# Numerical homogenization procedures for materials with complex micro-structure

## Prof. Dr.-Ing. habil. Alexander DÃ¼ster

**April 29, 2011, 7 a.m. HF 9904**

Two different numerical procedures to homogenize foamed and heterogeneous materials will be presented. In the first part of the talk the FE approach will be considered and applied to compute foamed materials. The FE 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 micromodel, 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 FE 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 effciently. Several examples ranging from porous and reinforced materials to foams will demonstrate this approach.