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NASA NTRS1989

Evaluation of thermally induced non-Fourier stress wave disturbances via tailored hybrid transfinite element formulations

Accurate solutions have been obtained for a class of non-Fourier models in dynamic thermoelasticity which are relevant to the understanding of thermally-induced stress wave disturbances. The method employs tailored hybrid formulations based on the transfinite element approach. The results show that significant thermal stresses may arise due to non-Fourier effects, especially when the speeds of propagation of the thermal and stress waves are equal.

Tamma, Kumar K.↗

NASA NTRS1989

On the shape and albedo variegation of asteroids - Results from Fourier analysis of synthetic and observed asteroid lightcurves

Fourier analyses have been conducted of those asteroids in the Lagerkvist et al. (1987) lightcurves with substantially complete phase coverage and point density, and for which a unique period has been identified. Fourier coefficients are thereby obtained for 188 lightcurves of 91 asteroids, as well as for 868 synthetic lightcurves from a set of models different in shape and albedo markings; by comparing model and asteroid lightcurve coefficients, the possible shape and albedo variations for 18 asteroids are derived. An attempt is made to define the general distribution of shapes of all asteroid samples analyzed.

Barucci, M. Antonietta↗

NASA NTRS1989

Essentially nonoscillatory spectral Fourier methods for shock wave calculations

An essentially nonoscillatory spectral Fourier method for the solution of hyperbolic partial differential equations is presented. The method is based on adding a nonsmooth function to the trigonometric polynomials which are the usual basis functions for the Fourier method. The high accuracy away from the shock is enhanced by using filters. Numerical results confirm that essentially no oscillations develop in the solution.

Cai, Wei↗

NASA NTRS1989

Studies in astronomical time series analysis. III - Fourier transforms, autocorrelation functions, and cross-correlation functions of unevenly spaced data

This paper develops techniques to evaluate the discrete Fourier transform (DFT), the autocorrelation function (ACF), and the cross-correlation function (CCF) of time series which are not evenly sampled. The series may consist of quantized point data (e.g., yes/no processes such as photon arrival). The DFT, which can be inverted to recover the original data and the sampling, is used to compute correlation functions by means of a procedure which is effectively, but not explicitly, an interpolation. The CCF can be computed for two time series not even sampled at the same set of times. Techniques for removing the distortion of the correlation functions caused by the sampling, determining the value of a constant component to the data, and treating unequally weighted data are also discussed. FORTRAN code for the Fourier transform algorithm and numerical examples of the techniques are given.

Scargle, Jeffrey D.↗

NASA NTRS1989

Fourier transform methods in local gravity modeling

New algorithms were derived for computing terrain corrections, all components of the attraction of the topography at the topographic surface and the gradients of these attractions. These algoriithms utilize fast Fourier transforms, but, in contrast to methods currently in use, all divergences of the integrals are removed during the analysis. Sequential methods employing a smooth intermediate reference surface were developed to avoid the very large transforms necessary when making computations at high resolution over a wide area. A new method for the numerical solution of Molodensky's problem was developed to mitigate the convergence difficulties that occur at short wavelengths with methods based on a Taylor series expansion. A trial field on a level surface is continued analytically to the topographic surface, and compared with that predicted from gravity observations. The difference is used to compute a correction to the trial field and the process iterated. Special techniques are employed to speed convergence and prevent oscillations. Three different spectral methods for fitting a point-mass set to a gravity field given on a regular grid at constant elevation are described. Two of the methods differ in the way that the spectrum of the point-mass set, which extends to infinite wave number, is matched to that of the gravity field which is band-limited. The third method is essentially a space-domain technique in which Fourier methods are used to solve a set of simultaneous equations.

Harrison, J. C.↗

NASA NTRS1990

Fourier analysis of finite element preconditioned collocation schemes

The spectrum of the iteration operator of some finite element preconditioned Fourier collocation schemes is investigated. The first part of the paper analyses one-dimensional elliptic and hyperbolic model problems and the advection-diffusion equation. Analytical expressions of the eigenvalues are obtained with use of symbolic computation. The second part of the paper considers the set of one-dimensional differential equations resulting from Fourier analysis (in the tranverse direction) of the 2-D Stokes problem. All results agree with previous conclusions on the numerical efficiency of finite element preconditioning schemes.

