Convergence of a force-based hybrid method for atomistic and continuum models in three dimension

TL;DR

This paper proves that a force-based hybrid method coupling atomistic and continuum models in three dimensions converges quadratically to the atomistic solution as the lattice parameter becomes small, using stability and consistency analysis.

Contribution

It introduces a general convergence proof for a hybrid atomistic-continuum method in 3D, applicable to various short-ranged potentials and lattice types.

Findings

Quadratic convergence of the hybrid scheme to atomistic solutions.

Development of stability analysis tools using pseudo-difference operators.

Applicability to general short-ranged potentials and simple 3D lattices.

Abstract

We study a force-based hybrid method that couples atomistic models with nonlinear Cauchy-Born elasticity models. We show that the proposed scheme converges quadratically to the solution of the atomistic model, as the ratio between lattice parameter and the characteristic length scale of the deformation tends to zero. Convergence is established for general short-ranged atomistic potential and for simple lattices in three dimension. The convergence is based on consistency and stability analysis. General tools are developed in the framework of pseudo-difference operators for stability analysis in arbitrary dimension of the multiscale atomistic and continuum coupling methods.