Analysis of a Force-Based Quasicontinuum Approximation
  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 is 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.   We prove that the force-based quasicontinuum equations have a unique solution under suitable restrictions on the loads. For Lennard-Jones next-nearest-neighbor interactions, we show that unique solutions exist for loads in a symmetric region extending nearly to the 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. 
Matthew Dobson and Mitchell Luskin