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NASA NTRS1972

Effect of radiometric errors on accuracy of temperature-profile measurement by the spectral-scanning method.

The spectral-scanning method may be used to determine the temperature profile of a jet- or rocket-engine exhaust stream by measurements of gas radiation and transmittance at two or more wavelengths. A single, fixed line of sight is used, using immobile radiators and radiometers outside of the gas stream, and there is no interference with the flow. A given radiometric error causes an error in computed temperatures. The ratio between temperature error and radiometric error depends on profile shape, path length, temperature level, the strength of line absorption, and the absorption coefficient and its temperature coefficient. These influence the choice of wavelengths for any given gas. Conditions for minimum temperature error are derived. Numerical results are presented for a two wavelength measurement on a family of profiles that may be expected in a practical case of H2 - O2 combustion. Under favorable conditions, the fractional error in temperature approximates the fractional error in radiant-flux measurement.

Buchele, D. R.↗

NASA NTRS1988

Spectral element methods: Algorithms and architectures

Spectral element methods are high-order weighted residual techniques for partial differential equations that combine the geometric flexibility of finite element methods with the rapid convergence of spectral techniques. Spectral element methods are described for the simulation of incompressible fluid flows, with special emphasis on implementation of spectral element techniques on medium-grained parallel processors. Two parallel architectures are considered: the first, a commercially available message-passing hypercube system; the second, a developmental reconfigurable architecture based on Geometry-Defining Processors. High parallel efficiency is obtained in hypercube spectral element computations, indicating that load balancing and communication issues can be successfully addressed by a high-order technique/medium-grained processor algorithm-architecture coupling.

Fischer, Paul↗

NASA NTRS1988

Spectral element methods - Algorithms and architectures

Spectral element methods are high-order weighted residual techniques for partial differential equations that combine the geometric flexibility of finite element methods with the rapid convergence of spectral techniques. Spectral element methods are described for the simulation of incompressible fluid flows, with special emphasis on implementation of spectral element techniques on medium-grained parallel processors. Two parallel architectures are considered; the first, a commercially available message-passing hypercube system; the second, a developmental reconfigurable architecture based on Geometry-Defining Processors. High parallel efficiency is obtained in hypercube spectral element computations, indicating that load balancing and communication issues can be successfully addressed by a high-order technique/medium-grained processor algorithm-architecture coupling.

Fischer, Paul↗

NASA NTRS1996

Chebyshev polynomials in the spectral Tau method and applications to Eigenvalue problems

Chebyshev Spectral methods have received much attention recently as a technique for the rapid solution of ordinary differential equations. This technique also works well for solving linear eigenvalue problems. Specific detail is given to the properties and algebra of chebyshev polynomials; the use of chebyshev polynomials in spectral methods; and the recurrence relationships that are developed. These formula and equations are then applied to several examples which are worked out in detail. The appendix contains an example FORTRAN program used in solving an eigenvalue problem.

Johnson, Duane↗

NASA NTRSDate not supplied

Spectral Gap Filling Method Using the Spectral Information from CLARREO-Pathfinder Measured Earth-Reflected Solar Spectra to Support CPF-CERES Intercalibration

The high-accuracy Earth-reflected solar spectral radiances from CLARREO Pathfinder (CPF) will serve as in-orbit benchmark measurements for intercalibrating other satellite imagers. The CPF team is developing a state-of-the-art intercalibration method to calibrate the shortwave channel (300-5000 nm) of the CERES instrument against the CPF reference with an uncertainty of 0.3%. Because the CPF instrument only covers spectral range from 350 nm to 2300 nm, the team has developed a spectral extension method to fill in the gaps between the spectral domains of CPF and the CERES shortwave channel. The method solely relies on using the hyperspectral information from the CPF wavelengths to predict the radiances below 350 nm and above 2300 nm. To quantify the relationship between the CPF spectrum and the non-overlapping portions of the spectrum between CPF and CERES, the team has simulated a large ensemble of spectra that cover the spectral range for both CPF and CERES. A hyperspectral bidirectional reflectance model for land surface based on thousands of measured land surface reflectance has been developed. This land surface model as well as the Cox-Munk model for ocean surface are used in the spectral gap filling. Various water/ice clouds, aerosols, atmosphere profiles and surfaces were used to simulate Earth-reflected radiances from 200 nm to 15 um under different solar and sensor geometries. The targeted uncertainty for the CPF-extended broadband CERES radiances is less than 0.1%. The previously derived training spectra were updated to replace the Kurucz solar irradiance model with the TSIS-1 Hybrid Solar Reference Spectrum. The influence of the scalar assumption in radiative transfer model equations on the uncertainty in estimating the broadband radiances was also investigated.

