Showing papers on "Elliptic coordinate system published in 2008"

Analysis of Electromagnetic Fields and Waves: The Method of Lines

Abstract: Preface. 1 THE METHOD OF LINES. 1.1 INTRODUCTION. 1.2 MOL: FUNDAMENTALS OF DISCRETISATION. 1.2.1 Qualitative description. 1.2.2 Quantitative description of the discretisation. 1.2.3 Numerical example. 2 BASIC PRINCIPLES OF THE METHOD OF LINES. 2.1 INTRODUCTION. 2.2 BASIC EQUATIONS. 2.2.1 Anisotropic material parameters. 2.2.2 Relations between transversal electric and magnetic fields - generalised transmission line (GTL) equations. 2.2.3 Relation to the analysis with vector potentials. 2.2.4 GTL equations for 2D structures. 2.2.5 Solution of the GTL equations. 2.2.6 Numerical examples. 2.3 EIGENMODES IN PLANAR WAVEGUIDE STRUCTURES WITH ANISOTROPIC LAYERS. 2.3.1 Introduction. 2.3.2 Analysis equations for eigenmodes in planar structures. 2.3.3 Examples of systemequations. 2.3.4 Impedance/admittance transformation in multilayered structures. 2.3.5 System equation in transformed domain. 2.3.6 System equation in spatial domain. 2.3.7 Matrix partition technique: two examples. 2.3.8 Numerical results. 2.4 ANALYSIS OF PLANAR CIRCUITS. 2.4.1 Discretisation of the transmission line equations. 2.4.2 Determination of the field components. 2.5 FIELD AND IMPEDANCE/ADMITTANCE TRANSFORMATION. 2.5.1 Introduction. 2.5.2 Impedance/admittance transformation in multilayered and multisectioned structures. 2.5.3 Impedance/admittance transformation with finite differences. 2.5.4 Stable field transformation through layers and sections. 3 ANALYSIS OF RECTANGULAR WAVEGUIDE CIRCUITS. 3.1 INTRODUCTION. 3.2 CONCATENATIONS OF WAVEGUIDE SECTIONS. 3.2.1 LSM and LSE modes in circular waveguide bends. 3.2.2 LSM and LSE modes in straight waveguides. 3.2.3 Impedance transformation at waveguide interfaces. 3.2.4 Numerical results for concatenations. 3.2.5 Numerical results for waveguide filters. 3.3 WAVEGUIDE JUNCTIONS. 3.3.1 E-plane junctions. 3.3.2 H-plane junctions. 3.3.3 Algorithm for generalised scattering parameters. 3.3.4 Special junctions: E-plane 3-port junction. 3.3.5 Matched E-plane bend. 3.3.6 Analysis of waveguide bend discontinuities. 3.3.7 Scattering parameters. 3.3.8 Numerical results. 3.4 ANALYSIS OF 3D WAVEGUIDE JUNCTIONS. 3.4.1 General description. 3.4.2 Basic equations. 3.4.3 Discretisation scheme for propagation between A and B. 3.4.4 Discontinuities. 3.4.5 Coupling to other ports. 3.4.6 Impedance/admittance transformation. 3.4.7 Numerical results. 4 ANALYSIS OF WAVEGUIDE STRUCTURES IN CYLINDRICAL COORDINATES. 4.1 INTRODUCTION. 4.2 GENERALISED TRANSMISSION LINE (GTL) EQUATIONS. 4.2.1 Material parameters in a cylindrical coordinate system. 4.2.2 GTL equations for z -direction. 4.2.3 GTL equations for phi -direction. 4.2.4 Analysis of circular (coaxial) waveguides with azimuthally-magnetised ferrites and azimuthallymagnetised solid plasma. 4.2.5 GTL equations for r -direction. 4.3 DISCRETISATION OF THE FIELDS AND SOLUTIONS. 4.3.1 Equations for propagation in z -direction. 4.3.2 Equations for propagation in phi -direction. 4.3.3 Solution of the wave equations in z - and phi -direction. 4.3.4 Equations for propagation in r -direction. 4.4 SOLUTION IN RADIAL DIRECTION. 4.4.1 Discretisation in z -direction - circular dielectric resonators. 4.4.2 Discretisation in z -direction - propagation in phi -direction. 