Disk Packings Research Articles

Parallel generation of initial element assemblies for two‐dimensional discrete element simulations

AbstractThe discrete element method (DEM) enables modeling of the behavior of granular materials and rock masses via the interaction of elements, as the whole model deforms under the applied loading. As a starting point of any DEM simulation, the computational domain needs to be filled with discrete elements. Disks are often used for 2D simulations to generate the model. There are techniques developed to fill a domain with disks, but some of them take considerable time, comparable to the DEM simulation itself. This article presents a parallel algorithm that can fill domains with a tight packing of disks. The algorithm is scalable and its implementation was tested on both a shared memory parallel computer and a multiprocessor, multicore workstation. The applicability of the algorithm in filling domains with disks is demonstrated on a practical example of ore flow at a drawpoint in mining. Copyright © 2008 John Wiley & Sons, Ltd.

On near-wall effects in hard disk packing between two concentric cylinders

Monodisperse packing of disks in confined geometries is rarely addressed in the literature. We present some theoretical aspects of the microstructural properties of 2D static packing of disks between two concentric cylinders. In the core of the geometry, the packing is intact, while it is dilated in the vicinity of the walls as a result of the removal of overlapping disks. We have derived the formula for average packing density in concentric cylinders geometry with given ratio of the particle diameter to the gap between cylinders. It is verified and analyzed further by constructing ordered and disordered packs of monodisperse disks in computer. In this context, Voronoi tessellation is performed for the resulting packs to demonstrate the thickness of the static boundary layer where the packing is influenced by the presence of cylindrical walls.

Rigidity of packings

In Ludwig Danzer’s Habilitatiionsschrift [L. Danzer, Endliche Punktmengen auf der 2-Sphäre mit möglichst grossem minimalabstand, Habilitationsschrift, Göttingen, 1963] he initiated a study of the local nature of the packings from the point of view of whether their density can be increased by a small perturbation of the packing configuration. This is an abbreviated biased introduction to the theory of such locally maximally dense packings of disks in various spaces from the point of view of the theory of tensegrity structures. This has applications to jamming of granular materials as well as leading to a better understanding of jammed packings.

Rotating bearings in regular and irregular granular shear packings

For 2D regular dense packings of solid mono-size non-sliding disks there is a mechanism for bearing formation under shear that can be explained theoretically. There is, however, no easy way to extend this model to include random dense packings which would better describe natural packings. A numerical model that simulates shear deformation for both near-regular and irregular packings is used to demonstrate that rotating bearings appear roughly with the same density in random and regular packings. The main difference appears in the size distribution of the rotating clusters near the jamming threshold. The size distribution is well described by a scaling form with a large-size cut-off that seems to grow without bounds for regular packings at the jamming threshold, while it remains finite for irregular packings. At packing densities above the jamming transition there can be no shear, unless the disks are allowed to break. Breaking of disks induces a large number of small local bearings. Clusters of rotating particles may contribute to e.g. pre-rupture yielding in landslides, snow avalanches and to the formation of aseismic gaps in tectonic fault zones.

Packings induced by piecewise isometries cannot contain the arbelos

Planar piecewise isometries with convex polygonal atoms that arepiecewise irrational rotations can naturally generate a packing ofphase space given by periodic cells that are discs. We show thatsuch packings cannot contain certain subpackings of Apollonianpackings, namely those belonging to a family of Arbelossubpackings. We do this by showing that the unit complex numbersgiving the directions of tangency within such an isometric-generatedpacking lie in a finitely generated subgroup of the circle group,whereas this is not the case for the Arbelos subpackings. In theopposite direction, we show that, given an arbitrary disc packing ofa polygonal region, there is a piecewise isometry whose regularcells approximate the given packing to any specified precision.

Continuous Distillation Experiment with Rotating Packed Bed

The performance of a rotating packed bed (RPB) with three kinds of packings was investigated using alcohol/water solution under continuous distillation conditions at atmospheric pressure. The effects of average high gravity factor ( β), reflux ratio ( R), and feedstock flux ( F) on mass-transfer in distillation were examined separately. Experimental results indicated that the total number of theoretical units (NTU) of RPB increased with β, R, and F. Of the three kinds of packings, the wave thread packing of stainless steel (Packing III) had the best mass transfer efficiency with the height equivalent of a theoretical plate (HETP) of approximately 7.35 mm- 23.58 mm, whereas the corrugated disk packing of stainless steel (Packing I) had the worst one with the HETP of about 13.4 mm-48.07 mm. Correlations were developed to describe the mass transfer efficiency for packings. Comparing experimental data with the data calculated by correlation, the average deviate obtained for each packing was 0.72%, 1.98%, and 2.7%, respectively, implying that the accuracy of correlations developed was reasonable.

