Elliptic coordinate system

About: Elliptic coordinate system is a research topic. Over the lifetime, 670 publications have been published within this topic receiving 11135 citations. The topic is also known as: elliptical coordinate system & elliptic coordinates.

Algebro–geometric constructions of the discrete Ablowitz–Ladik flows and applications

Abstract: Resorting to the finite-order expansion of the Lax matrix, the elliptic coordinates are introduced, from which the discrete Ablowitz–Ladik equations and the (2+1)-dimensional Toda lattice are decomposed into solvable ordinary differential equations. The straightening out of the continuous flow and the discrete flow is exactly given through the Abel–Jacobi coordinates. As an application, explicit quasiperiodic solutions for the (2+1)-dimensional Toda lattice are obtained.

Describing Rigid-Flexible Multibody Systems Using Absolute Coordinates

Abstract: This paper deals with the dynamic description of interconnected rigid and flexible bodies. The absolute nodal coordinate formulation is used to describe the motion of flexible bodies and natural coordinates are used to describe the motion of the rigid bodies. The absolute nodal coordinate formulation is a nonincremental finite element procedure, especially suitable for the dynamic analysis of flexible bodies exhibiting rigid body motion and large deformations. Nodal coordinates, which include global position vectors and global slopes, are all defined in a global inertial coordinate system. The advantages of using the absolute nodal coordinate formulation include constancy in the mass matrix and the need for only a minimal set of nonlinear constraint equations when connecting different flexible bodies with kinematic joints. When bodies within the system can be considered rigid, the above-mentioned advantages of the equations of motion can be preserved, provided natural coordinates are used. In the natural coordinate method, the coordinates used to describe rigid bodies include global position vectors of basic points and global unit vectors. As occurs in absolute nodal coordinate formulation, rotational coordinates are avoided and the mass matrix is also constant. This paper provides computer implementation of this formulation that uses absolute coordinates for general two-dimensional multibody systems. The constraint equations needed to define kinematic joints between different bodies can be linear or nonlinear. The linear constraint equations, which include those needed to define rigid connections and revolute joints, are used to define constant connectivity matrices that reduce the size of the system coordinates. These constant connectivity matrices are also used to obtain the mass matrix and generalized forces of the system. However, the nonlinear constraint equations that account for sliding joints require the use of the Lagrange multipliers technique. Numerical examples are provided and compared to the results of other existing formulations.

Developments in determining the gravitational potential using toroidal functions

Abstract: Cohl & Tohline (1999) have shown how the integration/summation expression for the Green’s function in cylindrical coordinates can be written as an azimuthal Fourier series expansion, with toroidal functions as expansion coefficients. In this paper, we show how this compact representation can be extended to other rotationally invariant coordinate systems which are known to admit separable solutions for Laplace’s equation.

On the non-hydrostatic equations in pressure and sigma coordinates

Abstract: The non-hydrostatic equations governing the inviscid, adiabatic motion of a perfect gas are formulated using pressure as vertical coordinate; and M. J. Miller's 1974 approximate quasi-non-hydrostatic pressure coordinate equations are derived by applying a systematic scale analysis and power series expansion. The derivation makes clear that these equations are the pressure coordinate counterparts of the anelastic height coordinate equations obtained by Y. Ogura and N. A. Phillips in 1962. The two sets cannot be interconverted by coordinate transformation and so they are not physically equivalent; but the differences are small at the order of validity of both sets. Consideration of a quasi-hydrostatic approximation emphasizes the non-hydrostatic character of Miller's equations. Sigma coordinate quasi-non-hydrostatic equations are obtained by direct transformation of the pressure coordinate forms, and consistent energy equations are derived for both sets. Convenient diagnostic partial differential equations for the geopotential field are obtained for both pressure and sigma coordinate forms. As shown by Miller, the quasi-non-hydrostatic formulation does not permit vertically propagating acoustic waves. Horizontally propagating acoustic waves (Lamb waves) are in general allowed, but can be removed from the pressure coordinate system by applying suitable boundary conditions. Some aspects of the treatment of the Lamb wave problem are corrected in this study. The quasi-non-hydrostatic sigma coordinate system permits Lamb waves, but it may still be considered suitable for convective and (especially) mesoscale modelling with or without orography. The possible use of the quasi-non-hydrostatic system in large-scale theory and modelling is also discussed.

Methods for geometry optimization of large molecules. I. An O(N2) algorithm for solving systems of linear equations for the transformation of coordinates and forces

Abstract: The most recent methods in quantum chemical geometry optimization use the computed energy and its first derivatives with an approximate second derivative matrix. The performance of the optimization process depends highly on the choice of the coordinate system. In most cases the optimization is carried out in a complete internal coordinate system using the derivatives computed with respect to Cartesian coordinates. The computational bottlenecks for this process are the transformation of the derivatives into the internal coordinate system, the transformation of the resulting step back to Cartesian coordinates, and the evaluation of the Newton–Raphson or rational function optimization (RFO) step. The corresponding systems of linear equations occur as sequences of the form yi=Mixi, where Mi can be regarded as a perturbation of the previous symmetric matrix Mi−1. They are normally solved via diagonalization of symmetric real matrices requiring O(N3) operations. The current study is focused on a special approac...