problems, providing a framework that combines accuracy and efficiency for modeling crystal structures with defects.

Contribution

It generalizes the AtC coupling method to higher dimensions and arbitrary potentials, with conjectured error estimates and numerical validation.

Findings

Numerical results support the conjectured error estimates.

Method successfully applied to a 1D Lennard-Jones system.

Framework adaptable to multi-dimensional atomistic models.

Abstract

Atomistic-to-Continuum (AtC) coupling methods are a novel means of computing the properties of a discrete crystal structure, such as those containing defects, that combine the accuracy of an atomistic (fully discrete) model with the efficiency of a continuum model. In this note we extend the optimization-based AtC, formulated in arXiv:1304.4976 for linear, one-dimensional problems to multi-dimensional settings and arbitrary interatomic potentials. We conjecture optimal error estimates for the multidimensional AtC, outline an implementation procedure, and provide numerical results to corroborate the conjecture for a 1D Lennard-Jones system with next-nearest neighbor interactions.