Ball Packings Research Articles

Remarks on Soft Ball Packings in Dimensions 2 and 3

We study translative arrangements of centrally symmetric convex domains in the plane (resp., of congruent balls in the Euclidean 3-space) that neither pack nor cover. We define their soft density depending on a soft parameter and prove that the largest soft density for soft translative packings of a centrally symmetric convex domain with 3-fold rotational symmetry and given soft parameter is obtained for a proper soft lattice packing. Furthermore, we show that among the soft lattice packings of congruent soft balls with given soft parameter the soft density is locally maximal for the corresponding face centered cubic (FCC) lattice.

Local Atomic Configurations in Intermetallic Crystals: Beyond the First Coordination Shell.

We have used a combined geometrical-topological approach to analyze 21,697 intermetallic crystal structures stored in the Inorganic Crystal Structure Database. Following a geometrical scheme of close packing of balls, we have considered the three most typical polyhedral atomic environments of the icosahedral, cuboctahedral, or twinned cuboctahedral shape as well as multi-shell (up to four shells) local atomic configurations (LACs) based on these cores in 10,657 unique crystal structure determinations. In total, half of intermetallic structures have been found to contain one of these configurations, with the icosahedral LACs being the most frequent. We have revealed that even a two-shell configuration strongly predetermines the overall connectivity (topological type) of an intermetallic crystal structure. The chemical and stoichiometric composition of the multi-shell LACs generally obeys the close-packing model: the number of atoms in the subsequent shells (Nk) varies around the value Nk = 10k2 + 2, which is valid for the same size atoms, to reach the densest packing for the kth shell. Deviations from the revealed regularities often indicate inconsistencies in the crystallographic information, unusual features of the structure, or the existence of more stable phases that can be used for the validation of experimental and modeling data.

New lower bounds for optimal horoball packing density in hyperbolic n-space for 6 le n le 9

Koszul type Coxeter simplex tilings exist in hyperbolic n-space mathbb {H}^n up to n = 9, and their horoball packings achieve the highest known regular ball packing densities for n = 3, 4, 5. In this paper we determine the optimal horoball packing densities of Koszul simplex tilings in dimensions 6 le n le 9, which give new lower bounds for optimal packing density in each dimension. The symmetries of the packings are given by Coxeter simplex groups, and a parameter related to the Busemann function gives an isometry invariant description of different optimal horoball packing configurations.

Defects and frustration in the packing of soft balls

This work introduces the Hookean-Voronoi energy, a minimal model for the packing of soft, deformable balls. This is motivated by recent studies of quasi-periodic equilibria arising from dense packings of diblock and star polymers. Restricting to the planar case, we investigate the equilibrium packings of identical, deformable objects whose shapes are determined by an N-site Voronoi tessellation of a periodic rectangle. We derive a reduced formulation of the system showing at equilibria each site must reside at the “max-center” of its associated Voronoi region and construct a family of ordered “single-string” minimizers whose cardinality is O(N2). We identify sharp conditions under which the system admits a regular hexagonal tessellation and establish that in all cases the average energy per site is bounded below by that of a regular hexagon of unit size. However, numerical investigation of gradient flow of random initial data, reveals that for modest values of N the system preponderantly equilibrates to quasi-ordered states with low energy and large basins of attraction. For larger N the distribution of equilibria energies appears to approach a δ-function limit, whose energy is significantly higher than the ground state hexagon. This limit is possibly shaped by two mechanisms: a proliferation of moderate-energy disordered equilibria that block access of the gradient flow to lower energy quasi-ordered states and a rigid threshold on the maximum energy of stable states.