4.4.3 Discretisation in phi -direction - eigenmodes in circular multilayered waveguides. 4.4.4 Eigenmodes of circular waveguides with magnetised ferrite or plasma - discretisation in r -direction. 4.4.5 Waveguide bends - discretisation in r -direction. 4.4.6 Uniaxial anisotropic fibres with circular and noncircular cross-section - discretisation in phi -direction. 4.5 DISCONTINUITIES IN CIRCULAR WAVEGUIDES - ONE-DIMENSIONAL DISCRETISATION IN RADIAL DIRECTION. 4.5.1 Introduction. 4.5.2 Basic equations for rotational symmetry. 4.5.3 Solution of the equations for rotational symmetry. 4.5.4 Admittance and impedance transformation. 4.5.5 Open ending circular waveguide. 4.5.6 Numerical results for discontinuities in circular waveguides. 4.5.7 Numerical results for coaxial line discontinuities and coaxial filter devices. 4.5.8 Non-rotational modes in circular waveguides. 4.5.9 Numerical results and discussion. 4.6 ANALYSIS OF GENERAL AXIALLY SYMMETRIC ANTENNAS WITH COAXIAL FEED LINES. 4.6.1 Introduction. 4.6.2 Theory. 4.6.3 Regions with crossed lines. 4.6.4 Two special cases. 4.6.5 Port relations of section D. 4.6.6 Numerical results. 4.6.7 Further structures and remarks. 4.7 DEVICES IN CYLINDRICAL COORDINATES -TWO-DIMENSIONAL DISCRETISATION. 4.7.1 Discretisation in r - and phi -direction. 4.7.2 Numerical results. 4.7.3 Discretisation in r - and z -direction. 4.7.4 Discretisation in phi - and z -direction. 4.7.5 GTL equations for r -direction. 5 ANALYSIS OF PERIODIC STRUCTURES. 5.1 INTRODUCTION. 5.2 PRINCIPLE BEHAVIOUR OF PERIODIC STRUCTURES. 5.3 GENERAL THEORY OF PERIODIC STRUCTURES. 5.3.1 Port relations for general two ports. 5.3.2 Floquetmodes for symmetric periods. 5.3.3 Concatenation of N symmetric periods. 5.3.4 Floquet modes for unsymmetric periods. 5.3.5 Some further general relations in periodic structures. 5.4 NUMERICAL RESULTS FOR PERIODIC STRUCTURES IN ONE DIRECTION. 5.5 ANALYSIS OF PHOTONIC CRYSTALS. 5.5.1 Determination of band diagrams. 5.5.2 Waveguide circuits in photonic crystals. 5.5.3 Numerical results for photonic crystal circuits. 6 ANALYSIS OF COMPLEX STRUCTURES. 6.1 LAYERS OF VARIABLE THICKNESS. 6.1.1 Introduction. 6.1.2 Matching conditions at curved interfaces. 6.2 MICROSTRIP SHARP BEND. 6.3 IMPEDANCE TRANSFORMATION AT DISCONTINUITIES. 6.3.1 Impedance transformation at concatenated junctions. 6.4 ANALYSIS OF PLANAR WAVEGUIDE JUNCTIONS. 6.4.1 Main diagonal submatrices. 6.4.2 Off-diagonal submatrices - coupling to perpendicular ports. 6.5 NUMERICAL RESULTS. 6.5.1 Discontinuities in microstrips. 6.5.2 Waveguide junctions. 7 PRECISE RESOLUTION WITH AN ENHANCED AND GENERALISED LINE ALGORITHM. 7.1 INTRODUCTION. 7.2 CROSSED DISCRETISATION LINES AND CARTESIAN COORDINATES. 7.2.1 Theoretical background. 7.2.2 Lines in vertical direction. 7.2.3 Lines in horizontal direction. 7.3 SPECIAL STRUCTURES IN CARTESIAN COORDINATES. 7.3.1 Groove guide. 7.3.2 Coplanar waveguide. 7.4 CROSSED DISCRETISATION LINES AND CYLINDRICAL COORDINATES. 7.4.1 Principle of analysis. 7.4.2 General formulas for eigenmode calculation. 7.4.3 Discretisation lines in radial direction. 7.4.4 Discretisation lines in azimuthal direction. 7.4.5 Coupling to neighbouring ports. 7.4.6 Steps of the analysis procedure. 7.5 NUMERICAL RESULTS. 8 WAVEGUIDE STRUCTURES WITH MATERIALS OF GENERAL ANISOTROPY IN ARBITRARY ORTHOGONAL COORDINATE SYSTEMS. 