# 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

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topDobson, 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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## Citations in EuDML Documents

top- Christoph Ortner, Endre Süli, Analysis of a quasicontinuum method in one dimension
- Christoph Ortner, The role of the patch test in 2D atomistic-to-continuum coupling methods
- Christoph Ortner, The role of the patch test in 2D atomistic-to-continuum coupling methods
- Matthew Dobson, Mitchell Luskin, An analysis of the effect of ghost force oscillation on quasicontinuum error

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