Sphere Packing Research Articles

Nonlinear Optimization and Adaptive Heuristics for Solving Irregular Object Packing Problems

We review and present several challenging model classes arising in the context of finding optimized object packings (OP). Except for the smallest and/or simplest general OP model instances, it is not possible to find their exact (closed-form) solution. Most OP problem instances become increasingly difficult to handle even numerically, as the number of packed objects increases. Specifically, here we consider classes of general OP problems that can be formulated in the framework of nonlinear optimization. Research experience demonstrates that—in addition to utilizing general-purpose nonlinear optimization solver engines—the insightful exploitation of problem-specific heuristics can improve the quality of numerical solutions. We discuss scalable OP problem classes aimed at packing general circles, spheres, ellipses, and ovals, with numerical (conjectured) solutions of non-trivial model instances. In addition to their practical relevance, these models and their various extensions can also serve as constrained global optimization test challenges.

UoC-10: Exploring the Packings of Chiral Copper(II) Paddle-Wheel Based Metal-Organic Cages (MOCs).

The fluorination of the central ring of 1,3,5-benzene-tris-(meta-benzoate) (referred to as BTMB) leads to a twisted tritopic linker which reacts with copper(II) ions to assemble into octahedral (pseudospherical) metal-organic cages (MOCs) with paddle-wheel units at their vertices. In this work, the different sphere packings of these MOCs are explored in detail together with their material properties, which closely resemble those of copper-based metal-organic frameworks (MOFs). Theoretical investigations of the linkers are carried out to analyze the energetic barrier imposed by the fluorine substituents to form the observed atropisomers.

Regular polytopes, sphere packings and Apollonian sections

Regular polytopes, sphere packings and Apollonian sections

Mesoatomic Engineering for Programmable Spherical Packing Superlattices in Self-Assembly of Giant Molecular Clusters

Mesoatomic Engineering for Programmable Spherical Packing Superlattices in Self-Assembly of Giant Molecular Clusters

Structure of face-centred icosahedral quasicrystals with cluster close packing.

A 6D structure model for face-centred icosahedral quasicrystals consisting of so-called pseudo-Mackay and mini-Bergman-type atomic clusters is proposed based on the structure model of the Al69.1Pd22Cr2.1Fe6.8 3/2 cubic approximant crystal (with space group Pa3, a = 40.5 Å) [Fujita et al. (2013). Acta Cryst. A69, 322-340]. The cluster centres form an icosahedral close sphere packing generated by the occupation domains similar to those in the model proposed by Katz & Gratias [J. Non-Cryst. Solids (1993), 153-154, 187-195], but their size is smaller by a factor τ2 [τ = (1 + (5)1/2)/2]. The clusters cover approximately 99.46% of the atomic structure, and the cluster arrangement exhibits 15 and 19 different local configurations, respectively, for the pseudo-Mackay and mini-Bergman-type clusters. The occupation domains that generate cluster shells are modelled and discussed in terms of structural disorder and local reorganization of the cluster arrangements (phason flip).

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.

Nanoparticle transport in partially saturated porous media: Attachment at fluid interfaces

Like the solid-water interface (SWI), air-water and oil-water interfaces (AWI and OWI) also act as collectors for nano-sized particles in porous media. The attachment of hydrophobic nanoparticles, which is often favorable and irreversible, is of particular interest because the transport and retention of such particles is closely linked to the fate of nanoplastics in unsaturated subsurface environments and the success of nanoremediation practices. Here, we show how a pore-network model (PNM) can be used to upscale the kinetics and extent of irreversible nanoparticle attachment at a single fluid-fluid interface under conditions of advection and dispersion in a sphere packing. By focusing on a trapped (immobile) non-wetting phase, we highlight a fundamental difference between the single-collector contact efficiency of AWI/OWI and SWI. Namely, AWI/OWI collectors, which are largely by-passed by the flowing aqueous phase, are exposed to a hydrodynamic environment dominated by diffusion. This difference has profound implications for the modelling of nanoparticle transport in porous media at the continuum (Darcy) scale. This study reveals the potential of pore network modelling as an essential complement to continuum models for upscaling the behavior of nanocolloids in porous media.

