Ghost Force Influence of a Quasicontinuum Method in Two Dimension

TL;DR

This paper analytically investigates the impact of ghost forces in a two-dimensional quasicontinuum method, revealing they cause finite size errors and induce a wider interfacial layer than in one dimension.

Contribution

It provides an explicit analytical solution and estimates the interfacial layer width, advancing understanding of ghost force effects in 2D quasicontinuum models.

Findings

Ghost forces cause finite size errors in the solution gradient.

The interfacial layer width scales as √ε, wider than in 1D.

Analytical expression for the 2D quasicontinuum solution is derived.

Abstract

We derive an analytical expression for the solution of a two-dimensional quasicontinuum method with a planar interface. The expression is used to prove that the ghost force may lead to a finite size error for the gradient of the solution. We estimate the width of the interfacial layer induced by the ghost force is of $\co(\sqrt{\eps}\,)$ with $\eps$ the equilibrium bond length, which is much wider than that of the one-dimensional problem.