A geometric multiscale finite element method for the dynamic analysis of heterogeneous solids

•A geometric multiscale finite element formulation is developed.•The method employs multi-node elements with numerical shape functions.•Heterogeneities are modelled using a limited number of macroscopic information.•The method is applied to static and dynamic analyses.

This paper presents a geom etric multiscale finite element formulation developed for predicting the response of heterogeneous materials and structures. The method is based on multi-node elements whose shape functions are computed numerically by means of an auxiliary fine scale discretization of the element itself. The elements explicitly resolve the geometry of heterogeneities occurring at sub-elemental length scales, and ensure compatibility across the element boundaries. The local auxiliary mesh is only used at the elemental level to compute the shape functions and does not need to be retained as part of macroscale simulations. The formulation of 2D and 3D elements is illustrated through examples where error estimates are conducted, and which illustrate the effectiveness of the method for the static and dynamic analyses of solids with local heterogeneities.