Showing papers on "Elliptic coordinate system published in 1980"

Efficient Generation of Body-Fitted Coordinates for Cascades Using Multigrid

Abstract: A METHOD for the generation of body-fitted curvilinear coordinate systems is discussed and a technique using multigrid is proposed to reduce the computing time and to improve the accuracy. This technique is used to accelerate the convergence of point and line SOR relaxation schemes. The computing time is reduced, respectively, by factors of 2 and 3 over the usual point and line SOR. The accuracy for a given computational effort is improved by one and two orders of magnitude, respectively, when compared with point SOR. The multigrid scheme is then applied to the generation of cur- vilinear coordinates for turbine cascades, Contents The numerical computation of a flowfield requires an adequate treatment of the boundary conditions which can be quite difficult to incorporate for complex geometries en- countered in practical engineering problems. This can be resolved by the use of an appropriate coordinate system where the coordinate lines coincide with the boundaries. A second characteristic is the ability to stretch the grid in order to concentrate more nodes in regions where high gradients of flow properties are expected. The main advantage is that both the mesh generation and the solution of the problem of in- terest are solved on a rectangular mesh. This lends itself to a very simple discretization using finite differences or finite volumes particularly at the boundaries where all in- terpolations are avoided. Another possible application is the automatic generation of a finite element mesh. Transformations that map arbitrary physical regions into a rectangle with the above characteristics have been proposed by several authors1'2 and are called body-fitted curvilinear coordinate systems. The approach consists of solving a system of elliptic equations which yield the physical coordinates in terms of the transformed coordinates. The physical coor- dinates Z = Z(T,rj) and = 0( r,r)) are obtained by solving the following system of nonlinear elliptic coupled equations + yZTT - T = 0

Boundary-fitted coordinate systems for arbitrary computational regions

Abstract: The method of Smith and Wiegel was used to generate meshes for mixer lobes and subsonic inlets that are compatible with flow analysis codes requiring a boundary fitted coordinate system. Successful application of this mesh generator required development of procedures to distribute the mesh points along the boundaries, to regulate the dependence of the connecting function to the local boundary slope, to concentrate the mesh into regions of special interest, and to modify the mesh grid so that it possessed a smooth progression of cell metrics and cell volumes in all directions. The method of Smith and Wiegel when coupled with the extensions mentioned above has proven to be easy to use and control for the inlet and mixer lobe geometries investigated.

Some remarkable R-separable coordinate systems for the Helmholtz equation

Abstract: We present examples to show that the phenomenon ofR-separation occurs nontrivially for the Helmholtz equation on a pseudo-Riemannian manifold.R-separable coordinate systems can be both orthogonal and non-orthogonal and a given coordinate system mayR-separate in more than one way. A satisfactory theory of variable separation for the Helmholtz equation must incorporateR-separation.

Generation of orthogonal boundary-fitted coordinate systems

Abstract: A method is presented for computing orthogonal boundary fitted coordinate systems for geometries with coordinate distributions specified on all boundaries. The system which has found most extensive use in generating boundary fitted grids is made up of Poisson equations, of which the functions P and Q provide a means for controlling the spacing and density of grid lines in the coordinate system. While questions remain concerning the existence and uniqueness of orthogonal systems, the generating method presented adds to the available, useful techniques for constructing these systems.

Separation of variables in Einstein spaces. II. No ignorable coordinates

Abstract: We prove that the only Einstein spaces which admit a coordinate system with no ignorable coordinates which separates the Hamilton-Jacobi equation are certain symmetric spaces of Petrov typeD due to Kasner and the constant-curvature de Sitter spaces. We also show that a space admitting a coordinate system with no ignorable coordinates which separates the Helmholtz (Schrodinger) equation must be of Petrov type

Second-order derivatives in a local frame developed in spheroidal and spherical coordinates

Abstract: Second-order derivatives of a general scalar function of position (F) with respect to the length elements along a family of local Cartesian axes are developed in the spheroidal and spherical coordinate systems. A link between the two kinds of formulations is established when the results in spherical coordinates are confirmed also indirectly, through a transformation from spheroidal coordinates. IfF becomesW (earth's potential) the six distinct second-order derivatives—which include one vertical and two horizontal gradients of gravity—relate the symmetric Marussi tensor to the curvature parameters of the field. The general formulas for the second-order derivatives ofF are specialized to yield the second-order derivatives ofU (standard potential) and ofT (disturbing potential), which allows the latter to be modeled by a suitable set of parameters. The second-order derivatives ofT in which the property ΔT=0 is explicitly incorporated are also given. According to the required precision, the spherical approximation may or may not be desirable; both kinds of results are presented. The derived formulas can be used for modeling of the second-order derivatives ofW orT at the ground level as well as at higher altitudes. They can be further applied in a rotating or a nonrotating field. The development in this paper is based on the tensor approach to theoretical geodesy, introduced by Marussi [1951] and further elaborated by Hotine [1969], which can lead to significantly shorter demonstrations when compared to conventional approaches.

Relationship between disclinations and coordinate systems

Abstract: The geometry of wedge disclinations and several related configurations has been analyzed in both cylindrical as well as Cartesian coordinates. It is shown that both representations yield a well-defined dislocation content, i.e. non-vanishing torsion tensor. On the other hand, the curvature tensor is found to vanish in the cylindrical coordinate representation, but remain finite in the Cartesian formulation.

Generations of orthogonal surface coordinates

Abstract: Two generation methods were developed for three dimensional flows where the computational domain normal to the surface is small. With this restriction the coordinate system requires orthogonality only at the body surface. The first method uses the orthogonal condition in finite-difference form to determine the surface coordinates with the metric coefficients and curvature of the coordinate lines calculated numerically. The second method obtains analytical expressions for the metric coefficients and for the curvature of the coordinate lines.

Long atmospheric waves on the sphere and on the polar plane

Abstract: In earlier studies of long low-frequency atmospheric waves in the polar atmosphere a cyclindrical coordinate system had been used with the planez=0 tangential to the earth's surface at the pole It had been found by numerical examples that the results obtained with a cylindrical coordinate system approximate quite well those obtained with a spherical coordinate system Here, it is shown that the formulae for the wave frequencies in both coordinate systems are the same to a good degree of approximation