Математичні моделі та алгоритми оптимальної упаковки куль та кубів у сферичний та кубічний контейнери

Розглянуто математичні моделі та алгоритми оптимальної збалансованоїрозрідженої упаковки куль та кубів у сферичний та кубічний контейнери.Збалансованою розрідженою (задаються допустимі відстані між обʼєктами)упаковкою об’єктів у зовнішній контейнер є така їх упаковка, коли центрваги сімейства об’єктів збігається з центром зовнішнього контейнера, авідстані між об’єктами та відстані від них до зовнішнього контейнера булине менші за наперед задані величини. Наведено математичні моделі, послідовні та паралельні алгоритми розв’язання задач знаходження збалансованої розрідженої упаковки куль різних радіусів у сферичний та кубічнийконтейнери. Наведено математичну модель задачі знаходження збалансованої розрідженої упаковки кубів у куб мінімального обʼєму за умови, щосторони всіх кубів паралельні осям координат, та опис негладкої штрафноїфункції для пошуку локальних мінімумів задачі. Досліджувані задачівідносяться до класу NP-важких задач. Математичні моделі представленібагатоекстремальними задачами нелінійного програмування. Для пошукунайкращого допустимого розвʼязку застосовується метод мультистарту усполученні з r-алгоритмом Шора. Для цього задача зводиться до задачібезумовної оптимізації за допомогою штрафних функцій у вигляді функціймаксимуму, а для пошуку локальних мінімумів із набору стартових точокзастосовуються методи мінімізації негладких функцій, що базуються навикористанні програмних реалізацій r-алгоритму. Математичні моделі тапослідовні і паралельні алгоритми, що розглядаються, можна використатидля розробки програмних засобів розв’язування задач знаходження збалансованої розрідженої упаковки сферичних та кубічних обʼєктів у сферичніта кубічні контейнери. Матеріал представлено в трьох розділах. У розд. 1 наведено математичну модель та алгоритми розв’язання задачі знаходження збалансованої розрідженої упаковки куль різних радіусів у сферичнийконтейнер. Описано послідовний та паралельний алгоритми знаходженнянайкращого допустимого розв’язку задач. У розд. 2 наведено математичну модель та алгоритми розв’язання задачі знаходження збалансованої розрідженоїупаковки куль різних радіусів у кубічний контейнер. Описано послідовний тапаралельний алгоритми знаходження найкращого допустимого розв’язкузадач. У розд. 3 наведено математичну модель задачі знаходження збалансованої розрідженої упаковки кубів у кубічний контейнер. Наведено описнегладкої штрафної функції для пошуку локальних мінімумів задачі.

Optimal ball and horoball packings generated by 3-dimensional simply truncated Coxeter orthoschemes with parallel faces

In this paper, we consider the ball and horoball packings belonging to 3-dimensional Coxeter tilings that are derived by simply truncated orthoschemes with parallel faces. The goal of this paper is to determine the optimal ball and horoball packing arrangements and their densities for all above Coxeter tilings in hyperbolic 3-space ℍ3. The centers of horoballs are required to lie at ideal vertices of the polyhedral cells constituting the tiling, and we allow horoballs of different types at the various vertices. We prove that the densest packing of the above cases is realized by horoballs related to (∞, 3, 6, ∞) and (∞, 6, 3, ∞) tilings with density ≈ 0.8413392. The determined densest packing configurations give the second-highest density belonging to the Coxeter groups, and second largest locally optimal density are expected in higher dimensions for groups with similar structures.

Assessment of the influence of the structural characteristics of granular systems of microsilicon on the properties of thermal insulation materials

The article discusses experimental studies of the size and shape of structured particles of microsilica small angle x-ray scattering method and a photophonon theoretical description of the heat transfer process in complex heterogeneous structures to assessment of the structural characteristics of granular systems for the properties of thermal insulating materials. The mechanism of heat transfer in granular, porous systems is quite complex, since heat exchange occurs in a material consisting of two phases (solid and gas) and at the phase boundary. Heat transfer in liquid thermal insulation coatings can be carried out from one solid particle to another. In this case, the thermal conductivity will depend on: the chemical and elemental composition of the material; particle granulometry; surface topology - the presence of inhomogeneities, defects on the surface; the number of touches and the contact area between the particles. The heat transfer of gas in the pores is carried out when gas molecules collide. Thermal conductivity will be determined by the ratio of the free path of molecules and linear pore sizes, temperature and dynamic viscosity of the gas phase, the nature of the interaction of gas molecules with the solid phase. Heat transfer by radiation depends on the nature of the particles, the dielectric, magnetic permeability and the degree of blackness of the particle surface. Based on the analysis of possible mechanisms of heat transfer in granular systems, it can be argued that the effective thermal conductivity of the system depends, all other things being equal, on the structure of the pore space of granular materials, topology and the number of particle touches. Considering idealized models of the structure of granular materials in the form of ordered folds of perfectly smooth balls, we can obtain several variants of structures: with tetrahedral; hexagonal; cubic packing of balls.