8.1 GENERALISED TRANSMISSION LINE EQUATIONS. 8.1.1 Material properties. 8.1.2 Maxwell's equations in matrix notation. 8.1.3 Generalised transmission line equations in Cartesian coordinates for general anisotropic material. 8.1.4 Generalised transmission line equations for general anisotropic material in arbitrary orthogonal coordinates. 8.1.5 Boundary conditions. 8.1.6 Interpolationmatrices. 8.2 DISCRETISATION. 8.2.1 Two-dimensional discretisation. 8.2.2 One-dimensional discretisation. 8.3 SOLUTION OF THE DIFFERENTIAL EQUATIONS. 8.3.1 General solution. 8.3.2 Field relation between interfaces A and B. 8.4 ANALYSIS OF WAVEGUIDE JUNCTIONS AND SHARP BENDS WITH GENERAL ANISOTROPIC MATERIAL BY USING ORTHOGONAL PROPAGATING WAVES. 8.4.1 Introduction. 8.4.2 Theory. 8.4.3 Main diagonal submatrices. 8.4.4 Off-diagonal submatrices - coupling to other ports. 8.4.5 Steps of the analysis procedure. 8.5 NUMERICAL RESULTS. 8.6 ANALYSIS OF WAVEGUIDE STRUCTURES IN SPHERICAL COORDINATES. 8.6.1 Introduction. 8.6.2 Generalised transmission line equations in spherical coordinates. 8.6.3 Analysis of special devices - conformal antennas. 8.6.4 Analysis of special devices - conical horn antennas. 8.6.5 Numerical results. 8.7 ELLIPTICAL COORDINATES. 8.7.1 GTL equations for z -direction. 8.7.2 GTL equations for xi -direction. 8.7.3 GTL equations for eta -direction. 8.7.4 Hollow waveguides with elliptic cross-section. 9 SUMMARY AND PROSPECT FOR THE FUTURE. A. DISCRETISATION SCHEMES AND DIFFERENCE OPERATORS. A.1 DETERMINATION OF THE EIGENVALUES AND EIGENVECTORS OF P. A.1.1 Calculation of the matrices delta. A.1.2 Derivation of the eigenvalues of the Neumann problem from those of the Dirichlet problem. A.1.3 The component of epsilonr at an abrupt transition. A.1.4 Eigenvalues and eigenvectors for periodic boundary conditions. A.1.5 Discretisation for non-ideal places of the boundaries. A.2 ABSORBING BOUNDARY CONDITIONS (ABCs). A.2.1 Introduction 1 . A.2.2 Factorisation of the Helmholtz equation. A.2.3 Pad'e approximation. A.2.4 Polynomial approximations. A.2.5 Construction of the difference operator for ABCs. A.2.6 Special boundary conditions (SBCs). A.2.7 Numerical results. A.2.8 ABCs for cylindrical coordinates. A.2.9 Periodic boundary conditions. A.3 HIGHER-ORDER DIFFERENCE OPERATORS [11]. A.3.1 Introduction. A.3.2 Theory. A.3.3 Numerical results. A.4 NON-EQUIDISTANT DISCRETISATION. A.4.1 Introduction. A.4.2 Theory. A.4.3 Interpolation. A.4.4 Numerical results. A.5 REFLECTIONS IN DISCRETISATION GRIDS. A.5.1 Introduction. A.5.2 Dispersion relations. A.5.3 Reflections at discretisation transitions. A.6 FIELD EXTRAPOLATION FOR NEUMANN BOUNDARY CONDITIONS. A.7 ABOUT THE NATURE OF THE METHOD OF LINES. A.7.1 Introduction. A.7.2 Relation between shielded structures and periodic ones. A.7.3 Method of Lines and discrete Fourier transformation. A.7.4 Discussion. A.8 RELATION BETWEEN THE MODE MATCHING METHOD (MMM) AND THE METHOD OF LINES (MoL) FOR INHOMOGENEOUSMEDIA. A.9 RECIPROCITYAND ITS CONSEQUENCES. B TRANSMISSION LINE EQUATIONS. B.1 TRANSMISSION LINE EQUATIONS IN FIELD VECTOR NOTATION. B.2 DERIVATION OF THE MULTICONDUCTOR TRANSMISSION LINE EQUATIONS. C SCATTERING PARAMETERS. D EQUIVALENT CIRCUITS FOR DISCONTINUITIES. E APPROXIMATE METALLIC LOSS CALCULATION IN CONFORMAL STRUCTURES. Index.

