An efficient three-dimensional database driven approach for multi scale homogenization

The two-scale homogenization theory, commonly known as the FE2 method, is a well-established technique used to model structures made of heterogeneous materials. Capable of capturing the microscopic effects at the macro level, the FE2 method assigns a representative volume element (RVE) of the materials microstructure at points across the macroscopic sample. This process results in the realization of a fully nested boundary value problem, where macroscopic quantities, required to model the structure, are obtained by homogenizing the RVEs response to macroscopic deformations. A limitation of the FE2 method though is the high computational costs, whereby its reduction has been a topic of much research in recent years. In this research, a two-scale database (TSD) model is presented to address this limitation. Instead of homogenizing the RVEs response to macroscopic deformations, the macroscopic quantities are now approximated using a database of precomputed RVEs. The homogenized results of an RVE are stored in a macroscopic right Cauchy-Green strain space. Discretizing this strain space into a finite set of right Cauchy-Green deformation tensors yields a material database, where the components of each tensor represent the boundary conditions prescribed to the RVE. A continuous approximation of the macroscopic quantities is attained using the Moving Least Squares (MLS) approximation method. Subsequent attention is paid to the implementation of the FE2 method and TSD model, for solving structures made of hyperelastic heterogeneous materials. Both approaches are developed in the in-house simulation software SESKA. A qualitative comparison of results from the FE2 method to those previously published, for a laminated composite beam undergoing material degradation, is presented to verify its implementation. To assess the TSD models performance, an evaluation into the numerical accuracy and computational performance, against the conventional FE2 method, is undertaken. While a significant improvement on computational times was shown, the accuracies in the TSD model were still left to be desired. Various remedies to improve the accuracy of the TSD model are proposed