Showing papers on "Elliptic coordinate system published in 2009"

Laplace-transform analytic-element method for transient porous-media flow

Abstract: A unified theory of the Laplace-transform analytic-element method (LT-AEM) for solving transient porous-media flow problems is presented. LT-AEM applies the analytic-element method (AEM) to the modified Helmholtz equation, the Laplace-transformed diffusion equation. LT-AEM uses superposition and boundary collocation with Laplace-space convolution to compute flexible semi-analytic solutions from a small collection of fundamental elements. The elements discussed are derived using eigenfunction expansions of element shapes in their natural coordinates. A new formulation for a constant-strength line source is presented in terms of elliptical coordinates and complex-parameter Mathieu functions. Examples are given illustrating how leaky and damped-wave hydrologic problems can be solved with little modification using existing LT-AEM techniques.

Theory and application of plane elliptic multipoles for static magnetic fields

Abstract: Standard textbooks on beam dynamics study the impact of the magnetic field quality on the beam using field representations based on circular multipoles. Iron dominated magnets, however, typically provide a good field region with a non-circular aspect ratio (i.e. an ellipse whose axis a is significantly larger than the axis b); a boundary not ideal for circular multipoles. The development of superconductors, originally driven to reach fields above ≈ 2 T , allows using them today in completely different fields: iron dominated DC magnets, to save the energy for coil powering as well as repeatedly fast ramped magnets. The cold mass of magnets, housed in common cryostats sectors, makes it tedious to implement additional correction magnets at a later stage, as it requires to warm up the sections where the magnets should be installed as well as unwelding the cryostat. Thus the field homogeneity of the magnets and its influence on the beam has to be thoroughly studied during the project planning phase. Elliptic multipoles, a new type of field expansion for static or quasi-static (here magnetic) two-dimensional fields, are proposed and investigated, which are particular solutions of the potential equation in plane elliptic coordinates obtained by the method of separation. The proper subsets of these particular solutions appropriate for representing static real or complex fields regular within an ellipse are identified. Formulas are given for computing expansion coefficients from given fields. The advantage of this new approach is that the expansion is valid, convergent and accurate in a larger domain, namely in an ellipse circumscribed to the reference circle of the common circular multipoles in polar coordinates. Formulas are derived for calculating the circular multipoles from the elliptical ones. The effectiveness of the approach was tested on many different magnet designs and is illustrated here on the dipole design chosen for the core synchrotron (SIS 100) of the FAIR project as well as on measurement data obtained by rotating coil probes.

Solving the three-dimensional equations of the linear theory of elasticity

Abstract: The three-dimensional Lame equations are solved using Cartesian and curvilinear orthogonal coordinates. It is proved that the solution includes only three independent harmonic functions. The general solution of equations of elasticity for stresses is found. The stress tensor is expressed in both coordinate systems in terms of three harmonic functions. The general solution of the problem of elasticity in cylindrical coordinates is presented as an example. The three-dimensional stress–strain state of an elastic cylinder subjected, on the lateral surface, to arbitrary forces represented by a series of eigenfunctions is determined. An axisymmetric problem for a finite cylinder is solved numerically

Two-Dimensional Scattering by a Conducting Elliptic Cylinder Coated With a Homogeneous Anisotropic Shell

Abstract: A solution to the two-dimensional scattering properties of a conducting elliptic cylinder coated with a confocal homogeneous anisotropic elliptical shell is obtained. The transmitted field of the anisotropic shell is expressed as an integral equation based on waves with different wave numbers and different directions of propagation. The waves in all directions are represented as the eigenfunction expansion in elliptic coordinates in terms of Mathieu functions. In order to solve the nonorthogonality properties of Mathieu functions, Galerkin's method is applied and a matrix is required for the computation of unknown expansion coefficients of the scattered and transmitted fields. Only the transverse magnetic (TM) polarization is presented, while the transverse electric (TE) polarization can be obtained in the same way. Some numerical results are presented in graphical forms. The result is in agreement with that available as expected when a coated elliptic cylinder degenerates to the coated circular one.

A sector elliptic p-element applied to membrane vibrations

Abstract: A new sector p-element is derived and implemented in elliptic coordinates. The element is applied to the free vibration analysis of annular elliptic membranes. The internal shape functions are derived from the shifted Legendre orthogonal polynomials. The stiffness and mass matrices may be integrated exactly using symbolic computing. One-quarter of the annular elliptic membrane is modeled as one element. The solution of the whole membrane is obtained from the solution of one-quarter with appropriate boundary conditions along the symmetry lines. The accuracy of the solution is improved simply by increasing the polynomial degree. Values for the natural frequencies of annular elliptic membranes are obtained and compared with published results. Comparisons show good agreement. New highly accurate values for the natural frequencies of annular elliptic membranes with different aspect ratios and boundary conditions are presented. A case of a sector annular elliptic membrane is also shown.