Enhanced permeability prediction in porous media using particle swarm optimization with multi-source integration

Accurately and efficiently predicting the permeability of porous media is essential for addressing a wide range of hydrogeological issues. However, the complexity of porous media often limits the effectiveness of individual prediction methods. This study introduces a novel Particle Swarm Optimization-based Permeability Integrated Prediction model (PSO-PIP), which incorporates a particle swarm optimization algorithm enhanced with dynamic clustering and adaptive parameter tuning (KGPSO). The model integrates multi-source data from the Lattice Boltzmann Method (LBM), Pore Network Modeling (PNM), and Finite Difference Method (FDM). By assigning optimal weight coefficients to the outputs of these methods, the model minimizes deviations from actual values and enhances permeability prediction performance. Initially, the computational performances of the LBM, PNM, and FDM are comparatively analyzed on datasets consisting of sphere packings and real rock samples. It is observed that these methods exhibit computational biases in certain permeability ranges. The PSO-PIP model is proposed to combine the strengths of each computational approach and mitigate their limitations. The PSO-PIP model consistently produces predictions that are highly congruent with actual permeability values across all prediction intervals, significantly enhancing prediction accuracy. The outcomes of this study provide a new tool and perspective for the comprehensive, rapid, and accurate prediction of permeability in porous media.

Low permittivity germanate LaAlGe2O7 ceramics: Synthesis, sintering behavior, spectral characteristics and microwave dielectric properties

Novel low permittivity LaAlGe2O7 ceramics was prepared by a solid-state reaction method, and the relationship between microstructure and microwave properties was revealed. XRD and Rietveld refinement confirmed that the crystal structure of LaAlGe2O7 is monoclinic P21/c space group. Raman spectroscopy reflects the trend of internal structural order and phonon-damping behavior of LaAlGe2O7 ceramics. The ceramic sintered at 1250 °C exhibited the best surface morphology and optimum microwave dielectric properties of εr = 9.22, Q × f = 33417 GHz, τf = −71.68 ppm/°C. The lattice energy and packing fraction are used to explain the change of Q × f value. The relationship between phonon damping behavior and microwave dielectric properties was revealed. The temperature stability was studied by bond valence. The LaAlGe2O7 ceramics is a strong candidate for the substrate material of millimeter wave communication.

Solving clustered low-rank semidefinite programs arising from polynomial optimization

We study a primal-dual interior point method specialized to clustered low-rank semidefinite programs requiring high precision numerics, which arise from certain multivariate polynomial (matrix) programs through sums-of-squares characterizations and sampling. We consider the interplay of sampling and symmetry reduction as well as a greedy method to obtain numerically good bases and sample points. We apply this to the computation of three-point bounds for the kissing number problem, for which we show a significant speedup. This allows for the computation of improved kissing number bounds in dimensions 11 through 23. The approach performs well for problems with bad numerical conditioning, which we show through new computations for the binary sphere packing problem.

From Hard to Soft Dense Sphere Packings: The Cohesive Energy of Barlow Structures Using Exact Lattice Summations for a General Lennard-Jones Potential.

There are infinitely many distinguishable dense sphere packings in dimension three (Barlow structures) with maximum packing density of ρ = π/√18. While Hales [Hales, T. C. Ann. Math. 2005, 162, 1065-1185.] proved that the packing density of the most dense cubic packing cannot be surpassed, it is currently not known how the infinite possibilities of densely packed Barlow structures with different sequences of hexagonal close packed layers are energetically related compared to the well-known face centered cubic (fcc) and hexagonal close packed (hcp) structures. We demonstrate that, for a general Lennard-Jones potential, which includes the hard-sphere model as a limiting case, Barlow packings lie energetically between fcc and hcp. Exceptions to this energy sequence only occur when fcc and hcp are energetically quasi-degenerate.

