Cone Boundary Research Articles

Loss cone boundary measurement using diagnostic neutral beam and neutral particle analyzer in a compact helical system

A horizontally scannable diagnostic neutral beam (DNB) has been installed on the compact helical system (CHS) in order to study the confinement of energetic ions with different pitch angles by varying the injection angle. The DNB has been designed to provide energetic ions as a test particle source with (1) no heating to background plasma and (2) small divergence angle. A charge-exchange neutral particle analyzer (NPA) to measure energetic ions injected by the DNB is also scannable and varies the observation angle on the equatorial plane in CHS. A combination of horizontally scannable DNB and NPA provides information on whether the energetic ions with different pitch angle are confined in the plasma or immediately lost. The experimental results are consistent with the prediction of single particle full orbit calculation in CHS.

Green's Function of an Infinite Slot Printed Between Two Homogeneous Dieletrics—Part II: Uniform Asymptotic Solution

This second part of a two-paper sequence deals with the uniform asymptotic description of the Green's function of an infinite slot printed between two different homogeneous dielectric media. Starting from the magnetic current derived in Part I, the dyadic Green's function is first formulated in integral form in both spectral and spatial domains; next, the asymptotic solution for the vector potential is evaluated asymptotically. The asymptotic ray-field is structured in three contributions: a spherical wave radiated by the source (space wave), a conical leaky wave, and a lateral wave. These contributions are first introduced by the stationary phase method applied to the space-domain radiation integral. From this approach it is seen that the lateral wave contribution is negligible in actual configurations. Next, a rigorous uniform asymptotic evaluation of the radiated field is formulated in the spectral domain by using a steepest descent path deformation which accounts for the vicinity of the pole to the saddle point. Through this rigorous asymptotics, the domain of existence of the leaky-wave is found to be limited by a conical shadow boundary which deviates from that defined by the stationary phase regime. Along this conical shadow boundary, phase matching occurs between space and leaky wave, which facilitates the transition mechanism between the two wave types. This transition occurs inside a conical transition region with elliptical cross section. The interference between space and leaky waves from the near to the far zone are discussed by means of illustrative examples, which also confirm the accuracy of the asymptotics.

Reduction of orders in boundary value problems without transmission property

Given an algebra of pseudo-differential operators on a manifold, an elliptic element is said to be a reduction of orders, if it induces isomorphisms of Sobolev spaces with a corresponding shift of smoothness. Reductions of orders on a manifold with boundary refer to boundary value problems. We employ specific smooth symbols of arbitrary real orders and with parameters, and we show that the associated operators induce isomorphisms between Sobolev spaces on a given manifold with boundary. Such operators for integer orders have the transmission property and belong to the calculus of Boutet de Monvel [1], cf. also [9]. In general, they fit to the algebra of boundary value problems without the transmission property in the sense of [17] and [24]. Order reducing elements of the present kind are useful for constructing parametrices of mixed elliptic problems. We show that order reducing symbols have the Volterra property and are parabolic of anisotropy 1; analogous relations are formulated for arbitrary anisotropies. We then investigate parameter-dependent operators, apply a kernel cut-off construction with respect to the parameter and show that corresponding holomorphic operator-valued Mellin symbols reduce orders in weighted Sobolev spaces on a cone with boundary. We finally construct order reducing operators on a compact manifold with conical singularities and boundary.

Centrifugally driven convection in the rotating cylindrical annulus with modulated boundaries

Abstract. The effect of sinusoidally modulated conical end boundaries on convection in a rotating cylindrical annulus is investigated theoretically and experimentally. A quasiperiodic time dependence of convection in the form of thermal Rossby waves is found and semi-quantitative agreement between theory and measurements can be established. The results are relevant to convection in the Earth's outer core close to the tangent cylinder touching the inner core at its equator.

Unsteady-State Cone Flows of Ideal Incompressible Liquids

A mathematical model is suggested for the phenomenon that is observed when a circulation flow generates a meridional flow in a chemical reactor, separator, hydrocyclone, or free jet. The problem of a linear wave confined within a cone is solved in closed form. The flow of a liquid within distorted conical boundaries is considered. The Sternberg–Koiter and no-flow paradoxes are found to take place. The notion of quasi-cone flow is introduced. An exact solution of the problem is obtained for the particular case of irrotational flow.