Unjamming of granular packings due to local perturbations: Stability and decay of displacements

We study the mechanical response generated by local deformations in jammed packings of rigid disks. Based on discrete element simulations we determine the critical force of the local perturbation that is needed to break the mechanical equilibrium and examine the generated displacement field. Displacements decay as a power law of the distance from the perturbation point. The decay exponent and the critical force exhibit a nontrivial dependence on the friction: Both quantities are nonmonotonic and have a sharp maximum at the friction coefficient 0.1. We find that the mechanical response properties are closely related to the problem of force indeterminate where similar nonmonotonic behavior was observed previously. We establish a direct connection between the critical force and the ensemble of static force networks.

Simulation Study of Granular Compaction Dynamics under Vertical Tapping

We study by numerical simulation the compaction dynamics of frictional hard disks in two dimensions, subjected to vertical shaking. Shaking is modeled by a series of vertical expansions of the disk packing, followed by dynamical recompression of the assembly under the action of gravity. The second phase of the shake cycle is based on an efficient event−driven molecular−dynamics algorithm. We analyze the compaction dynamics for various values of friction coefficient and coefficient of normal restitution. We find that the time evolution of the density is described by ρ(t)=ρ∞ − ρEα[−(t/τ)α], where Eα denotes the Mittag−Leffler function of order 0<α<1. The parameter τ is found to decay with tapping intensity Γ according to a power law τ ∝ Γ−γ , where parameter γ is almost independent of the material properties of grains. Also, an expression for the grain mobility during compaction process has been obtained.

Force transmission in a packing of pentagonal particles

We perform a detailed analysis of the contact force network in a dense confined packing of pentagonal particles simulated by means of the contact dynamics method. The effect of particle shape is evidenced by comparing the data from pentagon packing and from a packing with identical characteristics, except for the circular shape of the particles. A counterintuitive finding of this work is that, under steady shearing, the pentagon packing develops a lower structural anisotropy than the disk packing. We show that this weakness is compensated by a higher force anisotropy, leading to enhanced shear strength of the pentagon packing. We revisit "strong" and "weak" force networks in the pentagon packing, but our simulation data also provide evidence for a large class of "very weak" forces carried mainly by vertex-to-edge contacts. The strong force chains are mostly composed of edge-to-edge contacts with a marked zigzag aspect and a decreasing exponential probability distribution as in a disk packing.

Model for estimation of critical packing density in polydisperse hard-disk packings

We introduce a geometric model for unjamming the polydisperse hard-disk packing. The unjamming corresponds to a critical packing density which is assumed to happen uniformly throughout the packing. However, this model can be used for prediction of local dilation in packings. Molecular dynamics (MD) simulations are also performed to create random packings of polydisperse hard disks. In MD simulations, a densification scheme is applied to the event-driven algorithm in which the size of particles increases with controlled rates. The packing densities upon the onset of the critical state are estimated in a series of bidisperse packings by feeding the textural information of the mixtures from MD simulations into the model. These results are relevant for studies of unjamming and shear banding phenomena in granular materials, colloids, metallic glasses and similar systems.

Density of invariant disk packings for planar piecewise isometries

Iteration of a planar piecewise isometry may generate an invariant disk packing, and understanding the properties of the disk packing is helpful for estimating the Lebesgue measure of the exceptional set for the planar piecewise isometry. If the disk packing is not dense, then the Lebesgue measure of the exceptional set is positive. But it is not easy to check the density of a disk packing. In this paper, the authors present necessary and sufficient conditions for the density of a general disk packing, and discuss some properties of disk packings for planar piecewise isometries.

Growing and reproducing particles evolving through space and time

In this paper we develop a spatially explicit marked point process model to generate spatial patterns of reproducing and moving cells evolving through continuous time. In this context, marked points (i.e. cells) can divide and move as a result of (a) their own growth and division motions, and (b) the division motions and the growth of their touching neighbours. To illustrate such dynamics we present several simulated examples. Our results suggest that by replacing the mathematically convenient growth and interaction functions by others more suited to mimic realistic cell reproduction dynamics, this model may provide a modelling framework to simulate cell aggregate (i.e. tissue) dynamics. Moreover, adding movement to dividing and growing points opens up new areas of application as well as new theoretical problems such as the analysis and generation of dense packing of discs (or spheres).

