Homogenization method for topology optmization of struc-tures built with lattice materials.

Abstract : Thanks to the recent developments of the additive manufacturing processes, structures built with modulated microstructures and featuring a complex topology are now manufacturable. This leads to a resurrection of the homogenization method for shape optimization, an approach developed in the 80’s but which progressively faded away because yielding too complex structures for manufacturing processes at this time.The goal of this thesis is to develop shape optimization methods for structures built with modulated locally periodic lattice microstructures.Three steps have been defined. The first consists in computing the homogenized, or effective, elastic properties of microstructures according to few parameters characterizing their geometry. In the second step, the geometric properties of the microstructure and its orientation are optimized in the working domain, yielding a homogenized optimized structure. Such a structure is nevertheless not straightforwardly manufacturable. Indeed, the homogenization is equivalent to have a structure featuring cells whose size is converging to zero. Hence, in the third and last step, a deshomogenization process is proposed. It consists in building a sequence of genuine structures converging to the homogenized optimal structures. The key point is to respect locally the orientation of the cells, which is performed thanks to a grid diffeomorphism.In this thesis, we present the details of the whole method, for isotropic and orthotropic microstructures, in 2D and in 3D.A coupling of this method with the level-set shape optimization method is also presented, thanks which the set of geometric constraints on the final structures may be enlarged.
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Perle Geoffroy Donders. Homogenization method for topology optmization of struc-tures built with lattice materials.. Analysis of PDEs [math.AP]. Université Paris-Saclay, 2018. English. ⟨NNT : 2018SACLX100⟩. ⟨tel-02011712⟩

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