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NASA NTRS1976

The delta-Eddington approximation for radiative flux transfer

Simple approximations, like the Eddington, are often incapable of coping with the highly asymmetric phase functions typical of particulate scattering. A simple yet accurate method called the delta-Eddington approximation is proposed for determining monochromatic radiative fluxes in an absorbing-scattering atmosphere. In this method, the governing phase function is approximated by a Dirac delta function forward scatter peak and a two-term expansion of the phase function. The fraction of scattering into the truncated forward peak is taken proportional to the square of the phase function asymmetry factor, which distinguishes the delta-Eddington approximation from others of similar nature. The transmission, reflection, and absorption predicted by the delta-Eddington approximation are compared with doubling method calculations for realistic ranges of optical depth, single-scattering albedo, surface albedo, sun angle and asymmetry factor. The approximation is shown to provide an accurate and analytically simple parameterization of radiation to replace the empirism currently encountered in many general circulation and climate models.

Joseph, J. H.↗

NASA NTRS1994

AutoClass: A Bayesian Approach to Classification

We describe a Bayesian approach to the untutored discovery of classes in a set of cases, sometimes called finite mixture separation or clustering. The main difference between clustering and our approach is that we search for the "best" set of class descriptions rather than grouping the cases themselves. We describe our classes in terms of a probability distribution or density function, and the locally maximal posterior probability valued function parameters. We rate our classifications with an approximate joint probability of the data and functional form, marginalizing over the parameters. Approximation is necessitated by the computational complexity of the joint probability. Thus, we marginalize w.r.t. local maxima in the parameter space. We discuss the rationale behind our approach to classification. We give the mathematical development for the basic mixture model and describe the approximations needed for computational tractability. We instantiate the basic model with the discrete Dirichlet distribution and multivariant Gaussian density likelihoods. Then we show some results for both constructed and actual data.

Stutz, John↗

NASA NTRS1976

Dynamics of flexible spinning satellites with radial wire antennas

A dynamic analysis is presented for a spin-stabilized spacecraft employing four radial wire antennas with tip masses. The satellite is modelled as a 14-degree-of-freedom system, and the linearized equations of motion are found. The lowest-order vibrational modes and natural frequencies of the gyroscopically coupled system are then determined. Because the satellite spin rate is decreased by antenna deployment, a spin-up maneuver is needed. The response of the time-varying mode equations during spin-up is found for the planar modes in terms of Bessel functions and a Struve function of order -0.25. The particular solution is expressed in various forms, including an infinite series of Bessel functions and a particularly useful asymptotic expansion. An error formula for the latter is derived, showing that it gives good accuracy. A simple approximation to the complementary function is obtained using the Wenzel-Kramers-Brillouin method, and the phase error in the approximation is shown to be small.

Longman, R. W.↗

NASA NTRS1976

A kinetic-theory approach to turbulent chemically reacting flows

The paper examines the mathematical and physical foundations for the kinetic theory of reactive turbulent flows, discussing the differences and relation between the kinetic and averaged equations, and comparing some solutions of the kinetic equations obtained by the Green's function method with those obtained by the approximate bimodal method. The kinetic method described consists essentially in constructing the probability density functions of the chemical species on the basis of solutions of the Langevin stochastic equation for the influence of eddies on the behavior of fluid elements. When the kinetic equations are solved for the structure of the diffusion flame established in a shear layer by the bimodal method, discontinuities in gradients of the mean concentrations at the two flame edges appear. This is a consequence of the bimodal approximation of all distribution functions by two dissimilar half-Maxwellian functions, which is a very crude approximation. These discontinuities do not appear when the solutions are constructed by the Green's function method described here.

Chung, P. M.↗

NASA NTRS1979

Variational formulation of Budyko-Sellers climate models

A class of simple climate models including those of the Budyko-Sellers type are formulated from a variational principle. A functional is constructed for the zonally averaged mean annual temperature field such that extrema of the functional occur when the climate satisfies the usual energy-balance equation. Local minima of the functional correspond to stable solutions while saddle points correspond to unstable solutions. The technique can be used to construct approximate solutions from trial functions and to carry out finite-amplitude stability analyses. A spectral example is given in explicit detail.

