Showing papers on "Spherical coordinate system published in 1982"

Functions of several variables

Abstract: This chapter discusses the functions of several variables. A rectangular coordinate system in three dimensions, also called a Cartesian coordinate system, is formed by three mutually perpendicular coordinate lines that intersect at their origins. The point of intersection of the axes is called the origin of the coordinate system. Each pair of coordinate axes determines a plane, called a coordinate plane. These are referred to as the xy plane , the xz plane , and the yz plane . To each point P in three-dimensional space, one can assign an ordered triple of numbers ( a, b, c ), called the coordinates of P . It can be done by passing three planes through P parallel to the coordinate planes and recording the coordinates a , b , and c of the intersections of the planes with the x , y , and z axis, respectively. For functions of two variables, there are analogous notions.

Method and apparatus for determining the location of points on a three dimensional thing

Abstract: This invention discloses a method and an apparatus in which coordinates of points on a plurality of sections of a three dimensional thing are measured based on a plurality of different coordinate systems and then transformed into those in a single reference coordinate system. The coordinates of the section adjacent to the reference section are transformed into coordinates in the reference coordinate system which is used to measure the coordinates of the points on the reference section through the following processes: measuring the coordinates of at least three reference points on the overlapped portion between the reference section and the adjacent section based on the reference coordinate system and the second coordinate system which is used to measure the coordinates of the points on the adjacent section; determining the positional relation between the two systems above using the measured coordinates of the reference points; and then, transforming the coordinates of the points on the adjacent section defined by the second coordinate system into those in the reference section using the above positional relation. The coordinates of points in the other sections are subjected to similar processes to express them in the reference system.

Generalization of the spherical harmonic method to radiative transfer in multi-dimensional space

Abstract: The basic radiative transfer equation in three-dimensional space is expressed in terms of three commonly used coordinate systems, namely, Cartesian, cylindrical and spherical coordinates The concept of a transformation matrix is applied to the transformation processes between the Cartesian system and two other systems The spherical harmonic method is then applied to decompose the radiative transfer equation into a set of coupled partial differential equations for all three systems in terms of partial differential operators By truncating the number of partial differential equations into four along with further mathematical analyses, we obtain a modified Helmholtz equation For each coordinate system, analytical solutions in terms of infinite series are obtained whenever the equation is solvable by the technique of separation of variables with proper boundary conditions Numerical computations are carried out for one dimensional radiative transfer to illustrate the applicability of the technique developed in the present study

Numerical studies of nonspherical carbon combustion models

Abstract: We have performed a set of axisymmetric hydrodynamic calculations of the core carbon flash, which incorporate the treatment of two-dimensional convective flow without the need of a phenomenological theory of convection. Our computations which were done using a spherical coordinate system with 100 radial and 30 angular grid points show that, in the beginning, the nuclear burning slowly (roughly-equal 0.2 x speed of sound) propagates outward from the center of the star in the form of a very nonspherical ''salt finger'' shaped combustion front.

The stationary coordinate systems in flat spacetime

Abstract: The stationary metrics in flat spacetime are derived using a recent classification of the timelike Killing vector field trajectories. The metrics fall into six classes. A ’’simple’’ coordinate system from each class is selected as representative. Three of these systems are rectangular Minkowski coordinates, pseudocylindrical (’’accelerating’’) coordinates, and rotating coordinates. The remaining three appear to be new coordinate types which will be useful in exploring coordinate‐dependent effects in quantum field theory.

Supersonic Nonlinear Potential Flow with Implicit Isentropic Shock Fitting

Abstract: A fully implicit nonconservative marching technique is utilized to solve the full potential flow equation in a spherical coordinate frame for supersonic inviscid flow. Transonic techniques form the basis for the numerical scheme in the cross flow plane which allows for capture of both bow and embedded shocks. A column relaxation scheme is presented which exhibits enhanced computational efficiency over ring relaxation schemes. An implicit bow shock fitting procedure satisfying the isentropic jump condition is also presented and compared to the shock capture method, shock fit Euler solutions, and experimental data for arbitrary conical surfaces.

