Analysis of a force-based quasicontinuum approximation

Matthew Dobson; Mitchell Luskin

ESAIM: Mathematical Modelling and Numerical Analysis (2008)

  • Volume: 42, Issue: 1, page 113-139
  • ISSN: 0764-583X

Abstract

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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.

How to cite

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Dobson, Matthew, and Luskin, Mitchell. "Analysis of a force-based quasicontinuum approximation." ESAIM: Mathematical Modelling and Numerical Analysis 42.1 (2008): 113-139. <http://eudml.org/doc/250331>.

@article{Dobson2008,
abstract = { 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. },
author = {Dobson, Matthew, Luskin, Mitchell},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Quasicontinuum; ghost force; atomistic to continuum.; Lennard-Jones potential; convergence; ghost force iteration method},
language = {eng},
month = {1},
number = {1},
pages = {113-139},
publisher = {EDP Sciences},
title = {Analysis of a force-based quasicontinuum approximation},
url = {http://eudml.org/doc/250331},
volume = {42},
year = {2008},
}

TY - JOUR
AU - Dobson, Matthew
AU - Luskin, Mitchell
TI - Analysis of a force-based quasicontinuum approximation
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2008/1//
PB - EDP Sciences
VL - 42
IS - 1
SP - 113
EP - 139
AB - 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.
LA - eng
KW - Quasicontinuum; ghost force; atomistic to continuum.; Lennard-Jones potential; convergence; ghost force iteration method
UR - http://eudml.org/doc/250331
ER -

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