A priori error analysis of two force-based atomistic/continuum models of a periodic chain

Abstract

A priori error analysis of two force-based atomistic/continuum models of a periodic chain. Abstract The force-based quasicontinuum (QCF) approximation is a nonconservative atomistic/continuum hybrid model for the simulation of defects in crystals. We present an a priori error analysis of the QCF method, applied to a one-dimensional periodic chain, that is valid for an arbitrary interaction range, large deformations, and takes coarse-graining into account. Our main tool in this analysis is a new concept of atomistic stress.Moreover, we formulate a new atomistic/continuum coupling mechanism based on coupling stresses instead of forces and extend the a priori analysis to this new method. We show that the new method has several theoretical advantages over the original QCF method.

A Computational and Theoretical Investigation of the Accuracy of Quasicontinuum Methods

“…Again, this is surprising since various analyses [11,13,15,28,34] suggest that the force based method should be more accurate than the QNL method. However, we note that a similar effect was observed in onedimensional numerical experiments in [28]: while for large atomistic regions the QCF method was considerably more accurate than the QNL method, for smaller atomistic regions the QNL method was clearly more accurate. The atomistic regions studied here in our numerical experiments for the Lomer dislocation dipole are very small.…”

Analysis of an energy-based atomistic/continuum approximation of a vacancy in the 2D triangular lattice

“…A rigorous estimate on ∇ȳ a − ∇y ac L ∞ could, in principle, be achieved using the inverse function theorem [24,17,21], but requires stability of δ 2 E ac (I h y a ) as an operator from (discrete variants of) W 1,∞ to W −1,∞ . For the discretized Laplace operator such results are classical for quasiuniform meshes [26], and have recently been extended to locally refined meshes by Demlow et al [3].…”

The role of the patch test in 2D atomistic-to-continuum coupling methods

“…This process yields a notion of stress for atomistic models, which is related to the virial stress (see [2] for a recent reference; this connection will be discussed in detail elsewhere). A variant of this result for pair interactions in 1D was developed in [32]. where the stress function Σ a (y) ∈ P # 0 (T ε ) 2×2 is defined as follows:…”

“…Finally, §6 establishes the main result of this paper, Theorem 10: if an a/c method is patch test consistent and satisfies other natural technical assumptions, then it is also first-order consistent. The proof depends on a novel construction of stress tensors for atomistic models, related to the virial stress [2] (generalizing the 1D construction in [32,33]), and a corresponding construction for the stress tensor associated with the a/c energy. Moreover, we discuss in detail to what extent the technical conditions of Theorem 10 are required.…”