Spherical Cone Research Articles

Applications and extensions of the Z-cone model for the energy of a foam

A bubble in an ordered foam can be split into sections: one for each neighbour. We approximate these sections as spherical cones and obtain an analytic form for the energy as a function of deformation, and use this to obtain a liquid fraction profile for a foam under gravity. • We compute an effective Hookean spring constant as a function of contact number for bubble–bubble interactions. • We use the analytic Z -cone model to obtain a liquid fraction profile for a foam under gravity. • We extend the Z-cone model to allow the computation of energies for bubbles of differing sizes. We present an extended analysis of the Z -cone model for the estimation of the energy of an ordered foam. In particular, we show that it results in an interaction potential with an exponent consistent with those previously reported, as well as an equilibrium liquid fraction profile under gravity consistent with those experimentally observed in ordered foams. We also extend the model to deal with curved interfaces, as a first step to modelling ordered bidisperse foams.

Aerodynamic Design Optimization of a Kind of Reentry Capsule Based on CFD and Multi-objective Genetic Algorithm

This paper presents a new kind of aerodynamic shape for reentry capsule, called spherical cap cubic curve segment(SCCS) shape, which is modeled by replacing the reversing cone segment of the traditional spherical cap segment-reversing cone(SCSC) shape with a more various cubic curve segment, however the spherical cap section is retained. In the assumed allowable axial range of centroid under a designed Mach number, a multi-point/objective design optimization mathematical programming model was established against aerodynamic static longitudinal stability characteristics, hypersonic lift-to-drag ratio characteristics under trim angle of attack and off-set location placement of gravity center for the above reentry capsule. Then the aerodynamic design optimization problem of the proposed reentry capsule was investigated through multi-objective genetic algorithm(MOGA) combined numeric parallel field simulation methods, and finally the optimal Pareto front was given. Comparing with the results of the SCSC shape in the literature[1], it shows that the optimal Pareto front of the current cubic curve shape has a wider distribution range relative to the counterparts of the segment-reversing cone shape, however both are relatively close in the overlapping area.

Slow axial drift in three-dimensional granular tumbler flow

Models of monodisperse particle flow in partially filled three-dimensional tumblers often assume that flow along the axis of rotation is negligible. We test this assumption, for spherical and double cone tumblers, using experiments and discrete element method simulations. Cross sections through the particle bed of a spherical tumbler show that, after a few rotations, a colored band of particles initially perpendicular to the axis of rotation deforms: particles near the surface drift toward the pole, while particles deeper in the flowing layer drift toward the equator. Tracking of mm-sized surface particles in tumblers with diameters of 8-14cm shows particle axial displacements of one to two particle diameters, corresponding to axial drift that is 1-3% of the tumbler diameter, per pass through the flowing layer. The surface axial drift in both double cone and spherical tumblers is zero at the equator, increases moving away from the equator, and then decreases near the poles. Comparing results for the two tumbler geometries shows that wall slope causes axial drift, while drift speed increases with equatorial diameter. The dependence of axial drift on axial position for each tumbler geometry is similar when both are normalized by their respective maximum values.

Simulation and Experimental Investigation of Jetting Penetrator Charge at Large Stand-off Distance

Abstract In order to study and apply the penetration performance of jetting penetrator charge at long stand-off distance, three jetting penetrator charges (JPC), including spherical cone liner, truncated wide-angle liner and spherical segment liner, are designed. The numerical simulation analysis of the formation, elongation and penetration processes of rod-like jet is conducted by using LS-DYNA software. And the penetrating test is carried out at long stand-off distance. The test results show that the rod-like jet formed by the optimized spherical segment liner can pierce through a 90 mm thick 45# steel target at 20 charge diameters (CD) Stand-off distance when the charge detonation mode is a central point initiation, and the penetration depth can be up to 1.6CD. It is concluded that, at 20 CD stand-off distance, the penetration performance of JPC with spherical segment liner is the best, that of truncated wide-angle liner takes second place, and that of spherical cone liner is the worst.