North, G. R.↗

NASA NTRS1992

High temperature ablative foam

An ablative foam composition is formed of approximately 150 to 250 parts by weight polymeric isocyanate having an isocyanate functionality of 2.6 to 3.2; approximately 15 to 30 parts by weight reactive flame retardant having a hydroxyl number range from 200-260; approximately 10 to 40 parts by weight non-reactive flame retardant; approximately 10 to 40 parts by weight nonhydrolyzable silicone copolymer having a hydroxyl number range from 75-205; and approximately 3 to 16 parts by weight amine initiated polyether resin having an isocyanate functionality greater than or equal to 3.0 and a hydroxyl number range from 400-800.

Liu, Matthew T.↗

NASA NTRS2004

Simplified computational approach for dual-probe heat-pulse method

Two equations are currently available for estimating soil volumetric heat capacity (pc) with the dual-probe heat-pulse (DPHP) method. One is simple but gives only approximate results because it assumes that the DPHP sensor releases an impulse of heat instantaneously. The other explicitly accounts for the finite duration of heating and gives exact results. Unfortunately, the equation that gives exact results involves the exponential integral function, which is not available in most computer spreadsheet software packages or data logger function libraries. In this note we introduce an approximation of the exact equation that contains only simple algebraic functions. The approximation consists of the first five terms of a Taylor series, which are written as a telescoped polynomial for computational purposes. For most applications of the DPHP method, the polynomial approximation gives estimates of pc that are at least an order of magnitude more accurate than estimates obtained from the simple equation based on instantaneous heating.

NASA Center JSC↗

NASA NTRS1973

Control law synthesis and sensor design for active flutter suppression.

Methods are presented for representing unsteady aerodynamic loadings, valid for arbitrary motion, for the two-dimensional typical section with a trailing-edge control surface in incompressible flow and a three-dimensional lifting surface with leading- or trailing-edge control surfaces in subsonic compressible flow. Loads for the two-dimensional incompressible case are obtained via analytic continuation of the Theodorsen function into the complex plane, with guidance from a time-domain approximation of the Wagner indicial function. For the three-dimensional case, it is proposed that oscillatory generalized aerodynamic forces be approximated by Pade fractions, thereby permitting an obvious continuation into the complex plane. A theoretical justification for this procedure is briefly outlined, and some practical aspects of its implementation are discussed.

Lyons, M. G.↗

NASA NTRS2023

A Theoretical and Computational Revisit of Conversions Between Whitham’s F-Function and Equivalent Area

This paper proves mathematically that the integral transforms between Whitham’s F-function and equivalent area are the inverse transforms of each other if and only if the slope of the equivalent area at the origin is zero. This mathematical fact contradicts the accepted unconditional inverse relation between Whitham’s F-function and equivalent area in the sonic boom research literature. Piecewise linear approximations of an F-function and of the second derivative of an equivalent area are used to derive numerical formulas for conversions between Whitham’s F-function and equivalent area. Numerical results are included to show convergence of the numerical conversions as the maximum length of the segments for piecewise linear approximations goes to zero. These numerical conversions are approximately the inverse transforms of each other when the second derivative of an equivalent area is continuous and the slope of the equivalent area at the origin is zero.

