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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

  • 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.

    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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  • 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.


  • This article has been cited by:

    1. Sébastien Boyaval, Reduced-Basis Approach for Homogenization beyond the Periodic Setting, 2008, 7, 1540-3459, 466, 10.1137/070688791
    2. Meiqun Jiang, Xingye Yue, Numerical homogenization of well singularities in the flow transport through heterogeneous porous media: fully discrete scheme, 2007, 41, 0764-583X, 945, 10.1051/m2an:2007044
    3. Antoine Gloria, An Analytical Framework for Numerical Homogenization. Part II: Windowing and Oversampling, 2008, 7, 1540-3459, 274, 10.1137/070683143
    4. Antoine Gloria, An Analytical Framework for the Numerical Homogenization of Monotone Elliptic Operators and Quasiconvex Energies, 2006, 5, 1540-3459, 996, 10.1137/060649112
    5. Antoine Gloria, Patrick Le Tallec, Marina Vidrascu, Foundation, analysis, and numerical investigation of a variational network-based model for rubber, 2014, 26, 0935-1175, 1, 10.1007/s00161-012-0281-6
    6. Roberto Alicandro, Marco Cicalese, Antoine Gloria, Integral Representation Results for Energies Defined on Stochastic Lattices and Application to Nonlinear Elasticity, 2011, 200, 0003-9527, 881, 10.1007/s00205-010-0378-7
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  • © 2006 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
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