Ricci curvature bounds for warped products

We prove generalized lower Ricci curvature bounds for warped products over complete Finsler manifolds. On the one hand our result covers a theorem of Bacher and Sturm concerning Euclidean and spherical cones (Bacher and Sturm [3]). On the other hand it can be seen in analogy to a result of Bishop and Alexander in the setting of Alexandrov spaces with curvature bounded from below (Alexander and Bishop, 2004 [2]). For the proof we combine techniques developed in these papers. Because the Finslerian warped product metric can degenerate we regard a warped product as metric measure space that is in general neither a Finsler manifold nor an Alexandrov space again but a space satisfying a curvature-dimension condition in the sense of Lott–Villani/Sturm.

The best constants for multidimensional modular inequalities over spherical cones

In this paper, we establish the Muckenhoupt-type estimation for the best constant associated with the following multidimensional modular inequality over a spherical cone:where and . Similar results are also derived for the complementary integral operator. Our results provide the -dimensional modular forms of the works of Andersen and Heinig. As consequences of our results, we give the -dimensional weighted extensions of Levinson modular inequality, extensions of Stepanov and Heinig results, generalizations of the Hardy–Knopp-type inequalities, and those for the Riemann–Liouville operator and the Weyl fractional operator. We also point out that our estimates are better than those given in the works of Drabek, Heinig, Kunfer, and Sinnamon.

Coupled Shape and Reentry Trajectory Optimization of Entry Vehicle for Lunar Return

This paper deals with the reentry trajectory optimization problem for lunar return with consideration of entry vehicle’s fore-body shape. Three performance objectives are applied in this work: cross range, peak heat flux and total heat load. Aerothermodynamic models are based on modified Newtonian impact theory and semi-empirical correlations for convective and radiative stagnation-point heat transfer. A population based evolutionary algorithm has been executed to optimize the multidisciplinary problem. At last the numerical example showed the Pareto frontiers for spherical segment and sphere cone respectively, one of optimal trajectory designs selected from the Pareto frontiers are showed in this paper. The mission requirements are satisfied through the aerothermodynamic balance.

Investigation of the common nose cone shapes in different gas mixtures in high Knudsen numbers

The numerical simulation of different types of nose cone shapes involving conical, Bi-conic, parabolic, spherical Blunt cone and tangent ogive in high Knudsen numbers flow and for three different mixtures O2–N2, He–Ar and He–Xe are investigated in this article. A computer code based on the method of Direct Simulation Monte Carlo (DSMC) has been developed to study numerically the supersonic flow around these cone shapes in rarefaction condition. In order to verify the computer code, the flow field in the slip regime is solved and compared with the results of Navier Stokes equation with the slip boundary condition. The results obtained in the supersonic flows regime show that the parabolic and tangent ogive nose shapes have the least mean shear stress distribution and lowest tip temperature.

Spherical cone symmetric families generated by Landau-Lifshitz equation and their evolution

Until 2008, we had not seen a paper about the explicit dynamic solution of the multi-dimensional Landau-Lifshitz equation with the external magnetic field and anisotropy field. In the absence of an external magnetic field or anisotropy field, many experts have tried Hirotas famous method, Lie algebra structure and Backlund transformations, bilinear transform, but the exact solution is still not constructed. This paper presents a method to solve the Landau-Lifshitz equations with the external magnetic field and anisotropy field, and a new family of symmetric solutions, which we call it as spherical cone symmetric solution. Then, we shall show how these solutions evolve as time. Finally, we give a series of explicit dynamic spherical cone symmetric solutions.

Sommerfeld enhancement of DM annihilation: resonance structure, freeze-out and CMB spectral bound

In the last few years there has been some interest in WIMP Dark Matter models featuring a velocity dependent cross section through the Sommerfeld enhancement mechanism, which is a non-relativistic effect due to massive bosons in the dark sector. In the first part of this article, we find analytic expressions for the boost factor for three different model potentials, the Coulomb potential, the spherical well and the spherical cone well and compare with the numerical solution of the Yukawa potential. We find that the resonance pattern of all the potentials can be cast into the same universal form.In the second part of the article we perform a detailed computation of the Dark Matter relic density for models having Sommerfeld enhancement by solving the Boltzmann equation numerically. We calculate the expected distortions of the CMB blackbody spectrum from WIMP annihilations and compare these to the bounds set by FIRAS. We conclude that only a small part of the parameter space can be ruled out by the FIRAS observations.

