Abstract
The aim of this paper is to recourse to topology optimization method to synthesize mechanical matematerials, based on the topologial derivative. Material symmetries play a major role in the very definition and expression of the homogenized properties; the search of optimal microstructures is done in given symmetry classes characterized by invariants of the homogenized moduli. We synthesize thanks to this methodology periodic microstructures prone to auxetic and anti-auxetic behaviors, or with a very large or small bulk to shear modulus ratio.