Computer Simulation of Random Sphere Packing in an Arbitrarily Shaped Container

Most simulations of random sphere packing concern a cubic or cylindric container with periodic boundary, containers of other shapes are rarely studied. In this paper, a new relaxation algorithm with pre-expanding procedure for random sphere packing in an arbitrarily shaped container is presented. Boundaries of the container are simulated by overlapping spheres which covers the boundary surface of the container. We find 0.4~0.6 of the overlap rate is a proper value for boundary spheres. The algorithm begins with a random distribution of small internal spheres. Then the expansion and relaxation procedures are performed alternately to increase the packing density. The pre-expanding procedure stops when the packing density of internal spheres reaches a preset value. Following the pre-expanding procedure, the relaxation and shrinking iterations are carried out alternately to reduce the overlaps of internal spheres. The pre-expanding procedure avoids the overflow problem and gives a uniform distribution of initial spheres. Efficiency of the algorithm is increased with the cubic cell background system and double link data structure. Examples show the packing results agree well with both computational and experimental results. Packing density about 0.63 is obtained by the algorithm for random sphere packing in containers of various shapes.     

Mono-sized sphere packing algorithm development using optimized Monte Carlo technique

In this paper, fuel cell catalyst layer was developed using the optimized sphere packing algorithm. An optimization technique named adaptive random search technique (ARSET) was employed in this packing algorithm. The ARSET algorithm will generate the initial location of spheres and allow them to move in the random direction with the variable moving distance, randomly selected from the sampling range (α), based on the Lennard–Jones potential and Morse potential of the current and new configuration. The solid fraction values obtained from this developed algorithm are in the range of 0.610–0.624 while the actual processing time can significantly be reduced by 5.58–34% based on the number of spheres. The initial random number sampling range (α) was investigated and the appropriate α value is equal to 0.5.     

An efficient dense and stable particular elements generation method based on geometry

In this paper, a new discrete elements generation method based on geometry is proposed to fill geometric domains with particles (disks or spheres). By generating particles each one with a random radius or with a radius calculated from the iteration to ensure no overlaps exist between particles and identifying unstable particles and changing them to stable ones, a dense and stable packing can be created. A partitioning particle radius interval method and a particle stability inspection and improvement method are introduced to guarantee the algorithm's success and the stability of the particles. Some packings were created to evaluate the performances of the new method. The results showed that the algorithm was very efficient and was able to create isotropic packings of low porosities and large coordinate numbers. The partitioning particle radius interval method improved the generation efficiency significantly and increased the packing densities. Through the comparisons with several existing methods proposed recently, the method proposed in this work is found to be more efficient and can fill geometric domains with the lowest porosities. In addition, the stability of the particles is guaranteed and no complex triangular or tetrahedral mesh is required in particle generation, thereby making the new method simpler. Copyright © 2016 John Wiley & Sons, Ltd.     

Concerning the geometric limit of the density of a loose medium modeled by identical spherical particles

Based on the introduced system of definitions, the statistical-geometric property of a random close packing (RCP) of identical solid spheres (SS) is found that determines the geometric limit of the packing density. The results of calculations using computer models of SS packings show that the magnitude of the geometric limit virtually coincides with a real limiting density of the RCP of SS.Notation _max limiting density of packing in an RCP of SS - V ₀ volume of packing before placing trial spheres - V volume of packing after placing trial spheres - N number of spheres in a packing - k quantity of trial spheres - density of packing - x distance from a given point to the nearest center of SS - inaccessible volume - x _max the largest value ofx in a packing - _g model estimate of the limiting density of packing Ural State Technical University, Ekaterinburg, Russia. Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 68, No. 4, pp. 564–568, July–August, 1995.     

Bulk and interior packing densities of random close packing of hard spheres

The packing densities of random close packing of equal hard spheres (RCPHS) are studied. The RCPHS is generated by a rearrangement algorithm with an optimization subroutine. Traditionally defined packing density, bulk density, is found to be 0.635 ± 0.002 by extrapolation to infinite number of spheres. We propose that there exist a characteristic packing density without boundary effects. This interior packing density is calculated by two methods, resulting in values without statically significant difference. Interior packing density deduced from Voronoi diagram is 0.6690 ± 0.0006. Local packing density for each sphere is defined as ratio of its volume to volume of its corresponding Voronoi cell and is sensitive to sphere's local configuration and overlapping.     