On the deformation of ball packings

In this paper, we study the geometric aspects of ball packings on (M,T), where T is a triangulation on a 3-manifold M. First, we introduce a combinatorial Yamabe invariant YT, depending on the topology of M and the combinatoric of T. Then we prove that YT is attainable if and only if there is a constant curvature packing, and the combinatorial Yamabe problem can be solved by minimizing the Cooper-Rivin-Glickenstein functional. We also study the combinatorial Yamabe flow introduced by Glickenstein [23–25]. First, we prove a small energy convergence theorem that the flow would converge to a constant curvature metric if the initial energy is close in a quantitative way to the energy of a constant curvature metric. We also show even in case that the flow develops singularities in finite time, there is a natural way to extend the flow through the singularities such that the flow exists for all time. Finally, if the triangulation T is regular (i.e., all the numbers of tetrahedrons surrounding each vertex are equal), the combinatorial Yamabe flow converges exponentially fast to a constant curvature packing.

STUDY OF HEAT TRANSFER IN A MONOLAYER BALL PACKING

The features of nonstationary heat transfer of one layer of balls of the same size were studied in a fixed layer of liquid. The experiments were carried out in a cylindrical thermostat with a diameter of 128.9 mm filled with glass balls of 8.9 and 16.0 mm diam and polyoxymethylene balls of 9.0 mm diam with the highest possible packing density. The ball packing was filled with distilled water to a level of 0.1-0.2 mm higher than the ball diameter. The temperature of the ball packing surface was measured using a NEC 7100 thermal imager with a step change in temperature at the outer boundary of the packing. It is shown that, for a monolayer of balls of the same size in a stationary liquid, the temperature field is symmetrical relative to the packing center and the value of the effective thermal conductivity coefficient is constant over the packing cross section, both during heating and cooling. It has been found that the heat transfer coefficient and the time for establishing thermal equilibrium of a ball packing in a layer of stationary liquid at the same temperature difference depend on the heat flux direction at the packing boundary. The numerical values of the characteristic time of thermal equilibrium establishment in the heating and cooling regimes differ from each other by several times; whereas in the water layer without balls, this difference does not exceed 30%.

Geodesic ball packings generated by regular prism tilings in Nil geometry

Geodesic ball packings generated by regular prism tilings in Nil geometry

Computer 3D-simulation of the temperature and diffusion kinetics of SHS in the closest packing of Ni@Al “core-shell” mesocells for modes with variable values of the key ignition parameters

The paper presents the results of computer 3D-simulation of the temperature and diffusion kinetics of SHS in a test model cluster of Ni-Al particles for modes with variable values of the key parameters of the SHS combustion wave ignition. The key parameters for the SHS combustion wave ignition were chosen as follows: the initial temperature for preliminary heating of the Ni-Al particles mixture, the ignition temperature of the combustion wave in the mixture of Ni-Al particles, the duration of the action of the heat pulse until the combustion wave ignition, and the thickness of the ignited layer in the mixture of particles. A program has been created to generate a test model cluster in the form of the closest ball packing of the Ni@Al “core-shell” mesocells (CBP-structure cluster of the Ni@Al “core-shell” mesocells). Using such a CBP-structure cluster, was continued a testing of created software package intended for 3D-simulation of SHS macrokinetics in a heterogeneous particle mixture, taking into account parallel MPI-calculations. In addition, the value ranges of the key parameters of the SHS combustion wave ignition for which the simulation results are in adequate agreement with the experimental data are determined as the parameters of the program model for SHS-simulation. The results of computational experiments have shown that diffusion kinetics is interrelated with temperature kinetics, and in mesocells with different locations within the CBP-structure cluster, the formation of intermetallic phases occurs inhomogeneously.