On exact analytical solutions for the few-particle Schrödinger equation. II. The ground state of helium

Abstract: Analytical methods for solving the Schrodinger equation directly can be applied to few-particle systems. This is illustrated by deriving a solution to the first-order perturbation equation for the ground state of helium. This solution is the in form of a partial wave expansion in spherical polar coordinates with Legendre polynomials as the angular functions. The radial functions include polynomials and exponential integral functions. Arbitrary parameters in the formal solution, associated with the square-integrability of the wavefunction, are identified. Their values are determined by a least-squares method. The same arbitrary parameters occur in formal solutions of the higher-order perturbation equations. It is evident that a similar treatment can be applied to these equations, and to the eigenvalue problem.

Full-implicit continuous eulerian scheme in the spherical coordinates and its applications to solar phenomena

Abstract: In this paper, the full-implicit-continuous-eulerian(FICE) scheme is extended to the spher-ical coordinates so as to study global problems in solar atmosphere. We apply the so-called line-by-line method to solving the difference equation of pressure and the projected characteristicequations to determining the boundary conditions on the top and bottom. Besides, the varia-tion of gravitational field with altitude is taken into account and a non-uniform mesh alongthe radial direction is adopted. As a numerical example, we discuss the physical process in thesolar atmosphere due to a radial mass ejection in the neighbourhood of the equator on the solarsurface, with the initial magnetic field being taken to be a dipole field The results may eluci-dato the ejecting prominence and the accompanying coronal transient.

Numerical grid generation; Proceedings of the Symposium on Numerical Generation of Curvilinear Coordinate Systems and Their Use in the Numerical Solution of Partial Differential Equations, Nashville, TN, April 13-16, 1982

Abstract: General curvilinear coordinate systems are considered along with the error induced by coordinate systems, basic differential models for coordinate generation, elliptic grid generation, conformal grid generation, algebraic grid generation, orthogonal grid generation, patched coordinate systems, and solid mechanics applications of boundary fitted coordinate systems. Attention is given to coordinate system control and adaptive meshes, the application of body conforming curvilinear grids for finite difference solution of external flow, the use of solution adaptive grids in solving partial differential equations, adaptive gridding for finite difference solutions to heat and mass transfer problems, and the application of curvilinear coordinate generation techniques to the computation of internal flows. Other topics explored are related to the solution of nonlinear water wave problems using boundary-fitted coordinate systems, the numerical modeling of estuarine hydrodynamics on a boundary-fitted coordinate system, and conformal grid generation for multielement airfoils.

R-separable solutions of Einstein's field equations

Abstract: R-separable coordinate systems are introduced as a new class of systems in which the equations of general relativity may be solved. The static, axially symmetric vacuum and electrovac Einstein field equations are solved for two such systems, tangent spheres and bispherical. The bispherical is found to be more versatile than any previously used coordinate system in that its eigensolutions can represent the exteriors of single point, double point and line sources. The tangent sphere eigensolutions are found to be generalisations of the Curzon solution. The relatively simple nature of the individual bispherical eigensolutions allows explicit integration of the field equations for a completely general, static, two-body source. The bodies are then charged, according to the Weyl formalism, and the conditions for balance obtained. Finally, it is shown that all vacuum Weyl solutions are either type I or type D.

Free-Space Green's Functions of the Reduced Wave Equation.

