Analysis of a force-based quasicontinuum approximation
M. Luskin and M. Dobson
ESAIM-Mathematical Modelling and Numerical Analysis,, 42, 113-139 (2008).
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.