Analysis of Blended Atomistic/Continuum Hybrid Methods

TL;DR

This paper provides a detailed error analysis of atomistic-to-continuum blending methods, including energy and force-based quasicontinuum approaches, applicable in multiple dimensions and defect scenarios.

Contribution

It introduces new error estimates and a novel stability proof technique for atomistic/continuum coupling methods, broadening their theoretical understanding.

Findings

Valid in 2D and 3D for many-body interactions

Applicable to lattice defects like point defects and dislocations

Provides new force and energy consistency error bounds

Abstract

We present a comprehensive error analysis of two prototypical atomistic-to-continuum coupling methods of blending type: the energy-based and the force-based quasicontinuum methods. Our results are valid in two and three dimensions, for finite range many-body interactions (e.g., EAM type), and in the presence of lattice defects (we consider point defects and dislocations). The two key ingredients in the analysis are (i) new force and energy consistency error estimates; and (ii) a new technique for proving energy norm stability of a/c couplings that requires only the assumption that the exact atomistic solution is a stable equilibrium.