An analysis of the field theoretic approach to quasi-continuum method

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 in-troduced in various formulations of the quasi-continuum method lead to inconsistencies— namely, appearance of ghost forces or residual forces, non-conservative nature of approx-imate 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 quadra-ture 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 varia-tional 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 re-solves the inconsistencies present in previous formulations of the quasi-continuum method, and show using numerical examples the remarkable improvement in the accuracy of solu-tions. Further, this field theoretic formulation of quasi-continuum method makes mathemat-ical analysis of the method more amenable using functional analysis and homogenization theories