Showing papers on "Elliptic coordinate system published in 1999"

Coordinate system for hexagonal pixels

Abstract: A coordinate system is described which provides a natural means for representing hexagonally-organized pixels. The coordinate system has the unusual property that its basis vectors are not orthogonal. Vector-space properties and operations are described in this coordinate system, and shown to be straightforward computations. Some image processing algorithms, including calculations of image gradients and variable-conductance diffusion, are expressed and analyzed.

Addition formulas and the Rayleigh identity for arrays of elliptical cylinders.

Abstract: We apply the Rayleigh method to solve the problem where a uniform electrostatic field is imposed upon a rectangular array of elliptical cylinders embedded in a matrix of unit dielectric constant. This new formulation overcomes geometric restrictions inherent in previous methods and is shown in principle and in various examples to converge for all possible geometries of the array and inclusion. Also presented are forms of both the interior and exterior addition formulas for harmonic functions in elliptical coordinates that possess optimal regions of convergence.

Collective excitations in bose-einstein condensates in triaxially anisotropic parabolic traps

Abstract: The wave equation of low-frequency density waves in Bose-Einstein condensates at vanishing temperature in arbitrarily anisotropic harmonic traps is separable in elliptic coordinates, provided the condensate can be treated in the Thomas-Fermi approximation. We present a complete solution of the mode functions, which are polynomials of finite order, and their eigenfrequencies, which are characterized by three-integer quantum numbers.

3D eddy current formulation for moving conductors with variable velocity of coordinate system using edge finite elements

Abstract: A general formulation for moving conductors using 3D edge finite elements is introduced. In the formulation, the velocity of the coordinate system can be selected arbitrarily. A proper gauge condition for this formulation is also indicated. The optimal velocities of the coordinate system are investigated by numerical experiments. It is clarified that the accuracy of the analysis depends on the time variation of electromagnetic fields, which are observed differently in each coordinate system. From this point of view, we introduce a novel method that uses several different coordinate systems in one conductive region.

Boundary conditions of high-order accuracy at the poles of curvilinear coordinate systems

Abstract: We develop a method of imposing improved difference boundary conditions at singularity points of polar, cylindrical, and spherical coordinate systems with the aim to apply them to high-order accuracy schemes for stationary and nonstationary problems. We consider two main types of boundary value problems, viz. symmetric and arbitrary problems that have no symmetry. We obtain difference equations, which are consistent in order of accuracy and are of simple form, along the axis (at the centre) of symmetry. These are special approximations of the original equations in Cartesian coordinates. We study a problem of realizing the boundary conditions in implicit schemes of approximate factorization and in iterative processes. The author [4,5] constructed schemes of fourth-order accuracy for second-order equations in any orthogonal coordinate systems. In order to apply these results to coordinate systems with singularities (i.e. with lines or ambiguous points of the transformations of coordinates, in the following for brevity they are referred to as poles) it is necessary to impose difference boundary conditions with sufficient accuracy on nodes corresponding to these singularities. Thus, for example, when formulating the, boundary value problem for a cylinder it is necessary to impose difference boundary conditions on its axis, which would correspond to the used scheme in order of approximation. For the symmetric case the region of singularity is the line r = 0 in the rectangular range of the radius and height (r, z\\ and for an arbitrary problem it is the plane r = 0 in the three-dimensional range (r, ζ, φ\\ where φ is a vectorial angle. In polar coordinates the set r = 0 is a point or a line depending on whether a problem is symmetric or nonsymmetric, and in spherical coordinates it is a point or a plane. A common feature is the ambiguity of coordinates transformation at the poles (i.e. on the set r = 0) and the absence of boundary conditions in initial boundary value problems for a differential equation, whereas it is necessary to impose boundary conditions for a difference problem. For symmetric problems in the case of second-order accuracy schemes the problems of imposing boundary conditions, convergence and implementation of algorithms are properly developed (see, e.g. the monographs [2,6]). In the case of high-order accuracy schemes it is rather difficult to find out satisfactory relations at the poles even for symmetric boundary value problems. In this respect nonsymmetric boundary value problems are much more complicated even for conventional schemes, to say nothing of high-order accuracy schemes. This paper deals with the solution of these problems. We develop methods of imposing boundary conditions of high-order accuracy at the poles of some orthogonal coordinate systems most generally employed. We also describe methods of realizing these * Institute of Computational Technologies, Siberian Branch of the Russian Academy of Sciences, Novosibirsk 630090, Russia The work was supported by the program Integration Basic Research', Siberian Branch of the Russian Academy of Sciences (43) and the Russian Foundation for the Basic Research (97-01-00819).

Vibrations of a plate in fluid and associated damping due to energy transmission by dilatational waves

Abstract: The paper presents a solution to the problem of harmonic vibrations of a plate submerged in an unbounded medium of inviscid compressible fluid. The solution is obtained, as a limiting case, by means of a solution to the problem of an infinite elliptic cylinder vibrating in the fluid. The latter problem is solved with the help of the Fourier method of separation of variables in the elliptic coordinate system. For comparison purposes, a similar problem of circular cylinder vibrating in the fluid is also investigated. From the discussion presented it follows, that the fluid compressibility is essential in estimating hydrodynamical forces, especially in calculating damping of plate vibrations for higher frequencies.

Relativistic Electromagnetic Boundary Conditions - Abstract

Abstract: In Part I, an analytical examination of electromagnetic boundary conditions at the interface between two different media is presented in an orthogonal curvilinear coordinate system which conforms locally and instantaneously to the interface. Subsequently, an original natural coordinate system is introduced to resolve the uncertainty in the choice of an orthogonal curvilinear coordinate system. This natural coordinate system is also appeared to reduce the number of the nonvanishing discontinuity relations, and to determine local and instantaneous natural aspects of the e.m. boundary conditions problem.

Centroid Theorem of Coordinate tetrahedron in the Volume coordinate System

Abstract: The volume coordinate system is used widely in finite element methods, computer machine demonstration, and so on. In this paper, the centroid theorem of coordinate tetrahedron in the volume coordinate system is put and proved. The proof method supplies one of the link points with the education in linear algebra and analytic geometry.