# American Institute of Mathematical Sciences

March  2006, 1(1): 109-141. doi: 10.3934/nhm.2006.1.109

## A direct approach to numerical homogenization in finite elasticity

 1 CERMICS, Ecole Nationale des Ponts et Chaussées & INRIA Rocquencourt, 6 & 8 Av. B. Pascal, 77455 Champs-sur-Marne, France

Received  August 2005 Revised  November 2005 Published  January 2006

We describe, analyze, and test a direct numerical approach to a homogenized problem in nonlinear elasticity at finite strain. The main advantage of this approach is that it does not modify the overall structure of standard softwares in use for computational elasticity. Our analysis includes a convergence result for a general class of energy densities and an error estimate in the convex case. We relate this approach to the multiscale finite element method and show our analysis also applies to this method. Microscopic buck- ling and macroscopic instabilities are numerically investigated. The application of our approach to some numerical tests on an idealized rubber foam is also presented. For consistency a short review of the homogenization theory in nonlinear elasticity is provided.
Citation: Antoine Gloria Cermics. A direct approach to numerical homogenization in finite elasticity. Networks & Heterogeneous Media, 2006, 1 (1) : 109-141. doi: 10.3934/nhm.2006.1.109
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