Prolate Spheroidal Surface Research Articles

Sparsified discrete wave problem involving radiation condition on a prolate spheroidal surface

We develop and analyze a high-order outgoing radiation boundary condition for solving three-dimensional scattering problems by elongated obstacles. This Dirichlet to Neumann (DtN) condition is constructed using the classical method of separation of variables that allows to define the scattered field in a truncated domain. It reads as an infinite series which is truncated for numerical purposes. The radiation condition is implemented in a finite element framework represented by a large dense matrix. Fortunately, the dense matrix can be decomposed into a full block matrix which involves the degrees of freedom on the exterior boundary and a sparse finite element matrix. The inversion of the full block is avoided by using a Sherman-Morrison algorithm which reduces the memory usage drastically. Despite being of high-order, this method has only a low memory cost. Many applications such as sonar, radar or geophysical exploration involve exterior Helmholtz problems which raise the question of handling the unbounded propagation domain. One accurate approach consists in using boundary element methods (BEM), see Bendali & Fares (2008) and their references. The linear partial differential equation is reformulated as an integral equation which is set on the boundary of the scatterer. BEMs have thus the advantage of providing an exact reformulation of the exterior problem in a bounded region without introducing any kind of approximation before discretization. However , boundary element discretizations lead to dense matrices which increase both storage requirements and computational costs of the solution algorithm, especially in three dimensions. A popular alternative is the usage of volume finite element methods (FEM) Harari & Hughes (1992b) once the computational domain has been truncated. The truncation is performed by introducing an artificial boundary surrounding the scatterer which requires transforming the exterior problem into a mixed one corresponding to the initial volume formulation of the problem set in the bounded domain and coupled with a suitable boundary condition set on the external boundary. To get an accurate representation of the wave field into the bounded domain, the external boundary should not generate any reflection. This point is managed with †

Sparsified discrete wave problem involving a radiation condition on a prolate spheroidal surface

Abstract We develop and analyse a high-order outgoing radiation boundary condition for solving three-dimensional scattering problems by elongated obstacles. This Dirichlet-to-Neumann condition is constructed using the classical method of separation of variables that allows one to define the scattered field in a truncated domain. It reads as an infinite series that is truncated for numerical purposes. The radiation condition is implemented in a finite element framework represented by a large dense matrix. Fortunately, the dense matrix can be decomposed into a full block matrix that involves the degrees of freedom on the exterior boundary and a sparse finite element matrix. The inversion of the full block is avoided by using a Sherman–Morrison algorithm that reduces the memory usage drastically. Despite being of high order, this method has only a low memory cost.

Eigenanalysis for a confocal prolate spheroidal resonator using the null-field BIEM in conjunction with degenerate kernels

In this paper, we employ the null-field boundary integral equation method (BIEM) in conjunction with degenerate kernels to solve eigenproblems of the prolate spheroidal resonators. To detect the spurious eigenvalues and the corresponding occurring mechanisms which are common issues while utilizing the boundary element method or the BIEM, we use angular prolate spheroidal wave functions and triangular functions to expand boundary densities. In this way, the boundary integral of a prolate spheroidal surface is exactly determined, and eigenequations are analytically derived. It is revealed that the spurious eigenvalues depend on the integral, representations and the shape of the inner boundary. Furthermore, it is interesting to find that some roots of the confocal prolate spheroidal resonator are double roots no matter that they are true or spurious eigenvalues. Illustrative examples include confocal prolate spheroidal resonators of various boundary conditions. To validate these findings and accuracy of the present approach, the commercial finite-element software ABAQUS is also applied to perform acoustic analyses. Good agreement is obtained between the acoustic results obtained by the null-field BIEM and those provided by the commercial finite-element software ABAQUS.

Chemically based mathematical model for development of cerebral cortical folding patterns.

