Showing papers on "Elliptic coordinate system published in 2007"

Interconversion Between Truncated Cartesian and Polar Expansions of Images

Abstract: In this paper, we propose an algorithm for lossless conversion of data between Cartesian and polar coordinates, when the data is sampled from a 2-D real-valued function (a mapping: ) expressed as a particular kind of truncated expansion. We use Laguerre functions and the Fourier basis for the polar coordinate expression. Hermite functions are used for the Cartesian coordinate expression. A finite number of coefficients for the truncated expansion specifies the function in each coordinate system. We derive the relationship between the coefficients for the two coordinate systems. Based on this relationship, we propose an algorithm for lossless conversion between the two coordinate systems. Resampling can be used to evaluate a truncated expansion on the complementary coordinate system without computing a new set of coefficients. The resampled data is used to compute the new set of coefficients to avoid the numerical instability associated with direct conversion of the coefficients. In order to apply our algorithm to discrete image data, we propose a method to optimally fit a truncated expression to a given image. We also quantify the error that this filtering process can produce. Finally the algorithm is applied to solve the polar-Cartesian interpolation problem.

First-principle calculation of reduced masses in vibrational analysis using generalized internal coordinates: some crucial aspects and examples.

Abstract: In this paper we present and analyze the most essential aspects of reduced masses along generalized internal coordinates. The definition of reduced masses in the internal coordinate formalism is established through the Wilson G-matrix concept and includes sophisticated relations between internal and Cartesian coordinates. Moreover, reduced masses in internal coordinates are, in general, no longer constant but coordinate-dependent. Based on the approach presented earlier [Stare, J.; Balint-Kurti, G. G. J. Phys. Chem. A 2003, 107, 7204-7214] and on our experience with reduced masses discussed in this paper, we have developed a robust program for the calculation of Wilson G-matrix elements and their functional coordinate dependence. The approach is based on the first principles and can be used in virtually any (internal) coordinate set. Since the program allows for projection of any kind of nuclear motion on the selected internal coordinates, the method is particularly suitable for ab initio or DFT potential energy functions calculated by partial geometry optimization. Moreover, reduced masses obtained by this program can be used as a decision tool for selecting the most appropriate internal coordinates for the considered vibrational problem and for the inclusion or omission of the kinetic coupling terms in the vibrational Hamiltonian.

Elliptic curve point transformations

Abstract: In an elliptic curve cryptographic system, point coordinates in a first coordinate system are transformed into a second coordinate system. The transformed coordinates are processed by field operations, which have been modified for operating on the transformed point coordinates. In some implementations, the point coordinates are transformed using a linear transformation matrix having coefficients. The coefficients can be fixed, variable or random. In some implementations, the transformation matrix is invertible.

An Adaptive Vertical Coordinate Formulation for a Nonhydrostatic Model with Flux-Form Equations

Abstract: The concept of adaptive vertical coordinates is used to upgrade conventional terrain-following σ coordinates to arbitrary hybrid coordinates. Compared with previous approaches for implementing adaptive coordinates, the method presented here combines unrestricted applicability to nonhydrostatic models with the capability to integrate the atmospheric equations in flux form. The coordinate is based on a three-dimensional field carrying the vertical position of the coordinate surfaces, which is made time dependent by introducing a prognostic equation. As a specific example, the adaptive coordinate is used to emulate a hybrid isentropic system. Idealized tests in which the coordinate surfaces are artificially moved reveal that the ensuing spurious motions are small enough to be negligible in realistic applications. Mountain wave tests demonstrate that the hybrid coordinate remains numerically stable under strong forcing. However, the model layer distribution established with the hybrid isentropic coor...

Prestack depth migration in elliptic coordinates

Abstract: Riemannian wavefield extrapolation is extended to prestack migration through the use of 2D elliptic coordinate systems. The corresponding 2D elliptic coordinate extrapolation wavenumber is demonstrated to introduce only a slowness model stretch to the single-square-root operator, enabling the use of existing Cartesian implicit finite-difference extrapolators to propagate wavefields. A poststack migration example illustrates the advantages of elliptic coordinates in imaging overturning wavefields. Imaging tests of the SMAART JV Pluto 1.5 data set illustrate that the RWE migration algorithm generates high-quality prestack migration images comparable to, or better than, the corresponding Cartesian coordinate systems.

Multipole structure and coordinate systems

Abstract: Multipole expansions depend on the coordinate system, so that coefficients of multipole moments can be set equal to zero by an appropriate choice of coordinates. Therefore, it is meaningless to say that a physical system has a nonvanishing quadrupole moment, say, without specifying which coordinate system is used. (Except if this moment is the lowest non-vanishing one.) This result is demonstrated for the case of two equal like electric charges. Specifically, an adapted coordinate system in which the potential is given by a monopole term only is explicitly found, the coefficients of all higher multipoles vanish identically. It is suggested that this result can be generalized to other potential problems, by making equal coordinate surfaces adapt to the potential problem's equipotential surfaces.

Calculation of the Streaming Potential Near a Rotating Disk with Rotational Elliptic Coordinates

Abstract: The calculation with rotational elliptic coordinates of the streaming potential in the vicinity of a disk-shaped sample rotating in an electrolytic solution is presented. The measurement of this streaming potential is used to determine the zeta potential of planar surfaces. Rotational elliptic coordinates are favored in relation to integral transform methods because only simple mathematical methods are employed to explain the theory of this technique.

On Generating Isochoric Finite Deformations

Abstract: New isochoric finite deformations may be generated from any such deformation described in rectangular Cartesian coordinates by changing coordinate systems.