Electromagnetic beam modulation through transformation optical structures

Abstract: The transformation media concept based on the form-invariant Maxwell's equations under coordinate transformations has opened up new possibilities of manipulating electromagnetic (EM) fields. In this paper, we report on application of the finite-embedded coordinate transformation method to the design of EM beam-modulating devices both in Cartesian coordinates and in cylindrical coordinates. By designing the material constitutive tensors of the transformation optical structures through different kinds of coordinate transformations, either the beamwidth of an incident Gaussian plane wave could be modulated by a slab or the wave-propagating direction of an omnidirectional source could be modulated through a cylindrical shell. We present the design procedures and the full-wave EM simulations that clearly confirm the good performance of the proposed beam-modulating devices.

Laplace Transform Analytic Element Method for Transient Groundwater Flow Simulation

Abstract: The Laplace transform analytic element method (LT-AEM), applies the traditionally steady-state analytic element method (AEM) to the Laplace-transformed diffusion equation (Furman and Neuman, 2003). This strategy preserves the accuracy and elegance of the AEM while extending the method to transient phenomena. The approach taken here utilizes eigenfunction expansion to derive analytic solutions to the modified Helmholtz equation, then back-transforms the LT-AEM results with a numerical inverse Laplace transform algorithm. The two-dimensional elements derived here include the point, circle, line segment, ellipse, and infinite line, corresponding to polar, elliptical and Cartesian coordinates. Each element is derived for the simplest useful case, an impulse response due to a confined, transient, single-aquifer source. The extension of these elements to include effects due to leaky, unconfined, multi-aquifer, wellbore storage, and inertia is shown for a few simple elements (point and line), with ready extension to other elements. General temporal behavior is achieved using convolution between these impulse and general time functions; convolution allows the spatial and temporal components of an element to be handled independently. Comparisons are made between inverse Laplace transform algorithms; the accelerated Fourier series approach of de Hoog et al. (1982) is found to be the most appropriate for LT-AEM applications. An application and synthetic examples are shown for several illustrative forward and parameter estimation simulations to illustrate LT-AEM capabilities. Extension of LT-AEM to three-dimensional flow and non-linear infiltration are discussed.

On spectral approximations in elliptical geometries using Mathieu functions

Abstract: We consider in this paper approximation properties and applications of Mathieu functions. A first set of optimal error estimates are derived for the approximation of periodic functions by using angular Mathieu functions. These approximation results are applied to study the Mathieu-Legendre approximation to the modified Helmholtz equation and Helmholtz equation. Illustrative numerical results consistent with the theoretical analysis are also presented.

Handling of Singularity in Finite-Difference Time-Domain Procedure for Solving Maxwell's Equations in Cylindrical Coordinate System

Abstract: A consistent approach to determine the components of electric and magnetic fields at r = 0 for a cylindrical coordinate system in a finite-difference time-domain calculation procedure is proposed. It is based on the superposition of cylindrical and Cartesian coordinate systems and transformation between them. Simple formulas have been derived for calculating the electromagnetic field components at a coordinate center; they can be used for solving two-dimensional and three-dimensional problems in a cylindrical coordinate system. The proposed method is used for calculating the outflow of an electromagnetic wave from a circular waveguide into free space.

