Geometrically consistent approximations of the energy for the transition between nonlocal and local discrete models

This paper addresses the issue of consistent and generic energy-based passing from a discrete model with finite-range interactions to a surrogate discrete model with nearest neighbor interactions only. For this purpose, we introduce a transition zone that defines a volume interface between the two models and enables us to deal with incompatibilities between these models. Over the transition zone, a geometrically consistent approximation (in the sense of [1]) of the energy of the discrete system is defined; it sets up energies in an automatic manner and for a large set of configurations. In this framework, two possible schemes are then presented and analyzed, and performances of these schemes are compared on several numerical experiments in statics or dynamics regimes.