Deville, Michel O.↗

NASA NTRS1990

Using a fast Fourier method to model sound propagation in a stratified atmosphere over a stratified porous-elastic ground

Using a Fast Fourier integration method and a global matrix method for solution of the boundary condition equations at all interfaces simultaneously, a useful tool for predicting acoustic propagation in a stratified fluid over a stratified porous-elastic solid was developed. The model for the solid is a modified Biot-Stoll model incorporating four parameters describing the pore structure corresponding to the Rayleigh-Attenborough rigid-porous structure model. The method is also compared to another Fast Fourier code (CERL-FFP) which models the ground as an impedance surface under a horizontally stratified air. Agreement with the CERL FFP is good. The effects on sound propagation of a combination of ground elasticity, complex ground structure, and atmospheric conditions are demonstrated by theoretical results over a snow layer, and experimental results over a model ground surface.

Tooms, S.↗

NASA NTRS1991

A Fourier-Chebyshev pseudospectral method for solving steady 3-D Navier-Stokes and heat equations in cylindrical cavities

A Fourier-Chebyshev pseudospectral method for solving steady 3D Navier-Stokes equations in cylindrical cavities is presented and discussed. The general method is pseudo-unsteady and uses a semi-implicit finite difference scheme for the time discretization. The generalized ADI (Alternating Direction Implicit) procedure is then applied to reduce the problem to a successive solution of one-dimensional problems. The spatial approximation uses a Fourier-Galerkin approximation in the periodic direction and a Chebyshev-collocation approximation in the other directions. Difficulties related to the pressure are surmounted by using the artifical compressibility method. A suitable variable change was chosen to avoid the problem of singularity at the axis generated by cylindrical coordinates. The method is first tested on an advection-diffusion equation and then on the Navier-Stokes equations. Finally, the method is illustrated by a convection problem of a differentially heated fluid.

Pulicani, J. P.↗

NASA NTRS1992

On the Gibbs phenomenon 1: Recovering exponential accuracy from the Fourier partial sum of a non-periodic analytic function

It is well known that the Fourier series of an analytic or periodic function, truncated after 2N+1 terms, converges exponentially with N, even in the maximum norm, although the function is still analytic. This is known as the Gibbs phenomenon. Here, we show that the first 2N+1 Fourier coefficients contain enough information about the function, so that an exponentially convergent approximation (in the maximum norm) can be constructed.

Gottlieb, David↗

NASA NTRS1992

A discrete Fourier transform for virtual memory machines

An algebraic theory of the Discrete Fourier Transform is developed in great detail. Examination of the details of the theory leads to a computationally efficient fast Fourier transform for the use on computers with virtual memory. Such an algorithm is of great use on modern desktop machines. A FORTRAN coded version of the algorithm is given for the case when the sequence of numbers to be transformed is a power of two.

Galant, David C.↗

NASA NTRS1992

A VLSI architecture for simplified arithmetic Fourier transform algorithm

The arithmetic Fourier transform (AFT) is a number-theoretic approach to Fourier analysis which has been shown to perform competitively with the classical FFT in terms of accuracy, complexity, and speed. Theorems developed in a previous paper for the AFT algorithm are used here to derive the original AFT algorithm which Bruns found in 1903. This is shown to yield an algorithm of less complexity and of improved performance over certain recent AFT algorithms. A VLSI architecture is suggested for this simplified AFT algorithm. This architecture uses a butterfly structure which reduces the number of additions by 25 percent of that used in the direct method.

Reed, Irving S.↗

NASA NTRS1992

An RR Lyrae period shift in terms of the Fourier parameter Phi sub 31

The Fourier phase parameter Phi sub 31 has been determined for RRc stars in five globular clusters, NGC 6171, M5, M3, M53, and M15. The results indicate that the RRc stars in a given cluster show a sequence of Phi sub 31 increasing with period, and that the higher the cluster metallicity, the higher the sequence lies in a plot of Phi sub 31 with period. The Phi sub 31 values for the stars in NGC 6171 and M5 presented here are based on observations made with the University of Toronto 0.61 m telescope at Las Campanas, Chile, while those for M3, M53, and M15 are based on published data. A bootstrap procedure has been used to establish the uncertainties in the Fourier parameters. The physical significance of the relationship among Phi sub 31, period, and metallicity is not yet understood. It will need to be tested with hydrodynamic pulsation models computed with new opacities.