Capillarity-induced resonance of blobs in porous media: Analytical solutions, Lattice–Boltzmann modeling, and blob mobilization

Theoretical considerations and experiments in capillary tubes suggest that blobs exhibit resonance in porous media when they are trapped because of interfacial tension. Here, we investigate the hypothesis that such blobs can be mobilized by exploiting a phenomenon entitled capillarity-induced resonance, that is, by exciting the blobs at their resonant frequency. We used Lattice–Boltzmann (LB) modeling to perform numerical experiments, and we validated the LB model using analytical solutions that approximate the linear response of blobs with pinned menisci in straight and polygonal pore channels to an oscillatory body force. The LB simulations agree well with the quasistatic response, which the analytical solutions describe correctly. Furthermore, the frequency response, particularly the resonant frequency, agrees well, even though the analytical solutions do not accurately estimate viscous pressure drops. Numerical experiments in polygonal and sinusoidal pore channels, as well as disc packings, show that blobs, which are trapped even though a constant body force is applied, can indeed be mobilized by exploiting capillarity-induced resonance. Moreover, the resonant frequency can be estimated in numerical experiments by determining the dominant frequency in the blob amplitude in response to a force pulse. This is of great practical relevance for complex geometries, for which the resonant frequency cannot be easily predicted theoretically.

Frequency distribution of mechanically stable disk packings

Relative frequencies of mechanically stable (MS) packings of frictionless bidisperse disks are studied numerically in small systems. The packings are created by successively compressing or decompressing a system of soft purely repulsive disks, followed by energy minimization, until only infinitesimal particle overlaps remain. For systems of up to 14 particles, most of the MS packings were generated. We find that the packings are not equally probable as has been assumed in recent thermodynamic descriptions of granular systems. Instead, the frequency distribution, averaged over each packing-fraction interval Deltaphi , grows exponentially with increasing phi. Moreover, within each packing-fraction interval, MS packings occur with frequencies f{k} that differ by many orders of magnitude. Also, key features of the frequency distribution do not change when we significantly alter the packing-generation algorithm; for example, frequent packings remain frequent and rare ones remain rare. These results indicate that the frequency distribution of MS packings is strongly influenced by geometrical properties of the multidimensional configuration space. By adding thermal fluctuations to a set of the MS packings, we were able to examine a number of local features of configuration space near each packing. We measured the time required for a given packing to break to a distinct one, which enabled us to estimate the energy barriers that separate one packing from another. We found a gross positive correlation between the packing frequencies and the heights of the lowest energy barriers {0}; however, there is significant scatter in the data. We also examined displacement fluctuations away from the MS packings to assess the size and shape of the local basins near each packing. The displacement modes scale as d{i} approximately epsilon{0}{gamma{i}} with gamma{i} ranging from approximately 0.6 for the largest eigenvalues to 1.0 for the smallest ones. These scalings suggest that the packing frequencies are not determined by the local volume of configuration space near each packing, which would require that the dependence of f{k} on epsilon{0} is much stronger than the dependence we observe. The scatter in our data implies that in addition to epsilon{0} there are also other, as yet undetermined variables that influence the packing probabilities.

The [formula omitted]-convergence of SG circle patterns to the Riemann mapping

Thurston conjectured that the Riemann mapping function from a simply connected region onto the unit disk can be approximated by regular hexagonal packings. Schramm introduced circle patterns with combinatorics of the square grid (SG) and showed that SG circle patterns converge to meromorphic functions. He and Schramm proved that hexagonal disk packings converge in C ∞ to the Riemann mapping. In this paper we show a similar C ∞ -convergence for SG circle patterns.

Simulation study of granular compaction dynamics under vertical tapping

We study, by numerical simulation, the compaction dynamics of frictional hard disks in two dimensions, subjected to vertical shaking. Shaking is modeled by a series of vertical expansion of the disk packing, followed by dynamical recompression of the assembly under the action of gravity. The second phase of the shake cycle is based on an efficient event-driven molecular-dynamics algorithm. We analyze the compaction dynamics for various values of friction coefficient and coefficient of normal restitution. We find that the time evolution of the density is described by rho(t)=rho{infinity}-DeltarhoE{alpha}[-(ttau){alpha}], where E{alpha} denotes the Mittag-Leffler function of order 0<alpha<1 . The parameter tau is found to decay with tapping intensity Gamma according to a power law tau proportional, variantGamma{-gamma}, where parameter gamma is almost independent on the material properties of grains. Also, an expression for the grain mobility during the compaction process has been obtained. We characterize the local organization of disks in terms of contact connectivity and distribution of the Delaunay "free" volumes. Our analysis at microscopic scale provides evidence that compaction is mainly due to a decrease of the number of the largest pores. An interpretation of the memory effects observed for a discontinuous shift in tapping intensity Gamma is provided by the analysis of the time evolution of connectivity numbers and volume distribution of pores.