A two-component area coordinate method for quadrilateral elements

Abstract: In order to construct quadrilateral elements insensitive to mesh distortion,a new kind of quadrilateral area coordinate method,denoted as QAC-2,has been successfully developed. And related differential and integral formulae are also presented. As a natural coordinate system,QAC-2 has explicit physical meanings. It includes only two independent components,Z1 and Z2,which make it easier to communicate with Cartesian coordinates and isoparametric coordinates. Furthermore,since Z1 and Z2 are linear functions of Cartesian coordinates x and y,it is convenient to establish a polynomial with high order completeness in Cartesian coordinates by using Z1 and Z2,and this polynomial will keep its completeness order invariable for mesh distortion cases. QAC-2 is a simple and novel tool for developing more accurate and robust quadrilateral element models.

Construction and performance analysis of local DtN absorbing boundary conditions for exterior Helmholtz problems. Part I : Elliptical shaped boundaries

Abstract: We propose a new class of approximate {\it local} DtN boundary conditions to be applied on prolate spheroid-shaped exterior boundaries when solving acoustic scattering problems by elongated obstacles. These conditions are : (a) exact for the first modes, (b) easy to implement and to parallelize, (c) compatible with the local structure of the computational finite element scheme, and (d) applicable to exterior elliptical-shaped boundaries that are more suitable in terms of cost-effectiveness for surrounding elongated scatterers. Moreover, these conditions coincide with the classical local DtN condition designed for spherical-shaped boundaries. We investigate analytically and numerically the effect of the frequency regime and the slenderness of the boundary on the accuracy of these conditions when applied for solving radiators and scattering problems. We also compare their performance to the second order absorbing boundary condition (BGT2) designed by Bayliss, Gunzburger and Turkel when expressed in prolate spheroidal coordinates. The analysis reveals that, in the low frequency regime, the new second order DtN condition (DtN2) retains a good level of accuracy regardless of the slenderness of the boundary. In addition, the DtN2 boundary condition outperforms the BGT2 condition. Such superiority is clearly noticeable for large eccentricity values.

Electron energy spectrum in core-shell elliptic quantum wire

Abstract: The electron energy spectrum in core-shell elliptic quantum wire and elliptic semiconductor nanotubes are investigated within the effective mass approximation. The solution of Schrodinger equation based on the Mathieu functions is obtained in elliptic coordinates. The dependences of the electron size quantization spectrum on the size and shape of the core-shell nanowire and nanotube are calculated. It is shown that the ellipticity of a quantum wire leads to the break of degeneration of quasiparticle energy spectrum. The dependences of the energy of odd and even electron states on the ratio between semiaxes are of a nonmonotonous character. The anticrosing effects are observed at the dependences of electron energy spectrum on the transversal size of the core-shell nanowire.

Discretization of Macroscopic Transport Equations on Non-Cartesian Coordinate Systems

Abstract: We discuss discretization schemes for the Poisson equation, the isothermal drift-diffusion equations, and higher order moment equations derived from the Boltzmann transport equation for general coordinate systems. We briefly summarize the method of dimension reduction when the problem does not depend on one coordinate. Discretization schemes for dimension-reduced coordinate systems are introduced, which provide curvilinear coordinate systems. In addition to the reduction of the dimensionality, another benefit of these curved coordinate systems is that the domain approximation is more accurate, and therefore, the mesh point density can be kept smaller compared to the original problem. We obtain a discretization scheme for the isothermal drift-diffusion equation in closed from. For higher order transport equations, we use the approximation method of optimum artificial diffusivity and generalize it for non-Cartesian coordinate systems. For the special case of cylindrical coordinates, we can show that it is not necessary to introduce special discretization schemes apart from the standard Scharfetter-Gummel scheme.

Analysis of stresses in an infinite elastic conductive plate with an elliptic hole subjected to a uniform magnetic field

Abstract: Stress distribution is determined in a thin conductive plate containing an elliptic hole under the external electromagnetic loads The aim of this study is to explore the concentrating effects of both the electric current density and the stresses in the plate due to the existence of the elliptic hole Use is made of the elliptic coordinates for the ease of treating ellipse shaped boundary, while the analytical results for the components in the Cartesian coordinates are provided for convenience of application After the distribution of the current density is derived, an anti-plane shear problem is formulated whose solution is obtained in closed form The mode Ⅲ stress intensity factor is deduced by considering the case in which the minor axis vanishes when the ellipse degenerates to a Griffith crack

Soliton Families in Highly Nonlocal Nonlinear Media

Abstract: We explore two families of self-trapped modes in highly nonlocal nonlinear media. The first family relates interaction of solitons with the linear nondiffracting beams. The second family describes pure soliton solutions using elliptic coordinates.

FVM-and FEM-solution of elliptical boundary value problems in different coordinate systems

Abstract: In this paper several numerical concepts for the solution of elliptical boundary value problems will be compared with regard to their efficiency, computational load and accuracy. Especially regions which are images of rectangles and cuboids of coordinate transformations will be considered. I argue that it is thus possible to construct well structured discretisations which imply equation systems with very high solution properties. The approaches discussed will be tested with instructive examples in the light of their advantages and disadvantages.

Formulation of Integral Equations Using Analytic Green's Functions in Elliptic Cylindrical and Prolate Spheroidal Coordinates

Abstract: It is desired to develop and present techniques, using analytic Green’s functions in elliptic cylindrical coordinates and in prolate spheroidal coordinates, for formulating integral equations for physically practicable structures and sources. Equivalent models for two structures are introduced which serve as guides in the derivation of analytic Greens functions, comprising special functions and satisfying required boundary conditions. Expressions for needed field components are represented in terms of integrals of Greens functions times unknowns, and integral equations follow from the proper relationships among sources and field components needed to satisfy Maxwell’s equations.