Block diagonalisation of four-dimensional metrics

Abstract: It is shown that, in four dimensions, it is possible to introduce coordinates so that an analytic metric locally takes block diagonal form. i.e. one can find coordinates such that $g_{\alpha\beta} = 0$ for $(\alpha, \beta) \in S$ where $S = {(1, 3), (1, 4), (2, 3), (2, 4)}$. We call a coordinate system in which the metric takes this form a 'doubly biorthogonal coordinate system'. We show that all such coordinate systems are determined by a pair of coupled second-order partial differential equations.

Time-Dependent Lattice Methods for Ion-Atom Collisions in Cartesian and Cylindrical Coordinate Systems

Abstract: Time-dependent lattice methods in both Cartesian and cylindrical coordinates are applied to calculate excitation cross sections for p+H collisions at 40 keV incident energy. The time-dependent Schroedinger equation is solved using a previously formulated Cartesian coordinate single-channel method on a full 3D lattice and a newly formulated cylindrical coordinate multichannel method on a set of coupled 2D lattices. Cartesian coordinate single-channel and cylindrical coordinate five-channel calculations are found to be in reasonable agreement for excitation cross sections from the 1s ground state to the 2s, 2p, 3s, 3p, and 3d excited states. For extension of the time-dependent lattice method to handle the two electron dynamics found in p+He collisions, the cylindrical coordinate multichannel method appears promising due to the reduced dimensionality of its lattice.

Coordinate input system

Abstract: PROBLEM TO BE SOLVED: To provide a coordinate detection system which includes a coordinate computation method with an advantage of a diagonal scheme for a coordinate input panel having a coordinate input area with a shape of not only a square or a rectangle but also a parallelogram including a diamond shape, and includes a coordinate computation method of a diagonal scheme with which, even in the case where resistance values of sides adjacent to each other in a resistive surrounding electrode per length are different or an external resistance component having a resistance value that is too large to be ignorable, relative to a resistance value of the resistive surrounding electrode is present between four apexes and current detection means or potential output means of a control board, effects of those external resistance portions can be corrected.SOLUTION: In an orthogonal coordinate system having an inclination bisecting an interior angle of a parallelogram for a coordinate input area of the parallelogram and having such a coordinate axis as to pass through a center of the parallelogram, coordinates are computed on the basis of diagonal distribution or diagonal drive of a current, the coordinates are corrected using a ratio between a resistance value between the opposite angles and a resistance value of an external resistance component and further transformed into any arbitrary orthogonal coordinate system, thereby obtaining coordinates.

Hyperspherical Coulomb spheroidal representation in the Coulomb three-body problem

Abstract: The new representation of the Coulomb three-body wave function via the well-known solutions of the separable Coulomb two-centre problem $\phi_j(\xi,\eta)=X_j(\xi)Y_j(\eta)$ is obtained, where $X_j(\xi)$ and $Y_j(\eta)$ are the Coulomb spheroidal functions. Its distinguishing characteristic is the coordination with the boundary conditions of the scattering problem below the three-particle breakup. That is, the wave function of the scattering particles in any open channel is the asymptotics of the single, corresponding to that channel, term of the expansion suggested. The effect is achieved due to the new relation between three internal coordinates of the three-body system and the parameters of $\phi_j(\xi,\eta)$. It ensures the orthogonality of $\phi_j(\xi,\eta)$ on the sphere of a constant hyperradius, $\rho=const$, in place of the surface $R=|\bi{x}_2-\bi{x}_1|=const$ appearing in the traditional Born-Oppenheimer approach. The independent variables $\xi$ and $\eta$ are the orthogonal coordinates on that sphere with three poles in the coalescence points. They are connected with the elliptic coordinates on the plane by means of the stereographic projection. For the total angular momentum $J\ge 0$ the products of $\phi_j$ and the Wigner $D$-functions form the hyperspherical Coulomb spheroidal (HSCS) basis on the five-dimensional hypersphere, $\rho$ being a parameter. The system of the differential equations and the boundary conditions for the radial functions $f^J_i(\rho)$, the coefficients of the HSCS decomposition of the three-body wave function, are presented.

Telegraphist's equations for rectangular waveguides and analysis in nonorthogonal coordinates

Abstract: In our previous works, we have presented one differential method for the efficient calculation of the modal scattering matrix of junctions in rectangular waveguides. The formalism proposed relies on the Maxwell's equations under their covariant form written in a nonorthogonal coordinate system fitted to the structure under study. On the basis of a change of variables, we show in this paper that the curvilinear method and the generalized telegraphist's method lead to the same system of coupled differential equations.