Dirichlet problem for linear elliptic equations degenerating at a conical boundary point

Dirichlet problem for linear elliptic equations degenerating at a conical boundary point

Domain sensitivity analysis of the acoustic far‐field pattern

AbstractWe consider acoustic scattering problems described by the mixed boundary value problem for the scalar Helmholtz equation in the exterior of a 2D bounded domain or in the exterior of a crack. The boundary of the domain is assumed to have a finite set of corner points where the scattered wave may have singular behaviour. The paper is concerned with the sensitivity of the far‐field pattern with respect to small perturbations of the shape of the scatterer. Using a modification of the method of adjoint problems, we obtain an integral representation for the Gâteaux derivative which contains only boundary values of functions easily computable by standard BEM and which depends explicitly on the perturbation of the boundary. In some cases, we show the direct influence of the singularities of the solution on the sensitivity of the far‐field pattern. In this way, we generalize the domain sensitivity analysis developed earlier for smooth domains by Hettlich, Kirsch, Kress, Potthast and others. Finally, we show that the same approach can be applied to scattering from 3D domains with smooth edges. Copyright © 2002 John Wiley & Sons, Ltd.

The Stokes resolvent in 3D domains with conical boundary points: nonregularity in $L^p$-spaces

It is shown that solutions of the 3D Stokes resolvent problem in domains with conical boundary points, under homogeneous Dirichlet boundary conditions, do not satisfy the usual resolvent estimate in $L^p$-spaces if $p $ is close to infinity or close to $1.$

Rotary honing: a variant of the Taylor paint-scraper problem

The three-dimensional flow in a corner of fixed angle α induced by the rotation in its plane of one of the boundaries is considered. A local similarity solution valid in a neighbourhood of the centre of rotation is obtained and the streamlines are shown to be closed curves. The effects of inertia are considered and are shown to be significant in a small neighbourhood of the plane of symmetry of the flow. A simple experiment confirms that the streamlines are indeed nearly closed; their projections on planes normal to the line of intersection of the boundaries are precisely the ‘Taylor’ streamlines of the well-known ‘paint-scraper’ problem. Three geometrical variants are considered: (i) when the centre of rotation of the lower plate is offset from the contact line; (ii) when both planes rotate with different angular velocities about a vertical axis and Coriolis effects are retained in the analysis; and (iii) when two vertical planes intersecting at an angle 2β are honed by a rotating conical boundary. The last is described by a similarity solution of the first kind (in the terminology of Barenblatt) which incorporates within its structure a similarity solution of the second kind involving corner eddies of a type familiar in two-dimensional corner flows.

Influence of boundary conditions on the free vibrations of rotating truncated circular multi-layered conical shells

Abstract Considering the effects of the Coriolis and centrifugal accelerations and the initial hoop tension, this paper studies the influences of boundary conditions on the frequency characteristics of a rotating truncated circular multi-layered conical shell. The present study is based on the Love first approximation theory and Galerkin procedure. The obtained results include the relations of the frequency parameter, circumferential wave number and rotating velocity for different cone angles and boundary conditions. Variation of frequency characteristics is also discussed for different vibrational modes. In order to validate the present analysis, several numerical comparisons are made with those published in open literature and they show very good agreement.

Periodic Orbit Theory Analysis of a Family of Deformed Hemispherical Billiard Systems

We present a periodic orbit theory analysis of a novel three-dimensional billiard system, namely a quasispherical cavity with infinite walls along the conical boundary defined by θ=Θ, where θ is the standard polar angle; for Θ=π/2 this reduces to the special case of a hemispherical infinite well, while for Θ=π it is a spherical well with points along the negative z axis excluded. We focus especially on the connections between subsets of the energy eigenvalue space and their contributions to qualitatively different classes of closed orbits.

PERIODIC ORBIT THEORY ANALYSIS OF A FAMILY OF DEFORMED HEMISPHERICAL BILLIARD SYSTEMS

We present a periodic orbit theory analysis of a novel three-dimensional billiard system, namely a quasispherical cavity with infinite walls along the conical boundary defined by θ=Θ, where θ is the standard polar angle; for Θ=π/2 this reduces to the special case of a hemispherical infinite well, while for Θ=π it is a spherical well with points along the negative z axis excluded. We focus especially on the connections between subsets of the energy eigenvalue space and their contributions to qualitatively different classes of closed orbits.

