Numerical homogenization procedures for materials with complex micro-structure

Numerical homogenization procedures for materials with complex micro-structure

Two different numerical procedures to homogenize foamed and heterogeneous materials will be presented.  In the first part of the talk the FE² approach will be considered and applied to compute  foamed materials. The FE² method is a two-scale (micro/macro) approach where both the macro-model and the micro-model are discretized with finite elements. The constitutive model is replaced by attaching to  each integration point on the macro-level a micro-model, representing the micro-structure of the material.  The homogenization of the micro-model yields the constitutive answer, i.e. the stress-strain relation. Several numerical examples will demonstrate the properties of this two-scale approach. Since the FE²  method is computationally very demanding we will consider in the second part of the presentation  an alternative approach which is based on the computation of effective material properties. This approach  utilizes the Finite Cell Method (FCM) which can be interpreted as a combination of a fictitious domain method with finite elements of high-order. Starting from a computed tomography of the material of interest  the FCM can be applied to compute its effective material properties very efficiently. Several examples  ranging from porous and reinforced materials to foams will demonstrate this approach.