Rigidity of generalized Thurston’s sphere packings on 3-dimensional manifolds with boundary

Rigidity of generalized Thurston’s sphere packings on 3-dimensional manifolds with boundary

Treating produced water - crude oil emulsion using concentric packed UV- bioelectrochemical cell (CPUBEC)

This research paper comes in the context of the growing global scientific path to develop clean renewable energy production methods that are commensurate with the required environmental standards, reduce carbon emissions, and treat pollutants in the water produced with the production of crude oil from oil fields, which are large quantities and are constantly increasing with the growing global demand for oil and gas. A bioelectrochemical cell was manufactured from available capacitive materials (GQDs/TiO2) with biofilm spread in the parts that are not directly exposed to UV ray in an innovative installation consisting of a cylindrical shell with spherical packing anode and a cathode in the middle of membrane that separates the tow electrodes. Both electrodes are arranged in a Pyrex pot and connected to external resistance. The packed cell (with and without uv-ray) performance was compared with unpacked cell to determined the effect of the packing and UV ray by studying the efficiency of removing pollutants TPH and COD. the best results in the packed cell with UV ray as follows:97.1 %, 97 % respectively, while the best results obtained in the case of using an unpacked cell was 86.6 %, 84.8 %. the change in the pH during the operating cycle was also studied and it was found that the change in the packed cell was slightly greater than the change in the unpacked cell as a result of the complex interactive on the surface of the anode. The highest current density produced was 1316 mA/m2 on the fifth day in packed cell and presence of UV ray.

Algebraic curves and sphere packings

Algebraic curves and sphere packings

Optimizers of three-point energies and nearly orthogonal sets

This paper is devoted to spherical measures and point configurations optimizing three-point energies. Our main goal is to extend the classic optimization problems based on pairs of distances between points to the context of three-point potentials. In particular, we study three-point analogues of the sphere packing problem and the optimization problem for p p -frame energies based on three points. It turns out that both problems are inherently connected to the problem of nearly orthogonal sets by Erdős. As the outcome, we provide a new solution of the Erdős problem from the three-point packing perspective. We also show that the orthogonal basis uniquely minimizes the p p -frame three-point energy when 0 > p > 1 0>p>1 in all dimensions. The arguments make use of multivariate polynomials employed in semidefinite programming and based on the classical Gegenbauer polynomials. For p = 1 p=1 , we completely solve the analogous problem on the circle. As for higher dimensions, we show that the Hausdorff dimension of minimizers is not greater than d − 2 d-2 for measures on S d − 1 \mathbb {S}^{d-1} .

Direct Visualization of Polystyrene Sphere Packing in Solution with Cryo-FIB

Direct Visualization of Polystyrene Sphere Packing in Solution with Cryo-FIB

Assessment of Models for Nonlinear Oscillatory Flow Through a Hexagonal Sphere Pack

We review models for unsteady porous media flow in the volume-averaging framework and we discuss the theoretical relations between the models and the definition of the model coefficients (and the uncertainty therein). The different models are compared against direct numerical simulations of oscillatory flow through a hexagonal sphere pack. The model constants are determined based on their definition in terms of the Stokes flow, the potential flow and steady nonlinear flow. Thus, the discrepancies between the model predictions and the simulation data can be attributed to shortcomings of the models’ parametrisation. We found that an extension of the dynamic permeability model of Pride et al. (PRB 47(9):4964–4978, 1993) with a Forchheimer-type nonlinearity performs very well for linear flow and for nonlinear flow at low and medium frequencies, but the Forchheimer term with a coefficient obtained from the steady-state overpredicts the nonlinear drag at high frequencies. The model reduces to the unsteady Forchheimer equation with an acceleration coefficient based on the static viscous tortuosity for low frequencies. The unsteady Forchheimer equation with an acceleration coefficient based on the high-frequency limit of the dynamic tortuosity has large errors for linear flow at medium and high frequencies, but low errors for nonlinear flow at all frequencies. This is explained by an error cancellation between the inertial and the nonlinear drag.

