Collection Of Spheres Research Articles

Properties governing the flow of solution through crushed ore for heap leaching

Heap leaching of low grade ores and waste rock is often limited by the hydrology of the bed, which determines the rate of dissolution of the target mineral. The lead author has performed physical and hydraulic testing of a large number of ore samples considered for heap leaching. The data was reviewed to develop relationships between the hydrology and the physical properties of the ore. Compressibility of the agglomerates under load was found to increase with silt + clay (−75 μm) content as high contents of silt and clay result in the destruction of porosity under load. The residual moisture held up in the bed after draining under gravity was found to be proportional to the surface area calculated from the PSD, assuming the bed is a collection of spheres. The residual moisture was also proportional to the sand content (−4.75 mm), as a result of the domination of capillary forces over gravity forces at lower particle diameters and smaller interstitial spaces. The pressure drop versus flow relationship during saturated flow was modelled with a modified form of the Carman-Kozeny equation, whereby the particle radius is replaced with a hydraulic radius. The ratio between the experimentally determined hydraulic radius and the hydraulic radius calculated from the PSD surface area was used to calculate the tortuosity. Since the minimum particle size of the PSD is not exactly known, minimum diameters of 0.002 mm, 0.02 mm and 0.2 mm were used to calculate the PSD surface area. A value of 0.2 mm was found to provide the most physically realistic tortuosity values of between 1.48 and 4.44.

Approximate shape factors for soft magnetic composites

A general theory of the shape tensor is outlined, providing estimates of the shape or demagnetisation factors for either single or collections of magnetic particles. In the general case of multiple interacting magnetic particles, shape factors are insufficient to completely characterise the magnetic field. This paper provides approximate expressions for the shape factors for isolated magnetic cylinders, isolated blocks, and collections of spheres. The key assumption is that the stray field is approximately spatially constant over each magnetic particle, leading to a system of equations for the shape factors. These equations are linear for magnetic particles of very high susceptibility, and imply shape factors have non-local properties. Non-random particle alignment can lead to very small shape factors, and corresponding high susceptibility, while random particle alignments tend to produce shape factors approximating their isolated values. The special case, when the matrix system for the shape factors is singular, corresponds to the threshold when the composite susceptibility changes from depending on, to being independent of, the particle susceptibilities. An appendix summarises the magnetic fields from uniformly magnetised point dipoles, spheres, cylinders, ellipsoids and blocks.

Translational and rotational velocities in shear-driven jamming of ellipsoidal particles.

We study shear-driven jamming of ellipsoidal particles at zero temperature with a focus on the microscopic dynamics. We find that a change from spherical particles to ellipsoids with aspect ratio α=1.02 gives dramatic changes of the microscopic dynamics with much lower translational velocities and a new role for the rotations. Whereas the velocity difference at contacts-and thereby the dissipation-in collections of spheres is dominated by the translational velocities and reduced by the rotations, the same quantity is in collections of ellipsoids instead totally dominated by the rotational velocities. By also examining the effect of different aspect ratios we find that the examined quantities show either a peak or a change in slope at α≈1.2, which thus gives evidence for a crossover between different regions of low and high aspect ratio.

On the Complexity of Closest Pair via Polar-Pair of Point-Sets

Every graph $G$ can be represented by a collection of equi-radii spheres in a $d$-dimensional metric $\Delta$ such that there is an edge $uv$ in $G$ if and only if the spheres corresponding to $u$ and $v$ intersect. The smallest integer $d$ such that $G$ can be represented by a collection of spheres (all of the same radius) in $\Delta$ is called the sphericity of $G$, and if the collection of spheres are nonoverlapping, then the value $d$ is called the contact-dimension of $G$. In this paper, we study the sphericity and contact-dimension of the complete bipartite graph $K_{n,n}$ in various $L^p$-metrics and consequently connect the complexity of the monochromatic closest pair and bichromatic closest pair problems.

