Conical Vertex Research Articles

On a Family of Polyhedral Singular Vertices

We provide a local description of the curves with minimal length based at singularities in a family of polyhedral surfaces. These singularities are accumulation points of vertices with conical angles equal to π and 4π (or 3π, in a variation). While a part of the minimizing curves behaves quite like the ones reaching conical vertices, the singularities present features such as being connected to points arbitrarily close to them by exactly two minimizing curves. The spaces containing such singularities are constructed as metric quotients of an euclidean half-disk by certain identification patterns along its edge. These patterns are examples of what is known as paper-folding schemes, and we provide the foundational aspects about them which are necessary for our analysis. The arguments are based on elementary metric geometry and calculus.

Cup-cone structure in spallation of bulk metallic glasses

A special spallation morphology in bulk metallic glass, named as the “cup-cone” structure, is of particular interest since it manifests a unique “ductile–brittle” transition. To gain insights into the underlying mechanism for the formation of a cup-cone structure, we conduct planar impact experiments at various impact velocities, as well as finite element method analysis. Spall strength increases with increasing impact velocity. Scanning electron microscopy and X-ray computed tomography are performed on postmortem samples to characterize cup-cone structures; their average size and spacing decrease as impact velocity increases, and they dominate fracture morphology at high impact velocities. Cups and cones are generally distributed on the side away from and on the side closer to the target free surface, respectively. The initial nucleation sites of voids become the conical vertices of cup-cones, and the subsequent nucleation sites form along the conical surface and coalesce into the cracks and fracture surfaces.

Quantum State Transfer in Coronas

We study state transfer in quantum walks on graphs relative to the adjacency matrix. Our motivation is to understand how the addition of pendant subgraphs affect state transfer. For two graphs $G$ and $H$, the Frucht-Harary corona product $G \circ H$ is obtained by taking $|G|$ copies of the cone $K_{1} + H$ and by identifying the conical vertices according to $G$. Our work explores conditions under which the corona $G \circ H$ exhibits state transfer. We also describe new families of graphs with state transfer based on the corona product. Some of these constructions provide a generalization of related known results.

Effect of Ligament Morphology on Electrical Conductivity of Porous Silver

We investigate the effect of ligament morphology on electrical conductivity of open cell porous silver (Ag). Porous Ag was formed when silver nanoparticles in an organic phase were annealed at 150°C for durations ranging from 1 to 5 min. Electrical conductivity of porous Ag was about 20% of bulk value after 5 min annealing. Porous Ag was modeled as a collection of Kelvin cell (truncated octahedrons) structures comprised of conjoined conical ligaments and spherical vertices. An analytical expression for electrical conductivity was obtained. Electrical conductivity normal to hexagonal faces of the unit cell was computed. Our model indicates contribution of grain boundary to electrical resistance increases significantly after the first minute of annealing and plateaus thereafter. Using experimental electrical conductivity data as an input, the model suggests that the ratio, n, of surfaces of one half of a conjoined cone ligament is between 0.7 and 1.0. Average deviation from experimentally determined relative electrical conductivity, Δσ r, was minimal when n = 0.9.

A Novel Method of Cone Fitting Based on 3D Point Cloud

The accuracy and stability of circular conical surface is easily influenced by the different distribution of noise and points in point cloud, when 3D point cloud data fits conical surface. For this reason, this paper proposes a novel method of establishing and optimizing the objective function with conical vertex. First, calculate the axis direction and half-apex angle of the cone; second, calculate conical vertex according to the result got in the first step; third, establish objective function with axis direction and half-apex angle and vertex, and optimize it with Levenberg-Marquardt. The method’s feasibility and robustness are verified by simulating and real experiments. The proposed algorithm has been successfully applied to a self-development 3D measuring system (TN 3DOMS) of Mianyang Saint Buffalo Technologies Limited Company.

Electromagnetic scattering by a circular impedance cone: diffraction coefficients and surface waves

This paper is devoted to the study of electromagnetic scattering of a plane wave by a circular cone with impedance boundary conditions on its surface. The technique developed in the previous works is extended and applied to the electromagnetic diffraction problem with the aim of computing the farfield. It is known that by means of the Kontorovich–Lebedev integral representations for the Debye potentials and a ‘partial’ separation of variables, the problem is reduced to coupled functional difference equations for the relevant spectral functions. For a circular cone, the functional-difference equations are then further reduced to integral equations which are shown to be of Fredholm type. Certain useful integral representations for the solution of ‘Watson–Bessel’ and Sommerfeld types are exploited, which gives a theoretical basis for subsequent evaluation of the far-field (high-frequency) asymptotics for the diffracted field. To that end, we study analytic properties of the integrands in the Sommerfeld integrals. We also discuss the asymptotic expressions for the surface waves propagating from the conical vertex to infinity and give new expressions of the diffraction coefficients for the spherical wave in the domain illuminated by the rays reflected from the cone (in the so-called ‘non-oasis’ domain M ′′).

