Topology design of 2D and 3D elastic material microarchitectures with crystal symmetries displaying isotropic properties close to their theoretical limits

This paper evaluates the effect that different imposed crystal symmetries have on the topology design of two-phase isotropic elastic composites ruled by the target of attaining extreme theoretical properties. Extreme properties are defined by the Cherkaev–Gibiansky bounds, for 2D cases, or the Hashin–Shtrikman bounds, for 3D cases. The topology design methodology used in this study is an inverse homogenization technique which is mathematically formulated as a topology optimization problem. The crystal symmetry is imposed on the material configuration within a predefined design domain, which is taken as the primitive cell of the underlying Bravais lattice of the crystal system studied in each case. The influence of imposing crystal symmetries to the microstructure topologies is evaluated by testing five plane groups of the hexagonal crystal system for 2D problems and four space groups of the cubic crystal systems for 3D problems. A discussion about the adequacy of the tested plane or space groups to attain elastic properties close to the theoretical bounds is presented. The extracted conclusions could be meaningful for more general classes of topology design problems in the thermal, phononic or photonic fields.