Hyperspectral↗

NASA NTRS2016

Modified Spectral Fatigue Methods for S-N Curves With MIL-HDBK-5J Coefficients

The rainflow method is used for counting fatigue cycles from a stress response time history, where the fatigue cycles are stress-reversals. The rainflow method allows the application of Palmgren-Miner's rule in order to assess the fatigue life of a structure subject to complex loading. The fatigue damage may also be calculated from a stress response power spectral density (PSD) using the semi-empirical Dirlik, Single Moment, Zhao-Baker and other spectral methods. These methods effectively assume that the PSD has a corresponding time history which is stationary with a normal distribution. This paper shows how the probability density function for rainflow stress cycles can be extracted from each of the spectral methods. This extraction allows for the application of the MIL-HDBK-5J fatigue coefficients in the cumulative damage summation. A numerical example is given in this paper for the stress response of a beam undergoing random base excitation, where the excitation is applied separately by a time history and by its corresponding PSD. The fatigue calculation is performed in the time domain, as well as in the frequency domain via the modified spectral methods. The result comparison shows that the modified spectral methods give comparable results to the time domain rainflow counting method.

Irvine, Tom↗

NASA NTRS1992

A spectral overlay method for dissimilar materials

The spectral overlay method is applied to examine the stress distribution along the interface between the two different materials. The essential feature of the spectral overlay method is that the high resolution of localized steep gradients can be achieved by overlaying a spectral interpolant on a standard finite element mesh. In order to evaluate this method and compare it with conventional finite element method, a test problem with high gradients in Poisson equation has been devised. The results show that this method is very powerful in capturing the shape of the gradient field and its peak value. Solutions for two problems with interfaces between dissimilar materials are then given to illustrate the effectiveness of this method.

Belytschko, T.↗

NASA NTRS1986

A spectral multidomain method for the solution of hyperbolic systems

A multidomain Chebyshev spectral collocation method for solving hyperbolic partial differential equations were developed. Though spectral methods are global methods, an attractive idea is to break a computational domain into several domains, and a way to handle the interfaces is described. The multidomain approach offers advantages over the use of a single Chebyshev grid. It allows complex geometries to be covered, and local refinement can be used to resolve important features. For steady state problems it reduces the stiffness associated with the use of explicit time integration as a relaxation scheme. Furthermore, the proposed method remains spectrally accurate. Results showing performance of the method on one dimensional linear models and one and two dimensional nonlinear gas dynamics problems are presented.

Kopriva, D.↗

NASA NTRS1983

A spectral numerical method for the Navier-Stokes equations with applications to Taylor-Couette flow

A new spectral method for solving the incompressible Navier-Stokes equations in a plane channel and between concentric cylinders is presented. The method uses spectral expansions which inherently satisfy the boundary conditions and the continuity equation and yield banded matrices which are efficiently solved at each time step. In addition, the number of dependent variables is reduced, resulting in a reduction in computer memory requirements. Several test problems have been computed for the channel flow and for flow between concentric cylinders, including Taylor-Couette flow with axisymmetric Taylor vortices and wavy vortices. In all cases, agreement with available experimental and theoretical results is very good.

Moser, R. D.↗

NASA NTRS1995

Solution of acoustic workshop problems by a spectral multidomain method

We use a new staggered grid Chebyshev spectral multidomain method to solve three of the Workshop benchmark problems. The method defines solution unknowns at the nodes of the Chebyshev Gauss quadrature, and the fluxes at the nodes of the Chebyshev Gauss-Lobatto quadrature. The Chebyshev spectral method gives exponentially convergent phase and dissipation errors. The multidomain approximation gives the method flexibility. Using the method, we solve problems in Categories 1 and 5 of the benchmark problems.