ON CONTACT NUMBERS OF LOCALLY SEPARABLE UNIT SPHERE PACKINGS

The contact number of a packing of finitely many balls in Euclidean $d$-space is the number of touching pairs of balls in the packing. A prominent subfamily of sphere packings is formed by the so-called totally separable sphere packings: here, a packing of balls in Euclidean $d$-space is called totally separable if any two balls can be separated by a hyperplane such that it is disjoint from the interior of each ball in the packing. Bezdek, Szalkai and Szalkai (Discrete Math. 339(2): 668-676, 2016) upper bounded the contact numbers of totally separable packings of $n$ unit balls in Euclidean $d$-space in terms of $n$ and $d$. In this paper we improve their upper bound and extend that new upper bound to the so-called locally separable packings of unit balls. We call a packing of unit balls a locally separable packing if each unit ball of the packing together with the unit balls that are tangent to it form a totally separable packing. In the plane, we prove a crystallization result by characterizing all locally separable packings of $n$ unit disks having maximum contact number.

Ball packings for links

The ball number of a link L, denoted by ball(L), is the minimum number of solid balls (not necessarily of the same size) needed to realize a necklace representing L. In this paper, we show that ball(L)≤5cr(L) where cr(L) denotes the crossing number of a nontrivial nonsplittable link L. To this end, we use the connection of the Lorentz geometry with the ball packings. The well-known Koebe–Andreev–Thurston circle packing Theorem is also an important brick for the proof. Our approach yields an algorithm to construct explicitly the desired necklace representation of L in R3.

On Multifold Packings of Radius-1 Balls in Hamming Graphs

A $\lambda$-fold $r$-packing (multiple radius-$r$ covering) in a Hamming metric space is a code $C$ such that the radius-$r$ balls centered in $C$ cover each vertex of the space by not more (not less, respectively) than $\lambda$ times. The well-known $r$-error-correcting codes correspond to the case $\lambda=1$, while in general multifold $r$-packing are related with list decodable codes. We (a) propose asymptotic bounds for the maximum size of a $q$-ary $2$-fold $1$-packing as $q$ grows; (b) prove that a $q$-ary distance-$2$ MDS code of length $n$ is an optimal $n$-fold $1$-packing if $q\ge 2n$; (c) derive an upper bound for the size of a binary $\lambda$-fold $1$-packing and a lower bound for the size of a binary multiple radius-$1$ covering (the last bound allows to update the small-parameters table); (d) classify all optimal binary $2$-fold $1$-packings up to length $9$, in particular, establish the maximum size $96$ of a binary $2$-fold $1$-packing of length $9$; (e) prove some properties of $1$-perfect unitrades, which are a special case of $2$-fold $1$-packings. Keywords: Hamming graph, multifold ball packings, two-fold ball packings, list decodable codes, multiple coverings, completely regular codes, linear programming bound

RETRACTED: Analysis of Particle Damping Characteristics on Steel Vertical Machining Centre Column with Epoxy Reinforced Granite

Retracted paper: In high speed machining, performance is generally influenced by the dynamic behaviour of the machine tool structures. The machine tool structure is required to be rigid in order to remove the undesirable vibration and to improve the work piece quality. The most conventional material used in machine tool structure is cast iron which has both stiffness and dynamic characteristics to perform at varying speeds. The objective of this work is to improve damping capacity of vertical machining centre column. The damping capacity of column can be increased further by using passive damping method of ball packing. Damping capacity is a crucial factor which makes the dissipation of vibration happens at faster rate. As compared with cast iron established studies shows that epoxy granite a composite material improves damping capacity. Epoxy granite though could be a good choice for improving the machine tool performance at high speeds but is poor in static stiffness compared to cast iron. In this investigation it was observed that, the static stiffness of epoxy granite composite vertical machining centre column could be increased by using steel reinforcements. The final results reveal that, steel balls with epoxy granite provide faster dissipation time of 15ms at 70% packing ratio as compared to glass balls that showed dissipation time of 35ms. Also it was seen that, the steel balls offer the better damping capacity at optimum packing ratio of 50% mainly due to its specific gravity and mass of the balls.

Contact graphs of ball packings

A contact graph of a packing of closed balls is a graph with balls as vertices and pairs of tangent balls as edges. We prove that the average degree of the contact graph of a packing of balls (with possibly different radii) in R3 is not greater than 13.92. We also find new upper bounds for the average degree of contact graphs in R4 and R5.

