Volume 43, Number 3, May-June 2009
|Page(s)||591 - 604|
|Published online||08 April 2009|
An analysis of the effect of ghost force oscillation on quasicontinuum error
School of Mathematics, University of Minnesota, 206 Church Street SE Minneapolis,
MN 55455, USA. firstname.lastname@example.org; email@example.com
Revised: 26 October 2008
The atomistic to continuum interface for quasicontinuum energies exhibits nonzero forces under uniform strain that have been called ghost forces. In this paper, we prove for a linearization of a one-dimensional quasicontinuum energy around a uniform strain that the effect of the ghost forces on the displacement nearly cancels and has a small effect on the error away from the interface. We give optimal order error estimates that show that the quasicontinuum displacement converges to the atomistic displacement at the rate O(h) in the discrete and w1,1 norms where h is the interatomic spacing. We also give a proof that the error in the displacement gradient decays 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 the atomistic to continuum coupling error in terms of a general underlying interatomic potential and gives the estimates described above in the discrete and w1,p norms.
Mathematics Subject Classification: 65Z05 / 70C20
Key words: Quasicontinuum / atomistic to continuum / ghost force.
© EDP Sciences, SMAI, 2009
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