Many materials problems require the accuracy of atomistic modeling in small regions, such as the neighborhood of a crack tip. However, these localized defects typically interact through long ranged elastic fields with a much larger region that cannot be computed atomistically. Many methods have recently been proposed to compute solutions to these multiscale problems by coupling atomistic models near a localized defect with continuum models where the deformation is nearly uniform. During the past several years, we have given a theoretical structure to the description and formulation of atomistic-to-continuum coupling that has clarified the relation between the various methods and the sources of error. Our theoretical analysis and benchmark simulations have guided the development of optimally accurate and efficient coupling methods.
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