Two-scale topology optimisation of cellular materials under mixed boundary conditions
Résumé
This work proposes a theoretical/numerical framework for the topology optimisation of anisotropic architected cellular materials at different scales. In particular, the topological variable (i.e., the pseudo-density field) is defined at both the scale of the representative volume element (i.e., the unit cell) of the material and at the macroscopic scale of the structure. The two-scale topology optimisation problem is formulated in the most general sense, i.e., by considering non-zero Neumann-Dirichlet boundary conditions. The proposed method is based on: (a) non-uniform rational basis spline hyper-surfaces to represent the topological variable at each scale, (b) the solid isotropic material with penalisation approach, (c) a general numerical homogenisation scheme based on the strain energy to establish the link between scales. The proposed formulation exploits the properties of non-uniform rational basis spline entities to determine the relationships occurring among the topological variables defined at different scales to correctly state the optimisation problem and to satisfy the hypotheses at the basis of the homogenisation method. In particular, scale separation (a necessary condition to be met in order to apply the homogenisation method) and manufacturing requirements are implicitly ensured by introducing minimum length scale constraints on the topological variables defined at both macroscopic scale and unit cell scale, respectively. Furthermore, the sensitivity of the optimised topology (at each scale) to the applied boundary conditions and to the elastic symmetry group of the representative volume element is investigated by founding new and original results. The effectiveness of the approach is tested on 2D and 3D benchmark problems taken from the literature.
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