Analysis of blended atomistic/continuum hybrid methods

Abstract

Abstract. We present a comprehensive error analysis of two prototypical atomistic-tocontinuum 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.

Analysis of Boundary Conditions for Crystal Defect Atomistic Simulations

“…This extends previous works such as [31,33,35] and presents a framework for ongoing and future development of a/c methods and their analysis; see for example [23,24,26,36], and references therein, for works in this direction.…”

“…To demonstrate the applicability of our framework we analyzed simple variants of some of the most commonly employed schemes: Dirichlet boundary conditions, periodic boundary conditions, far-field approximation via linearised lattice elasticity and via nonlinear continuum elasticity (Cauchy-Born, atomistic-tocontinuum coupling). In parallel works [12,23,24,36] this framework has already been exploited resulting in new and improved formulations of atomistic/continuum and quantum/atomistic coupling schemes.…”

“…In these approximate models, we solve a far-field problem concurrently with the atomistic core problem in order to improve the boundary conditions placed on the atomistic core. We also mention that, based on the approach advocated in the present paper, improved a/c schemes with superior optimal convergence rates have recently been developed in [24,36].…”

“…In order to demonstrate how to apply our framework to this setting, we shall restrict ourselves to QNL type schemes [14,35,41], but our discussion applies essentially verbatim to other force-consistent energy-based schemes such as [28,38,39]. For other types of a/c couplings the general framework is still applicable; see in particular [24] for a complete analysis of blending-type a/c methods.…”

A Posteriori Error Estimate and Adaptive Mesh Refinement Algorithm for Atomistic/Continuum Coupling with Finite Range Interactions in Two Dimensions

“…The construction and analysis of different a/c coupling methods have attracted considerable attention in the research community in recent years. Rigorous a prior analysis and systematic benchmark has been done in, for example, [14,24,12,29,30,15]. We refer readers to [18,16] for a review.…”