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An analysis of the effect of ghost force oscillation on quasicontinuum error

Published online by Cambridge University Press:  08 April 2009

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

The atomistic to continuum interface for quasicontinuum energiesexhibits nonzero forces under uniform strain that have beencalled ghost forces.In this paper,we prove for a linearization of a one-dimensional quasicontinuum energyaround a uniform strainthat the effect of the ghost forces on the displacementnearly cancels and has a small effect on the error away from the interface.We give optimal order error estimatesthat show that the quasicontinuum displacementconverges to the atomistic displacement at the rate O(h)in the discrete $\ell^\infty$ andw 1,1 norms where h is the interatomic spacing.We also give a proof that the error in the displacement gradientdecays 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 theatomistic to continuum coupling error in terms of a general underlying interatomic potential and gives the estimates described above in the discrete $\ell^\infty$ and w1,p norms.

Keywords

Quasicontinuumatomistic to continuumghost force.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 43 , Issue 3 , May 2009 , pp. 591 - 604
Copyright
© EDP Sciences, SMAI, 2009

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