Tangents-free property in invariant disk packings generated by some general planar piecewise isometries

In this paper we investigate properties of invariant disk packings in the dynamics of some general planar piecewise isometries. By introducing codings underlying the map operations, we can give explicit expressions for the centers of the disks by analytic functions of the parameters, and then show that tangents free in invariant disk packing is a common property of the dynamics of piecewise isometries arising from various backgrounds. This supports the long-standing conjecture that the complements of the packings have positive Lebesgue measure.

Burning of Ammonium-Perchlorate Ellipses and Spheroids in Fuel Binder

Our simulations of heterogeneous propellant combustion have always assumed that the oxidizer particles (ammonium perchlorate) are disks (two dimensional) or spheres (three dimensional). Here the effects on the burning rate are examined when the disks/spheres are replaced by ellipses/ellipsoids. In two dimensions, it is shown that an area-preserving deformation of a pack of disks, generating a pack of ellipses, can lead to significant variations in the burning rate. However, if the ellipses are randomly packed, so that the alignment of their axes is random, the shape effect is small. In three dimensions, volume-preserving deformation, which generates ellipsoids, leads to burning rate changes no greater than those in two dimensions. Absent a random packing algorithm for ellipsoids, it is speculated that here also a random alignment of the axes would eliminate the effect. ECENT years have seen significant advances in the modeling of heterogeneous propellant combustion, specifically in the context of ammonium‐perchlorate (AP)/hydroxyl-terminatedpolybutadiene propellants. In Ref. 1, a two-dimensional treatment, a numerical code is described in which the combustion field is fully coupled to heat conduction in the solid via a nonuniformly regressing nonplanar surface. The corresponding three-dimensional problem is described in Refs. 2 and 3, the first using two-step kinetics and the second three-step kinetics. Comparisons between the numerical results and experimental results of Miller 4 show that the effects of propellant morphology on burning rates can be satisfactorily predicted over a wide pressure range. At fixed pressure, the variations in burning rate due to different morphologies are as much as threefold, so that these are nontrivial predictions. In this connection, note how we chose the values of the numerous parameters for this problem, parameters identified in Refs. 1‐3. Most are typical values defined by the community at large. Some are uncertain, of course, particularly the surface pyrolysis parameters, but once a choice was made no post hoc adjustments were made to better fit the three-dimensional data of Miller. An important parameter subset is those of the gas kinetics, the reaction rates for the two-step and three-step kinetics. There, we used one-dimensional burning rate data for pure AP and for homogeneous blends of AP and binder. The use of a three-step mechanism enabled us to fit blend burning rates over a wide range of mixture ratios, and it is presumably for this reason that the three-step results are more accurate than the two-step results. Most important, our calculations are not a massive exercise in curve fitting. This does not mean that better choices for some of our parameter values are not possible. Although we reasonably capture qualitative aspects of sandwich burning, 3 most notably the surface topography, the surface topography in our three-dimensional simulations does not show concavity in the surfaces of large AP particles at pressures for which it is observed experimentally. It seems probable that this

Note on: An improved algorithm for the packing of unequal circles within a larger containing circle

This note proposed an improved version of the algorithm proposed by Wang et al. [Wang, H. Q., Huang, W. Q., Zhang, Q., & Xu, D. M. (2002). An improved algorithm for the packing of unequal circles within a larger containing circle. European Journal of Operational Research, 141, 339 347] for solving the disk packing problem with equilibrium constraints. An efficient strategy of accelerating the search process is introduced in the gradient method to shorten the execution time. A number of computational results are presented, showing the effectiveness of the proposed method.

On the geometric dilation of closed curves, graphs, and point sets

Let G be an embedded planar graph whose edges are curves. The detour between two points p and q (on edges or vertices) of G is the ratio between the length of a shortest path connecting p and q in G and their Euclidean distance | p q | . The maximum detour over all pairs of points is called the geometric dilation δ ( G ) . Ebbers-Baumann, Grüne and Klein have shown that every finite point set is contained in a planar graph whose geometric dilation is at most 1.678, and some point sets require graphs with dilation δ ⩾ π / 2 ≈ 1.57 . They conjectured that the lower bound is not tight. We use new ideas like the halving pair transformation, a disk packing result and arguments from convex geometry, to prove this conjecture. The lower bound is improved to ( 1 + 10 −11 ) π / 2 . The proof relies on halving pairs, pairs of points dividing a given closed curve C in two parts of equal length, and their minimum and maximum distances h and H. Additionally, we analyze curves of constant halving distance ( h = H ), examine the relation of h to other geometric quantities and prove some new dilation bounds.