Analysis of scattering from arbitrary configuration of elliptical obstacles using T-matrix representation

Abstract: A new method of analysis of an arbitrary configuration of elliptical cylinders is presented. The treatment of such a problem is based on a combination of the cylindrical and elliptical coordinates systems. This approach is applied to the formulation of the scattered field at the surface of equivalent cylinder surrounding dielectric or metal elliptical object. To analyse the arbitrary set of elliptical cylinders, the iterative scattering procedure is used. A good agreement of the proposed method with the commercial FD simulator is achieved.

Ray tracing and inhomogeneous dynamic ray tracing for anisotropy specified in curvilinear coordinates

Abstract: SUMMARY Ray tracing has recently been expressed for anisotropy specified in a local Cartesian coordinate system, which may vary continuously in a model specified by elastic parameters. It takes advantage of the fact that anisotropy is often of a simpler nature locally (and is thus specified by a smaller number of elastic parameters) and that the orientation of its symmetry elements may vary. Here we extend this approach by replacing the local Cartesian coordinate system with a curvilinear coordinate system of global extent and by applying the new approach to ray tracing and inhomogeneous dynamic ray tracing. The curvilinear coordinate system is orthogonal and is constructed so that the coordinate axes are consistent with the considered anisotropy of the medium. Our formulation allows for computation of ray attributes (e.g. ray velocity vector and paraxial ray attributes) in the curvilinear coordinate system, while rays are computed in global Cartesian coordinates. Compared to the classic formulation in terms of 21 elastic moduli in global Cartesian coordinates, the main advantages are improved efficiency, lower computer-memory requirements, and conservation of anisotropic symmetry throughout the model.

Stability of the Bloch wall via the Bogomolnyi decomposition in elliptic coordinates

Abstract: We consider the one-dimensional anisotropic XY model in the continuum limit. Stability analysis of its Bloch wall solution is hindered by the non-diagonality of the associated linearized operator and the Hessian of energy. We circumvent this difficulty by showing that the energy admits a Bogomolnyi bound in elliptic coordinates and that the Bloch wall saturates it—that is, the Bloch wall renders the energy minimum. Our analysis provides a simple but nontrivial application of the BPS (Bogomolnyi–Prasad–Sommerfield) construction in one dimension, where its use is often believed to be limited to reproducing results obtainable by other means.

N-mode linear-composite coordinate representation and its applications

Abstract: Using the technique of integration within an ordered product of operators, a unitary transformation matrix and N linear-composite coordinate operators are constructed for an N-particle system, and the complete and orthonormal common eigenvectors of the N linear-composite coordinate operators are examined. The N-mode linear-composite coordinate representation is proposed, and its application to a general N-mode coupling quantum harmonic system is studied for solving some dynamic problems.

Thin Wire Modeling for FDTD Electromagnetic Calculations in the Two-Dimensional Cylindrical Coordinate System

Abstract: In this paper, the equivalent radius of a thin wire represented using the FDTD method in the two-dimensional (2D) cylindrical coordinate system is identified as 0.135Δr, where Δr is the lateral side length of the rectangular cells, while that of a thin wire represented in the 3D Cartesian coordinate system is known to be 0.230Δr. Furthermore, it is shown that the technique proposed by Noda and Yokoyama to represent a thin wire having an arbitrary radius in the 3D Cartesian coordinate system can be applied successfully to representing such a thin wire in the 2D cylindrical coordinate system if 0.135Δr is used for the equivalent radius instead of 0.230Δr.

