An Analysis of the Effect of Ghost Force Oscillation on Quasicontinuum Error

TL;DR

This paper analyzes how ghost forces at the atomistic-continuum interface affect quasicontinuum error, providing optimal convergence rates and decay estimates for displacement errors in a one-dimensional model.

Contribution

It offers a rigorous analysis of ghost force effects on quasicontinuum errors, including explicit decay estimates and optimal convergence rates in various norms.

Findings

Ghost forces have a small effect on displacement error away from the interface.

Quasicontinuum displacement converges to atomistic displacement at rate O(h).

Displacement gradient error decays away from the interface at O(h|log h|).

Abstract

The atomistic to continuum interface for quasicontinuum energies exhibits nonzero forces under uniform strain that have been called ghost forces. In this paper, we prove for a linearization of a one-dimensional quasicontinuum energy around a uniform strain that the effect of the ghost forces on the displacement nearly cancels and has a small effect on the error away from the interface. We give optimal order error estimates that show that the quasicontinuum displacement converges to the atomistic displacement at the optimal rate O($h$) in the discrete $\ell^\infty$ norm and O($h^{1/p}$) in the $w^{1,p}$ norm for $1 \leq p < \infty.$ where $h$ is the interatomic spacing. We also give a proof that the error in the displacement gradient decays away from the interface to O($h$) at distance O($h|\log h|$) in the atomistic region and distance O($h$) in the continuum region. E, Ming, and Yang previously gave a counterexample to convergence in the $w^{1,\infty}$ norm for a harmonic interatomic potential. Our work gives an explicit and simplified form for the decay of the effect of the atomistic to continuum coupling error in terms of a general underlying interatomic potential and gives the estimates described above in the discrete $\ell^\infty$ and $w^{1,p}$ norms.