Growth model for filamentary streamers in an ambient field

We have simulated the fast streamer stage of liquid dielectric breakdown as stochastic growth of a branching fractal tree. Breakdown and threshold properties of the fluid are represented in the random filter procedure. A range of fractal densities, from sparse to bushy, is approximated by the choice of power-law (4th-power to linear). The choice of threshold (cutoff) voltage also significantly affects the growth form. These parameters combine with the shape and concentration of the electric field, to regulate the distribution and directedness of the local discharge growth pattern. Inclusion of a voltage gradient along the streamer tree produces a secondary narrowing effect on the growth. A large grid (128 cubed) is used for the discretization. Diagonal growth paths to neighbor-vertices are included, increasing the choice of available directions for each discharge event. We use a combination of data-parallel programming and three-dimensional visualization. Complete growth histories, evolving from the voltage distribution, can be displayed in animation or in color banding against the "trials" variable, which simulates a time tick. Side views of the structures provide comparison against sub-microsecond snapshots from experiment. Results include sparse, directed trees evolving from a 4th-power-law filter; also dense trees from a linear filter, whose conical upper-envelope boundary is strongly influenced by the choice of threshold (cutoff) potential.

Ambiguity in the evaluation of the effective action on the cone

An ambiguity in the computation of the one-loop effective action for fields living on a cone is illustrated. It is shown that the ambiguity arises due to the noncommutativity of the regularization of ultraviolet and (conical) boundary divergencies.

Asymptotic behaviour of solutions of the first boundary-value problem for strongly hyperbolic systems near a conical point at the boundary of the domain

An existence and uniqueness theorem for generalized solutions of the first initial-boundary-value problem for strongly hyperbolic systems in bounded domains is established. The question of estimates in Sobolev spaces of the derivatives with respect to time of the generalized solution is discussed. It is shown that the smoothness of generalized solutions with respect to time is independent of the structure of the boundary of the domain but depends on the coefficients of the right-hand side. Results on the smoothness of the generalized solution and its asymptotic behaviour in a neighbourhood of a conical boundary point are also obtained.

Multispectral and hyperspectral image analysis with convex cones

A new approach to multispectral and hyperspectral image analysis is presented. This method, called convex cone analysis (CCA), is based on the bet that some physical quantities such as radiance are nonnegative. The vectors formed by discrete radiance spectra are linear combinations of nonnegative components, and they lie inside a nonnegative, convex region. The object of CCA is to find the boundary points of this region, which can be used as endmember spectra for unmixing or as target vectors for classification. To implement this concept, the authors find the eigenvectors of the sample spectral correlation matrix of the image. Given the number of endmembers or classes, they select as many eigenvectors corresponding to the largest eigenvalues. These eigenvectors are used as a basis to form linear combinations that have only nonnegative elements, and thus they lie inside a convex cone. The vertices of the convex cone will be those points whose spectral vector contains as many zero elements as the number of eigenvectors minus one. Accordingly, a mixed pixel can be decomposed by identifying the vertices that were used to form its spectrum. An algorithm for finding the convex cone boundaries is presented, and applications to unsupervised unmixing and classification are demonstrated with simulated data as well as experimental data from the hyperspectral digital imagery collection experiment (HYDICE).

Local solvability and regularity results for a class of semilinear elliptic problems in nonsmooth domains

We consider a class of semilinear elliptic problems in two- and three-dimensional domains with conical points. We introduce Sobolev spaces with detached asymptotics generated by the asymptotical behaviour of solutions of corresponding linearized problems near conical boundary points. We show that the corresponding nonlinear operator acting between these spaces is Frechet differentiable. Applying the local invertibility theorem we prove that the solution of the semilinear problem has the same asymptotic behaviour near the conical points as the solution of the linearized problem if the norms of the given right hand sides are small enough. Estimates for the difference between the solution of the semilinear and of the linearized problem are derived.

Estimates of generalized solutions of the Dirichlet problem for quasilinear elliptic equations of the second order in a domain with conical boundary point

We obtain a priori estimates for generalized second derivatives (in the Sobolev weighted norm) of solutions of the Dirichlet problem for the elliptic equation $$\frac{d}{{dx_i }}a_i (x,u,u_x ) + a(x,u,u_x ) = 0,x \in G,$$ in the neighborhood of a conical boundary point of the domain G. We give an example that demonstrates that the estimates obtained are almost exact.

L^p-theory for the stokes system in 3d domains with conical boundary points

L^p-theory for the stokes system in 3d domains with conical boundary points

Transfinite interpolation of conic triangles using quadric patches

The use of quadric surface patches to form a continuous approximating surface to given curved surfaces was discussed in a paper by Baart and McLeod. A method was proposed for the construction of continuous patches on a covering mesh of conic segments, where each patch consists of three quadric subpatches, and each conic boundary segment interpolates three points on the given surface. This construction is not unique, and in this paper we investigate the effect of various combinations of the parameters on the resulting quadric patches. We also propose a method that will ensure that, were the given sculptured patch itself quadric, the approximation will be exact.