Topology optimization of modulated and oriented periodic microstructures by the homogenization method

Grégoire Allaire (1, 2) , Perle Geoffroy-Donders (1) , Olivier Pantz (3)

1 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique
2 DeFI - Shape reconstruction and identification
3 LJAD - Laboratoire Jean Alexandre Dieudonné

This paper is concerned with the topology optimization of structures made of periodically perforated material, where the microscopic periodic cell can be macroscopically modulated and oriented. The main idea is to optimize the homogenized formulation of this problem, which is an easy task of parametric optimization, then to project the optimal microstruc-ture at a desired lengthscale, which is a delicate issue, albeit computa-tionally cheap. The main novelty of our work is, in a plane setting, the conformal treatment of the optimal orientation of the microstructure. In other words, although the periodicity cell has varying parameters and orientation throughout the computational domain, the angles between its members or bars are conserved. The main application of our work is the optimization of so-called lattice materials which are becoming increasingly popular in the context of additive manufacturing. Several numerical examples are presented for compliance minimization in 2-d.

Domaines: Analyse numérique [math.NA]

Grégoire Allaire, Perle Geoffroy-Donders, Olivier Pantz. Topology optimization of modulated and oriented periodic microstructures by the homogenization method. 2018. ⟨hal-01734709v1⟩