Coupled atomistic–continuum simulations using arbitrary overlapping domains

We present a formulation for coupling atomistic and continuum simulation methods for application to both quasistatic and dynamic analyses. In our formulation, a coarse-scale continuum discretization is assumed to cover all parts of the computational domain with atomistic crystals introduced only in regions of interest. The geometry of the discretization and crystal are allowed to overlap arbitrarily. Our approach uses interpolation and projection operators to link the kinematics of each region, which are then used to formulate a system potential energy from which we derive coupled expressions for the forces acting in each region. A hyperelastic constitutive formulation is used to compute the stress response of the defect-free continuum with constitutive properties derived from the Cauchy–Born rule. A correction to the Cauchy–Born rule is introduced in the overlap region to minimize fictitious boundary effects. Features of our approach will be demonstrated with simulations in one, two and three dimensions.