Search NASA⌕ Search

SEARCH · Search NASA

Results for “fourier expansion”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 19 records
All sourcesNASA NTRSDOE OSTI
NASA NTRS1983

Number of terms required in the Fourier expansion of the reflection function for optically thick atmospheres

Computational results are presented for the separate terms in the Fourier expansion of the phase function and the reflection function of a semiinfinite, conservatively scattering atmosphere composed of cloud particles. The calculations involve successive applications of invariant imbedding, doubling, and asymptotic fitting methods to cover the range from very thin to very thick atmospheres. From the results, the ratio of the total reflection function to the first-order reflection function is determined as well as the number of terms required to describe the reflection function to an accuracy of 0.1 percent. The number of terms required depends strongly on the zenith angles of incidence and reflection as well as on details of the phase function. These results are compared with similar results obtained for a Henyey-Greenstein phase function having the same asymmetry factor as in the cloud model.

King, M. D.↗
NASA NTRS1977

A note of expansions of functions of velocity in the two-body problem

Fourier expansions based on both the true anomaly and the mean anomaly are obtained for the functions of velocity in the two-body problem; the series of coefficients is written from classical formulae involving associated Legendre polynomials, Gegenbauer polynomials, or Bessel functions. The Fourier expansions are compared with the expansions in powers of eccentricity developed by Broucke (1974) through use of computerized Poisson series manipulation.

Kinoshita, H.↗
NASA NTRSDate not supplied

Development of An Improved BRDF Hotspot Model and its Use in VLIDORT to Study the Impact of Atmospheric Scattering on Hotspot Directional Signatures in the Atmosphere

The term “hotspot” refers to the sharp increase of reflectance occurring when incident (solar) and reflected (viewing) directions almost coincide in the backscatter direction. The accurate simulation of hotspot directional signatures is important for many remote sensing applications. The RossThick-LiSparse-Reciprocal (RTLSR) Bidirectional Reflectance Distribution Function (BRDF) model is widely used in radiative transfer simulations, and the hotspot model mostly used is from Maignan- Bréon but it typically requires large values of numerical quadrature and Fourier expansion terms in order to represent the hotspot accurately. To improve its use in atmospheric radiative transfer (RT) model simulations, in this paper we have developed a modified version based on the Maignan-Bréon’s hotspot BRDF model that converge much faster numerically, making it more practical for use in RT models that require Fourier expansion of BRDF to simulate the top-of-atmosphere (TOA) hotspot signatures. Using the vector linearized discrete ordinate radiative transfer model (VLIDORT), we found that reasonable TOA hotspot accuracy can be obtained with just 23 Fourier terms for clear atmospheres, and 63 Fourier terms for atmospheres with aerosol scattering. One advantage of this modified model is that the new hotspot model agrees very well with the original RossThick model away the hotspot region, making it is very convenient to use in the condition with and without hotspot in applications. This model can calculate the amplitude of hot spot accurately, and has been added in the most recent version of VLIDORT. However, there are some difference of this modified model with the original model for scattering angle close the hot spot, and it may not be appropriate for those who need an exact representation of the hot spot angular signature close to hot spot.

Xiaozhen (Shawn) Xiong↗
NASA NTRSDate not supplied

A New BRDF Hotspot Model and its Use in Improving Radiative Transfer Model Efficiency and Application to VLIDORT-based PCRTM Model

The term “hotspot” refers to the sharp increase of reflectance occurring when incident (solar) and reflected (viewing) directions almost coincide in the backscatter direction. The accurate simulation of hotspot directional signatures is important for many remote sensing applications. The RossThick-LiSparse-Reciprocal (RTLSR) Bidirectional Reflectance Distribution Function (BRDF) model is widely used in radiative transfer simulations, and the hotspot model mostly used is from Maignan- Bréon but it typically requires large values of numerical quadrature and Fourier expansion terms in order to represent the hotspot accurately. To improve its use in atmospheric radiative transfer (RT) model simulations, we have developed a modified version based on the Maignan-Bréon’s hotspot BRDF model that converge much faster numerically, making it more practical for use in RT models that require Fourier expansion of BRDF to simulate the top-of-atmosphere (TOA) hotspot signatures. On basis of the vector linearized discrete ordinate radiative transfer model (VLIDORT), we have built a line-by-line based simulation system that can simulate the TOA radiance for atmospheres with 26 gases, different aerosols, dust and clouds, and surface with different BRDF. Using simulated results under diverse atmospheric conditions, we have developed VLIDORT-PCRTM model.

