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NASA NTRS1993

On the Gibbs phenomenon 3: Recovering exponential accuracy in a sub-interval from a spectral partial sum of a piecewise analytic function

The investigation of overcoming Gibbs phenomenon was continued, i.e., obtaining exponential accuracy at all points including at the discontinuities themselves, from the knowledge of a spectral partial sum of a discontinuous but piecewise analytic function. It was shown that if we are given the first N expansion coefficients of an L(sub 2) function f(x) in terms of either the trigonometrical polynomials or the Chebyshev or Legendre polynomials, an exponentially convergent approximation to the point values of f(x) in any sub-interval in which it is analytic can be constructed.

Gottlieb, David↗

NASA NTRS1996

Shape and Orientation of Mercury from Radar Ranging Data

If Mercury's radius is expanded in Legendre functions to the second degree and order, the systematic error in radar ranging can be reduced significantly. We interpret the expansion coefficients in terms of a best-fit ellipsoid displaced with respect to the center of mass in the equatorial plane. The ellipsoid's principal axes are rotated in the equatorial plane such that the long axis is aligned with cartographic longitude 15.3 deg +/- 2.9 deg (west). The pole location is consistent with the IAU pole, normal to Mercury's orbital plane. There is a significant equatorial ellipticity (a - b)/a = (540 +/- 54) x 10(exp -6). The center of figure is offset from the center of mass (C.F.-C.M.) by 640 +/- 78 m in the equatorial plane in the direction of cartographic longitude 319.5 deg +/- 6.9 deg. The magnitude of the equatorial center of figure offset implies an excess crustal thickness of 12 km or less, comparable to the Moon's excess. By comparing the equatorial ellipticity with the Mariner 10 gravity coefficient C(sub 22) and assuming Airy isostatic compensation, we conclude that Mercury's crustal thickness is in the range from 100 to 300 km.

Anderson, J. D.↗

NASA NTRS1986

Inference of stress and texture from angular dependence of ultrasonic plate mode velocities

The theory for the angular dependence of the ultrasonic wave velocity in a symmetry plane of an orthorhombic, stressed material is presented. The two waves having polarizations in this plane are shown to have velocities which can be estimated from measurements of the SH sub 0 and S sub 0 guided modes of a thin plate: the relationship being exact for the SH sub 0 mode and requiring a 10% correction for the S sub 0 mode at long wavelength. It is then shown how stress and texture can be independently inferred from various features of the angular dependence of these two velocities. From the SH sub 0 data, the ability to determine the directions and differences in magnitudes of principal stresses is described and supported by experimental data on several materials. From a combination of the SH sub 0 and S sub 0 data, a procedure is proposed for determining the coefficients W sub 400, W sub 420 and W sub 440 of an expansion of the crystallite orientation distribution function in terms of generalized Legendre functions. Possible applications in process control are indicated.

Thompson, R. B.↗

NASA NTRS1974

Analysis of acoustic radiation in a jet flow environment

The analysis of sound fields from arbitrary source distributions in terms of Legendre and spherical Hankel functions is well known. The purpose of this paper is to extend this classical method of analysis to environments such as jet flows where flow and flow gradients are inherently present. The wave-equation governing the radiation of sound in such an environment is derived. The steady state flow and flow gradients in the axial and transverse directions appear as coefficients in the terms of the wave-equation. A semi-numerical method is used to solve the wave-equation in terms of modified spherical harmonics yielding the phase velocities and the directivities of an infinite set of modes. The directivity of each mode is obtained in terms of modified Legendre functions by numerical integration. Some results of these directivity and phase-velocity calculations are presented for a limited number of frequency and flow parameters. Both convective and shear refraction are shown to be important.

Mungur, P.↗

NASA NTRS1983

Spectral methods for time dependent partial differential equations

The theory of spectral methods for time dependent partial differential equations is reviewed. When the domain is periodic Fourier methods are presented while for nonperiodic problems both Chebyshev and Legendre methods are discussed. The theory is presented for both hyperbolic and parabolic systems using both Galerkin and collocation procedures. While most of the review considers problems with constant coefficients the extension to nonlinear problems is also discussed. Some results for problems with shocks are presented.

