Showing papers on "Elliptic coordinate system published in 2012"

SH-wave scattering by a semi-elliptical hill using a null-field boundary integral equation method and a hybrid method

Abstract: SUMMARY Following the success of seismic analysis of a semi-circular hill, the problem of SH-wave scattering by a semi-elliptical hill is revisited by using the null-field boundary integral equation method (BIEM). To fully use the analytical property in the null-field boundary integral equation approach in conjunction with degenerate kernels for solving the semi-elliptical hill scattering problem, the problem is decomposed to two regions to produce elliptical boundaries by using the technique of taking free body. One is the half-plane problem containing a semi-elliptical boundary. This semi-infinite problem is imbedded in an infinite plane with an artificial elliptical boundary such that degenerate kernel can be fully applied. The other is an interior problem bounded by an elliptical boundary. The degenerate kernel in the elliptic coordinates for two subdomains is used to expand the closed-form fundamental solution. The semi-analytical formulation in companion with matching boundary conditions yields six constraint equations. Instead of finding admissible wave-expansion bases, our null-field BIEM in conjunction with degenerate kernels has the five features over the conventional BIEM/BEM, (1) free of calculating principal values, (2) exponential convergence, (3) elimination of boundary-layer effect, (4) meshless and (5) well-posed system. All numerical results are compared well with those of using the hybrid method which is also described in this paper. It is interesting to find that a focusing phenomenon is also observed in this study.

Coordinate reduction for exploring chemical reaction paths

Abstract: The potential energy surface for the reaction of a typical molecular system composed of N atoms is defined uniquely by 3N-6 coordinates. These coordinates can be defined by the Cartesian coordinates of the atomic centers (minus overall translation and rotation), or a set of internally defined coordinates such as bond stretches, angle bends, and torsions. By applying principal component analysis to the geometries along a reaction path, a reduced set of coordinates, d ≪ 3N-6, can be obtained. This reduced set of coordinates can reproduce the changes in geometry along the reaction path with chemical accuracy and may help improve the efficiency of reaction path optimization algorithms.

A Numerical Study of Two Coordinates for Energy Balance Equations by Wave Model

Abstract: For the numerical simulations of the ocean wave model, the coordinates for the operation of wave prediction model need to be studied due to the high resolutions affecting the domain boundaries. In the present study, the two coordinates for the wave energy balance equations required for operation and computation are proposed. The two-dimensional models, the spherical coordinate and Cartesian coordinate propagations with deep and shallow water conditions, are used to test the operation of the wave model from the eye storm generation at the Pacific Ocean entering into the South China Sea (SCS) and the Gulf of Thailand (GoT) respectively. The results suggest that the spherical coordinate propagations with deep water conditions are acceptable with similar values of the observational data.

Analytical and numerical investigation for true and spurious eigensolutions of an elliptical membrane using the real-part dual BIEM/BEM

Abstract: The paper performs analytical and numerical investigation of the true and spurious eigensolutions of an elliptical membrane using the real-part boundary integral equation method (BIEM) following the successful work on a circular case by using the dual boundary element method (BEM) (Kuo et al. in Int. J. Numer. Methods Eng. 48:1401–1422, 2000). We extend to the elliptical case in this paper. To analytically study the eigenproblems of an elliptical membrane, the elliptical coordinates and Mathieu functions are adopted. The fundamental solution is expanded into the degenerate kernel by using the elliptical coordinates and the boundary densities are expanded by using the eigenfunction expansion. The Jacobian terms may exist in the degenerate kernel, boundary density and boundary contour integration but they can cancel each other out. Therefore, the orthogonal relations are reserved in the boundary contour integral. It is interesting to find that the BIEM using the real or the imaginary-part kernel to deal with an elliptical membrane yields spurious eigensolutions. This finding agrees with those corresponding to the circular case. The spurious eigenvalues in the real-part BIEM are found to be the zeros of the mth-order (even or odd) modified Mathieu functions of the second kind or their derivatives. To verify this finding, the BEM is implemented. Furthermore, the commercial finite-element code ABAQUS is also utilized to provide eigensolutions for comparisons. It is found that good agreement is obtained.

Coordinate Transformation Methods

Abstract: The C-method was born in the eighties in Clermont-Ferrand , France, from the need to solve rigorously diffraction problems at corrugated periodic surfaces in the resonance regime. The main difficulty of such problems is the matching of boundaries conditions. For that purpose, Chandezon et al introduced the so called translation coordinate system in which the boundary of the physical problem coincides with coordinate surfaces. The second ingredient of C-method is to write Maxwell's equation under the covariant form. This formulation comes from relativity where the use of curvilinear non orthogonal coordinate system is essential and natural. The main feature of this formalism is that Maxwell's equations remain invariant in any coordinate system, the geometry being shifted into the constitutive relations. The third ingredient of C-method is that it is a modal method. This nice property is linked with the translation coordinate system in which a diffraction problem may be expressed as an eigenvalue eigenvector problem with periodic boundary conditions. The key point of C-method is the joint use of curvilinear coordinates and covariant formulation of Maxwell's equations. All the new developments in the modeling of gratings like Adaptive Spatial Resolution, and Matched Coordinates derive from this fundamental observation.