Multiscale modeling of organic molecules contamination in head-disk interface under high thermal stress

Heat assisted magnetic recording (HAMR) which locally reduces coercivity with a laser pulse, is currently the most promising technology to achieve areal density beyond 10 Tbit/in2. However, due to the extreme operating condition of HAMR, the head-disk interface (HDI) suffers from extensive depletion and contamination of organic molecules. Our previous studies indicate that the conventional linear perfluoropolyether (PFPE) lubricants/grease molecules are not suitable for the severe thermal stress especially when coupled with external fields as the lubricant layer can be depleted and damaged. To simulate the molecular evaporation phenomena, we introduced novel multiscale modeling scheme, “Collection Of Spheres” model, which can predict mesoscale phenomena based on the atomistic/molecular level details via reduced order method. We have examined how the molecular architect affects evaporation under various degree of thermal stress (e.g., peak temperature and duration). Specifically, we examined several PFPEs (e.g., Z, Zdol, and Ztetraol) to study the effect of back-bone and end group properties on evaporation. Our multiscale model can provide holistic simulation of the HDI and provide molecular design criteria for HAMR device.

A Computational Model for E. coli Cytoplasm: Diffusion and Hydrodynamics

The dynamics of proteins is essential for the quantification of various cellular processes like rates of enzymatic reactions, signal transduction and protein association reactions. However, understanding the structure and dynamics of macromolecules in a cell is complicated by the highly crowded nature of the cell. It is likely that properties of macromolecules in cell may differ significantly to that measured in dilute solution. Diffusion plays important roles in many processes occurring inside the cell. The estimation of diffusion coefficient of macromolecules in a cell can be considered as a first step in understanding the complex nature of the heterogeneous environment of the cell.In this current work we developed a computational model of E. coli cytoplasm and performed extensive Brownian dynamics simulation to calculate diffusivity of proteins. Our model differs from some of the previous models of E. coli cytoplasm in the following way; (1) The proteins modeled as flexible units by considering them as a collection of spheres. (2) hydrodynamic interaction (HI), which is essential to get accurate diffusion coefficient, was considered using a mean field approach.The model predicts accurately the diffusion coefficient of Green Fluorescent Protein (GFP) in E.coli cell. We have found that HI is essential to get correct diffusion coefficient for this highly crowded system. The presence of anomalous diffusion has also been observed for short time (∼1 micro sec), which was identified using fractional Brownian motion (FBM) analysis. It was found that repulsive interaction between different proteins is the main reason for the anomalous diffusion. To understand the anomalous diffusion observed in simulations, we also formulated a one dimensional random walk model in which successive steps are biased and correlated. This analytical model can explain some of the findings from our simulation.

Random light scattering by collections of ellipsoids

Theory of weak scattering of random optical fields from deterministic collections of particles with soft ellipsoidal scattering potentials of arbitrary shapes and orientations is developed. Far-field intensity distribution produced on scattering is shown to be influenced by source correlation properties as well as by a number, shapes and orientations of scatterers. The theory extends previous results on scattering from collections of spheres with soft Gaussian potentials and is applicable to analysis of a wide range of media including blood cells.

The influence of hydrodynamic interactions on protein dynamics in confined and crowded spaces—assessment in simple models

We consider several systems that are confined within a softly repulsive sphere. The first one is a model protein, crambin, which is described by a structure-based coarse grained model. We demonstrate that the folding process is accelerated by the hydrodynamic interactions (HI) in a way that depends on the radius of the sphere. The tighter the encompassing sphere, the smaller the effect, independent of the nature of the starting conformations. The second system is a protein surrounded by protein-like softly repulsive spheres that make the confined space crowded. In this case, the HI shorten the folding times in a way which depends on the degree of crowdedness only weakly. The third system is a collection of spheres that are meant to represent molecules. We show that confinement increases association times. We also observe that the HI either facilitate or obstruct association of two spheres depending on the crowding conditions. The dependence of the association time on crowdedness in the confining sphere is qualitatively distinct from that derived by Wieczorek and Zielenkiewicz for a cube with the periodic boundary conditions.