A three dimensionally periodic,eleven coordinated,dense packing of symmetry equivalent spheres

Symmetry equivalent equal size spheres can be packed in an array filling 71.87% of space. Each sphere in this packing has eleven neighbors at unit distance. The density of this tetragonal close packing is second only to closest packing with twelve coordination. Anions in numerous compounds, including some which are solid electrolytes, arrange themselves in this pattern. It occurs among others in rutiles, β-BeO, Li₄GeO₄ and CaSO₄.     

Experimental study on the packing densification of mixtures of spherical and cylindrical particles subjected to 3D vibrations

To identify the dense packing of cylinder–sphere binary mixtures (spheres as filling objects), the densification process of such binary mixtures subjected to three-dimensional (3D) mechanical vibrations was experimentally studied. Various influential factors including vibration parameters (such as vibration time t, vibration amplitude A, frequency ω, vibration acceleration Γ) as well as particle size ratio r (small sphere vs. large cylinder), composition of the binary mixtures X_L (volume fraction of cylinders), and container size D (container diameter) on the packing density ρ were systematically investigated. The results show that the optimal vibration parameters for different binary cylinder–sphere mixtures are different. The smaller the size ratio, the less vibration acceleration is needed to form a stable dense packing. For each binary mixture, high packing density can be obtained when the volume fraction of large cylindrical particles is dominant. Meanwhile, increasing the container size can decrease the container wall effect and get higher packing density. The proposed analytical model has been proved to be valid in predicting the packing densification of current cylinder–sphere binary mixtures.     

Improved compaction of ZnO nano-powder triggered by the presence of acetate and its effect on sintering

The retention of nanocrystallinity in dense ceramic materials is still a challenge, even with the application of external pressure during sintering. The compaction behavior of high purity and acetate enriched zinc oxide (ZnO) nano-powders was investigated. It was found that acetate in combination with water plays a key role during the compaction into green bodies at moderate temperatures. Application of constant pressure resulted in a homogeneous green body with superior packing density (86% of theoretical value) at moderate temperature (85 °C) in the presence of water. In contrast, no improvement in density could be achieved if pure ZnO powder was used. This compaction behavior offers superior packing of the particles, resulting in a high relative density of the consolidated compact with negligible coarsening. Dissolution accompanying creep diffusion based matter transport is suggested to strongly support reorientation of ZnO particles towards densities beyond the theoretical limit for packing of ideal monosized spheres. Finally, the sintering trajectory reveals that grain growth is retarded compared to conventional processing up to 90% of theoretical density. Moreover, nearly no radial shrinkage was observed after sinter-forging for bodies performed with this advanced processing method.     

Improved compaction of ZnO nano-powder triggered by the presence of acetate and its effect on sintering

Abstract

The retention of nanocrystallinity in dense ceramic materials is still a challenge, even with the application of external pressure during sintering. The compaction behavior of high purity and acetate enriched zinc oxide (ZnO) nano-powders was investigated. It was found that acetate in combination with water plays a key role during the compaction into green bodies at moderate temperatures. Application of constant pressure resulted in a homogeneous green body with superior packing density (86% of theoretical value) at moderate temperature (85 °C) in the presence of water. In contrast, no improvement in density could be achieved if pure ZnO powder was used. This compaction behavior offers superior packing of the particles, resulting in a high relative density of the consolidated compact with negligible coarsening. Dissolution accompanying creep diffusion based matter transport is suggested to strongly support reorientation of ZnO particles towards densities beyond the theoretical limit for packing of ideal monosized spheres. Finally, the sintering trajectory reveals that grain growth is retarded compared to conventional processing up to 90% of theoretical density. Moreover, nearly no radial shrinkage was observed after sinter-forging for bodies performed with this advanced processing method.     

A statistical analysis of void size distribution in a simulated narrowly graded packing of spheres

The void microstructure of a simulated packing of polydisperse spheres has been investigated by means of a radical Delaunay tessellation. We have focused on creating sphere packings by mimicking processes involved in the construction of embankment dams: the polydisperse spheres are collectively released under gravity and denser states are mainly obtained by means of shearing cycles. This study has been performed on a narrowly graded material for four porosities ranging from 0.42 to 0.36. The void structure is quantified in terms of probability density functions of pore and constriction sizes, cumulative distributions and connectivity functions. We emphasize the implications of the sample construction technique on the geometric packing arrangements, among them a well disordered medium where tetrahedra remain the most represented unit void structure. We point out that when porosity decreases, void distributions become narrower but the initial structure is never destroyed. Nevertheless, the densification modifies significantly the computed mean void quantities. In this study, usual geometric arrangements obtained for very dense materials are not encountered.     