Whitham's F-function↗

NASA NTRS2001

2-D Modeling of Nanoscale MOSFETs: Non-Equilibrium Green's Function Approach

We have developed physical approximations and computer code capable of realistically simulating 2-D nanoscale transistors, using the non-equilibrium Green's function (NEGF) method. This is the most accurate full quantum model yet applied to 2-D device simulation. Open boundary conditions and oxide tunneling are treated on an equal footing. Electrons in the ellipsoids of the conduction band are treated within the anisotropic effective mass approximation. Electron-electron interaction is treated within Hartree approximation by solving NEGF and Poisson equations self-consistently. For the calculations presented here, parallelization is performed by distributing the solution of NEGF equations to various processors, energy wise. We present simulation of the "benchmark" MIT 25nm and 90nm MOSFETs and compare our results to those from the drift-diffusion simulator and the quantum-corrected results available. In the 25nm MOSFET, the channel length is less than ten times the electron wavelength, and the electron scattering time is comparable to its transit time. Our main results are: (1) Simulated drain subthreshold current characteristics are shown, where the potential profiles are calculated self-consistently by the corresponding simulation methods. The current predicted by our quantum simulation has smaller subthreshold slope of the Vg dependence which results in higher threshold voltage. (2) When gate oxide thickness is less than 2 nm, gate oxide leakage is a primary factor which determines off-current of a MOSFET (3) Using our 2-D NEGF simulator, we found several ways to drastically decrease oxide leakage current without compromising drive current. (4) Quantum mechanically calculated electron density is much smaller than the background doping density in the poly silicon gate region near oxide interface. This creates an additional effective gate voltage. Different ways to. include this effect approximately will be discussed.

Svizhenko, Alexei↗

NASA NTRS2007

Direct Calculation of the Scattering Amplitude Without Partial Wave Decomposition: Inclusion of Correlation Effects - III

In the first two papers in this series, we developed a method for studying electron-hydrogen scattering that does not use partial wave analysis. We constructed an ansatz for the wave function in both the static and static exchange approximations and calculated the full scattering amplitude. Here we go beyond the static exchange approximation, and include correlation in the wave function via a modified polarized orbital. This correlation function provides a significant improvement over the static exchange approximation: the resultant elastic scattering amplitudes are in very good agreement with fully converged partial wave calculations for electron-hydrogen scattering. A fully variational modification of this approach is discussed in the conclusion of the article Popular summary of Direct calculation of the scattering amplitude without partial wave expansion. III ....." by J. Shertzer and A. Temkin. In this paper we continue the development of In this paper we continue the development of a new approach to the way in which researchers have traditionally used to calculate the scattering cross section of (low-energy) electrons from atoms. The basic mathematical problem is to solve the Schroedinger Equation (SE) corresponding the above physical process. Traditionally it was always the case that the SE was reduced to a sequence of one-dimensional (ordinary) differential equations - called partial waves which were solved and from the solutions "phase shifts" were extracted, from which the scattering cross section was calculated.

Shertzer, Janine↗

NASA NTRS1986

Sensitivity analysis and approximation methods for general eigenvalue problems

Optimization of dynamic systems involving complex non-hermitian matrices is often computationally expensive. Major contributors to the computational expense are the sensitivity analysis and reanalysis of a modified design. The present work seeks to alleviate this computational burden by identifying efficient sensitivity analysis and approximate reanalysis methods. For the algebraic eigenvalue problem involving non-hermitian matrices, algorithms for sensitivity analysis and approximate reanalysis are classified, compared and evaluated for efficiency and accuracy. Proper eigenvector normalization is discussed. An improved method for calculating derivatives of eigenvectors is proposed based on a more rational normalization condition and taking advantage of matrix sparsity. Important numerical aspects of this method are also discussed. To alleviate the problem of reanalysis, various approximation methods for eigenvalues are proposed and evaluated. Linear and quadratic approximations are based directly on the Taylor series. Several approximation methods are developed based on the generalized Rayleigh quotient for the eigenvalue problem. Approximation methods based on trace theorem give high accuracy without needing any derivatives. Operation counts for the computation of the approximations are given. General recommendations are made for the selection of appropriate approximation technique as a function of the matrix size, number of design variables, number of eigenvalues of interest and the number of design points at which approximation is sought.

Murthy, D. V.↗

NASA NTRS1997

A Multi-Resolution Nonlinear Mapping Technique for Design and Analysis Application

This report describes a nonlinear mapping technique where the unknown static or dynamic system is approximated by a sum of dimensionally increasing functions (one-dimensional curves, two-dimensional surfaces, etc.). These lower dimensional functions are synthesized from a set of multi-resolution basis functions, where the resolutions specify the level of details at which the nonlinear system is approximated. The basis functions also cause the parameter estimation step to become linear. This feature is taken advantage of to derive a systematic procedure to determine and eliminate basis functions that are less significant for the particular system under identification. The number of unknown parameters that must be estimated is thus reduced and compact models obtained. The lower dimensional functions (identified curves and surfaces) permit a kind of "visualization" into the complexity of the nonlinearity itself.