The mechanism for cortical folding pattern formation is not fully understood. Current models represent scenarios that describe pattern formation through local interactions, and one recent model is the intermediate progenitor model. The intermediate progenitor (IP) model describes a local chemically driven scenario, where an increase in intermediate progenitor cells in the subventricular zone correlates to gyral formation. Here we present a mathematical model that uses features of the IP model and further captures global characteristics of cortical pattern formation. A prolate spheroidal surface is used to approximate the ventricular zone. Prolate spheroidal harmonics are applied to a Turing reaction-diffusion system, providing a chemically based framework for cortical folding. Our model reveals a direct correlation between pattern formation and the size and shape of the lateral ventricle. Additionally, placement and directionality of sulci and the relationship between domain scaling and cortical pattern elaboration are explained. The significance of this model is that it elucidates the consistency of cortical patterns among individuals within a species and addresses inter-species variability based on global characteristics and provides a critical piece to the puzzle of cortical pattern formation.

Analytical and numerical computation of prolate spheroidal shell capacitance

AbstractThis article presents analytical and numerical methods of computing the capacitance of a prolate spheroidal shell. Detailed and comprehensive equations for determining the exact results were developed. Also, an explicit finite difference (FD) numerical technique for solving Laplace's equation in prolate spheroidal coordinates systems for an axially symmetric geometry has been developed. Consequently, the FD results were used to compute the capacitance of the spheroidal shell in an easy to understand manner. This was achieved using the Dirichlet boundary conditions on the prolate spheroidal surfaces. The capacitance results obtained using FD did agree with those obtained using analytical method. © 2009 Wiley Periodicals, Inc. Microwave Opt Technol Lett 51: 2361–2365, 2009; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.24630

Diffusion fields associated with prolate spheroids in size and shape coarsening

The diffusion field or solute concentration distributed around a prolate spheroidal particle simulating a rod- or needle-shaped precipitate has been solved for varying aspect ratios of the precipitate and varying concentrations along the precipitate surface owing to the curvature effect. With the prolate spheroidal coordinates, the principal curvatures of the prolate spheroidal surface are derived as functions of the angular variable, and the Laplace field equation is separated into two Legendre equations either on the radial variable or on the angular variable. The analytical solution fitting the present boundary conditions is secured as the sum of a Legendre function of the second kind of order zero and a Legendre series. The Legendre function of the second kind gives the concentration distribution when the curvature effect can be ignored, whereas the Legendre series represents the concentration contributed from the curvature effect. Numerical results of normalized concentrations are presented as functions of the radial and angular variables for selected aspect ratios. The tangent component of the concentration gradients, contributed from the curvature effect, may cause complicated mass transfer, which is responsible for shape coarsening.

Axisymmetric solutions of the Jeans equation

We present axisymmetric solutions of the Jeans equations for a tracer density |$\rho\ell(R,z)$| in a chosen potential Φ(R, z) assuming that (1) the velocity ellipsoid aligns along prolate spheroidal coordinate surfaces, and (2) the axial ratio of the velocity ellipsoid in the meridional plane |$\sigma_\lambda/\sigma_v$| is known at every point. The solutions are general, and do not require that the potential be separable, nor that the density and potential be related to each other. The solutions can display a diverse range of properties. In particular, the specified anisotropy can be chosen smoothly to increase or to decrease with radius, to pass through a minimum or a maximum, or to remain constant.

Numerical Approach to Far Field Acoustic Wave Propagation from a Prolate Spheroidal Surface

A method is described for calculating the far field, transient, three-dimensional pressure field generated by acoustic waves emanating from a prolate spheroidal surface. The pressure field is expanded in a series of spherical harmonics, which leads to a system of linear, tridiagonal, one dimensional wave equations that can be integrated efficiently by numerical techniques based on the method of characteristics. Integrals involving orthogonal functions are approximated by a numerically robust scheme where exact orthogonality is obtained (in the absence of round-off errors) in terms of weighted sums over a discrete variable.