Generation of three-dimensional delaunay meshes from weakly structured and inconsistent data

A method is proposed for the generation of three-dimensional tetrahedral meshes from incomplete, weakly structured, and inconsistent data describing a geometric model. The method is based on the construction of a piecewise smooth scalar function defining the body so that its boundary is the zero isosurface of the function. Such implicit description of three-dimensional domains can be defined analytically or can be constructed from a cloud of points, a set of cross sections, or a “soup” of individual vertices, edges, and faces. By applying Boolean operations over domains, simple primitives can be combined with reconstruction results to produce complex geometric models without resorting to specialized software. Sharp edges and conical vertices on the domain boundary are reproduced automatically without using special algorithms. Refs. 42. Figs. 25.

Vertex singularities associated with conical points for the 3D laplace equation

The solution u to the Laplace equation in the neighborhood of a vertex in a three-dimensional domain may be described by an asymptotic series in terms of spherical coordinates . For conical vertices, we derive explicit analytical expressions for the eigenpairs νi and fi(θ, φ), which are required as benchmark solutions for the verification of numerical methods. Thereafter, we extend the modified Steklov eigen-formulation for the computation of vertex eigenpairs using p/spectral finite element methods and demonstrate its accuracy and high efficiency by comparing the numerically computed eigenpairs to the analytical ones. Vertices at the intersection of a crack front and a free surface are also considered and numerical eigenpairs are provided. The numerical examples demonstrate the efficiency, robustness, and high accuracy of the proposed method, hence its potential extension to elasticity problems. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011

The Fourier Singular Complement Method for the Poisson problem. Part II: axisymmetric domains

This paper is the second part of a threefold article, aimed at solving numerically the Poisson problem in three-dimensional prismatic or axisymmetric domains. In the first part of this series, the Fourier Singular Complement Method was introduced and analysed, in prismatic domains. In this second part, the FSCM is studied in axisymmetric domains with conical vertices, whereas, in the third part, implementation issues, numerical tests and comparisons with other methods are carried out. The method is based on a Fourier expansion in the direction parallel to the reentrant edges of the domain, and on an improved variant of the Singular Complement Method in the 2D section perpendicular to those edges. Neither refinements near the reentrant edges or vertices of the domain, nor cut-off functions are required in the computations to achieve an optimal convergence order in terms of the mesh size and the number of Fourier modes used.

Moffatt eddies in the cone

Consider Stokes flow in a cone of half-angle , a length scale, affect the flow fields. The fields in conical containers exhibit toroidal corner vortices whose structure is different from those at a conical vertex; their growth and evolution to primary vortices is briefly examined.

The Fourier Singular Complement Method for the Poisson problem. Part I: prismatic domains

This is the first part of a threefold article, aimed at solving numerically the Poisson problem in three-dimensional prismatic or axisymmetric domains. In this first part, the Fourier Singular Complement Method is introduced and analysed, in prismatic domains. In the second part, the FSCM is studied in axisymmetric domains with conical vertices, whereas, in the third part, implementation issues, numerical tests and comparisons with other methods are carried out. The method is based on a Fourier expansion in the direction parallel to the reentrant edges of the domain, and on an improved variant of the Singular Complement Method in the 2D section perpendicular to those edges. Neither refinements near the reentrant edges of the domain nor cut-off functions are required in the computations to achieve an optimal convergence order in terms of the mesh size and the number of Fourier modes used.

Error estimates for the Fourier-finite-element approximation of the Lam� system in nonsmooth axisymmetric domains

This paper is concerned with the effective implementation of the Fourier-finite-element method, which combines the approximating Fourier and the finite-element methods, for treating the Dirichlet problem for the Lame equations in axisymmetric domains Open image in new window with conical vertices and reentrant edges. The partial Fourier decomposition reduces the three-dimensional boundary value problem to an infinite sequence of decoupled two-dimensional boundary value problems on the plane meridian domain Open image in new window The asymptotic behavior of the solutions of the reduced problems near angular points of Ωa is described by suitable singular functions and treated numerically by linear finite elements on locally graded meshes. For Open image in new window it is proved that the rate of convergence of the combined approximations in Open image in new window is of the order Open image in new window where h and N are the parameters of the finite-element- and Fourier-approximation, respectively, with h→0 and N→∞.

