Robust algorithm to generate a diverse class of dense disordered and ordered sphere packings via linear programming

Abstract

We have formulated the problem of generating dense packings of nonoverlapping, non-tiling nonspherical particles within an adaptive fundamental cell subject to periodic boundary conditions as an optimization problem called the Adaptive Shrinking Cell (ASC) formulation [S. Torquato and Y. Jiao, Phys. Rev. E 80, 041104 (2009)]. Because the objective function and impenetrability constraints can be exactly linearized for sphere packings with a size distribution in d-dimensionalEuclidean space R d , it is most suitable and natural to solve the corresponding ASC optimization problem using sequential linear programming (SLP) techniques. We implement an SLP solution to produce robustly a wide spectrum of jammed sphere packings in R d for d = 2, 3, 4, 5 and 6 with a diversity of disorder and densities up to the respective maximal densities. A novel feature of this deterministic algorithm is that it can produce a broad range of inherent structures (locally maximally dense and mechanically stable packings), besides the usual disordered ones (such as the maximally random jammed state), with very small computational cost compared to that of the best known packing algorithms by tuning the radius of the influence sphere. For example, in three dimensions, we show that it can produce with high probability a variety of strictly jammed packings with a packing density anywhere in the wide range [0.6, 0.7408 . . .], where π/ √ 18 = 0.7408 . . .corresponds to the density of the densest packing. We also apply the algorithm to generate various disordered packings as well as the maximally dense packings for d = 2, 4, 5 and 6. Our jammed sphere packings are characterized and compared to the corresponding packings generated by the well-known Lubachevsky-Stillinger molecular-dynamics packing algorithm. Compared to the LS procedure, our SLP protocol is able to ensure that the final packings are truly jammed, produces disordered jammed packings with anomalously low densities, and is appreciably more robust and computationally faster at generating maximally dense packings, especially as the space dimension increases.

Existence of isostatic, maximally random jammed monodisperse hard-disk packings

“…In addition, polydisperse disk packings in 2D have been a popular way to elicit disordered packings, which are often isostatic (13,14,17,40). However, an isostatic packing of monodisperse disks has proved to be elusive until now.…”

“…Packings are compressed using the TJ algorithm with an influence sphere of diameter γ = D=40 (40), where D is the diameter of a sphere. For a single LP iteration, box deformations (both normal and shear movements) are limited in magnitude to less than D=200 and sphere translations are limited to kΔr i k ≤ D=200.…”

Jamming in finite systems: Stability, anisotropy, fluctuations, and scaling

“…We can also consider the situation where the shape of the container or simulation box is allowed to relax along with the particle positions [16,33]. This introduces d(d + 1)/2 − 1 additional degrees of freedom, independent of system size, which are associated with the shape of the box.…”

“…1). Note that changing the shape of the simulation box can be interpreted as changing the metric tensor of the space in which the particles live [33].…”

Inherent secure communications using lattice based waveform design

“…Very recently, the statistical mechanics community has been interested in finding the most dense lattices of a given dimension. Several algorithms have been proposed that, while not finding the densest lattice, find very dense lattices, in some cases the densest known lattice for a given dimension ( [2,23,31,35,46,47]). Due to time, these algorithms unfortunately have not been implemented to try to find good lattices for our problem, but it is believed that these algorithms could be successful, and thus a brief description of them is included.…”