Numerical studies of a coarse-grained approximation for dynamics of an atomic chain

In many applications, materials are modeled by a large number of particles (or atoms) where each particle interacts with all others. Near or nearest-neighbor interaction is considered to be a good simplification of the full interaction in the engineering community. However, the resulting system is still too large to be solved under the existing computer power. In this paper we shall use the finite element and/or quasicontinuum idea to both position and velocity variables in order to reduce the number of degrees of freedom. The original and approximate particle systems are related to the discretization of the virtual internal bond model (continuum model). We focus more on the discrete system since the continuum description may not be physically complete because the stress-strain relation is not monotonically increasing and thus not necessarily well posed. We provide numerical justification on how well the coarse-grained solution is close to the fine grid solution in either a viscosity-demping or a temporal-average sense