A particle packing algorithm for packed beds with size distribution

A particle packing algorithm for simulating realistic packed beds of spheres with size distribution is described. The algorithm used the Monte Carlo method combined with the simulated annealing minimisation algorithm to solve the packed bed simulations. The objective function which was minimised was a combination of two functions, one describing the deviation from the target mean coordination number of the spheres in each size interval and the other the average fraction of overlapping volume of the spheres per contact. In this way a realistic bed structure was maintained while at the same time controlling the coordination number of the spheres. The algorithm used an experimentally validated model to predict the mean coordination number of the spheres in each size interval.     

DEM simulation of binary sphere packing densification under vertical vibration

The densification of random binary sphere packings subjected to vertical vibration was modeled by using the discrete element method (DEM). The influences of operating parameters such as the vibration conditions, sphere size ratio (diameter ratio of larger versus small spheres), and composition (volume fraction of large spheres) of the binary mixture on the fractional packing density φ (defined as the volume of spheres divided by the volume of container) were studied. Two packing states, i.e., random loose packing (RLP) and random close packing (RCP), were reproduced and their micro properties such as the coordination number (CN), radial distribution function (RDF), and force structure were characterized and compared. The results indicate that properly controlling vibration conditions can realize the transition of binary packing structure from the RLP to RCP state when the sphere size ratio and composition are fixed, and the fractional packing density for RCP after vibration can reach φ_RCP?≈?0.86. Different packing characteristics from RLP and RCP identify that RCP shows much denser and more uniform structure than RLP. The current modeling results are in good agreement with those obtained from both the physical experiments and the proposed empirical models in the literature.     

CFD-DEM numerical study on air impacted packing densification of equiaxed cylindrical particles

A CFD-DEM model was developed to reproduce the packing densification process of mono-sized equiaxed cylindrical particles under air impact. The effects of operating parameters on packing density were firstly studied. Then various microscopic properties of packing structures such as coordination number (CN), contact types, particle orientations, pore features were characterized and compared. And corresponding densification mechanisms were analysed based on particle motion behaviour, local structure evolution, and forces. Results indicate that the air impact can realize the packing densification of cylindrical particles under appropriate conditions. The pore size distribution in the packing of cylindrical particles shows a tail at larger pore sizes compared with that in the packing of equal spheres. Both the size and the sphericity of the pores decrease in the final dense packing; also, more surface-surface and less surface-edge contacts between two particles therein can be formed. More cylindrical particles tend to be in parallel or perpendicular contact with each other to form more stable local structures during air impact. Most particles at higher position move down (direction of gravity/air impact) with about one particle length during the densification process and most particles exhibit translational motion to realize the local rearrangement for pore filling through air impact induced inter-particle forces.     

Computer simulation of morphology and packing behaviour of irregular particles, for predicting apparent powder densities

This paper describes a methodology for prediction of powder packing densities which employs a new approach, designated as random sphere construction (RSC), for modelling the shape of irregular particles such as those produced by water atomization of iron. The approach involves modelling an irregular particle as a sphere which incorporates smaller corner spheres located randomly at its surface. The RSC modelling technique has been combined with a previously developed particle packing algorithm (the random build algorithm), to provide a computer simulation of irregular particle packings. Analysis of the simulation output data has allowed relationships to be established between the particle modelling parameters employed by the RSC algorithm, and the density of the simulated packings. One such parameter is η, which is the number of corner spheres per particle. A relationship was established between η (which was found to have a profound influence on packing density), and the fractional density of the packing, fd. Vision system techniques were used to measure the irregularity of the simulated particles, and this was also related to η. These two relationships were then combined to provide a plot of fractional density for a simulated packing against irregularity of the simulated particles. A comparison was made of these simulated packing densities and observed particle packing densities for irregular particles, and a correlation coefficient of 0.96 was obtained. This relatively good correlation indicates that the models developed are able to realistically simulate packing densities for irregular particles. There are a considerable number of potential applications for such a model in powder metallurgy (PM), process control. In combination with on-line particle image analysis, the model could be used to automatically predict powder densities from particle morphology.     

