A numerical methodology developed for the microarchitecture design of 3D elastic two-phase periodic composites with effective isotropic properties close to the theoretical bounds is here presented and analyzed. This methodology is formulated as a topology optimization problem and is implemented using a level-set approach jointly with topological derivative. The most salient characteristic of this methodology is the imposition of preestablished crystal symmetries to the designed topologies; we integrate a topological optimization formulation with crystal symmetries to design mechanical metamaterials. The computational homogenization of the composite elastic properties is determined using a Fast Fourier Transform (FFT) technique. Due to the design domains are the primitive cells of Bravais lattices compatible with the space group imposed to the material layout, we have adapted the FFT technique to compute the effective properties in 3D parallelepiped domains. In this work, to find the topologies satisfying the proposed targets, we test four space groups of the cubic crystal system. Thus, the achievement of composites with effective elasticity tensor having cubic symmetry is guaranteed, and the isotropic response is then enforced by adding only one scalar constraint to the topology optimization problem. To assess the methodology, the following microarchitectures are designed and reported: two auxetic composites, three pentamode materials, and one maximum stiffness composite. With only one exception, all the remaining topologies display effective elastic properties with Zener coefficients approximating to 1.

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