Abstract: : The governing differential equations are solved by the method of separation of variables for the separate cases of Cartesian, cylindrical and spherical coordinate systems. The resulting integral/series expansions of the Green's functions for outgoing waves are then reduced to their standard and simple form. In addition to bringing together, in tabular form, a number of frequently used Green's functions this memorandum also provides an introduction to their derivation. (Author)

A General Diffraction Theory: A New Approach

Abstract: Several years ago we published some results concerning the structure of geometrical wavefronts in a homogeneous medium. A description of a wavefront was obtained as a general solution of the eikonal equation. By means of some standard techniques in differential geometry this led to the description of the general caustic. This in turn led us to the study of the geometric aberrations of an optical system from a completely different point of view. These results also suggest an approach to the physics of. the Propagation of light in a homogeneous medium which ought to lead to a proper vector diffraction theory. Recall Hertz' approach to the problem of the spherical wavefront in which he transformed the Maxwell equations into wave equations for the vector and scalar potential functions. To apply these to the spherical wavefront he transformed the arguments of the aradient, divergence and curl operators to a spherical coordinate system. The near-field solution of the wave equation predicted the existence of the dipole oscillator. The farfield solution yielded the now well-known description of the polarization and energy distribution on a spherical wavefront. Present work involves applying these same techniques to the general wavefront obtained as a solution of the eikonal equation. The vector differential operators have been transformed to an appropriate generalized coordinate system. The wave equation for the Potential functions have been obtained and an intermediate integral has been found. As is stands at the present time the electric and magnetic vectors can be expressed in terms of this intermediate integral.© (1982) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

Calculating method of three-dimensional coordinate value

Abstract: PURPOSE:To reduce the time for calculation of coordinate values with simple constitution by corresponding beforehand rotation numbers and storing the local coordinate system that defines a shape when said system is not inclined with respect to the entire coordinate system. CONSTITUTION:The element of a rotary matrix is any among 0, 1, -1 when a local coordinate axis and the entire coordinate axis coincide. Rotation numbers are provided in correspondence to 24 kinds of the rotary matrix when the local coordinate system is not inclined with respect to the entire coordinate system. Values of x1, y1, z1 are inputted to the register RG-EX of the local coordinate axis and the signals of 1-24 and 25 are generated from a rotation number device RN. A switch SW is changed over so as to pass the output through a code inverter INV when a storage/generation device RVN which emits a control signal in correspondence to the numbers 1-24 outputs -1. Input is made to the register RG-GN of the entire coordinate system through a matrix calculation circuit MTP in the case of the number 25. The coordinate value in the entire coordinate is then calculated by substitution calculations.

Full implicit continuous Eulerian scheme in the spherical coordinates and its applications to solar phenomena.

Abstract: In this paper, the full-implicit-continuous-eulerian(FICE) scheme is extended to the spher-ical coordinates so as to study global problems in solar atmosphere. We apply the so-called line-by-line method to solving the difference equation of pressure and the projected characteristicequations to determining the boundary conditions on the top and bottom. Besides, the varia-tion of gravitational field with altitude is taken into account and a non-uniform mesh alongthe radial direction is adopted. As a numerical example, we discuss the physical process in thesolar atmosphere due to a radial mass ejection in the neighbourhood of the equator on the solarsurface, with the initial magnetic field being taken to be a dipole field The results may eluci-dato the ejecting prominence and the accompanying coronal transient.

A general system of polar coordinates with applications

Abstract: We present in an expository way a general method of introducing certain “polar coordinates” which can be easily applied to handle some interesting problems in the fields of singular integral operators, differentiation theory,... by means of a technique which follows the steps of the rotation method of Calderon and zygmund. A more complete technical exposition will be published elsewhere.

Numerical solution of potential flow about arbitrary 2-dimensional multiple bodies

Abstract: A procedure for the finite-difference numerical solution of the lifting potential flow about any number of arbitrarily shaped bodies is given. The solution is based on a technique of automatic numerical generation of a curvilinear coordinate system having coordinate lines coincident with the contours of all bodies in the field, regardless of their shapes and number. The effects of all numerical parameters involved are analyzed and appropriate values are recommended. Comparisons with analytic solutions for single Karman-Trefftz airfoils and a circular cylinder pair show excellent agreement. The technique of application of the boundary-fitted coordinate systems to the numerical solution of partial differential equations is illustrated.