Hyperspherical Coulomb spheroidal representation in the Coulomb three-body problem

Abstract: A new representation of the Coulomb three-body wavefunction via the well-known solutions of the separable Coulomb two-centre problem j(ξ, η) = Xj(ξ)Yj(η) is obtained, where Xj(ξ) and Yj(η) are the Coulomb spheroidal functions. Its distinguishing characteristic is the coordination with the asymptotic conditions of the scattering problem below the three-particle breakup. That is, the wavefunction of two colliding clusters in any open channel is the asymptotics of the single, corresponding to that channel, term of the suggested expansion. The effect is achieved due to a new relation between three internal coordinates of a three-body system and the parameters of j(ξ, η). It ensures the orthogonality of j(ξ, η) on a sphere of constant hyperradius, ρ = const, in place of the surface R = |x2 − x1| = const appearing in the traditional Born–Oppenheimer approach. The independent variables ξ and η are the orthogonal coordinates on this sphere with three poles in the coalescence points. They are connected with the elliptic coordinates on the plane by means of a stereographic projection. For the total angular momentum J ≥ 0 the products of j and the Wigner D-functions form a hyperspherical Coulomb spheroidal (HSCS) basis on a five-dimensional hypersphere, ρ being a parameter. The system of the differential equations and the boundary conditions for the radial functions fJi(ρ), the coefficients of the HSCS decomposition of the three-body wavefunction, are presented.

Angle-domain Common-Image Gathers In Generalized Coordinates

Abstract: The theory of angle-domain common-image gathers (ADCIGs) is extended to migrations performed in generalized 2D coordinate systems. I have developed an expression linking the definition of reflection opening angle to differential traveltime operators and spatially varying weights derived from the non-Cartesian geometry. Generalized-coordinate ADCIGs can be calculated directly using Radon-based offset-to-angle approaches for coordinate systems satisfying the Cauchy-Riemann differentiability criteria. The canonical examples of tilted-Cartesian, polar, and elliptical coordinates can be used to illustrate the ADCIG theory. I have compared analytically and numerically generated image volumes for a set of elliptically shaped reflectors. Experiments with a synthetic data set showed that elliptical-coordinate ADCIGs better resolve the reflection opening angles of steeply dipping structure, relative to conventional Cartesian image volumes, because of improved large-angle propagation and enhanced sensitivity to stee...

Algebro-geometric solutions for some (2+1)-dimensional discrete systems

Abstract: Starting from a discrete spectral problem, a discrete soliton hierarchy is derived. Some ( 2 + 1 ) -dimensional discrete systems related to the hierarchy are proposed. The elliptic coordinates are introduced and the equations in the discrete soliton hierarchy are decomposed into solvable ordinary differential equations. The straightening out of the continuous flow and the discrete flow are exactly given through the Abel–Jacobi coordinates. As an application, explicit algebro-geometric solutions for the ( 2 + 1 ) -dimensional discrete systems are obtained.

Volume coordinate method for hexahedral elements

Abstract: Based on the successful applications of the area coordinate method for quadrilateral elements in 2D problems, a new volume coordinate method is proposed for hexahedral elements in 3D problems, including: 1) the definition of characteristic parameters of a convex hexahedron, and related characteristic conditions under which a hexahedron degenerates into other special solids; 2) the definition of the volume coordinates for hexahedral elements; 3) the transformations between the volume coordinates and the Cartesian and isoparametric coordinates; 4) differential formulas for hexahedral volume coordinates. It can be readily observed that besides keeping the advantages of local natural coordinate system, the new coordinate system has the linearity with the global coordinate system. The volume coordinate method provides a new tool for developing high performance hexahedral elements which are insensitive to mesh distortion.

Hyperspherical elliptic coordinates: New approximate symmetry of three‐body coulomb problem

Abstract: New approximate symmetry of the three‐body Coulomb problem is discussed The symmetry reveals itself on the level of the hyperspherical adiabatic Hamiltonian, giving two new quantum numbers nξ and nη On the basis of these quantum numbers, we introduce the hyperspherical elliptic ansatz for representing a stationary state wave function of a three‐body system with arbitrary masses, which gives direct generalization of the well‐known Born‐Oppenheimer formula Such an approach provides a simple and accurate computational scheme for study of correlation and mass‐polarization effects in a wide class of three‐body systems

Prestack 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 post-stack migration example illustrates the advantages of elliptic coordinates in imaging overturning wavefields. Imaging tests of BP Velocity Benchmark data set illustrate that the RWE migration algorithm generates high-quality prestack migration images comparable to, or better than, the corresponding Cartesian coordinate systems.

