Optimal Packings of Superballs

Dense hard-particle packings are intimately related to the structure of low-temperature phases of matter and are useful models of heterogeneous materials and granular media. Most studies of the densest packings in three dimensions have considered spherical shapes, and it is only more recently that nonspherical shapes (e.g., ellipsoids) have been investigated. Superballs (whose shapes are defined by |x1|^2p + |x2|^2p + |x3|^2p <= 1) provide a versatile family of convex particles (p >= 0.5) with both cubic- and octahedral-like shapes as well as concave particles (0 < p < 0.5) with octahedral-like shapes. In this paper, we provide analytical constructions for the densest known superball packings for all convex and concave cases. The candidate maximally dense packings are certain families of Bravais lattice packings. The maximal packing density as a function of p is nonanalytic at the sphere-point (p = 1) and increases dramatically as p moves away from unity. The packing characteristics determined by the broken rotational symmetry of superballs are similar to but richer than their two-dimensional "superdisk" counterparts, and are distinctly different from that of ellipsoid packings. Our candidate optimal superball packings provide a starting point to quantify the equilibrium phase behavior of superball systems, which should deepen our understanding of the statistical thermodynamics of nonspherical-particle systems.Comment: 28 pages, 16 figure

Volumes of solids swept tangentially around general surfaces

In Part I (Forum Geom., 15 (2015) 13-44) the authors introduced solid tangent sweeps and solid tangent clusters produced by sweeping a planar region S tangentially around cylinders. This paper extends Part I by sweeping S not only along cylinders but also around more general surfaces, cones for example. Interesting families of tangentially swept solids of equal height and equal volume are constructed by varying the cylinder or the planar shape S. For most families in this paper the solid tangent cluster is a classical solid whose volume is equal to that of each member of the family. We treat many examples including familiar quadric solids such as ellipsoids, paraboloids, and hyperboloids, as well as examples obtained by puncturing one type of quadric solid by another, all of whose volumes are obtained with the extended method of sweeping tangents. Surprising properties of their centroids are also derived

Volumes of solids swept tangentially around cylinders

In earlier work ([1]-[5]) the authors used the method of sweeping tangents to calculate area and arclength related to certain planar regions. This paper extends the method to determine volumes of solids. Specifically, take a region S in the upper half of the xy plane and allow the plane to sweep tangentially around a general cylinder with the x axis lying on the cylinder. The solid swept by S is called a solid tangent sweep. Its solid tangent cluster is the solid swept by S when the cylinder shrinks to the x axis. Theorem 1: The volume of the solid tangent sweep does not depend on the profile of the cylinder, so it is equal to the volume of the solid tangent cluster. The proof uses Mamikon's sweeping-tangent theorem: The area of a tangent sweep to a plane curve is equal to the area of its tangent cluster, together with a classical slicing principle: Two solids have equal volumes if their horizontal cross sections taken at any height have equal areas. Interesting families of tangentially swept solids of equal volume are constructed by varying the cylinder. For most families in this paper the solid tangent cluster is a classical solid of revolution whose volume is equal to that of each member of the family. We treat forty different examples including familiar solids such as pseudosphere, ellipsoid, paraboloid, hyperboloid, persoids, catenoid, and cardioid and strophoid of revolution, all of whose volumes are obtained with the extended method of sweeping tangents. Part II treats sweeping around more general surfaces

Finite Element Analysis of the Effect of Acoustic Wavelength to Hierarchical Side Length and Facet Area for Elastic Scattering from Polygonal Rings and Geodesic Spheres