Isoperimetric regions in spherical cones and Yamabe constants of M × S 1

We study isoperimetric regions on Riemannian manifolds of the form (M n × (0, π), sin2(t)g + dt 2) where g is a metric of positive Ricci curvature ≥ n − 1. When g is an Einstein metric we use this to compute the Yamabe constant of $${(M \times \mathbb{R}, g+ dt^2 )}$$ and so to obtain lower bounds for the Yamabe invariant of M × S 1.

A new kind of spherical gear and its application in a robot's wrist joint

The meshing principles and the manufacturing methods for spherical gears with concave cone surface teeth are presented in this paper. The profile of the convex teeth of this kind of gear is analyzed and calculated. The theoretical feasibility of this kind of gear is proven by the analysis of the profile, contact ratios and undercut. It makes the process of manufacturing easier than before, improving the manufacturing precision and simplifying the manufacturing process. This paper introduces the theory and manufacturing method of concave tooth profile, that is spherical cone gearing. The analysis and calculations on the convex tooth profile engaging with the concave cone tooth is also proved.

Cone kinetics model for two-phase film silicon deposition

We study quantitatively the morphology of nanocrystalline silicon (nc-Si:H) cones that appear during amorphous silicon (a-Si:H) film growth by chemical vapor deposition from hydrogen-diluted silane. The shapes of the nc-Si:H inclusions are found to be spherical cones, consistent with our “cone kinetics” model for silicon film growth. This model predicts cone development when growth is isotropic and there is sparse nucleation of a second material phase with a higher growth rate. Application of the cone kinetics model provides insights into the deposition of technologically important thin film materials, including protocrystalline silicon, highly P-doped nc-Si:H, silicon heterojunctions, and silicon epitaxy.

Calculation of the viscous-interaction effects in the Euler/boundary-layer methods

A simple approach is proposed to incorporate the effect of viscous interaction in the calculation of aerodynamic heating of three-dimensional geometries by Euler/boundary-layer methods. In the present study, aerodynamic heating is calculated from the solution of a set of approximate convective-heating equations along the streamlines. These heating equations are derived using axisymmetric analogue. To determine streamline paths and flow properties along them, Euler equations are solved around the three-dimensional body. The effect of viscous interaction on the aerodynamic heating can be significant, especially if the Reynolds number is low. To incorporate this effect, the whole process should be repeated having considered the boundary-layer thickness all around the body. However, this approach causes different difficulties, and is not cost efficient as well. In the present work, the effect of boundary-layer thickness on the aerodynamic heating is incorporated through the correction of pressure distribution on the surface of the body. This approach of correcting pressure distribution is applied to spherical and elliptical blunted cones in laminar hypersonic flows. It is shown that the heating rates are improved in low Reynolds numbers, where viscous-interaction effect is significant. The present approach can result in a reduction of computational cost, when viscous interaction is significant in calculation of aerodynamic heating.

Local Isoperimetric Inequalities for Sectors on Surfaces and Cones

On surfaces we give conditions under which the solution of a restricted local isoperimetric problem for sectors with small solid angle is the circular sector and we characterize these surfaces. Also we study this problem for general spherical cones on hypersurfaces in higher dimensional Riemannian manifolds.

On testing a class of restricted hypotheses

The aim of this paper is to provide the asymptotic distribution function, under the null and the alternative hypothesis, of statistics based on the likelihood, when the parameter space is restricted by a cone C such that D C is a spherical cone or the polyhedral cone { t ∈ R N + 1 : t 1 ⩾ 0 , … , t N + 1 ⩾ 0 } where D is a fixed known positive definite matrix. The results obtained ensure the calculation of threshold and power of such restricted tests. Our study is not limited to i.i.d. observations. Numerical calculations show that the likelihood ratio test restricted by a cone C 1 ⊂ C 2 is uniformly more powerful than the likelihood ratio test restricted by C 2 on all the alternatives that belong to C 1 . Particularly, a restricted test is uniformly more powerful than a non-restricted one on all the restricted alternatives.