Xiaozhen Xiong↗
NASA NTRS1974

Contoured patterns from reflector systems - A spherical wave expansion solution

Any electromagnetic field - e.g., a contoured antenna pattern may be expanded into a sum of spherical waves, similar to a one-dimensional Fourier expansion. This expanded wave could then be scattered from a source reflector to determine a resulting required feed field in the vicinity of the focal region. This is of questionable value, and the more significant information to produce would be the required feed pattern itself. Hence the required contour pattern is instead scattered from the back of the reflector, and the result is the required feed pattern with origin at the focus of the chosen reflector.

Williams, W. F.↗
NASA NTRS1992

Resolution properties of the Fourier method for discontinuous waves

In this paper we discuss the wave-resolution properties of the Fourier approximations of a wave function with discontinuities. It is well known that a minimum of two points per wave is needed to resolve a periodic wave function using Fourier expansions. For Chebyshev approximations of a wave function, a minimum of pi points per wave is needed. Here we obtain an estimate for the minimum number of points per wave to resolve a discontinuous wave based on its Fourier coefficients. In our recent work on overcoming the Gibbs phenomenon, we have shown that the Fourier coefficients of a discontinuous function contain enough information to reconstruct with exponential accuracy the coefficient of a rapidly converging Gegenbauer expansion. We therefore study the resolution properties of a Gegenbauer expansion where both the number of terms and the order increase.

Gottlieb, David↗
NASA NTRS1990

Task reports on developing techniques for scattering by 3D composite structures and to generate new solutions in diffraction theory using higher order boundary conditions

There are two tasks described in this report. First, an extension of a two dimensional formulation is presented for a three dimensional body of revolution. With the introduction of a Fourier expansion of the vector electric and magnetic fields, a coupled two dimensional system is generated and solved via the finite element method. An exact boundary condition is employed to terminate the mesh and the fast fourier transformation is used to evaluate the boundary integrals for low O(n) memory demand when an iterative solution algorithm is used. Second, the diffraction by a material discontinuity in a thick dielectric/ferrite layer is considered by modeling the layer as a distributed current sheet obeying generalized sheet transition conditions (GSTC's).

Volakis, John L.↗
NASA NTRS1990

A finite element conjugate gradient FFT method for scattering

An extension of a two dimensional formulation is presented for a three dimensional body of revolution. With the introduction of a Fourier expansion of the vector electric and magnetic fields, a coupled two dimensional system is generated and solved via the finite element method. An exact boundary condition is employed to terminate the mesh and the fast fourier transformation (FFT) is used to evaluate the boundary integrals for low O(n) memory demand when an iterative solution algorithm is used. By virtue of the finite element method, the algorithm is applicable to structures of arbitrary material composition. Several improvements to the two dimensional algorithm are also described. These include: (1) modifications for terminating the mesh at circular boundaries without distorting the convolutionality of the boundary integrals; (2) the development of nonproprietary mesh generation routines for two dimensional applications; (3) the development of preprocessors for interfacing SDRC IDEAS with the main algorithm; and (4) the development of post-processing algorithms based on the public domain package GRAFIC to generate two and three dimensional gray level and color field maps.

Collins, Jeffery D.↗
NASA NTRS1974

Nonlinear cellular motions in Poiseuille channel flow

In a number of nonlinear solutions to the equations of channel flow the velocity is decomposed into a mean part plus a nonlinear disturbance. The idea that nonlinear effects place a limitation on the amplitudes of the disturbance flow is considered. In the reported investigation the disturbance flow is represented by drastically limited Fourier expansions in the downstream coordinate. The resulting equations are solved numerically with high accuracy to obtain a good representation of the cross-stream structure of the solution. The results of the investigation show that indeed the nonlinear terms always limit the amplitude of the disturbance flow in this approximation.

