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

Dobson, Matthew ; Luskin, Mitchell

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 43 (2009) no. 3, pp. 591-604.

Résumé

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 ${\ell }^{\infty }$ and $w^{1,1}$ 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 ${\ell }^{\infty }$ and $w^{1,p}$ norms.

Détail

DOI : 10.1051/m2an/2009007

Classification : 65Z05, 70C20
Mots-clés : quasicontinuum, atomistic to continuum, ghost force