# Numerical homogenization procedures for materials with complex micro-structure

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

**April 29, 2011, 9 a.m. HF 9904**

Two dierent numerical procedures to homogenize foamed and heteroge-

neous materials will be presented. In the rst 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 nite 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 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 eective material properties. This approach uti-

lizes the Finite Cell Method (FCM) which can be interpreted as a combination

of a ctitious domain method with nite elements of high-order. Starting from

a computed tomography of the material of interest the FCM can be applied

to compute its eective material properties very eciently. Several examples

ranging from porous and reinforced materials to foams will demonstrate this

approach.