Showing papers on "Elliptic coordinate system published in 2018"

Analytical solution for converging elliptic shear wave in a bounded transverse isotropic viscoelastic material with nonhomogeneous outer boundary.

Abstract: Dynamic elastography methods—based on optical, ultrasonic, or magnetic resonance imaging—are being developed for quantitatively mapping the shear viscoelastic properties of biological tissues, which are often altered by disease and injury. These diagnostic imaging methods involve analysis of shear wave motion in order to estimate or reconstruct the tissue's shear viscoelastic properties. Most reconstruction methods to date have assumed isotropic tissue properties. However, application to tissues like skeletal muscle and brain white matter with aligned fibrous structure resulting in local transverse isotropic mechanical properties would benefit from analysis that takes into consideration anisotropy. A theoretical approach is developed for the elliptic shear wave pattern observed in transverse isotropic materials subjected to axisymmetric excitation creating radially converging shear waves normal to the fiber axis. This approach, utilizing Mathieu functions, is enabled via a transformation to an elliptic coordinate system with isotropic properties and a ratio of minor and major axes matching the ratio of shear wavelengths perpendicular and parallel to the plane of isotropy in the transverse isotropic material. The approach is validated via numerical finite element analysis case studies. This strategy of coordinate transformation to equivalent isotropic systems could aid in analysis of other anisotropic tissue structures.

Ground motions around a deep semielliptic canyon with a horizontal edge subjected to incident plane SH waves

Abstract: We study scattering of antiplane shear waves induced by a deep semielliptic canyon with a horizontal edge We employ the region-point-matching technique to cope with the problem considered Through an auxiliary boundary, a part of the circumference of a semiellipse, the whole analyzed region is divided into two subregions We express the displacement fields in terms of Mathieu functions We unify two distinct elliptic coordinates via a simple coordinate transformation relation Integration of the coordinate transformation relation into the region-point-matching technique simplifies the procedure for constructing simultaneous equations Imposing the continuity conditions and traction-free ones, we obtain the expansion coefficients Frequency-domain results demonstrate ground motion variability based on several key factors Ground surface responses under seismic shaking are also simulated in the time domain

Invariance property of coordinate transformation

Abstract: Is coordinate transformation invariant with respect to coordinate systems? Intuitively one assumes so, but the answer is more complex as in some cases it is invariant and in some cases it is not. I...

Generation of second-harmonic Ince-Gaussian beams

Abstract: As a continuous transition between the Hermite-Gaussian and Laguerre-Gaussian beams, the Ince-Gaussian beams form a family of exact orthogonal solutions of the free-space paraxial wave equation in elliptic coordinates. Ince-Gaussian beams have multiple transverse mode patterns, which make them unique in terms of application in the fields of bioengineering, particle manipulation, and quantum entanglement. Here, based on binary nonlinear computer-generated holograms with a domain structure (realized via electric field poling at room temperature), we generate a second-harmonic Ince-Gaussian beam pumped with a fundamental Gaussian beam. In this process, the transverse part of the phase-matching condition is satisfied, which is called the Raman–Nath-type nonlinear diffraction. Both frequency conversion and beam shaping can be realized simultaneously, thereby offering the advantage of integration of both functions into a single device.As a continuous transition between the Hermite-Gaussian and Laguerre-Gaussian beams, the Ince-Gaussian beams form a family of exact orthogonal solutions of the free-space paraxial wave equation in elliptic coordinates. Ince-Gaussian beams have multiple transverse mode patterns, which make them unique in terms of application in the fields of bioengineering, particle manipulation, and quantum entanglement. Here, based on binary nonlinear computer-generated holograms with a domain structure (realized via electric field poling at room temperature), we generate a second-harmonic Ince-Gaussian beam pumped with a fundamental Gaussian beam. In this process, the transverse part of the phase-matching condition is satisfied, which is called the Raman–Nath-type nonlinear diffraction. Both frequency conversion and beam shaping can be realized simultaneously, thereby offering the advantage of integration of both functions into a single device.

The rotated Cartesian coordinate method to remove the axial singularity of cylindrical coordinates in finite‐difference schemes for elastic and viscoelastic waves

Abstract: SUMMARY When modelling the propagation of 3-D non-axisymmetric viscoelastic waves in cylindrical coordinates using the finite-difference time-domain (FDTD) method, one encounters a mathematical singularity due to the presence of 1/r terms in the viscoelastic wave equations For many years this issue has been impeding the accurate numerical solution near the axis In this paper, we propose a simple but effective method for the treatment of this numerical singularity problem By rotating the Cartesian coordinate (RCC) system around the z-axis in cylindrical coordinates, the numerical singularity problems in both 2-D and 3-D cylindrical coordinates can be removed This algorithm has three advantages over the conventional treatment techniques: 1) the excitation source can be directly loaded at r = 0; 2) the central difference scheme with second-order accuracy is maintained; 3) the stability condition at the axis is consistent with the FDTD in Cartesian coordinates This method is verified by several 3-D numerical examples Results show that the method is accurate and stable at the singularity point The improved FDTD algorithm is also applied to sonic logging simulations in non-axisymmetric formations and sources This article is protected by copyright All rights reserved

Elliptic basis for the Zernike system: Heun function solutions

Abstract: The differential equation that defines the Zernike system, originally proposed to classify wavefront aberrations of the wavefields in the disk of a circular pupil, had been shown to separate in three distinct coordinate systems obtained from polar coordinates on a half-sphere. Here we find and examine the separation in the generic elliptic coordinate system on the half-sphere and its projected disk, where the solutions, separated in Jacobi coordinates, contain Heun polynomials.

New Method for Large Deflection Analysis of an Elliptic Plate Weakened by an Eccentric Circular Hole

Abstract: The bending analysis of moderately thick elliptic plates weakened by an eccentric circular hole has been investigated in this article. The nonlinear governing equations have been presented by considering the von-Karman assumptions and the first-order shear deformation theory in cylindrical coordinates system. Semi-analytical polynomial method (SAPM) which had been presented by the author before has been used. By applying SAPM method, the nonlinear partial differential equations have been transformed to the nonlinear algebraic equations system. Then, the nonlinear algebraic equations have been solved by using Newton–Raphson method. The obtained results of this study have been compared with the results of other references and the accuracy of the results has been shown. The effect of some important parameters on the results such as the location of the circular hole, the ratio of major to minor radiuses of elliptical plate, the size of circular hole and boundary conditions have been studied. It is concluded that applying the presented method is very convenient and efficient. So, it can be used for analyzing the mechanical behavior of elliptical plates, instead of relatively complicated formulations in elliptic coordinates system.

Fourier transform in elliptic coordinates: Case of axial symmetry

Abstract: We perform the Fourier transform in two dimensional elliptic coordinates. The case of axial symmetry allows to reduce significantly the final transformation. The obtained integral formulae generalize the Fourier-Bessel transform due to relation between elliptic and polar coordinates. We show also the alternating definition using Mathieu functions. The application of the formulae obtained is the transitions of operator representations in Quantum Mechanics.

Integral Relations for Bessel Functions and Analytical Solutions for Fourier Transform in Elliptic Coordinates

Abstract: The work continues the development of Fourier transform in elliptic coordinates. Several analytical results for elementary and special functions have been obtained. In fact these are the new relations for the integral representation for the multiplication of two Bessel functions of zero order. We analyze the natural restrictions of application of these formulae. Also we provide the recommendations based on the numerical analysis for the using of obtained results.