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Homogenization and optimization

J. F Ganghoffer. LEMTA, Université de Lorraine, Nancy, France
'Homogenization and optimization'
The purpose of this lecture is to expose discrete homogenization methods developed to construct the effective continuum behavior of initially discrete structures or materials. Many architectured materials have such a structure, for instance foams, reticulated structures, fibrous media, or biological structures such as bones. Those media can be represented as repetitive truss lattices, with a more or less complex topology. The developed homogenization method is systematic and versatile as it can handle any lattice; the choice of the more appropriate effective continuum is dictated by the evaluation of internal lengths scales (bending and torsion lengths for a Cosserat medium). Anisotropic micropolar continua are obtained, the properties of which reflecting the topology of the architecture and its micromechanical properties. The underlying methodology has been written in algorithmic format and has been implemented in a dedicated code, using as an input the geometrical parameters inherent to the discrete topology, and delivering as an output the effective mechanical moduli. Examples are given for various kinds of architectured materials, including foams, textiles, biomaterials and bone structures. In the second part of the lecture, we expose the notion of topological derivative as a mathematical tool (introduced by Sokolowski and coworkers) to find optimum structures. A domain decomposition method is introduced, based on the replacement of the initial singular perturbation of the geometrical domain by a regular perturbation of a boundary nonlocal operator (Steklov-Poincaré). The method is applied to find formally the optimal effective compliance of solid bodies undergoing an internal evolution due to growth phenomena. It provides a methodology to compute optimal shapes in future developments.

References
  • J.F. Ganghoffer, J. Sokolowski. Growth of solid bodies in the framework of shape and topology optimization. Computer Methods in Materials Science. 12, N°1, 2012.
  • F. Dos Reis, J.F. Ganghoffer. Equivalent mechanical properties of auxetic lattices from discrete homogenization. Comput. Mat. Science. 2012, 51, 314-321.
  • F. Dos Reis, J.F. Ganghoffer. Construction of micropolar models from lattice homogenization. Computers Struct. 2012, 112-113, 354-363.
  • I. Goda, M. Assidi, J.F. Ganghoffer. A 3D elastic micropolar model for vertebral trabecular bone from lattice homogenization of the trabecular structure. Journal of the Mechanical Behavior of Biomedical Materials. 2012. In print.
   
© Realizacja Zbigniew Kacprzyk