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.

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.

DOE-generated laser beams with given orbital angular moment: application for micromanipulation

Abstract: A countable set of linearly independent solutions of the paraxial wave (Schroedinger-type) equation is derived and given the name hyper-geometric modes. These solutions describe pure optical vortices that can be generated when a spiral phase plate is illuminated with a plane wave. The distinction between these modes and the familiar paraxial modes is that in propagation the radius of the former increases as a square root of distance and the phase velocity is the same for all modes. In the present work experimental results on trapping and rotation of 5-10 micron-sized biological objects (yeast cells) and polysty rene beads of diameter 5 P m using various laser beams are discussed. Keywords: diffractive optical elements, pure op tical vortices, orbital angular moment, hyper-geometric modes, optical microparticle manipulation 1. INTRODUCTION The higher-order Bessel and Laguerre-Gaussian (LG) modes contain optical vortices providing screw character and presence of orbital angular moment. A microparticle, trapped in such a beam, receives a rotary movement. The new types of laser beams having orbital angular moment - optical vortices "imbedded" in a plane or a Gaussian beam, are considered. After passing some distance, such fields get rather stable configuration, reminding of LG modes, and are distributed under the similar law. In optics, the Hermite-Gauss (HG) and LG modes, which are partial solutions of the paraxial wave equation (PWE) or Schroedinger equation in the Cartesian or cylindrical coordinates, have long been in wide use [1]. They represent the transverse modes of stable laser resonators. Such modes preserve their structure (cross-section intensity distribution), changing only the scale along the propagation axis. Because these modes form an orthogonal basis it is possible to use their linear combinations for constr ucting other solutions of the PWE. In the cylindrical coordinates, the PWE has other modal solutions that, similar to the HG and LG modes, preserve their structure, changing only in scale. These are referred to as paraxial diffracted Bessel modes [2] and should be distinguished from the paraxial diffraction-free Bessel beams [3 ], which will be reffered to as the Durnin-Bessel modes, to distinguish them from the diffracting Bessel modes. As di stinct from the Gaussian mode s, both Bessel modes possess the infinite energy (their intensity being finite at every space point). The effective diameter of the diffracted Bessel beam increases linearly along the optical axis with increa sing distance from the initial plane. The Durnin-Bessel (DB) beam have a constant diameter. Recently introduced [4-8] new modal solutions of the PWE have been studied theoretically [4-7] and experimentally [8]. These are the Ince-Gaussian modes derived as a solution of the PWE in the elliptic coordinates. In these coordinates, the PWE is solved via separation of variab les, with the solution found as a product of the Gaussian function by the Ince polynomials. Note that the Ince pol ynomials are properly a solution of the Whitteker-Hill equation [2]. The Ince-Gaussian (IG) modes represent an orthogonal basis that generalizes the HG and LG modes. When the elliptic coordinates change to cylindrical (the ellipses change to the circumferences) the IG modes change to the LG modes. With the ellipse eccentric ity tending to infinity (the ellipse changing to a line segment), the IG modes change to the HG modes.

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

On the stress concentration factor of circular/elliptic hole and rigid inclusion under the remote anti-plane shear by using degenerate kernels

Abstract: The stress concentration factor (SCF) along the boundary of a hole and a rigid inclusion in an infinite isotropic solid under the anti-plane shear is revisited by using degenerate kernels in the boundary integral equation (BIE) although this result was obtained by invoking the extended circle theorem of Milne-Thomson as well as the complex variable approach The degenerate kernel of series form for the closed-form fundamental solution is used for the circle and the ellipse in terms of polar and elliptic coordinates, respectively The slender ratio of the ellipse and the orientation are two parameters for our study The strain energy density along the boundary is increased or decreased due to the different types of loading and various aspect ratios of the ellipse An analytical solution for the SCF is then derived for any orientation of the ellipse relative to the applied load The reciprocal relation for the SCF between a hole and a rigid inclusion with respect to different loading is also addressed Besides, this analytical derivation can clearly show the appearing mechanism why the BEM/BIEM suffers the degenerate scale in the rigid inclusion

Full modal analysis of confocal coaxial elliptical waveguides

Abstract: An efficient method for analysing confocal coaxial elliptical waveguides is presented. Using elliptical coordinates, the differential Helmholtz equation is transformed into a linear matrix eigenvalue problem by means of the method of moments. The expressions of the vector mode functions for the full spectrum of these guides are constructed, including the TEM, TM and TE modes. The convergence of the method is very good, giving an efficient and accurate code. Comparisons with numerical results found in the technical literature validate the presented theory.