Mixed finite element and atomistic formulation for complex crystals

A general formulation for the analysis of complex Bravais crystals using atomic energy functionals embedded within a finite element framework is presented. The method uses atomistic potentials to determine the constitutive response of the system. Unlike traditional finite element methods, the nonlinear elastic effects, the symmetries of the underlying crystal, and the possibility of uniform structural phase transformations are naturally included in this formulation. Explicit expressions for empirical energy functionals with separable two- and three-body potentials, and semiempirical tight-binding energy functionals with two-center integrals are presented. A simple application to silicon underscores the importance of including internal relaxation in a finite element treatment of a complex crystal. In a forthcoming companion paper, the method presented here is applied to the nanoindentation of silicon.