Direct FE2 for concurrent multilevel modelling of heterogeneous structures

FE2 methods can be used for the analysis of heterogeneous materials at the continuum macroscale while simultaneously accounting for the microstructural details. FE2 analyses typically comprise two levels of Finite Element (FE) simulations that are performed concurrently. At the macroscale level, the entire heterogeneous material or structure is discretized into homogenized continuum finite elements. Homogenized constitutive relations are not required for the macroscale calculations. Instead they are obtained from microscale level FE simulations on representative volume elements (RVE) of the material where the different phases of the heterogeneous material are explicitly modelled. The paper presents how the two levels of simulations can be collapsed into one by combining the equations governing both levels of FE analyses. The result is a single system of equations in terms of only the microscale level degrees of freedom, d̃. The equations take the familiar form of Kd̃=f̃. It is shown that K comprises the stiffness contributions from the RVE meshes scaled by an amount that is dependent on the relative sizes of the macroscale finite element and the RVE mesh, and the geometry of and choice of shape functions for the macroscale finite element. The derived force vector, f̃, is a direct outcome of the usual kinematic relations used to bridge the nodal displacements across the macroscale and microscale. We also show how this Direct FE2 can be carried out as a single simulation on any commercial FE software that supports multipoint constraints (MPC). The Direct FE2models are shown to give similar results to full FE meshes of heterogeneities throughout the entire domain with significantly less degrees of freedom. We further demonstrate that in-built capabilities of the commercial codes are naturally available with Direct FE2 through examples involving large deformation, plasticity and viscoelasticity.