Regular Polyprism Parallel Coordinate Plot as a Statistical Graphics Tool

Abstract: The parallel coordinate plot is a graphical data analysis technique for plotting multivariate data. The parallel coordinate plot overcomes the visualization problem of the Cartesian coordinate system for dimensions greater than 4. But, using different ordering of coordinate axes in the parallel coordinate plot of the same data may make different interpretations. Hence, we can use the regular polyprism parallel coordinate plot as an alternative for overcoming the variable arrangement problem of the parallel coordinate plot.

Coordinate Transformation Between New Forms of Geodetic Coordinate Systems

Abstract: This paper presents a transformation method between two different new forms of geodetic coordinate systems on the same ellipsoidal surface.The method not only can be applied to the transformation between two new form of geodetic coordinate systems which are located in different areas,but can realize the computation on the ellipsoidal surface simply and feasibly in a more extensive region.Calculation instances indicate that the method is feasible and accurate.

Laplace Transform Analytic Element Method

Abstract: The Laplace transform analytic element method (LT-AEM), applies the traditionally steady-state analytic element method (AEM) to the Laplace-transformed diffusion equation (Furman and Neuman, 2003). This strategy preserves the accuracy and elegance of the AEM while extending the method to transient phenomena. The approach taken here utilizes eigenfunction expansion to derive analytic solutions to the modified Helmholtz equation, then back-transforms the LT-AEM results with a numerical inverse Laplace transform algorithm. The two-dimensional elements derived here include the point, circle, line segment, ellipse, and infinite line, corresponding to polar, elliptical and Cartesian coordinates. Each element is derived for the simplest useful case, an impulse response due to a confined, transient, single-aquifer source. The extension of these elements to include effects due to leaky, unconfined, multi-aquifer, wellbore storage, and inertia is shown for a few simple elements (point and line), with ready extension to other elements. General temporal behavior is achieved using convolution between these impulse and general time functions; convolution allows the spatial and temporal components of an element to be handled independently.

Stability of the Bloch wall via the Bogomolnyi decomposition in elliptic coordinates

Abstract: We consider the one-dimensional anisotropic XY model in the continuum limit. Stability analysis of its Bloch wall solution is hindered by the nondiagonality of the associated linearised operator and the hessian of energy. We circumvent this difficulty by showing that the energy admits a Bogomolnyi bound in elliptic coordinates and that the Bloch wall saturates it -- that is, the Bloch wall renders the energy minimum. Our analysis provides a simple but nontrivial application of the BPS (Bogomolnyi - Prasad - Sommerfield) construction in one dimension, where its use is often believed to be limited to reproducing results obtainable by other means.

On Some Properties of Infinite-dimensional Elliptic Coordinates

Abstract: A number of the equations in classical mechanics is integrable in Jacobi elliptic coordinates. In recent years a generalization of elliptic coordinates to the infinite case has been offered. We consider this generalization and indicate the connection of infinite-dimensional elliptic coordinates with some inverse spectral problems for infinite Jacobi matrices and sturm-Liouville operators (two spectra inverse problems). Also a link with the rank one perturbation theory is shown.

An Efficient Dynamics Analysis using a Parametric Generalized Coordinate

Abstract: A parametric generalized coordinate formulation for the equations of motion and kinematic constraint equations is presented. The equations of motion and the constraint equations are simplified by introducing parametric generalized coordinates in addition to the Cartesian generalized coordinates. The proposed method has been demonstrated to be very effective for the implicit implementation of the equations of motion and constraints. Sparse matrix solver combined with the proposed formulation plays a key role in the implicit implementation. An automobile example is solved to demonstrate the efficiency of the proposed method.

Evaluation of two-center electric field gradient integrals over STOs using elliptical coordinates

Abstract: In this study, analytical expressions are obtained for two different types of two-center EFG integrals and the integrals are calculated using these expressions. Analytical expressions for the integrals are given in terms of auxiliary J and K functions defined by Ozmen et al. [Int. J. Quantum Chem. 91, 13 (2003)] for the evaluation of two-center Coulomb integrals using elliptical coordinates. Results are compared and found to be in good agreement for small quantum numbers with the existing results from the literature. Numerical stability of the proposed method is very good and therefore we also present results for higher quantum numbers, however there is no previous results found in the literature to compare.