M. Iyer and V. Gavini
Journal of the Mechanics and Physics of Solids, 59, 1506–1535 (2011).
The quasi-continuum method has provided many insights into the behavior of lattice defects in the past decade. However, recent numerical analysis suggests that the approximations introduced in various formulations of the quasi-continuum method lead to inconsistencies namely, appearance of ghost forces or residual forces, non-conservative nature of approximate forces, etc. which affect the numerical accuracy and stability of the method. In this work, we identify the source of these errors to be the incompatibility of using quadrature rules, which is a local notion, on a non-local representation of energy. We eliminate these errors by first reformulating the extended interatomic interactions into a local variational problem that describes the energy of a system via potential fields. We subsequently introduce the quasi-continuum reduction of these potential fields using an adaptive finite-element discretization of the formulation. We demonstrate that the present formulation resolves the inconsistencies present in previous formulations of the quasi-continuum method, and show using numerical examples the remarkable improvement in the accuracy of solutions. Further, this field theoretic formulation of quasi-continuum method makes mathematical analysis of the method more amenable using functional analysis and homogenization theories.