In this work, the frequency response from two-dimensional polygon and three-dimensional geodesic spheres is numerically simulated using a coupled structural acoustic finite element model. The model is composed of a submerged thin-walled elastic shell structure surrounded by an infinite acoustic air domain. Infinite elements are used to simulate the far-field acoustic radiation condition. Results for the faceted polygon and geodesic sphere are compared to their canonical counterpart viz. a circle/ring and a spherical shell. A unique feature of this study is to compare results as the number of facets in the polygon or geodesic is increased, such that the surface area converges in the limit of a large number of facet sides approaching the geometry of a circle or sphere. In this work the ratio of acoustic wavelength to the local geometric parameter of edge length in 2-D, and facet area in 3-D is proposed and varied to quantify the comparison between the faceted shapes with that of the corresponding reference circle or sphere. A threshold ratio is proposed, up to which scattering response of a polygonal/geodesic spherical scatterer matches the scattering response of a circle/sphere which has the same diameter as the circumscribing circle/sphere of the polygon/geodesic sphere. This ratio is an approximation and can be considered as a guide rule for design. Conversely, this ratio can be used for the inverse scattering problem, where from a known scattering response, the faceted geometry can be predicted without prior knowledge. The geodesic sphere was invented by Buckminster Fuller in the early 1950’s, has been of interest in architecture due to the larger open interior spaces which can be constructed. Of particular interest in this work is the hierarchical geometric structure of the geodesic sphere which increasingly approximates a spherical surface as the hierarchy (degree) increases. The geodesic sphere has been modelled by taking an icosahedron and projecting the triangular faces onto a surface of the sphere using vector geometry. The scattering response of elastic structures in the mid-frequency resonance band depends strongly on the total mass. For comparisons, the natural frequency of the hierarchical geometries generated in 2-D and 3-D, are designed to have the same total mass. Using this approach, differences in natural frequency and scattering response are driven primarily by changes in overall stiffness and stiffness distribution, and to a lesser degree, by changes in mass distribution. To give a wide range of frequency response, natural vibration frequencies for the different elastic shells have been extracted up to 3000 Hz corresponding to the nondimensional frequency ka = 55, where k is the wavenumber defined by the circular frequency over the acoustic wave speed (speed of sound in air), and a is the diameter of the circle/shell which circumscribes the scatterer. Convergence with the natural frequencies of ring/sphere is observed as the hierarchy in polygons (number of sides) and the geodesic sphere (degree) increases. The target strength is calculated at the important front and back locations on the surface of the elastic scatterer subject to an incoming plane acoustic wave along the major axis aligned with the geometry. More frequency data points near the natural frequencies are used to provide increased resolution needed to capture the peak amplitudes in the response at resonance. Target strength at the same location, calculated for the circle/spherical scatterer is compared and quantified by the ratio of wavelength to facet dimension. Scattering from rigid bodies has been studied to validate the elastic scattering response in air

Derivative relationships between volume and surface area of compact regions in R^d

We explore the idea that the derivative of the volume, V, of a region in R^d with respect to r equals its surface area, A, where r = d V/A. We show that the families of regions for which this formula for r is valid, which we call homogeneous families, include all the families of similar regions. We determine equivalent conditions for a family to be homogeneous, provide examples of homogeneous families made up of non-similar regions, and offer a geometric interpretation of r in a few cases.Comment: 15 page

A collision avoidance system for a spaceplane manipulator arm

Part of the activity in the area of collision avoidance related to the Hermes spaceplane is reported. A collision avoidance software system which was defined, developed and implemented in this project is presented. It computes the intersection between the solids representing the arm, the payload, and the objects. It is feasible with respect to the resources available on board, considering its performance

Stress-strain behavior and geometrical properties of packings of elongated particles

We present a numerical analysis of the effect of particle elongation on the quasistatic behavior of sheared granular media by means of the Contact Dynamics method. The particle shapes are rounded-cap rectangles characterized by their elongation. The macroscopic and microstructural properties of several packings subjected to biaxial compression are analyzed as a function of particle elongation. We find that the shear strength is an increasing linear function of elongation. Performing an additive decomposition of the stress tensor based on a harmonic approximation of the angular dependence of branch vectors, contact normals and forces, we show that the increasing mobilization of friction force and the associated anisotropy are key effects of particle elongation. These effects are correlated with partial nematic ordering of the particles which tend to be oriented perpendicular to the major principal stress direction and form side-to-side contacts. However, the force transmission is found to be mainly guided by cap-to-side contacts, which represent the largest fraction of contacts for the most elongated particles. Another interesting finding is that, in contrast to shear strength, the solid fraction first increases with particle elongation, but declines as the particles become more elongated. It is also remarkable that the coordination number does not follow this trend so that the packings of more elongated particles are looser but more strongly connected.Comment: Submited to Physical Review