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.

Solving the three-dimensional equations of the linear theory of elasticity

Abstract: The three-dimensional Lame equations are solved using Cartesian and curvilinear orthogonal coordinates. It is proved that the solution includes only three independent harmonic functions. The general solution of equations of elasticity for stresses is found. The stress tensor is expressed in both coordinate systems in terms of three harmonic functions. The general solution of the problem of elasticity in cylindrical coordinates is presented as an example. The three-dimensional stress–strain state of an elastic cylinder subjected, on the lateral surface, to arbitrary forces represented by a series of eigenfunctions is determined. An axisymmetric problem for a finite cylinder is solved numerically

System using parallel coordinates for automated line detection in noisy images

Abstract: A system for detecting lines in images using line neighborhoods and a parallel coordinate transformation. The process introduces the concept of line neighborhoods to accommodate the uncertainty in line detection arising from image noise. Because line neighborhoods in Cartesian coordinates have ambiguous and unbounded regions and always overlap one another, a parallel coordinate transform is used to transform Cartesian coordinate image plane line segments to points in a bounded and nonambiguous region of the parallel coordinate transform plane. Line detection then becomes a simple problem of detecting point clusters in the parallel coordinate transform plane.

A semi-analytical method for near-trapped mode and fictitious frequencies of multiple scattering by an array of elliptical cylinders in water waves

Abstract: In this paper, we focus on the water wave scattering by an array of four elliptical cylinders. The null-field boundary integral equation method (BIEM) is used in conjunction with degenerate kernels and eigenfunctions expansion. The closed-form fundamental solution is expressed in terms of the degenerate kernel containing the Mathieu and the modified Mathieu functions in the elliptical coordinates. Boundary densities are represented by using the eigenfunction expansion. To avoid using the addition theorem to translate the Mathieu functions, the present approach can solve the water wave problem containing multiple elliptical cylinders in a semi-analytical manner by introducing the adaptive observer system. Regarding water wave problems, the phenomena of numerical instability of fictitious frequencies may appear when the BIEM/boundary element method (BEM) is used. Besides, the near-trapped mode for an array of four identical elliptical cylinders is observed in a special layout. Both physical (near-trapped mode) and mathematical (fictitious frequency) resonances simultaneously appear in the present paper for a water wave problem by an array of four identical elliptical cylinders. Two regularization techniques, the combined Helmholtz interior integral equation formulation (CHIEF) method and the Burton and Miller approach, are adopted to alleviate the numerical resonance due to fictitious frequency.

Level crossings and commuting observables for the quantum elliptic billiard

Abstract: The energy spectrum of the quantum elliptic billiard is obtained by solving the Schrodinger equation in elliptic coordinates. The corresponding eigenstates also diagonalise an operator B which commutes with H. A numerical search of the exact eigenvalues of B and H permits one to follow each state as a function of the deformation parameter mu . Geometrical arguments, also valid for the simpler problem of the rectangular box, allow one to understand the results obtained.

Laplace Transform Analytic Element Method for Transient Groundwater Flow Simulation

Abstract: The Laplace transform analytic element method (LT-AEM), applies the traditionally steady-state analytic element method (AEM) to the Laplace-transformed diffusion equation (Furman and Neuman, 2003). This strategy preserves the accuracy and elegance of the AEM while extending the method to transient phenomena. The approach taken here utilizes eigenfunction expansion to derive analytic solutions to the modified Helmholtz equation, then back-transforms the LT-AEM results with a numerical inverse Laplace transform algorithm. The two-dimensional elements derived here include the point, circle, line segment, ellipse, and infinite line, corresponding to polar, elliptical and Cartesian coordinates. Each element is derived for the simplest useful case, an impulse response due to a confined, transient, single-aquifer source. The extension of these elements to include effects due to leaky, unconfined, multi-aquifer, wellbore storage, and inertia is shown for a few simple elements (point and line), with ready extension to other elements. General temporal behavior is achieved using convolution between these impulse and general time functions; convolution allows the spatial and temporal components of an element to be handled independently. Comparisons are made between inverse Laplace transform algorithms; the accelerated Fourier series approach of de Hoog et al. (1982) is found to be the most appropriate for LT-AEM applications. An application and synthetic examples are shown for several illustrative forward and parameter estimation simulations to illustrate LT-AEM capabilities. Extension of LT-AEM to three-dimensional flow and non-linear infiltration are discussed.