Blending Methods for Coupling Atomistic and Continuum Models

In this paper we review recent developments in blending methods for atomistic-to-continuum (AtC) coupling in material statics problems. Such methods tie together atomistic and continuum models by using a bridge domain that connects the two models. There are several reasons why AtC coupling methods (of which blending methods are a subset) are important and have been subject to an increased interest in the recent years. Despite tremendous increases in computational power, fully atomistic simulations on an entire model domain remain computationally infeasible for many applications of interest. As a result, attention has focused on hybrid schemes where in all regions with well-behaved solutions, the atomistic (microscopic) model is replaced by a (macroscopic) continuum model enabling a more efficient computational scheme (see [14, 29, 9] for general information). The main challenge is the synthesis of the two distinct models in a manner that minimizes, or altogether eliminates, undesirable artifacts such as ghost forces, unphysical solutions let alone supporting mathematical analysis. Notable AtC coupling methods include the quasicontinuum method [39], the bridging scale decomposition [41], and [24] where atomistic and continuum models are overlapped (see the latter two references, and those mentioned above for numerous citations to the literature). The numerical analysis of AtC methods has lagged in comparison to the number of methods proposed; see the recent papers [2, 3, 17, 31, 26, 27] for analyses of the quasicontinuum method. Blending methods couple atomistic/continuum modes via a dedicated blending , or bridge, region inserted between the atomistic and continuum subdomains. The atomistic and continuum models are tied together by using a suitable " continuity " condition (or balance law) for the atomistic and continuum positions or displacements in this region. A complicating factor is that the atomistic and (classical) continuum elastic models rely on nonlocal and local models of force interaction, respectively. In the (classical) elastic context, the " local force " is that exerted on a body by contact forces occurring on the surface (of the body). In contrast, in the atomistic model, forces are summed from atoms separated by a finite distance. The incompatibility arising from coupling local and nonlocal force models is intrinsic. The goal of our paper is threefold. First, we review how blending approaches attempt to ameliorate the various negative effects by relying on an interface between the two models. Second, we discuss the beginnings of a JFISH: " CHAP06 " — 2009/6/4 — 17:33 — …