Analysis of a Force-Based Quasicontinuum Approximation

TL;DR

This paper analyzes a force-based quasicontinuum method for a one-dimensional atomic system, proving solution uniqueness, convergence properties, and extending understanding of ghost force correction techniques.

Contribution

It introduces a force-based quasicontinuum approximation that corrects ghost forces and provides rigorous analysis of solution existence and iterative convergence.

Findings

Unique solutions exist under certain load conditions.

Ghost force iteration is a contraction, ensuring convergence.

Solutions extend nearly to the tensile limit for Lennard-Jones interactions.

Abstract

We analyze a force-based quasicontinuum approximation to a one-dimensional system of atoms that interact by a classical atomistic potential. This force-based quasicontinuum approximation is derived as the modification of an energy-based quasicontinuum approximation by the addition of nonconservative forces to correct nonphysical ``ghost'' forces that occur in the atomistic to continuum interface. We prove that the force-based quasicontinuum equations have a unique solution under suitable restrictions on the loads. For Lennard-Jones next-nearest-neighbor interactions, we show that unique solutions exist for loads in a symmetric region extending nearly to the tensile limit. We give an analysis of the convergence of the ghost force iteration method to solve the equilibrium equations for the force-based quasicontinuum approximation. We show that the ghost force iteration is a contraction and give an analysis for its convergence rate.