Zahn, J.-P.↗
NASA NTRS1975

On the Laplace relation among the satellites of Uranus

Three of Uranus' satellites are involved in an interaction similar to the Laplace relation among the Galilean satellites of Jupiter. However, there are no pair-wise commensurabilities in the Uranus system as there are among the Galilean satellites. Thus analysis by Fourier expansion of disturbing functions becomes too unwieldy for practical calculation. Instead, the unexpanded perturbations are numerically averaged over a time scale short compared to the important variations in the Laplace relation. Perturbations of Miranda's orbit are estimated to be of observable magnitude for reasonable masses of Ariel and Umbriel.

Greenberg, R.↗
NASA NTRS1979

The seasonal and interannual variability of total ozone as revealed by the BUV Nimbus-4 experiment

The BUV/Nimbus-4 total ozone data is analyzed with emphasis on the seasonal and interannual variability for the period April 1970 to April 1972. An objective analysis using a Fourier expansion shows the annual wave dominates at mid and high latitudes where the semiannual wave becomes significant in the tropics. A small interannual difference is detected and is most likely due to changes in the general circulation.

Hilsenrath, E.↗
NASA NTRS1979

Analysis of solar radiation pressure induced coupled librations of a gravity stabilized axisymmetric satellite

This paper presents an analysis of solar radiation pressure induced coupled librations of gravity stabilized cylindrical spacecraft with a special reference to geostationary communication satellites. The Lagrangian approach is used to obtain the corresponding equations of motion. The solar induced torques are assumed to be free of librational angles and are represented by their Fourier expansion. The response and periodic solutions are obtained through linear and nonlinear analyses, using the method of harmonic balance in the latter case. The stability conditions are obtained using Routh-Hurwitz criteria. To establish the ranges of validity the analytic response is compared with the numerical solution. Finally, values of the system parameters are suggested to make the satellite behave as desired. Among these is a possible approach to subdue the solar induced roll resonance. It is felt that the approximate analysis presented here should significantly reduce the computational efforts involved in the design and stability analysis of the systems.

Shrivastava, S. K.↗
NASA NTRS1981

Large-eddy simulation of the shear-free turbulent boundary layer

The shear-free turbulent boundary layer is calculated by the large-eddy simulation technique. The filtered Navier-Stokes equations are used; the method of integration employs Fourier expansions in the homogeneous directions and finite differences in the cross-stream direction. Results indicate that the simulation is capable of predicting the primary Reynolds-number effects.

Biringen, S.↗
NASA NTRS1982

A new numerical method for the simulation of three dimensional flow in a pipe

A new numerical technique for simulating three dimensional, unsteady, incompressible pipe flows is presented and its utility and accuracy is shown. Each vector function in the expansion of the velocity field is divergence free and satisfies the boundary conditions for viscous flow. Some of the benefits of the expansion technique are that pressure is eliminated from the dynamics, only two unknowns per mesh point are required, implicit treatment of the viscous terms is provided at no extra computational cost, and no fractional time steps are required. The method uses spectral expansions: Fourier series in the azimuthal and streamwise directions, and Jacobi polynominals in the radial direction.

Leonard, A.↗
NASA NTRS1982

A new numerical method for the simulation of three-dimensional flow in a pipe

A new numerical technique for simulating three dimensional, unsteady, incompressible pipe flows is presented and its utility and accuracy is shown. Each vector function in the expansion of the velocity field is divergence free and satisfies the boundary conditions for viscous flow. Some of the benefits of the expansion technique are that pressure is eliminated from the dynamics, only two unknowns per mesh point are required, implicit treatment of the viscous terms is provided at no extra computational cost, and no fractional time steps are required. The method uses spectral expansions: Fourier series in the azimuthal and streamwise directions, and Jacobi polynominals in the radial direction. Previously announced in STAR as N82-31644

Leonard, A.↗