Comparison of several staggered atomistic-to-continuum concurrent coupling strategies

In this contribution several staggered schemes used to couple continuum mechanics (CM) and molecular mechanics (MM) are proposed. The described approaches are based on the atomistic-to-continuum correspondence, obtained by spatial averaging in the spirit of Irving and Kirkwood, and Noll. Similarities between this and other concurrent coupling schemes are indicated, thus providing a broad overview of different approaches in the field. The schemes considered here are decomposed into the surface-type (displacement or traction boundary conditions) and the volume-type. The latter restricts the continuum displacement field (and possibly its gradient) in some sense to the atomistic (discrete) displacements using Lagrange multipliers. A large-strain CM formulation incorporating Lagrange multipliers and a strategy to solve the resulting coupled linear system using an iterative solver is presented.Finally, the described coupling methods are numerically examined using two examples: uniaxial deformation and a plate with a hole relaxed under surface tension. Accuracy and convergence rates of each method are reported. It was found that the displacement (surface) coupling scheme and the Lagrangian (volume) scheme based on either discrete displacements or the H1H1 norm derived from continuous displacement fields provide the best performance.