The analytic form of Green’s function in elliptic coordinates

Abstract: The purpose of this study is the derivation of a closed-form formula for Green’s function in elliptic coordinates that could be used for achieving an analytic solution for the second-order diffraction problem by elliptical cylinders subjected to monochromatic incident waves. In fact, Green’s function represents the solution of the so-called locked wave component of the second-order velocity potential. The mathematical analysis starts with a proper analytic formulation of the second-order diffraction potential that results in the inhomogeneous Helmholtz equation. The associated boundary-value problem is treated by applying Green’s theorem to obtain a closed-form solution for Green’s function. Green’s function is initially expressed in polar coordinates while its final elliptic form is produced through the proper employment of addition theorems.

Application of discrete variable representation to planar H2+ in strong xuv laser fields.

Abstract: We present an efficient and accurate grid method to study the strong field dynamics of planar H(2)(+) under Born-Oppenheimer approximation. After introducing the elliptical coordinates to the planar H(2)(+), we show that the Coulomb singularities at the nuclei can be successfully overcome so that both bound and continuum states can be accurately calculated by the method of separation of variables. The time-dependent Schrodinger equation (TDSE) can be accurately solved by a two-dimensional discrete variable representation (DVR) method, where the radial coordinate is discretized with the finite-element discrete variable representation for easy parallel computation and the angular coordinate with the trigonometric DVR which can describe the periodicity in this direction. The bound states energies can be accurately calculated by the imaginary time propagation of TDSE, which agree very well with those computed by the separation of variables. We apply the TDSE to study the ionization dynamics of the planar H(2)(+) by short extreme ultra-violet (xuv) pulses, in which case the differential momentum distributions from both the length and the velocity gauge agree very well with those calculated by the lowest order perturbation theory.

Algebro-Geometric Solutions to Fokas-Lenells Equation

Abstract: The Fokas-Lenells equation is given by the Lenard gradient sequence for a given spectral problemThen,this equation is decomposed into solvable ordinary differential equationsBased on the finite-order expansion of the Lax matrix,elliptic coordinates are introducedSo,the flow can be straighten out by the Abel-Jacobi coordinatesAt the end, algebro-geometric solutions of the Fokas-Lenells equation are presented by means of the Riemannθfunction

Position function in mathematics

Abstract: The Position Function is a new and original mathematical function introduced by the author into the mathematical domain; it can be used in numerous fields and domains as signal theory, signal processing, mathematics, physics and many others. The main goal of this function is to give as an output two values which are (+1 and -1) whatever is the input and whatever is the coordinate system we are working on, for example the same function can be used in Cartesian coordinate system, in Spherical and Cylindrical coordinate system or any other system. This function is similar to the sign function in which it gives 3 values (+1, 0 and -1) in the Cartesian coordinate system only, but the difference is that the Position function gives two values (+1 and -1) not only in the Cartesian Coordinate system but in any other Coordinate system. The Position Function will by widely used due to its importance. It has many advantages that are discussed in this paper.

Restricted Three-Body Problem in Different Coordinate Systems

Abstract: In this paper of the series, the equations of motion for the spatial circular restricted three-body problem in sidereal spherical coordinates system were established. Initial value procedure that can be used to compute both the spherical and Cartesian sidereal coordinates and velocities was also developed. The application of the procedure was illustrated by numerical example and graphical representations of the variations of the two sidereal coordinate systems.

Nonlinear Accelerator with Transverse Motion Integrable in Normalized Polar Coordinates

Abstract: Several families of nonlinear accelerator lattices with integrable transverse motion were suggested recently. One of the requirements for the existence of two analytic invariants is a special longitudinal coordinate dependence of fields. This paper presents the particle motion analysis when a problem becomes integrable in the normalized polar coordinates. This case is distinguished from the others: it yields an exact analytical solution and has a uniform longitudinal coordinate dependence of the fields (since the corresponding nonlinear potential is invariant under the transformation from the Cartesian to the normalized coordinates). A number of interesting features are revealed: while the frequency of radial oscillations is independent of the amplitude, the spread of angular frequencies in a beam is absolute. A corresponding spread of frequencies of oscillations in the Cartesian coordinates is evaluated via the simulation of transverse Schottky noise.