A New and Efficient Poisson-Boltzmann Solver for Interaction of Multiple Proteins.

We derive a new numerical approach to solving the linearized Poisson Boltzmann equation (PBE) by representing the protein surface as a collection of spheres in which the surface charges can then be iteratively solved by new analytical multipole methods previously introduced by us [Lotan & Head-Gordon, 2006]. We show that our Poisson Boltzmann semi-analytical method, PB-SAM, realizes better accuracy, more flexible memory management, and at reduced cost relative to either finite difference or boundary element method PBE solvers. We provide two new benchmarks of PBE solution accuracy to test the numerical PBE solutions based on (1) arrays of up to hundreds of spherical low dielectric geometries with asymmetric charges in which mutual polarization is treated exactly, and (2) two overlapping spheres with increasing charge asymmetry by solving the PB-SAM method to very high pole order. We illustrate the strength of the PB-SAM approach by computing the potential profile of an array of 60 T1-particle forming monomers of the bromine mosaic virus.

Acoustic repropagation technique and practical source characterization for simulation noise model databases.

By utilizing the acoustic repropagation technique in conjunction with an open-source, platform-independent database, a robust noise source can be constructed that is well suited for use in simulation noise models. This paper describes a widely applicable methodology for creating a three dimensional noise source from measurements that can account for source directivity. It will be shown how the resulting data, known as noise spheres, may contain a variety of noise metrics representing the source at a fixed distance for all spherical angles. Such spheres can be combined to represent complex noise sources and depending on their metric content can be used to accurately predict the near and far field environment about the source. A case utilizing narrow band spectra with phase information will be explored. It will be shown how a collection of spheres can be used to represent different operating states of a source and how the use of an independent parameter such as speed or power can be used to interpolate non-exact states in a noise model. By way of example, an approach to incorporating non-linear sound propagation from a high amplitude noise source is explored in conjunction with the advanced acoustic model.

Genus Expansion for Real Wishart Matrices

We present an exact formula for moments and cumulants of several real compound Wishart matrices in terms of an Euler characteristic expansion, similar to the genus expansion for complex random matrices. We consider their asymptotic values in the large matrix limit: as in a genus expansion, the terms which survive in the large matrix limit are those with the greatest Euler characteristic, that is, either spheres or collections of spheres. This topological construction motivates an algebraic expression for the moments and cumulants in terms of the symmetric group. We examine the combinatorial properties distinguishing the leading order terms. By considering higher cumulants, we give a central-limit-type theorem for the asymptotic distribution around the expected value.

A211 三次元微視的数値計算に基づく充填層の流動抵抗と熱分散の予測(OS-3:多孔質体内の伝熱(I))

Three-dimensional numerical simulation has been conducted to determine the permeability, Forchheimer coefficient, thermal dispersion coefficient and interfacial heat transfer coefficient within packed bed in porous medium. A macroscopically uniform flow is assumed to pass through a collection of spheres placed regularly in an infinite space, where a temperature gradient is imposed to flow direction. Due to the periodicity of the model, only one structural unit can be taken for a calculation domain to resolve an entire domain of porous medium. The numerical results thus obtained are integrated over a unit structure to determine the permeabilty and dispersion coefficient. These macroscopic values agree well with empirical Ergun equation and Wakao-Kaugei equation.

Advancing front techniques for filling space with arbitrary separated objects

A review is given of advancing front techniques for filling space with arbitrary separated objects. Over the last decade, these techniques have reached a considerable degree of maturity and are being used to generate clouds of points for SPH and FPM simulations, as well as spheres, ellipsoids, objects defined by a collection of spheres or polyhedral objects for DEM simulations. Algorithmic as well as implementational aspects are discussed. Techniques to obtain maximum packing, such as closest object placement (during generation) and move/enlarge (after generation) are also considered. Several examples are included that demonstrate the capabilities developed.

