Blending Methods for Coupling Atomistic and Continuum Models

We review some recent developments in the coupling of atomistic and continuum models based on the blending of the two models in an overlap, or bridge, subdomain. These coupling schemes resemble overlapping domain decomposition methods. However, their analysis and development is complicated by the non-local force model employed by the atomistic model. We present an abstract framework for atomistic-to-continuum (AtC) coupling methods and formulate precise mathematical notions of patch and consistency tests for the methods. The framework admits both forcebased or energy-based coupling methods and allows us to identify four general classes of blending methods. We subject each class to patch and consistency tests and discuss important implementation issues such as: the enforcement of displacement continuity constraints in the bridge region; internal vs. external blending; the role of ghost forces, or forces that arise from coupling nonlocal and local models of force, and how they can be mitigated by blending methods.