Plane elliptic or toroidal multipole expansions for static fields: Applications within the gap of straight and curved accelerator magnets

Abstract: Purpose – The purpose of this paper is to present new basis functions suitable to parameterize two‐dimensional static potentials or (magnetic) fields and to show their application in practical cases.Design/methodology/approach – Regular multipole solutions of the potential equation in plane elliptic coordinates are found by separation. The resulting set of functions is reduced to complete subsets suitable for expanding regular potentials or irrotational source‐free fields. Approximate regular plane solutions of the potential equation in local toroidal coordinates are computed by R‐separation and power series expansions in the inverse aspect ratio. The harmonic signals induced in a coil rotating in such a toroidal multipole field are computed from the induction law by similar expansions.Findings – The elliptic expansions are useful in a larger area than circular multipole expansions and give better fits. The toroidal expansions permit one to estimate the effect of the curvature of magnets on the field and ...

Three-dimensional scattering by an infinite homogeneous anisotropic elliptic cylinder in terms of Mathieu functions

Abstract: A solution to the problem of three-dimensional scattering by an infinite homogeneous anisotropic elliptic cylinder for an obliquely incident plane wave of an arbitrary linear polarization is proposed The axial components of the electromagnetic fields inside an anisotropic elliptic cylinder are represented as two coupled integrals of suitable eigenfunctions in elliptic coordinates in terms of Mathieu functions Scattering by an anisotropic elliptic cylinder is different from scattering by a sphere or a circular cylinder because of the nonorthogonality properties of Mathieu functions at the interface between two different media In order to solve this problem, Galerkin's method is applied to the boundary conditions to solve the unknown coefficients Numerical results are presented, discussed, and compared with available data

A Comparative Study of Modeling the Magnetostatic Field in a Current-Carrying Plate Containing an Elliptic Hole

Abstract: The presence of crack-like defects can cause an uneven distribution of the electric current density in a cracked conductor. To investigate the perturbation of the magnetic field resulting from the disturbed electric current, computational modeling of the magnetostatics is attempted on an infinite conductive plate, which contains an elliptic hole and is subjected to uniform current flow at infinity. Both 2-D and 3-D analyses are considered in this study. The 2-D analysis requires certain crucial assumptions and the governing Maxwell's equations are solved analytically in elliptic coordinates. The 3-D numerical computation is based on superposition of the elementary solution, whose derivation utilizes the Biot-Savart law. To improve the efficiency of the 3-D calculation, an adaptive mesh refinement algorithm is implemented in the numerical discretization. Finally, through a comparative study, the validity of the introduced simplifications in the 2-D analysis is benchmarked with the 3-D computational results. The present study shows that the 2-D solution predicts the upper bound for the out-of-plane component of the magnetic field perturbed by the elliptical hole, whose semi-major axis does not exceed ten times the thickness of the plate.

Calibration of a multi-plane X-ray unit

Abstract: For calibrating a multi-plane X-ray unit, an internal matrix mapping coordinate system internal to the unit onto coordinate system of a first radiographic plane is predefined in a reference projection geometry. An external matrix mapping coordinate system external to the unit onto coordinate system of the first plane is determined from image data captured by the first plane at calibration points and coordinates of the calibration points. An external matrix mapping coordinate system external to the unit onto coordinate system of a second radiographic plane is determined from image data captured by the second plane at the calibration points and the coordinates of the calibration points. A measure of position of the second plane with respect to coordinate system internal to the unit or with respect to the first plane is determined from the internal and external matrix of the first plane and the external matrix of the second plane.