Frequencies of resonance of axisymmetric modes of open prolate spheroidal shells

Governing differential equations are derived for the axisymmetric motion of an open prolate spheroidal shell with inner and outer surfaces defined by prolate spheroidal coordinate surfaces. Transverse shear and rotary inertia terms are included in the shell equations. The open end of the shell, defined by a circumferential coordinate line, is assumed clamped. Approximate solutions for the axisymmetric frequencies of resonance of the in-vacuo and fluid-loaded shell are derived using the Galerkin method. For the in-vacuo shell, good agreement with published results for the exact frequencies of resonance of a spherical shell is obtained. Results are presented for the frequencies of resonance of open prolate spheroidal shells as a function of shell eccentricity, mode number, shell thickness, and opening angle, with and without fluid loading.

Steady conduction of heat in prolate spheroidal systems

The steady local temperature distribution has been determined for arbitrary prolate spheroidal systems with various boundary conditions. The temperature was derived for all possible systems, such as when the temperature was kept constant on the hyperbolic surfaces and variable on the prolate spheroidal surfaces, and vice versa. In'the latter case, roots with respect to the degree of the associated Legendre function for arguments larger than unity had to be determined. Some of the presented solutions have been evaluated numerically, and the isothermal surfaces have been presented.

Resonant frequencies and mode shapes of open prolate spheroidal shells

Governing differential equations are derived for an open prolate spheroidal shell with inner and outer surfaces defined by prolate spheroidal coordinate surfaces. Transverse shear and rotary inertia are included in the shell equations. The open end of the shell, defined by an axisymmetric coordinate line, is assumed clamped. Approximate solutions for the axisymmetric resonant frequencies and mode shapes of the in‐vacuo shell are derived using Galerkin's variational method.

Acoustic radiation from prolate spheroidal lenses

This study deals with the mathematical analysis of acoustic radiation from multiple prolate spheroidal surfaces. It forms the basis for developing a better model to predict a submarine's sonar performance. The sonar array and its dome were modeled as a prolate spheroidal array confocal to a prolate spheroidal dome. The dome was represented by a mass only 0.25‐in.‐thick steel shell, with a major axis equal to twice the length of the bow sonar cavity and a minor axis dimension typical for a submarine. Four loading conditions were imposed on the array: (a) free‐field radiation, (b) acoustic lens, (c) mass‐loaded acoustic lens, and (d) mass‐loaded non‐lens. The influence of the dome mass on the radiated pressure of a free‐field array was found to be significant and must be included in analyses of sonar systems. The influences of the wall mass on the performance of the acoustic lens was not significant. Significant focusing by the lens was not observed. This was due to the low frequency of the array and the internal acoustic scattering by the array itself. [Doctorial dissertation.]

Radiation Loading of Prolate Spheroidal Surfaces

Prolate spheroidal coordinates and associated prolate spheroidal wave functions are used to study the acoustic radiation from arbitrarily vibrating prolate spheroidal surfaces. The theory was originally developed to study the farfield sound propagation from submerged ellipsoids of revolution with high fineness ratios (L/R). Boundary conditions are adjusted to the general wave solutions to obtain an expression for the sound pressure. The results show that a single prolate spheroidal vibration mode excites an infinite number of pressure modes. A local radiation impedance is defined and discussed. This impedance shows the effect of the strong locally variable nearfield and that the spheroid generates (L/2R) more acoustic power per unit area at the equator than at the poles. The application of the theory is demonstrated through numerical calculations of the farfield radiation from a particular submerged body of revolution in which experimental data was available. The agreement between the predicted and measured sound-pressure levels is good.

Anisotropy of the Ultrasonic Attenuation in Metals

The anisotropy of ultrasonic attenuation is calculated by using Pippard's formula for a model in which the Fermi surface consists of a single prolate spheroidal surface. For longitudinal ultrasounds the attenuation coefficient α shows two maxima in the regions δ=35^°∼45^° and 145°∼135°, but for transverse ultrasounds α shows a maximum at δ=90°, where δ is the angle between the major axis of the spheroid and the direction of propagation. Next, the method is applied to bismuth for longitudinal sound waves. In this case, however, because of a large deformation potential, the general features of the anisotropy are very different from the former case. Moreover, in this case the anisotropy of the deformation potential itself seems to play a very important role.