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 can be 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 during constant strain. The algorithmic simplicity and consistency with the purely atomistic model at constant strain has made the force-based quasicontinuum approximation popular for large-scale quasicontinuum computations. We prove that the force-based quasicontinuum equations have a unique solution when the magnitude of the external forces satisfy explicit bounds. For Lennard-Jones next-nearest-neighbor interactions, we show that unique solutions exist for external forces that extend the system nearly to its 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.
Mots-clés : quasicontinuum, ghost force, atomistic to continuum
@article{M2AN_2008__42_1_113_0,
author = {Dobson, Matthew and Luskin, Mitchell},
title = {Analysis of a force-based quasicontinuum approximation},
journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
pages = {113--139},
publisher = {EDP-Sciences},
volume = {42},
number = {1},
year = {2008},
doi = {10.1051/m2an:2007058},
mrnumber = {2387424},
zbl = {1140.74006},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an:2007058/}
}
TY - JOUR AU - Dobson, Matthew AU - Luskin, Mitchell TI - Analysis of a force-based quasicontinuum approximation JO - ESAIM: Modélisation mathématique et analyse numérique PY - 2008 SP - 113 EP - 139 VL - 42 IS - 1 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an:2007058/ DO - 10.1051/m2an:2007058 LA - en ID - M2AN_2008__42_1_113_0 ER -
%0 Journal Article %A Dobson, Matthew %A Luskin, Mitchell %T Analysis of a force-based quasicontinuum approximation %J ESAIM: Modélisation mathématique et analyse numérique %D 2008 %P 113-139 %V 42 %N 1 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an:2007058/ %R 10.1051/m2an:2007058 %G en %F M2AN_2008__42_1_113_0
Dobson, Matthew; Luskin, Mitchell. Analysis of a force-based quasicontinuum approximation. ESAIM: Modélisation mathématique et analyse numérique, Tome 42 (2008) no. 1, pp. 113-139. doi : 10.1051/m2an:2007058. http://www.numdam.org/articles/10.1051/m2an:2007058/
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