A Gaussian quadrature method for total energy analysis in electronic state calculations

Abstract: This article reports studies by Fukushima and coworkers since 1980 concerning their highly accurate numerical integral method using Gaussian quadratures to evaluate the total energy in electronic state calculations. Gauss-Legendre and Gauss-Laguerre quadratures were used for integrals in the finite and infinite regions, respectively. Our previous article showed that, for diatomic molecules such as CO and FeO, elliptic coordinates efficiently achieved high numerical integral accuracy even with a numerical basis set including transition metal atomic orbitals. This article will generalize straightforward details for multiatomic systems with direct integrals in each decomposed elliptic coordinate determined from the nuclear positions of picked-up atom pairs. Sample calculations were performed for the molecules O3 and H2O. This article will also try to present, in another coordinate, a numerical integral by partially using the Becke's decomposition published in 1988, but without the Becke's fuzzy cell generated by the polynomials of internuclear distance between the pair atoms. Instead, simple nuclear weights comprising exponential functions around nuclei are used. The one-center integral is performed with a Gaussian quadrature pack in a spherical coordinate, included in the author's original program in around 1980. As for this decomposition into one-center integrals, sample calculations are carried out for Li2. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2009

From Jacobi Problem of Separation of Variables to Theory of Quasipotential Newton Equations

Abstract: Our solution to the Jacobi problem of finding separation variables for natural Hamiltonian systems H = ½p 2 + V(q) is explained in the first part of this review. It has a form of an effective criterion that for any given potential V(q) tells whether there exist suitable separation coordinates x(q) and how to find these coordinates, so that the Hamilton-Jacobi equation of the transformed Hamiltonian is separable. The main reason for existence of such criterion is the fact that for separable potentials V(q) all integrals of motion depend quadratically on momenta and that all orthogonal separation coordinates stem from the generalized elliptic coordinates. This criterion is directly applicable to the problem of separating multidimensional stationary Schrodinger equation of quantum mechanics. Second part of this work provides a summary of theory of quasipotential, cofactor pair Newton equations $$ \ddot q $$ = M(q) admitting n quadratic integrals of motion. This theory is a natural generalization of theory of separable potential systems $$ \ddot q $$ = −∇(q). The cofactor pair Newton equations admit a Hamilton-Poisson structure in an extended 2n + 1 dimensional phase space and are integrable by embedding into a Liouville integrable system. Two characterizations of these systems are given: one through a Poisson pencil and another one through a set of Fundamental Equations. For a generic cofactor pair system separation variables have been found and such system have been shown to be equivalent to a Stackel separable Hamiltonian system. The theory is illustrated by examples of driven and triangular Newton equations.

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...

Is Possible to Plot Matrices into a Multi-Dimensional Coordinate System?

Abstract: This paper proposes two types of multi-dimensional coordinate systems to plot matrices, these two multi-dimensional coordinate systems are the surface mapping coordinate system and the four dimensional physical space. The idea is to represent graphically a large system of simultaneous equations into the same graphical space and time.

Computation of orthogonal Fourier-Mellin moments in two coordinate systems.

Abstract: The computing method for orthogonal Fourier-Mellin moments in a polar coordinate system is presented in detail The image expressed in a Cartesian system has to be transformed into a polar coordinate system first when we calculate the orthogonal Fourier-Mellin moments of the image in a polar coordinate system, which will increase both computational complexity and error To solve the problem, a new direct computing method for orthogonal Fourier-Mellin moments in a Cartesian coordinate system is proposed, which can avoid the image transformation between two coordinate systems and eliminate the rounding error in coordinate transformation and decrease the computational complexity

The quadrilateral fully-parametrized plate elements based on the absolute nodal coordinate formulation

Abstract: Summary. The article provides a review of the quadrilateral fully-parametrized plate elements based on absolute nodal coordinate formulation that can be used in the dynamic analysis of large deformations in multibody applications. The absolute nodal coordinate formulation is a recently proposed approach to the analysis of multibody systems that can take into account nonlinearities, including large deflections and plasticity. In the absolute nodal coordinate formulation, finite elements are defined in the global coordinate system using position coordinates together with independent global gradient vectors that are, in fact, partial derivatives of the position vector with respect to the element coordinates. This leads to a constant mass matrix in two and threedimensional applications and is a unique feature among the beam and plate elements based on the absolute nodal coordinate formulation.

Calculation of complex shapes in the Fourier modal method through the concept of coordinate transformations

Abstract: In a recent publication [1] we showed that complex shapes can be calculated efficiently in the Fourier modal method (FMM) through the concept of coordinate transformations. The new coordinate system has to be aligned in such a way that the lines of constant coordinates match the interfaces. Thus, the approach of adaptive spatial resolution (ASR) can be included easily to increase the convergence in the case of metallic materials and to simplify the derivation of appropriate coordinate systems. We are going to present the fundamental ideas of the method and show our latest examples of coordinate transformations to match such common structures as cylinders, triangles, and rotated squares.

