CHIN.PHYS.LETT.
Vol. 25, No. 5 (2008) 1724
Numerical Simulation of Random Close Packing with Tetrahedra
∗
LI Shui-Xiang(李水乡)∗∗ , ZHAO Jian(赵健), ZHOU Xuan(周璇)
College of Engineering, Peking University, Beijing 100871
State Key Laboratory for Turbulence and Complex System Study, Peking University, Beijing 100871
(Received 9 January 2008)
The densest packing of tetrahedra is still an unsolved problem. Numerical simulations of random close packing
of tetrahedra are carried out with a sphere assembly model and improved relaxation algorithm. The packing
density and average contact number obtained for random close packing of regular tetrahedra is 0.6817 and 7.21
respectively, while the values of spheres are 0.6435 and 5.95. The simulation demonstrates that tetrahedra can be
randomly packed denser than spheres. Random close packings of tetrahedra with a range of height are simulated
as well. We find that the regular tetrahedron might be the optimal shape which gives the highest packing density
of tetrahedra.
PACS: 45. 70. Cc, 81. 05. Rm, 05. 10. Ln
The problem of finding the densest packing of
tetrahedra was first suggested by Hilbert (1901) and
it still remains an unsolved problem. Hoylman[1]
found that the density of optimal lattice packing for
any tetrahedron is 18/49 = 0.3673 · · ·. Conway and
Torquato[2] found the density of uniform packing for
regular tetrahedra is 2/3 = 0.6666 · · ·. Recently,
they presented that the regular tetrahedron is a counterexample to Ulam’s conjecture which stated that
the optimal density for packing congruent spheres is
smaller than that for any other convex body. Instead,
they suggested that the regular tetrahedron might be
the convex body having the smallest packing density.
The best packing fraction they found was under 0.72,
considerably less than the densest sphere packing of
0.7405. However, Chaikin et al.[3] declared afterwards
that the best packing density of random packing tetrahedra which they measured in experiments is above
0.75. Dong and Ye[4] concluded from experiments
that the random close packing density of tetrahedra
is less than 0.5. With numerical simulation, Latham
et al.[5] found the random loose packing density of
tetrahedra is 0.416, which is less than that of spheres
(0.586). These figures are quite inconsistent and confused. Much more effort should be taken to clarify the
behaviour of tetrahedra packings.
The research of tetrahedra packings are of industrial importance in geotechnical, mining and transportation engineering. Additionally, the tetrahedron
is a basic geometric element, any other shapes can be
discretized into tetrahedra using Delaunay triangulation technique. In this Letter, the random close packings of tetrahedra are simulated with sphere assembly
model and improved relaxation algorithm. We try to
answer the following questions from the simulations in
this work. Can tetrahedra be randomly packed denser
than spheres or not? What is the optimal shape of a
∗ Supported
tetrahedron which gives the highest packing density
of tetrahedra?
Fig. 1. Sphere assembly model of a regular tetrahedron.
Generally, non-spherical representation model can
be classified into analytical model or assembly model.
Analytical model describes the particle shape contour
with mathematical equations. Contact detection of
particles turns to the solution of a set of equations.
Although the analytical model gives a precise representation, the robustness and efficiency problem remains. Moreover, different models should be built for
different shapes. Assembly model is another direction
which represents the non-spherical particles with an
assembly of basic and simple shapes. Sphere assembly
model[6] represents the non-spherical particles with
overlapping or tangent spheres. Therefore, the contacts between non-spherical particles can be treated
as contacts of spheres. An assembly model composed
of 11-layer, 294 overlapping spheres is constructed for
by the National Natural Science Foundation of China under Grant No 10772005.
Email: [email protected]
c 2008 Chinese Physical Society and IOP Publishing Ltd
°
∗∗
No. 5
LI Shui-Xiang et al.
a tetrahedron, as shown in Fig. 1. Each sphere has
equal size with radius of 1.0. To increase the sharpness around vertices, two small spheres with radii of
0.5 and 0.25 are added to each vertex of the tetrahedron. Although this model has round edges, it is
more similar to the real objects than the analytical
one. The exact volume of the tetrahedron composed
of spheres is crucial to the precision of packing density, and it is obtained from an AutoCAD system in
this work.
The packing algorithm in numerical simulation can
be classified into four categories,[7] sequential addition, collective rearrangement, advancing front and
optimization approach. An improved relaxation algorithm which falls into collective rearrangement is
applied to the random close packing of tetrahedra.