Kopriva, Davis A.↗

NASA NTRS1983

Spectral multigrid methods for elliptic equations 2

A detailed description of spectral multigrid methods is provided. This includes the interpolation and coarse-grid operators for both periodic and Dirichlet problems. The spectral methods for periodic problems use Fourier series and those for Dirichlet problems are based upon Chebyshev polynomials. An improved preconditioning for Dirichlet problems is given. Numerical examples and practical advice are included.

Zang, T. A.↗

NASA NTRS1984

Spectral multigrid methods for elliptic equations II

Zang, T. A.↗

NASA NTRS1988

Nonconforming mortar element methods: Application to spectral discretizations

Spectral element methods are p-type weighted residual techniques for partial differential equations that combine the generality of finite element methods with the accuracy of spectral methods. Presented here is a new nonconforming discretization which greatly improves the flexibility of the spectral element approach as regards automatic mesh generation and non-propagating local mesh refinement. The method is based on the introduction of an auxiliary mortar trace space, and constitutes a new approach to discretization-driven domain decomposition characterized by a clean decoupling of the local, structure-preserving residual evaluations and the transmission of boundary and continuity conditions. The flexibility of the mortar method is illustrated by several nonconforming adaptive Navier-Stokes calculations in complex geometry.

Maday, Yvon↗

NASA NTRS1984

A spectral collocation method for the Navier-Stokes equations

A Fourier-Chebyshev spectral method for the incompressible Navier-Stokes equations is described. It is applicable to a variety of problems including some with fluid properties which vary strongly both in the normal direction and in time. In this fully spectral algorithm, a preconditioned iterative technique is used for solving the implicit equations arising from semi-implicit treatment of pressure, mean advection and vertical diffusion terms. The algorithm is tested by applying it to hydrodynamic stability problems in channel flow and in external boundary layers with both constant and variable viscosity.

Malik, M. R.↗

NASA NTRS1985

A spectral collocation method for the Navier-Stokes equations

Malik, M. R.↗

NASA NTRS1993

Relaxation schemes for Chebyshev spectral multigrid methods

Two relaxation schemes for Chebyshev spectral multigrid methods are presented for elliptic equations with Dirichlet boundary conditions. The first scheme is a pointwise-preconditioned Richardson relaxation scheme and the second is a line relaxation scheme. The line relaxation scheme provides an efficient and relatively simple approach for solving two-dimensional spectral equations. Numerical examples and comparisons with other methods are given.

Kang, Yimin↗

NASA NTRS1989

Spectral element methods for the incompressible Navier-Stokes equations

Spectral element methods are high-order weighted-residual techniques for partial differential equations that combine the geometric flexibility of finite element techniques with the rapid convergence rate of spectral schemes. The theoretical foundations and numerical implementation of spectral element methods for the incompressible Navier-Stokes equations are presented, considering the construction and analysis of optimal-order spectral element discretizations for elliptic and saddle (Stokes) problems, as well as the efficient solution of the resulting discrete equations by rapidly convergent tensor-product-based iterative procedures. Several examples of spectral element simulation of moderate Reynolds number unsteady flow in complex geometry are presented.

Maday, Yvon↗

NASA NTRS1961

Applications of Power Spectral Analysis Methods to Maneuver Loads Obtained on Jet Fighter Airplanes During Service Operations

Power spectral densities of normal load factor have been obtained for two service operational training flights of a Republic F-84G airplane and three service operational training flights of a North American F-86A airplane in order to indicate the load-factor frequency content and possible uses of power spectral methods in analyzing maneuver load data. It was determined that the maneuvering load-factor time histories appeared to be described by a truncated normal distribution. The power spectral densities obtained were relatively level at frequencies below 0.03 cycle per second and varied inversely with approximately the cube of the frequency at the higher frequencies. In general, the frequency content was very low above 0.2 cycle per second. The load-factor peak distributions were estimated fairly well from the spectrum analysis. In addition, peak load data obtained during service operations of fighter-type airplanes with flight time totaling about 24,000 hours were examined and appeared to agree reasonably well with the type of equations obtained from spectrum peak-load distributions.

Mayer, John P.↗