Showing papers on "Elliptic coordinate system published in 1987"

Two-dimensional, fully numerical solutions of second-order Dirac equations for diatomic molecules. part 3

Abstract: The exact, four-component second-order Dirac equation in transformed elliptical coordinates is derived for a diatomic system. Numerical solutions are obtained using a point-relaxation approach. The relativistic correction to the total energy of H2+ at R = 2a.u. is found to be −7.365 × 10−6a.u. The deviation from an earlier value of −7.383 × 10−6a.u. by Luke et al. or by Bishop is shown to arise from a spuriously large c−4 term in their Hamiltonian.

Electronic resonances of HeT+ resulting from the beta decay of the tritium molecule

Abstract: Calculations for the three lowest resonant states of the molecular ion HeT+ are presented. These states are known to be produced with high probabilities in the beta decay of the T2 molecule. Therefore, their positions and widths are needed for an accurate interpretation of the neutrino mass experiments employing T2 molecules as a source of radiation. These quantities may also be useful in interpreting low-energy He++H scattering and far-ultraviolet spectroscopy of HeH+. The resonance positions and widths have been obtained using the method of analytic continuation of the real valued stabilisation graphs calculated with a basis set of explicitly correlated functions in elliptic coordinates. A discussion of the complex scaling method in these coordinates is given, and an appropriate stabilisation transformation pertinent to the use of elliptic coordinates is derived. As a test of the method, the lowest resonant states of the H2 molecule and Li+ ion have also been calculated, and the results agree very well with data in the literature obtained with much more laborious methods.

Position control method in which a cartesian point is transformed through two cartesian coordinate systems

Abstract: When NC data inputted from an NC data supply unit (1) are data (G68) indicative of a coordinate transformation, a discriminating unit (2a) applies rotational axis vectors, the coordinates of centers of rotation and the angles of rotation, all of which are commanded following the data G68, to a transformation matrix generator (2b) and instructs the latter to generate transformation matrices. In response, the transformation matrix generator (2b) generates coordinate transformation matrices [M 1 ], [M 2 ] using the given data. When the NC data are path data, on the other hand, the discriminating unit inputs these data to a coordinate transformation unit (2c), which proceeds to subject the positional coordinates contained in the path data to a coordinate transformation using the transformation matrices [M 1 ], [M 2 ] and apply the results to an axis controller (3). The axis controller (3) performs machining by moving a tool along a path obtained by rotations through angles θ, φ about first and second rotational axes, respectively, based on the position data obtained by the coordinate transformation.

Natural convection from a flat plate oriented at an arbitrary angle in an infinite medium

Abstract: Steady two-dimensional natural convection around a heated finite flat plate oriented at an arbitrary angle, α, with respect to the gravity force is formulated in terms of an elliptic coordinate system. The method of spectral series expansion is used to reduce the governing coupled partial differential equations to three sets of second-order ordinary differential equations. These equations are truncated and then numerically integrated by the finite-element method of collocation. The effect of boundary conditions on the outer boundary of the computational field is discussed. The flows for Ra = 1000.0, Pr = 0.7 and α = 0,2, 45, and 90 degrees are analyzed using illustrations of fluid flow patterns, graphs of surface vorticity and local Nusselt number, and contour plots of isotherms.

Toroidal coordinates for a three-body problem

Abstract: A three-body problem is considered in toroidal coordinates. It is shown that the distance between two particles placed at the foci of the toroidal coordinate system can be expressed in terms of the hyperradius of a three-body system; the variables in the kinetic energy operator are separable for zero total angular momentum. An equivalent treatment of the original problem is also possible in four-dimensional space in the cylindrical coordinate system. In this approach the free motion of three particles is characterised by three quantum numbers: hypermoment, poloidal and azimuthal.

Solution of the laminar transport equations in complex geometries by means of curvilinear coordinates

Abstract: The standard coordinate systems (i.e. cartesian, cylindrical-polar, spherical-polar or combinations theoreof) which are usually employed in the numerical solution of the transport differential equations, are suitable for simple geometries only. In the present study the momentum, mass and energy conservation equations are expressed in terms of general curvilinear-orthogonal coordinates, which enable the irregularly shaped solution domains, encountered in practice, to be completely mapped i.e. all boundaries to coincide with, coordinate surfaces. A mixed coordinate system is also introduced, which, is curvilinear-orthogonal in, the two of the three directions and rectilinear in the third direction. The latter system gives simpler equations and is suitable for straight flow passages of arbitrary cross sectional shape. The momentum, mass and energy conservation, differential, equations are transformed to finite-difference ones by integration over six-sided control volumes formed by coordinate surfaces and are then solved by an iterative procedure. The method is tested successfully in various flow and heat transfer cases.