Stability of a force-based hybrid method with planar sharp interface

TL;DR

This paper analyzes the stability and convergence of a force-based hybrid atomistic-continuum method with a planar interface, ensuring second order accuracy under specific conditions.

Contribution

It establishes stability conditions and convergence proofs for a hybrid scheme coupling atomistic models with continuum elasticity at a sharp interface.

Findings

Proves second order convergence as lattice parameter ratio tends to zero.

Identifies stability conditions for hybrid schemes with planar sharp interface.

Applies results to 2D triangular lattice atomistic-to-continuum scheme.

Abstract

We study a force-based hybrid method that couples atomistic model with Cauchy-Born elasticity model with sharp transition interface. We identify stability conditions that guarantee the convergence of the hybrid scheme to the solution of the atomistic model with second order accuracy, as the ratio between lattice parameter and the characteristic length scale of the deformation tends to zero. Convergence is established for hybrid schemes with planar sharp interface for system without defects, with general finite range atomistic potential and simple lattice structure. The key ingredient of the proof is regularity and stability analysis of elliptic systems of difference equations. We apply the results to atomistic-to-continuum scheme for a 2D triangular lattice with planar interface.