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Numerical simulation results show that the upper bound order of random packing densities of basic 3D objects is cube (0.78) > ellipsoid (0.74) > cylinder (0.72) > spherocylinder (0.69) > tetrahedron (0.68) > cone (0.67) > sphere (0.64), while the upper bound order of ordered packing densities of basic 3D objects is cube (1.0) > cylinder and spherocylinder (0.9069) > cone (0.7854) > tetrahedron (0.7820) > ellipsoid (0.7707) > sphere (0.7405); these two orders are significantly different. The random packing densities of ellipsoid, cylinder, spherocylinder, tetrahedron and cone are closely related to their shapes. The optimal aspect ratios of these objects which give the highest packing densities are ellipsoid (axes ratio = 0.8:1:1.25), cylinder (height/diameter = 0.9), spherocylinder (height of cylinder part/diameter = 0.35), tetrahedron (regular tetrahedron) and cone (height/bottom diameter = 0.8).packing, particle, nonspherical particle, cylinder, cone, spherocylinder, tetrahedron Citation:Li S X, Zhao J, Lu P, et al. Maximum packing densities of basic 3D objects.To pursue the densest packing has never lost its attraction to human beings. The earliest history of studies on packing problem can be traced back to the famous Kepler Conjecture (the problem of maximum packing density of identical spheres, 1661) and the debate between Newton and Gregory (the problem of maximum coordinate number of identical spheres, 1694). In 1900, Hilbert further presented the packing problem, especially the densest packing of spheres and regular tetrahedra, as the 18th problem in his celebrated list of 23 mathematical problems [1]. For centuries, packing problem has always been attractive since it is not only a basic problem of mathematics and physics, but also extensively applied to many branches of science, engineering and even in daily life. These applications range from the macroscale of celestial body motions to the microscale of molecular arrangements. According to the packing structures, packing problems can be classified into ordered packing and disordered packing. For ordered packing, Hales proposed a proof of the Kepler Conjecture in 1998 [2]. However, it still leaves a long way to the solution of the Hilbert's 18th problem. For disordered packing, random packing which is closely related to matter structure has been investigated extensively. The first systematic study on random packing was undertaken by Bernal in 1950s on the random packing of spheres [3]. Nowadays, numerical simulation has become the main means of random packing researches. Zhao et al.[4] gave a summarization and classification of numerical simulation approaches available on random packing. In respect of particle shapes, sphere is the most comprehensively studied particle shape on random packing, and the packing results are accepted widely within the academic community. Nonspherical particles are often simplified to equivalent spheres in engineering applications. However, recent investigations indicated that the packing properties of nonspherical particles ...

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Regular tetrahedra have been demonstrated recently giving high packing density in random configurations. However, it is unknown whether the random-packing density of tetrahedral particles with other shapes can reach an even higher value. A numerical investigation on the random packing of regular and irregular tetrahedral particles is carried out. Shape effects of rounded corner, eccentricity, and height on the packing density of tetrahedral particles are studied. Results show that altering the shape of tetrahedral particles by rounding corners and edges, by altering the height of one vertex, or by lateral displacement of one vertex above its opposite face, all individually have the effect of reducing the random-packing density. In general, the random-packing densities of irregular tetrahedral particles are lower than that of regular tetrahedra. The ideal regular tetrahedron should be the shape which has the highest random-packing density in the family of tetrahedra, or even among convex bodies. An empirical formula is proposed to describe the rounded corner effect on the packing density, and well explains the density deviation of tetrahedral particles with different roundness ratios. The particles in the simulations are verified to be randomly packed by studying the pair correlation functions, which are consistent with previous results. The spherotetrahedral particle model with the relaxation algorithm is effectively applied in the simulations.

Sphere packing is an attractive way to generate high quality mesh. Several algorithms have been proposed in this topic, however these algorithms are not sufficiently fast for large scale problems. The paper presents an efficient sphere packing algorithm which is much faster and appears to be the most practical among all sphere packing methods presented so far for mesh generation. The algorithm packs spheres inside a domain using advancing front method. High efficiency has resulted from a concept of 4R measure, which localizes all the computations involved in the whole sphere packing process.