Processing of fully dense and high conductivity W[sbnd]20Cu composite via copper melt infiltration into a pressed tungsten wire skeleton

The current study proposed melt infiltration into entangled pressed wire as a proper method for rapid and energy-efficient production of tungsten‑copper composites. The process includes wire packing through different strategies, pressing, and infiltration steps. First, a predesigned wire structure through different strategies of simple, spiral, cylindrical, and spherical is packed and pressed to form a permeable tungsten skeleton. Then, by infiltrating molten copper under a dry hydrogen atmosphere into the porous preforms, the W20Cu composite is fabricated and characterized. An almost fully dense sample with excellent hardness of 238.1 HV1 and high conductivity of 58.64% IACS in W20Cu composite was achieved. These values are almost 25% and 50% higher than that from the conventional melt infiltration method, respectively. It was seen that the packing strategy is so effective in achieving the best properties where the processed composite by spherical packing strategy exhibits better density, conductivity, and hardness than the sample produced via the other three packing strategies. However, all the properties of the melt infiltration into entangled pressed wire processed composite via all packing strategies are higher than those processed via conventional methods. These excellent properties could be attributed to the mostly bonded tungsten phase, high purity, pore-free, and suitable distribution of tungstens which are a result of this process characteristics. Besides, lower cycle time and energy consumption are other advantages of the new process compared to conventional melt infiltration via powder metallurgy. This approach presents a promising prospect for future industrial applications.

Deformation, shape transformations, and stability of elastic rod loops within spherical confinement

Mechanical insight into the packing of slender objects within confinement is essential for understanding how polymers, filaments, or wires organize and rearrange in limited space. Here we combine theoretical modeling, numerical optimization, and experimental studies to reveal spherical packing behavior of thin elastic rod loops of homogeneous or inhomogeneous stiffness. Across varying loop lengths, a rich array of configurations including circle, saddle, figure-eight, and more intricate patterns are identified. A theoretical framework rooted in the local equilibrium of force and moment is proposed for the rod loop deformation, facilitating the determination of internal and contact forces experienced by the rods during deformation. For the confined homogeneous rod loops, their stable and metastable configurations are well described using proposed Euler rotation curves, which offer a concise and effective approach for configuration prediction. Moreover, formulated analysis on the stability and critical force for homogeneous rod loops on great circles of the spherical confinement are performed. For inhomogeneous rod loops with two segments of differing stiffness, the stiffer segment exhibits less deviation from the great circle, while the softer segment undergoes more pronounced deformation. These findings not only enhance our understanding of buckling and post-buckling phenomena but also offer insights into filament patterning within confining environments.

Resolving Flow Structures Inside Three Dimensional Packed Beds With Gas Flow By Ray Tracing Particle Image Velocimetry

Two different configurations are investigated in this study by ray tracing Particle Image Velocimetry to access the interior of different spherical packing structures. Configuration 1 consists of a regular bidisperse packed bed of 10mm and 40mm spheres based on a body-centered cubic arrangement. Ray tracing Particle Image Velocimetry was performed behind two spheres in the 40mm layer for particle Reynolds numbers from 200 to 700. The averaged velocity flow fields show the jet structures according to the pores between the underlying smaller spheres and the influence of the large spheres. Since ray-tracing Particle Image Velocimetry suffers from experimental and numerical limitations, planar 2D Particle Image Velocimetry measurements are performed to obtain spatially highly resolved validation data in the optically accessible void pores. These measurements were performed at different height positions in the bed showcasing that similar flow fields appear in the different pores. Configuration 2 deals with the jet flow (Re=5900) around an individual sphere, part of a body-centred cubic arrangement of d=40mm spheres. The emphasis lays on the temporally resolved snapshot data of the turbulent jet flow obtained by ray tracing Particle Image velocimetry. Data obtained from both configurations is used for a comprehensive discussion of the chances and challenges of ray tracing Particle Image Velocimetry. Limitations appear on the experimental side, like generation of depth of field, illumination and reflections, sufficient measurement signal, and accuracy of the geometry, but also on the numerical side, like the reproduction of the geometry, material properties, and real-world conditions. Since all these phenomena are interlinked, performing the ray tracing method in more complex systems is questionable.