Gottlieb, D.↗

NASA NTRS1979

Analysis of the vibratory excitation of gear systems. II - Tooth error representations, approximations, and application

The second part of a theory for predicting the vibratory excitation of gear systems from fundamental descriptions of gear tooth elastic properties and deviations of tooth faces from perfect involute surfaces is presented. The first part of the theory provides expressions for the Fourier-series coefficients of the vibratory excitation, and this paper gives expressions for these Fourier-series coefficients in terms of easily interpreted gear tooth metrics that are readily evaluated from tooth-face measurements. Results are given for rectangular tooth-face contact regions using two-dimensional Legendre polynomial expansions of local tooth-pair stiffnesses and stiffness-weighted deviations of tooth faces from perfect involute surfaces. A rigorous transfer function approach is developed that permits separation of the effects of gear tooth errors and gear design parameters; the theory is applicable to helical and spur gears and is illustrated with measurements of tooth-spacing errors and tooth profiles obtained from a pair of spur gears.

Mark, W. D.↗

NASA NTRS1994

A Galerkin method for linear PDE systems in circular geometries with structural acoustic applications

A Galerkin method for systems of PDE's in circular geometries is presented with motivating problems being drawn from structural, acoustic, and structural acoustic applications. Depending upon the application under consideration, piecewise splines or Legendre polynomials are used when approximating the system dynamics with modifications included to incorporate the analytic solution decay near the coordinate singularity. This provides an efficient method which retains its accuracy throughout the circular domain without degradation at singularity. Because the problems under consideration are linear or weakly nonlinear with constant or piecewise constant coefficients, transform methods for the problems are not investigated. While the specific method is developed for the two dimensional wave equations on a circular domain and the equation of transverse motion for a thin circular plate, examples demonstrating the extension of the techniques to a fully coupled structural acoustic system are used to illustrate the flexibility of the method when approximating the dynamics of more complex systems.

Smith, Ralph C.↗

NASA NTRS1994

On the Gibbs phenomenon 4: Recovering exponential accuracy in a sub-interval from a Gegenbauer partial sum of a piecewise analytic function

We continue our investigation of overcoming Gibbs phenomenon, i.e., to obtain exponential accuracy at all points (including at the discontinuities themselves), from the knowledge of a spectral partial sum of a discontinuous but piecewise analytic function. We show that if we are given the first N Gegenbauer expansion coefficients, based on the Gegenbauer polynomials C(sub k)(sup mu)(x) with the weight function (1 - x(exp 2))(exp mu - 1/2) for any constant mu is greater than or equal to 0, of an L(sub 1) function f(x), we can construct an exponentially convergent approximation to the point values of f(x) in any subinterval in which the function is analytic. The proof covers the cases of Chebyshev or Legendre partial sums, which are most common in applications.

Gottlieb, David↗

NASA NTRS1987

Sensitivity studies and a simple ozone perturbation experiment with a truncated two-dimensional model of the stratosphere

The 1-1/2-D model of Holton (1986), which is actually a highly truncated two-dimensional model, describes latitudinal variations of tracer mixing ratios in terms of their projections onto second-order Legendre polynomials. The present study extends the work of Holton by including tracers with photochemical production in the stratosphere (O3 and NOy). It also includes latitudinal variations in the photochemical sources and sinks, improving slightly the calculated global mean profiles for the long-lived tracers studied by Holton and improving substantially the latitudinal behavior of ozone. Sensitivity tests of the dynamical parameters in the model are performed, showing that the response of the model to changes in vertical residual meridional winds and horizontal diffusion coefficients is similar to that of a full two-dimensional model. A simple ozone perturbation experiment shows the model's ability to reproduce large-scale latitudinal variations in total ozone column depletions as well as ozone changes in the chemically controlled upper stratosphere.

Stordal, Frode↗

NASA NTRS1986

Stability analysis of spectral methods for hyperbolic initial-boundary value systems

A constant coefficient hyperbolic system in one space variable, with zero initial data is discussed. Dissipative boundary conditions are imposed at the two points x = + or - 1. This problem is discretized by a spectral approximation in space. Sufficient conditions under which the spectral numerical solution is stable are demonstrated - moreover, these conditions have to be checked only for scalar equations. The stability theorems take the form of explicit bounds for the norm of the solution in terms of the boundary data. The dependence of these bounds on N, the number of points in the domain (or equivalently the degree of the polynomials involved), is investigated for a class of standard spectral methods, including Chebyshev and Legendre collocations.

Gottlieb, D.↗