The algorithm is a geometrically based approach, and
the gravity is not concerned. The original relaxation algorithm of He[8] simulated the random packing of spheres successfully. However, the original algorithm is unsuitable for non-spherical particles since
only translation of particles is involved. In this work,
torque and rotation are introduced to the original algorithm to simulate the motion of non-spherical particles. The algorithm begins with randomly placed
large overlapping configuration of particles. Afterwards, iterations of relaxation procedure are carried
out to gradually reduce the overlaps of the particles.
Displacement of each sphere is computed in terms of
the overlaps with nearby spheres. Displacement and
rotation angle of a particle are derived from the summation of displacements and torques of all spheres in
the assembly model. The boundaries of the packing
region are enlarged at the end of each iteration. The
final packing is achieved when the maximum overlap
rate of spheres is below a predefined value. The efficiency of the algorithm is considerably increased by
means of the background cell method.[9]
The motive of the simulation for regular tetrahedra
packing is to see whether tetrahedra can be randomly
packed denser than spheres. Numerical simulation can
be applied to pure geometric packings, but the experiment can not. Mathematical analysis of optimal
packing can hardly be applied to a random system so
far. Latham et al.[5] studied the random loose packing
of tetrahedra with numerical simulation, the packing
density they derived is less than that of spheres. However, the algorithm they employed involves only translation but no rotation of tetrahedron. Convergence of
the algorithm was achieved by ‘reject and throw’ operations. It seems that the algorithm is unsuitable
for random close packing. Furthermore, contact number of tetrahedra is not included in their work. In
this study, a sphere assembly model and improved relaxation algorithm described above are employed to
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simulate random close packing of tetrahedra, and the
results are compared with spheres. Cubic region and
periodic boundary are applied to all simulations in
this work. There are 2000 spheres and 2000 regular
tetrahedra randomly packed in this simulation, respectively.
Fig. 2. Random close packing of 2000 regular tetrahedra.
Fig. 3. Distribution of contact number in random close
packing of regular tetrahedra.
In the simulation of sphere packing, the packing density (PD) obtained is 0.6435, while the average contact number (CN) is 5.95. These results are
very close to the well-known figures[10] (PD ≈ 0.64,
CN≈ 6). In the simulation of regular tetrahedron
packing, results obtained is PD = 0.6817, CN = 7.21.
These values are considerably higher than that of the
sphere packing in both packing density and contact
number. Simulations also show that more precise
sphere assembly model has the tendency to increase
the packing density. Therefore, we believe that tetrahedra can be randomly packed denser than spheres.
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LI Shui-Xiang et al.
Figure 2 shows the random close packing of 2000 regular tetrahedra. Figure 3 shows the distribution of
contact number, the maximum contact number in this
simulation is 13.
Vol. 25
of contact number when the height of tetrahedron increases.
Fig. 5. Average contact number versus the height of tetrahedron.
Fig. 4. Packing density versus the height of tetrahedron.
The purpose of the simulation for various heights
of tetrahedron is to find the optimal shape of tetrahedron which gives the highest packing density. A range
of height from 5 to 15 with increment of 1 is studied
by numerical simulation. Note that the height of 10
is the shape of a regular tetrahedron. Base triangle of
tetrahedron is unchanged in this simulation, and the
length of edge is 12.2474. There are 200 tetrahedra involved in each simulation, and our observation shows
that there are no remarkable differences between the
packing results of 200 and 2000 tetrahedra. Each simulation is carried out at least three times, and the
values obtained are the averages. Figure 4 shows the
packing density results. There is a peak around the
height of 10 in the figure. This indicates that the regular tetrahedra might be the optimal shape to give
the highest packing density of tetrahedra. Both taller
and flatter tetrahedra pack less dense than the regular tetrahedra. Latham et al.[5] studied the change of
tetrahedron height in random loose packing as well.
Their results are also provides in Fig. 4 for comparison (data has been transformed). The same tendency
of decrease of the packing density when the height
increases can be found. Nevertheless, only tall tetrahedra are concerned in their research. Figure 5 gives
the contact number results, and we can see a decrease
In summary, random close packings of tetrahedra
are simulated with the sphere assembly model and
the improved relaxation algorithm. The main conclusions form the simulations are: (1) Tetrahedra can
be randomly packed denser than spheres; (2) Regular tetrahedra might be the optimal shape to give the
highest packing density of tetrahedra. It should be
mentioned that the results of this work are restricted
to the random packing, and only geometrical packing
with undeformed particles is concerned. Extension of
the conclusions to other kinds of packings needs to be
proven.
References
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[4] Dong Q and Ye D N 1993 Chin. Sci. Bull. 38 54
[5] Latham J P, Lu Y and Munjiza A A 2001 Geotechnique 51
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[7] Zhao L, Li S X and Liu Y W 2007 Chin. J. Computat.
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