1. Introduction
Atomistic-to-continuum coupling methods (a/c methods) are a class of computational multiscale schemes that combine the accuracy of atomistic models of defects with the computational efficiency of continuum models of elastic far-fields [1–5]. In the present article, we present the first successful implementation of a practical patch test consistent energy based a/c coupling scheme. Previously such schemes were only available for 2-body interactions [6,7]. In recent years a numerical analysis theory of a/c methods has emerged; we refer to [8] for a review. This theory has identified three prototypical classes of a/c schemes: patch test consistent energy-based coupling, force-based coupling (including force-based blending), and energy-based blending. The classical numerical analysis concepts of consistency and stability are applied to precisely quantify the errors committed in these schemes, and clear guidelines are established for their practical implementation including optimisation of approximation parameters. The results in [9,10,8,11,12] indicate that patch test consistent a/c couplings observe (quasi-)optimal error estimates in the energynorm. However, to this date, no general construction and implementation of such schemes has been presented. Instead, one normally compromises by either turning to patch test consistent force-based schemes [13–15,1] or to blending schemes [3,16] which have some control over the consistency error. Quasi-optimal implementations of such schemes are described in [16,15].
