  Three-dimensional discrete numerical simulation is used to investigate the
properties of close-packed frictionless granular assemblies as a function of
particle polydispersity and shape. Unlike some experimental results,
simulations show that disordered packings of pinacoids (eight-face convex
polyhedron) achieve higher solid fraction values than amorphous packings of
spherical or rounded particles, thus fulfilling the analogue of Ulam's
conjecture stated by Jiao and co-workers for random packings [Y. Jiao and S.
Torquato, Phys. Rev. E $\textbf{84}$, $041309$ ($2011$)]. This seeming
discrepancy between experimental and numerical results is believed to lie with
difficulties in overcoming interparticle friction through experimental
densification processes. Moreover, solid fraction is shown to increase further
with bidispersity and peak when the volume proportion of small particles
reaches $30\%$. Contrarywise, substituting up to $50\%$ of flat pinacoids for
isometric ones yields solid fraction decrease, especially when flat particles
are also elongated. Nevertheless, particle shape seems to play a minor role on
packing solid fraction compared to polydispersity. Additional investigations
focused on the packing microstructure confirm that pinacoid packings fulfill
the isostatic conjecture and that they are free of order except beyond $30$ to
$50\%$ of flat or flat \& elongated polyhedra in the packing. This order
increase progressively takes the form of a nematic phase caused by the
reorientation of flat or flat \& elongated particles to minimize the packing
potential energy. Simultaneously, this reorientation seems to increase the
solid fraction value slightly above the maximum achieved by monodisperse
isometric pinacoids, as well as the coordination number. Finally, partial
substitution of elongated pinacoids for isometric ones has limited effect on
packing solid fraction or order.