SU-FF-I-113: Anatomical Power Spectrum and Detectability: An Analytical and Experimental Basis

Purpose: Superposition of anatomical clutter in medical images is known to degrade detectability of underlying structures. This paper investigates background power spectrum (PS) in tomosynthesis and cone‐beam CT(CBCT) in a manner that is general to various applications (e.g., breast and chest imaging) and allows analysis of generalized noise‐equivalent quanta (GNEQ) and task‐specific detectability. Methods: An analytical basis for physical phantom design was established from principles of self‐similarity and fractal dimension. Random arrangements of equal volumes of spheres were found to present power‐law noise comparable to that of, for example, breast or chest imaging. Rectangular and cylindrical phantoms containing spheres of various diameter (3–16 mm) were configured on an experimental bench for tomosynthesis and CBCT. Background PS was analyzed as a function of orbital extent, ranging from ∼10°–360°. Results were fit to the empirical form K/f∧b and incorporated in the GNEQ to investigate detectability versus tomosynthesis arc, imaging task, and dose. Results: Clutter phantoms based on self‐similar collections of spheres provided background PS described well by power‐law noise. 3D background PS were highly asymmetric, subtending the tomosynthesis arc in the Fourier domain and scales with the number of backprojected views. Analysis of the fully 3D background PS yields K and b that vary with tomosynthesis angle in a manner consistent with expectations, is distinct from that previously analyzed in 2D slices, and highlights the effect of orbital extent on 3D detectability. Analysis of GNEQ identified total arc and dose required to achieve a given level of detectability, depending on the imaging task. Conclusions: An analytical basis drawn from fractal theory guided phantom design providing power‐law noise.Analysis of background PS, GNEQ, and detectability index over a broad range of system configurations provided a general basis pertinent to modality‐specific systems, such as breast and chest tomosynthesis and CBCT.

Positive curvature effects and interparticle capillary condensation during nitrogen adsorption in particulate porous materials

The adsorption of nitrogen in a collection of spheres that touch or merge in a sintering-like manner is modeled using a Derjaguin–Broeckhof–de Boer approach. The proposed model accounts for both positive curvature effects and for capillary condensation at the contact between two spheres. A methodology is proposed to fit the P / P 0 > 0.4 adsorption region with the coordination number of the spheres as the only adjustable parameter. The use of the model is illustrated on a series of silica aerogels. The suitability of various standard isotherms needed for the modeling is also discussed.

Packing spheres and fractal Strichartz estimates in $\mathbb {R}^d$ for $d\geq 3$

We prove an estimate for the spherical average operator in R d if d > 3. This leads to a lower bound for the Hausdorff dimension of unions of certain collections of spheres and to a Strichartz-type estimate for solutions of the wave equation.

Numerical simulation of rock behaviour through a discrete model

A common problem of excavation machinery based on mechanical actions is the unknown interaction of the cutting tools with various types of soils. Due to the involved non-linearities, the numerical analysis of such phenomena is very complex. To overcome these drawbacks some authors proposed to model the soil as a collection of spheres. In this paper we apply a strategy for soil modelling which is based on discretization of the soil with rigid disks and suitable contact models. The basic idea is to concentrate at the contact level the real mechanical behaviour of the soil. The goal is achieved by extending the general concept of contact as an unilateral constraint condition, through a suitable constitutive law: the contact laws have been implemented in the node-to-segment contact formulation within the framework of the penalty method. The aim of this work is to study the behaviour of some soils under different loading conditions, and to develop contact constitutive laws suitable for the discrete model. In order to carry out the proposed strategy a “macro” and a “micro” level are established, and macromechanical and micromechanical models are developed. In the micromechanical model the mutual contact interaction between two disks is studied, while the macromechanical model deals with the behaviour of a regular array of disks. The framework for the plastic behaviour of the contact element consists of a failure criterion; a one-dimensional, rate-independent elasto-plastic flow rule for normal and tangential force; two specific yield surfaces, and a hardening or softening law. In this paper we have focused our attention on the simulation of soils and rocks: a new constitutive contact law is developed and applied for the simulation of different soils with different testing conditions (such as uniaxial and shear tests).