Morse functions with sphere fibers

A smooth closed manifold is said to be an almost sphere if it admits a Morse function with exactly two critical points. In this paper, we characterize those smooth closed manifolds which admit Morse functions such that each regular fiber is a finite disjoint union of almost spheres. We will see that such manifolds coincide with those which admit Morse functions with at most three critical values. As an application, we give a new proof of the characterization theorem of those closed manifolds which admit special generic maps into the plane. We also discuss homotopy and diffeomorphism invariants of manifolds related to the minimum number of critical values of Morse functions; in particular, the Lusternik-Schnirelmann category and spherical cone length. Those closed orientable 3-manifolds which admit Morse functions with regular fibers consisting of spheres and tori are also studied.

Detailed molecular comparison between the inhibition mode of A/B-type carboxypeptidases in the zymogen state and by the endogenous inhibitor latexin.

Treatment of advanced stages of prostate carcinoma with histone-deacetylase inhibitors entails expression of human procarboxypeptidase-A4 (hPCPA4). The three-dimensional structure of hPCPA4 has been solved and shows the features of related metallocarboxypeptidase zymogens, with a preformed alpha/beta/-hydrolase active-enzyme moiety (hCPA4) and an inhibiting pro-domain (PD). The protease moiety recalls a sphere, out of which a spherical cone has been cut. This results in a funnel-like structure, at the bottom of which the active-site cleft resides. The border of this funnel is shaped by loops, which are responsible for the interaction with the PD, characterised by a large interface area and relatively few contacts. Such an inhibitory mode is evocative of the recently reported structure of the human inhibitor latexin in its complex with hCPA4. The main contacting structure of latexin is similar to the one employed for PD inhibition. In both cases, active-site blocking relies mainly on a loop provided by the central part of a beta sheet.

Three degrees-of-freedom joint for spatial hyper-redundant robots

We present a new mechanical design for a compact three degrees-of-freedom joint mechanism. This joint design is ideal for use in robotic devices, especially hyper-redundant or snake-like robots. This joint exhibits an unprecedented range of motion: the origin of a coordinate frame attached to the output link of the joint can move on a spherical cone whose apex angle is 120° and this coordinate frame can be rotated around an axis passing through its origin and the center of the sphere. Moreover, the joint is optimized for strength, compactness and accuracy. These characteristics are verified by exploring the magnitude of the wrenches and errors that the joint can produce in its effective workspace. Finally, we present a mechatronic integration of four of our joints to construct a 12 degrees-of-freedom spatial hyper-redundant robot.

Homogeneity and rigidity of the Brookes–Groves invariant for the non-commutative torus

In [4] Bieri and Strebel defined a geometric invariant Σ for finitely generated modules over the group algebras of finitely generated abelian groups. They used this to define a criterion for when metabelian groups are finitely presented. This invariant was further developed by Bieri, Strebel and Groves and has many interesting applications. In [2] Bieri and Groves showed that when the group algebra is defined over a Dedekind domain the complement of Σ must be a closed rational spherical polyhedral cone. In [6,7] Brookes and Groves defined a similar invariant Δ for modules over the crossed product of a division ring by a free finitely generated abelian group. Such a crossed product is often known as the (coordinate ring of) the non-commutative torus since in the special case where it is commutative it is the coordinate ring of an algebraic torus. If in the commutative case we take the complement of Δ and identify points that differ by a positive scalar multiple we obtain Σ . Brookes and Groves were unable to prove that their invariant must be a rational polyhedral cone, although using the methods of [2] they do prove a weaker version of the result; they show that for any finitely generated module M , Δ(M) must contain a rational polyhedral cone Δ∗(M) of dimension equal to the Gelfand–Kirillov dimension of M and moreover that the complement Δ(M)\Δ∗(M) must be contained inside a rational polyhedral cone of strictly smaller dimension.