An analysis of the effect of ghost force oscillation on quasicontinuum error
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 43 (2009) no. 3, p. 591-604

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 rate O($h$) in the discrete ${\ell }^{\infty }$ and ${w}^{1,1}$ norms 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($h|logh|$) in the atomistic region and distance O($h$) in the continuum region. 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.

DOI : https://doi.org/10.1051/m2an/2009007
Classification:  65Z05,  70C20
Keywords: quasicontinuum, atomistic to continuum, ghost force
@article{M2AN_2009__43_3_591_0,
author = {Dobson, Matthew and Luskin, Mitchell},
title = {An analysis of the effect of ghost force oscillation on quasicontinuum error},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
publisher = {EDP-Sciences},
volume = {43},
number = {3},
year = {2009},
pages = {591-604},
doi = {10.1051/m2an/2009007},
zbl = {1165.81414},
mrnumber = {2536250},
language = {en},
url = {http://www.numdam.org/item/M2AN_2009__43_3_591_0}
}

Dobson, Matthew; Luskin, Mitchell. An analysis of the effect of ghost force oscillation on quasicontinuum error. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 43 (2009) no. 3, pp. 591-604. doi : 10.1051/m2an/2009007. http://www.numdam.org/item/M2AN_2009__43_3_591_0/

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