Pressing technology and burning quality of spherical fuel briquettes made from autumn leaves

A process has been developed to press fuel briquettes from autumn leaves. It has been shown that a spherical shape of a briquette is optimal since even dense packing of balls provides air access to each individual ball. This is especially important in the initial phase of burning when briquettes touch each other and gain a certain temperature, because of the quality of burning increases. The technology of briquette pressing has been devised. We studied dispersion of particles of dry leaves of nutwood, maple, and oak after grinding and found that particle size distribution of powders of different leaves is not the same, and their bulk density is proportional to size with the highest content.A new method for pressing briquettes in a round closed matrix has been proposed. The difference of the method is in the fact that the process takes place at the creation of a scheme of all-embracing uniform compression with a spherical application of force and obtaining of a ball-shaped briquette in the final phase. The method makes it possible to compress ground leaves radially and evenly and thereby to ensure equal burning conditions for a briquette in a radial direction from any point on the periphery. We derived a mathematical model of the dependence of the density of dry briquettes on the bulk density of ground leaves and the degree of compression of a briquette. It has been shown that the bulk density of ground leaves (65 %) has the greatest influence on the density of dry briquettes. An increase in the bulk density of ground leaves leads to an increase in the density of dry briquettes. The influence of the degree of compression of a briquette is much smaller (35 %) but it is significant. Its increase leads to an increase in the density of dry briquettes. The density of dry briquettes was from 0.67 to 1.07 g/cm3 during studies. We proposed a design of a round closed matrix with a variable wall thickness, which makes it possible to reduce its metal consumption and cost by 20‒30 %. It has been shown that it is expedient to use spherical briquettes of leaves as an alternative environmentally friendly fuel

3-dimensional combinatorial Yamabe flow in hyperbolic background geometry

We study the 3-dimensional combinatorial Yamabe flow in hyperbolic background geometry. For a triangulation of a 3-manifold, we prove that if the number of tetrahedra incident to each vertex is at least 23, then there exist real or virtual ball packings with vanishing (extended) combinatorial scalar curvature, i.e., the total (extended) solid angle at each vertex is equal to 4 π 4\pi . In this case, if such a ball packing is real, then the (extended) combinatorial Yamabe flow converges exponentially fast to that ball packing. Moreover, we prove that there is no real or virtual ball packing with vanishing (extended) combinatorial scalar curvature if the number of tetrahedra incident to each vertex is at most 22.

NUMERICAL INTEGRATION ON GRAPHS: WHERE TO SAMPLE AND HOW TO WEIGH.

Let G = (V,E,w) be a finite, connected graph with weighted edges. We are interested in the problem of finding a subset W ⊂ V of vertices and weights aw such that for functions that are 'smooth' with respect to the geometry of the graph; here ~ indicates that we want the right-hand side to be as close to the left-hand side as possible. The main application are problems where f is known to vary smoothly over the underlying graph but is expensive to evaluate on even a single vertex. We prove an inequality showing that the integration problem can be rewritten as a geometric problem ('the optimal packing of heat balls'). We discuss how one would construct approximate solutions of the heat ball packing problem; numerical examples demonstrate the efficiency of the method.

Simulation of spatial ordering of powder mixture particles when packed in square matrices using an example of monodisperse balls

The peculiarities of self-ordering of large particles of powder mixture components with layer-by-layer pressing of long billets used in material production by the method of self-propagating high-temperature synthesis (SHS) are studied using the example of structure formation at compaction of thin layers of a model system of monodisperse calibrated solid balls (1 mm in diameter). In unilateral axial compaction, the case of packing of thin layers of balls (containing 1–5 monolayers) in a square matrix (with a ratio of the transverse dimension of the matrix to the diameter of the ball ∼14) is considered at low pressures that do not cause significant deformation of the balls. In the disperse system under study, a structural ordering of the particles occurs with a characteristic stepped view of the curve of the compact height dependence on the number of balls in the initial charge of balls. With an increase in the number of balls in the charge, the density of thin layers changes periodically during the compaction. In the compacts, fragments of structures in which the particles are arranged orderly like regular packing of ideal balls appear. The square shape of the matrix provides the formation of fragments of rhombohedral and cubic packing. The influence of nonuniform distribution of the initial charge of balls inside the matrix on the structuring of thin layers during compaction is considered.