Representation of a nonspherical ice particle by a collection of independent spheres for scattering and absorption of radiation: 2. Hexagonal columns and plates

A cloud of nonspherical ice particles may be represented in radiation models by a collection of spheres, in which the model cloud contains the same total volume of ice and the same total surface area as the real cloud but not the same number of particles. The spheres then have the same volume‐to‐area (V/A) ratio as the nonspherical particle. In previous work this approach was shown to work well to represent randomly oriented infinitely long circular cylinders for computation of hemispherical reflectance, transmittance, and absorptance. In this paper the results have been extended to hexagonal columns and plates using a geometric optics technique for large particles and finite‐difference‐time‐domain theory (FDTD) for small particles. The extinction efficiency and single‐scattering coalbedo for these prisms are closely approximated by the values for equal‐V/A spheres across the ultraviolet, visible, and infrared from 0.2 to 25 μm wavelength. Errors in the asymmetry factor can be significant where ice absorptance is weak, at visible wavelengths for example. These errors are greatest for prisms with aspect ratios close to 1. Errors in hemispheric reflectance, absorptance, and transmittance are calculated for horizontally homogeneous clouds with ice water paths from 0.4 to 200,000 g m−2 and crystal thicknesses of 1 to 400 μm, to cover the range of crystal sizes and optical depths from polar stratospheric clouds (PSCs) through cirrus clouds to surface snow. The errors are less than 0.05 over most of these ranges at all wavelengths but can be larger at visible wavelengths because of the ideal shapes of the prisms. The method was not tested for, and is not expected to be accurate for, angle‐dependent radiances.

Geometric integration on spheres and some interesting applications

Geometric integration theory can be employed when numerically solving ODEs or PDEs with constraints. In this paper, we present several one-step algorithms of various orders for ODEs on a collection of spheres. To demonstrate the versatility of these algorithms, we present representative calculations for reduced free rigid body motion (a conservative ODE) and a discretization of micromagnetics (a dissipative PDE). We emphasize the role of isotropy in geometric integration and link numerical integration schemes to modern differential geometry through the use of partial connection forms; this theoretical framework generalizes moving frames and connections on principal bundles to manifolds with nonfree actions.

Homotopy classes that are trivial mod ℱ

If F is a collection of topological spaces, then a homotopy class \alpha in [X,Y] is called F-trivial if \alpha_* = 0: [A,X] --> [A,Y] for all A in F. In this paper we study the collection Z_F(X,Y) of all F-trivial homotopy classes in [X,Y] when F = S, the collection of spheres, F = M, the collection of Moore spaces, and F = \Sigma, the collection of suspensions. Clearly Z_\Sigma (X,Y) \subset Z_M(X,Y) \subset Z_S(X,Y), and we find examples of finite complexes X and Y for which these inclusions are strict. We are also interested in Z_F(X) = Z_F(X,X), which under composition has the structure of a semigroup with zero. We show that if X is a finite dimensional complex and F = S, M or \Sigma, then the semigroup Z_F(X) is nilpotent. More precisely, the nilpotency of Z_F(X) is bounded above by the F-killing length of X, a new numerical invariant which equals the number of steps it takes to make X contractible by successively attaching cones on wedges of spaces in F, and this in turn is bounded above by the F-cone length of X. We then calculate or estimate the nilpotency of Z_F(X) when F = S, M or \Sigma for the following classes of spaces: (1) projective spaces (2) certain Lie groups such as SU(n) and Sp(n). The paper concludes with several open problems.