The Multi-Dimensional (MD) Coordinate Systems

Abstract: This paper shows different multi-dimensional coordinate systems that can be applied on science and social sciences This paper offers a serial of multi-dimensional coordinate systems to visualize and analyze from 4-Dimensions until Infinity dimensions into the same graphical space At the inception of multi-dimensional coordinate systems, the following new types of multi-dimensional coordinate systems are presented: Pyramid coordinate system, Diamond coordinate system, 4-Dimensional coordinate system, 5-Dimensional coordinate system, Infinity Dimensional coordinate system and Multi-functional Pictorial coordinate system These six Multi-Dimensional coordinate systems are constructed based on the traditional 3-D space concept, but they represent 4-D, 5-D, 8-D, 9-D and Infinity-Dimension

Two-Dimensional Scattering by a Conducting Elliptic Cylinder Coated With a

Abstract: A solution to the two-dimensional scattering proper- ties of a conducting elliptic cylinder coated with a confocal ho- mogeneous anisotropic elliptical shell is obtained. The transmitted field of the anisotropic shell is expressed as an integral equation based on waves with different wave numbers and different direc- tions of propagation. The waves in all directions are represented as the eigenfunction expansion in elliptic coordinates in terms of Mathieu functions. In order to solve the nonorthogonality proper- ties of Mathieu functions, Galerkin's method is applied and a ma- trix is required for the computation of unknown expansion coeffi- cients of the scattered and transmitted fields. Only the transverse magnetic (TM) polarization is presented, while the transverse elec- tric (TE) polarization can be obtained in the same way. Some nu- merical results are presented in graphical forms. The result is in agreement with that available as expected when a coated elliptic cylinder degenerates to the coated circular one.

Potential Coupling Through a Slot Backed by an Elliptical Cavity: Neumann and Dirichlet Boundary Conditions

Abstract: Exact, closed-form expressions are presented for the solution to Laplace's equation everywhere in a specific cavity-backed aperture problem. A symmetrically placed slot enables the static field in a half-space above a hard or soft ground plane to penetrate into the interior of an elliptically shaped cavity. Separation-of-variables in elliptic coordinates yields a summable series, whereupon the aperture field or its normal derivative appears naturally as a series of edge-condition weighted Chebyshev polynomials in the Cartesian coordinates. Analytic coefficients explicitly display simple dependence upon the cavity and point-source geometry.

Wave Elevation Between Elliptical Structures

Abstract: The hydrodynamic interaction of waves with arrays of vertical elliptical cylinders is considered. The present paper aims at developing of an efficient calculation method for predicting the extreme elevation of the free surface, in the fluid domain between ship-shaped structures in close proximity. Linear potential theory is employed and the solution method is based on the semi-analytical formulation of the various velocity potentials in elliptical coordinates, using series expansions of Mathieu functions and the so-called addition theorem for Mathieu functions.Copyright © 2009 by ASME

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-coordinateADCIGs 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 steep structural dips afforded by coordinate system transformations

Coordinate calculation method

Abstract: A touch panel detects a point in one of a plurality of unit areas at which an input was made, the unit areas being arranged in a matrix in an instruction plane. A game apparatus repeatedly acquires detection coordinates for locating a unit area detected by a pointing device. Also, the game apparatus repeatedly calculates, in response to the acquisition of the detection coordinates, detailed coordinates by which a point can be represented with accuracy in more detail than by the detection coordinates. The detailed coordinates indicate a point in the direction of a unit area indicated by previously acquired detection coordinates, as viewed from a predetermined reference point within a unit area indicated by currently acquired detection coordinates.

The Pyramidal Coordinate System

Abstract: This paper propose an alternative multidimensional coordinate system, it is called “The Pyramidal (P) Coordinate System”. The Pyramidal coordinate system is available to visualize five dimensions into the same graphical space. The idea is to visualize multi-variable economic modeling into the same graphical space. The Pyramidal coordinate system is working only with positive values. In our case, we suggest to use growth rates into each coordinate system. Additionally, all growth rates in each coordinate in the Pyramidal coordinate system request the application of absolute values.

Power flow of the guided mode of a waveguide

Abstract: The integral ∫ΩΦΨdΩ taken over an arbitrary plane region Ω where the scalar functions of the point Φ and Ψ are the solutions of the Helmholtz two-dimensional equation is presented as a contour, i.e., in the invariant view and in three main orthogonal coordinate systems on a plane, namely, in the Cartesian, polar, and elliptic coordinate systems. An invariant expression in the view of the contour integral for power flow of the guided mode through an arbitrary region of the cross section of a waveguide with constant permittivity has been obtained.