Phan, Minh Q.↗

NASA NTRS1998

A Multi-Resolution Nonlinear Mapping Technique for Design and Analysis Applications

Phan, Minh Q.↗

NASA NTRS1990

MAFT: The Multicomputer Architecture for Fault-Tolerance

Multicomputer Architecture for Fault-Tolerance (MAFT) is a loosely coupled multiprocessor system designed to achieve an unreliability of less than 10(exp -10)/hr in flight-critical real time applications. The MAFT design objectives and architecture are presented. The fault-tolerance implementation of major functions in MAFT is also presented, including communication; task scheduling; reconfiguration; clock synchronization; and data handling and voting. The need for Byzantine agreement or approximate agreement in various functions is discussed. Different methods were selected to achieve agreement in various subsystems. These methods are illustrated by a more detailed description of the task scheduling and error handling subsystems.

Kieckhafer, Roger M.↗

NASA NTRS2006

Comparison of the Radiative Two-Flux and Diffusion Approximations

Approximate solutions are sometimes used to determine the heat transfer and temperatures in a semitransparent material in which conduction and thermal radiation are acting. A comparison of the Milne-Eddington two-flux approximation and the diffusion approximation for combined conduction and radiation heat transfer in a ceramic material was preformed to determine the accuracy of the diffusion solution. A plane gray semitransparent layer without a substrate and a non-gray semitransparent plane layer on an opaque substrate were considered. For the plane gray layer the material is semitransparent for all wavelengths and the scattering and absorption coefficients do not vary with wavelength. For the non-gray plane layer the material is semitransparent with constant absorption and scattering coefficients up to a specified wavelength. At higher wavelengths the non-gray plane layer is assumed to be opaque. The layers are heated on one side and cooled on the other by diffuse radiation and convection. The scattering and absorption coefficients were varied. The error in the diffusion approximation compared to the Milne-Eddington two flux approximation was obtained as a function of scattering coefficient and absorption coefficient. The percent difference in interface temperatures and heat flux through the layer obtained using the Milne-Eddington two-flux and diffusion approximations are presented as a function of scattering coefficient and absorption coefficient. The largest errors occur for high scattering and low absorption except for the back surface temperature of the plane gray layer where the error is also larger at low scattering and low absorption. It is shown that the accuracy of the diffusion approximation can be improved for some scattering and absorption conditions if a reflectance obtained from a Kubelka-Munk type two flux theory is used instead of a reflection obtained from the Fresnel equation. The Kubelka-Munk reflectance accounts for surface reflection and radiation scattered back by internal scattering sites while the Fresnel reflection only accounts for surface reflections.

Spuckler, Charles M.↗

NASA NTRS1975

Thermodynamics and phase separation of dense fully-ionized hydrogen-helium fluid mixtures

The free energy of a hydrogen-helium fluid mixture is evaluated for the temperatures and densities appropriate to the deep interior of a giant planet such as Jupiter. The electrons are assumed to be fully pressure-ionized and degenerate. In this regime, an appropriate first approximation to the ionic distribution functions can be found by assuming hard sphere interactions. Corrections to this approximation are incorporated by means of the perturbation theory of Anderson and Chandler. Approximations for the three-body interactions and the nonlinear response of the electron gas to the ions are included. It is predicted that a hydrogen-helium mixture, containing 10% by number of helium ions, separates into hydrogen-rich and helium-rich phases below about 8000 K, at the pressures relevant to Jupiter (4-40 Megabars). It is also predicted that the alloy occupies less volume per ion than the separated phases. The equations of state and other thermodynamic derivatives are tabulated. Implications of these results are discussed.

Stevenson, D. J.↗

NASA NTRS1970

Quadratic Liapunov functions

Criteria for estimating errors in quadratic approximations to asymptotic stability involving Popov condition based on existence of quadratic Liapunov functions

Weissenberger, S.↗