Experimental Evaluation of Conical-Cylindrical Thickeners Operanting with Aqueous Barium Sulfate Suspensions

The objective of this work was to analyze the performance of three continuous thickeners of different geometries in steady-state operation: the conical-convergent, with a cross section area decreasing with the depth, the conical-divergent, in which the cross section area increases with the depth and the cylindrical. The experimental runs were made in prototypes, operating with aqueous suspensions of barium sulfate with a density of 4520 kg/m 3 . In order to verify how the thickeners capacity behaves as a function of the geometry. Computer simulations (Maple® software) based on the model proposed by in a previous article were compared to the experimental data. It was verified through both, the computer simulations and the experimental tests, that the cylindrical thickeners presented smaller capacities than the conical-divergent thickeners; however larger than the one presented by the conical-convergent thickener. It was also observed that the larger the absolute value of the conical vertex's angle, the bigger is the difference among the capacities of the conical and cylindrical systems. The model used describes well the experimental behavior of continuous thickeners in steady-state operation.

Graded Mesh Refinement and Error Estimates for Finite Element Solutions of Elliptic Boundary Value Problems in Non-smooth Domains

This paper is concerned with the effective numerical treatment of elliptic boundary value problems when the solutions contain singularities. The paper deals first with the theory of problems of this type in the context of weighted Sobolev spaces and covers problems in domains with conical vertices and non-intersecting edges, as well as polyhedral domains with Lipschitz boundaries. Finite element schemes on graded meshes for second-order problems in polygonal/polyhedral domains are then proposed for problems with the above singularities. These schemes exhibit optimal convergence rates with decreasing mesh size. Finally, we describe numerical experiments which demonstrate the efficiency of our technique in terms of ‘actual’ errors for specific (finite) mesh sizes in addition to the asymptotic rates of convergence.

Zeroth-order, electrohydrostatic solution for electrospraying in cone-jet mode

Liquid and charge emissions in steady regime from an electrified meniscus take place when its shape develops an almost conical point. A very thin and remarkably stable capillary jet emerging from the conical vertex eventually breaks up into droplets, forming the so-called electrospray. The emission process is an extremely complicated electrohydrodynamic phenomenon which may be solved by performing a perturbation analysis which requires the knowledge of an electrostatic, zeroth-order solution for the shape of the electrified meniscus with a conical tip. In this work, the shape of these tip-ended, axisymmetric, electrified menisci is solved for the first time assuming negligible space charge. The solution of this first-order, electrostatic problem yields not only the backbone result on a further perturbation scheme to solve the liquid and charge emission process, but also the old problem of that value of the potential difference at which the liquid is to be connected in order to achieve a tip-ended shape, necessary to have electrostatic atomization, as a function of the liquid properties and boundary geometry.

Corner Behavior of Solutions of Semilinear Dirichlet Problems

In recent years there has been considerable attention paid to the behavior of solutions of elliptic boundary value problems in domains with piecewise smooth boundary. In two dimensions the study concerns the behavior of a solution near a corner, and in three (or more) dimensions two cases have been given considerable attention: a conical vertex on the boundary, or an edge.The solution of such a problem may be singular at the nonsmooth boundary points. The standard example in two dimensions is a solution in polar coordinates of the Dirichlet problem near a corner of interior angle πα;u = r1/α sin θ/α is a function which is harmonic in the sector 0 < θ < πα, has zero boundary values near the corner, and yet at the origin has unbounded derivatives of order > 1/α unless 1/α is an integer.

On the behavior of electromagnetic fields at twisted edges and on conical vertexes in inhomogeneous medium

The behavior of electromagnetic fields near a twisted edge and near a conical vertex in inhomogeneous medium is considered, both starting from the conditon that the energy density must be integrable over any finite domain. Differential equations of Sturm-Liouville type are then set up, which, together with relevant boundary conditions, determine the eigenvalues and accordingly the field behavior. Static fields are shown to have the same behavior in the neighborhood of such geometrical singularities. Media with piecewise constant permeability and permittivity are discussed as application examples of the concept of “impedance” or “admittance” for solving the associated eigenvalue problems.