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Analysis of a force-based quasicontinuum approximation

Published online by Cambridge University Press:  12 January 2008

Matthew Dobson  and
Mitchell Luskin
Matthew Dobson
Affiliation:
School of Mathematics, University of Minnesota, 206 Church Street SE, Minneapolis, MN 55455, USA. dobson@math.umn.edu; luskin@umn.edu
Mitchell Luskin
Affiliation:
School of Mathematics, University of Minnesota, 206 Church Street SE, Minneapolis, MN 55455, USA. dobson@math.umn.edu; luskin@umn.edu
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Abstract

We analyze a force-based quasicontinuum approximation to aone-dimensional system of atoms that interact by a classicalatomistic potential. This force-based quasicontinuum approximationcan be derived as the modification of an energy-basedquasicontinuum approximation by the addition of nonconservativeforces to correct nonphysical “ghost” forces that occur in theatomistic to continuum interface during constant strain. The algorithmicsimplicity and consistency with the purely atomistic modelat constant strain has made the force-basedquasicontinuum approximation popular for large-scalequasicontinuum computations.We prove that the force-based quasicontinuum equations havea unique solution when the magnitude of the external forces satisfyexplicit bounds. For Lennard-Jones next-nearest-neighborinteractions, we show that unique solutions existfor external forces that extend the system nearly to its tensile limit.We give an analysis of the convergence of the ghost force iteration methodto 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 itsconvergence rate.

Keywords

Quasicontinuumghost forceatomistic to continuum.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 42 , Issue 1 , January 2008 , pp. 113 - 139
Copyright
© EDP Sciences, SMAI, 2008

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