Clement, Christine M.↗

NASA NTRS1993

Application of fast Fourier transforms to the direct solution of a class of two-dimensional separable elliptic equations on the sphere

An efficient, direct, second-order solver for the discrete solution of a class of two-dimensional separable elliptic equations on the sphere (which generally arise in implicit and semi-implicit atmospheric models) is presented. The method involves a Fourier transformation in longitude and a direct solution of the resulting coupled second-order finite-difference equations in latitude. The solver is made efficient by vectorizing over longitudinal wave-number and by using a vectorized fast Fourier transform routine. It is evaluated using a prescribed solution method and compared with a multigrid solver and the standard direct solver from FISHPAK.

Moorthi, Shrinivas↗

NASA NTRS1992

Fourier Transform Fabry-Perot Interferometer

We are developing a compact, rugged, high-resolution remote sensing instrument with wide spectral scanning capabilities. This relatively new type of instrument, which we have chosen to call the Fourier-Transform Fabry-Perot Interferometer (FT-FPI), is accomplished by mechanically scanning the etalon plates of a Fabry-Perot interferometer (FPI) through a large optical distance while examining the concomitant signal with a Fourier-transform analysis technique similar to that employed by the Michelson interferometer. The FT-FPI will be used initially as a ground-based instrument to study near-infrared atmospheric absorption lines of trace gases using the techniques of solar absorption spectroscopy. Future plans include modifications to allow for measurements of trace gases in the stratosphere using spectral lines at terahertz frequencies.

Snell, Hilary E.↗

NASA NTRS1992

Imaging solar flares in hard X-rays using Fourier telescopes

The sun emits hard X-rays (above 10 keV) during solar flares. Imaging hard X-ray sources on the sun with spatial resolutions on the order of 1-5 arcsec and integration times of 1 sec will provide greater insight into the energy release processes during a solar flare. In these events, tremendous amounts of energy stored in the solar magnetic field are rapidly released leading to emission across the electromagnetic spectrum. Two Fourier telescope designs, a spatial modulation collimator and a rotating modulation collimator, were developed to image the full sun in hard X-rays (10-100 keV) in an end-to-end simulation. Emission profiles were derived for two hard X-ray solar flare models taken from the current solar theoretical literature and used as brightness distributions for the telescope simulations. Both our telescope models, tailored to image solar sources, were found to perform equally well, thus offering the designer significant flexibility in developing systems for space-based platforms. Given sufficient sensitive areas, Fourier telescopes are promising concepts for imaging solar hard X-rays.

Campbell, Jonathan W.↗

NASA NTRS1993

Mapping agroecological zones and time lag in vegetation growth by means of Fourier analysis of time series of NDVI images

Examples are presented of applications of a fast Fourier transform algorithm to analyze time series of images of Normalized Difference Vegetation Index values. The results obtained for a case study on Zambia indicated that differences in vegetation development among map units of an existing agroclimatic map were not significant, while reliable differences were observed among the map units obtained using the Fourier analysis.

Menenti, M.↗

NASA NTRS1994

Solar Rotating Fourier Telescope

Proposed telescope based on absorbing Fourier-transform grids images full Sun at unprecedented resolution. Overcomes limitations of both conventional optical and pinhole cameras. Arrays of grids and detectors configured for sensitivity to selected fourier components of x-ray images.

Campbell, Jonathan↗

NASA NTRS1982

Integrated optics in an electrically scanned imaging Fourier transform spectrometer

An efficient, lightweight and stable, Fourier transform spectrometer was developed. The mechanical slide mechanism needed to create a path difference was eliminated by the use of retro-reflecting mirrors in a monolithic interferometer assembly in which the mirrors are not at 90 degrees to the propagation vector of the radiation, but rather at a small angle. The resulting plane wave fronts create a double-sided inteferogram of the source irradiance distribution which is detected by a charge-coupled device image sensor array. The position of each CCD pixel in the array is an indication of the path difference between the two retro-reflecting mirrors in the monolithic optical structure. The Fourier transform of the signals generated by the image sensor provide the spectral irradiance distribution of the source. For imaging, the interferometer assembly scans the source of irradiation by moving the entire instrument, such as would occur if it was fixedly mounted to a moving platform, i.e., a spacecraft. During scanning, the entrace slot to the monolithic optical structure sends different pixels to corresponding interferograms detected by adjacent columns of pixels of the image sensor.

Breckinridge, James B.↗