A direct approach to numerical homogenization in finite elasticity

Antoine Gloria Cermics; Antoine Gloria Cermics

doi:10.3934/nhm.2006.1.109

Networks and Heterogeneous Media

2006, Volume 1, Issue 1: 109-141. doi: 10.3934/nhm.2006.1.109

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A direct approach to numerical homogenization in finite elasticity

Antoine Gloria Cermics ^{1
,}

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

Received: 01 August 2005 Revised: 01 November 2005
Primary: 74Q05; Secondary: 74B20.

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.
- Homogenization,
- determination of effective properties,
- nonlinear elasticity
Citation: Antoine Gloria Cermics. A direct approach to numerical homogenization in finite elasticity[J]. Networks and Heterogeneous Media, 2006, 1(1): 109-141. doi: 10.3934/nhm.2006.1.109

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Abstract

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.
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