The present work deals with a computer-design strategy to obtain a novel class of random porous materials, called M-Voronoi, with smooth void shapes and variable intervoid ligament sizes that can reach very low relative densities. This is achieved via a numerical, large strain, nonlinear elastic, void growth mechanical process. This method has been developed to generate both 2D and 3D random geometries. The mechanical properties of the M-Voronoi has been examined experimentally and numerically at large compressive strains and they show an enhanced mechanical resilience as compared to the existing random and periodic geometries. In the numerical study, we compare the M-Voronoi geometries with random polydisperse porous materials with spherical voids (RSA) and the classical TPMS-like geometries. We perform simulations at very large strain compression loading to assess the mechanical response, while considering the matrix an elastic-perfectly plastic material without hardening. We observe an enhanced yield stress in the geometries with random tologies as opposed to the TPMS periodic structures. This behavior is explained by noting that deformation localizes in geometries with periodic pattern, contrary to the random geometries which exhibit a rather diffused localization. Furthermore, in contrast to the periodic topologies, providing sufficient number of voids in the random M-Voronoi and RSA geometries, results in an isotropic response in small and large strains. Nevertheless, achieving very low densities are possible only with M-Voronoi random geometries.