Light-scattering efficiency of starch acetate pigments as a function of size and packing density

We study theoretically the light-scattering efficiency of paper coatings made of starch acetate pigments. For the light-scattering code we use a discrete dipole approximation method. The coating layer is assumed to consists of roughly equal-sized spherical pigments packed either at a packing density of 50% (large cylindrical slabs) or at 37% or 57% (large spheres). Because the scanning electron microscope images of starch acetate samples show either a particulate or a porous structure, we model the coatings in two complementary ways. The material can be either inside the constituent spheres (particulate case) or outside of those (cheeselike, porous medium). For the packing of our spheres we use either a simulated annealing or a dropping code. We can estimate, among other things, that the ideal sphere diameter is in the range 0.25-0.4 microm.     

Contributions to the generalization of advancing front particle packing algorithms

The generation of a set of particles with high initial volume fraction is a major problem in the context of discrete element simulations. Advancing front algorithms provide an effective means to generate dense packings when spherical particles are assumed. The objective of this paper is to extend an advancing front algorithm to a wider class of particles with generic size and shape. In order to get a dense packing, each new particle is placed in contact with other two (or three in 3D) particles of the advancing front. The contact problem is solved analytically using wrapping intersection technique. The results presented herein will be useful in the application of these algorithms to a wide variety of practical problems. Examples of geometric models for applications to biomechanics and cutting tools are presented. Copyright © 2015 John Wiley & Sons, Ltd.     

全级配混凝土三维细观模型的建模方法研究

根据混凝土材料细观结构特点,采用混合同余算法生成随机数,模拟混凝土骨料在形状和空间分布方面具有的随机性特点。按照混凝土级配理论,分析了不同级配混凝土细观组成特点,研究了不同粒径骨料的含量和分布规律,提出了一种高效的三维骨料生长和凸性判定算法,研究了骨料生长过程中的形状控制技术,避免了尖角、薄片状骨料的出现;在骨料投放过程中采用综合投放算法,大幅度提高了骨料一次性投放成功率和骨料体积含量,缩短了计算时间。在三维细观有限元网格剖分方面,采用向量判别法,优化了材料属性判定,提高了计算效率。算例表明,该文提出的细观模型,在模拟混凝土静力特性和冲击破坏特性方面具有较高的可靠性。     

Effect of vibration direction on the packing of sphero-cylinders

Particle packing is widely encountered when coping with granular materials, while mechanical vibration is usually used for packing densification. Vibration direction has been proven to be crucial for the ordered packing of spherical particles, but there are few reports for non-spherical ones in this regard. In this study, the effect of vibration direction on the macroscopic and microscopic packing parameters of sphero-cylinders are systematically examined using discrete element method (DEM). Due to the anisotropic shapes of sphero-cylinders, their packing characteristics are much richer and also more complex than those of spheres. It is found that vibration direction affects both the packing density and the packing structure of sphero-cylinders through tuning their orientation distributions and contact modes. Moreover, vibration direction plays a significant role in determining the optimal vibration intensities for dense packing. When the sphericity of Voronoi cell decreases and/or the density increases, the Nematic order parameter increases accordingly. Besides, no obvious relationship between the packing density and the average contact number is observed.     

Structural shape and topology optimization based on level-set modelling and the element-propagating method

This article introduces the element-propagating method to structural shape and topology optimization. Structural optimization based on the conventional level-set method needs to solve several partial differential equations. By the insertion and deletion of basic material elements around the geometric boundary, the element-propagating method can avoid solving the partial differential equations and realize the dynamic updating of the material region. This approach also places no restrictions on the signed distance function and the Courant–Friedrichs–Lewy condition for numerical stability. At the same time, in order to suppress the dependence on the design initialization for the 2D structural optimization problem, the strain energy density is taken as a criterion to generate new holes in the material region. The coupled algorithm of the element-propagating method and the method for generating new holes makes the structural optimization more robust. Numerical examples demonstrate that the proposed approach greatly improves numerical efficiency, compared with the conventional level-set method for structural topology optimization.     

一刀切约束下的二维装箱问题高效求解算法

目的 为实现大规模物料的快速剪裁切割,对考虑一刀切约束的二维装箱问题进行研究,并构建相应的改进优先度算法IPH(Improved Priority Algorithm,IPH).方法 IPH能够在不需要任何迭代搜索下,直接进行剩余空间分割与填充.为此,发展PH算法中的优先度放置规则,并以最大化生成大空间面积和最小化生成小空间面积为基础,设计改进砌砖式空间分割策略.结果 针对标准数据集的对比实验表明,IPH能够在较短时间内完成大规模算例的高效求解,并首次获得了多个算例的最优填装效果.结论 基于概率较优的启发式求解方法,能够实现无迭代优选下的一刀切二维装箱问题直接求解,且运算效果令人满意.