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NASA NTRS1989

Wind profile estimation from point to point laser distortion data

The author's results on the problem of using laser distortion data to estimate the wind profile along the path of the beam are presented. A new model for the dynamics of the index of refraction in a non-constant wind is developed. The model agrees qualitatively with theoretical predictions for the index of refraction statistics in linear wind shear, and is approximated by the predictions of Taylor's hypothesis in constant wind. A framework for a potential in-flight experiment is presented, and the estimation problem is discussed in a maximum likelihood context.

Leland, Robert↗

NASA NTRS2008

Using PEACE Data from the four CLUSTER Spacecraft to Measure Compressibility, Vorticity, and the Taylor Microscale in the Magnetosheath and Plasma Sheet

We present determinations of compressibility and vorticity in the magnetosheath and plasma sheet using moments from the four PEACE thermal electron instruments on CLUSTER. The methodology used assumes a linear variation of the moments throughout the volume defined by the four satellites, which allows spatially independent estimates of the divergence, curl, and gradient. Once the vorticity has been computed, it is possible to estimate directly the Taylor microscale. We have shown previously that the technique works well in the solar wind. Because the background flow speed in the magnetosheath and plasma sheet is usually less than the Alfven speed, the Taylor frozen-in-flow approximation cannot be used. Consequently, this four spacecraft approach is the only viable method for obtaining the wave number properties of the ambient fluctuations. Our results using electron velocity moments will be compared with previous work using magnetometer data from the FGM experiment on Cluster.

Goldstein, Melvyn L.↗

NASA NTRS1988

An approximation method for configuration optimization of trusses

Two- and three-dimensional elastic trusses are designed for minimum weight by varying the areas of the members and the location of the joints. Constraints on member stresses and Euler buckling are imposed and multiple static loading conditions are considered. The method presented here utilizes an approximate structural analysis based on first order Taylor series expansions of the member forces. A numerical optimizer minimizes the weight of the truss using information from the approximate structural analysis. Comparisons with results from other methods are made. It is shown that the method of forming an approximate structural analysis based on linearized member forces leads to a highly efficient method of truss configuration optimization.

Hansen, Scott R.↗

NASA NTRS1986

On recent advances and future research directions for computational fluid dynamics

This paper highlights some recent accomplishments regarding CFD numerical algorithm constructions for generation of discrete approximate solutions to classes of Reynolds-averaged Navier-Stokes equations. Following an overview of turbulent closure modeling, and development of appropriate conservation law systems, a Taylor weak-statement semi-discrete approximate solution algorithm is developed. Various forms for completion to the final linear algebra statement are cited, as are a range of candidate numerical linear algebra solution procedures. This development sequence emphasizes the key building blocks of a CFD RNS algorithm, including solution trial and test spaces, integration procedure and added numerical stability mechanisms. A range of numerical results are discussed focusing on key topics guiding future research directions.

Baker, A. J.↗

NASA NTRS1993

Rational trigonometric approximations using Fourier series partial sums

A class of approximations (S(sub N,M)) to a periodic function f which uses the ideas of Pade, or rational function, approximations based on the Fourier series representation of f, rather than on the Taylor series representation of f, is introduced and studied. Each approximation S(sub N,M) is the quotient of a trigonometric polynomial of degree N and a trigonometric polynomial of degree M. The coefficients in these polynomials are determined by requiring that an appropriate number of the Fourier coefficients of S(sub N,M) agree with those of f. Explicit expressions are derived for these coefficients in terms of the Fourier coefficients of f. It is proven that these 'Fourier-Pade' approximations converge point-wise to (f(x(exp +))+f(x(exp -)))/2 more rapidly (in some cases by a factor of 1/k(exp 2M)) than the Fourier series partial sums on which they are based. The approximations are illustrated by several examples and an application to the solution of an initial, boundary value problem for the simple heat equation is presented.

Geer, James F.↗

NASA NTRS1972

Application of the adiabatic-nuclei theory to vibrational excitation.

The adiabatic-nuclei theory is shown to be quantitatively applicable to vibrational excitation of diatomic molecules by electron impact. When the dependence of the fixed-nuclei phase shifts on internuclear separation is approximated by a two-term Taylor series, the resulting formulas are transparent and easily evaluated. Application to e-H2 excitation is made, phase-shift derivatives are estimated, and the theory is found to fit experimental data.

Faisal, F. H. M.↗

NASA NTRS1973

Electric field corrections in the guidance-center plasma

Electric field autocorrelations for the two-dimensional electrostatic guiding-center plasma are calculated numerically. It is concluded that the autocorrelation, averaged over a thermal equilibrium ensemble, is damped in an approximately exponential fashion, as predicted by Taylor and McNamara. Oscillatory behavior of the type predicted by Taylor and Thompson is not observed.

Joyce, G.↗

NASA NTRS1974

Electric field correlations in the guiding-center plasma

Joyce, G.↗

NASA NTRS1982

Piezoelectric and pyroelectric coefficients for ferroelectric crystals with polarizable molecules

Expressions for piezoelectric and pyroelectric coefficients for a crystal of polarizable point dipoles are derived. The effect of crystal structure on the local electric field acting to polarize the molecules is included via the Lorentz-factor formalism. The derived expressions for the piezo- and pyroelectric coefficients are found to contain terms dependent on derivatives of the Lorentz factors. These terms reflect the changing of molecular dipole moments in response to the changing local electric field in the strained crystal. Inclusion of this effect results in predictions of coefficients substantially different from those obtained using the Lorentz field approximation.

Purvis, C. K.↗

NASA NTRS1991

Programs For Modeling Fault-Tolerant Computing Systems

Pade Approximation with Scaling, (PAWS) and Scaling Taylor Exponential Matrix (STEM) computer programs are software tools for design and validation. Provide flexible, user-friendly, language-based interface for input of Markov mathematical methods describing behaviors of fault-tolerant computer systems. Markov models include both recovery from faults via reconfiguration and behaviors of such systems when faults occur. PAWS and STEM produce exact solutions of probability of system failure and provide conservative estimate of number of significant digits in solution. Written in PASCAL and FORTRAN.

Butler, Ricky W.↗

NASA NTRS1993

Programs Help Create And Evaluate Markov Models

Pade Approximation With Scaling (PAWS) and Scaled Taylor Exponential Matrix (STEM) computer programs provide flexible, user-friendly, language-based interface for creation and evaluation of Markov models describing behaviors of fault-tolerant reconfigurable computer systems. Produce exact solution for probabilities of system failures and provide conservative estimates of numbers of significant digits in solutions. Also offer as part of bundled package with SURE and ASSIST, two other reliable analysis programs developed by Systems Validation Methods group at Langley Research Center.

Butler, Ricky W.↗

NASA NTRS1996

Evaluation of a vortex-based subgrid stress model using DNS databases

The performance of a SubGrid Stress (SGS) model for Large-Eddy Simulation (LES) developed by Misra k Pullin (1996) is studied for forced and decaying isotropic turbulence on a 32(exp 3) grid. The physical viability of the model assumptions are tested using DNS databases. The results from LES of forced turbulence at Taylor Reynolds number R(sub (lambda)) approximately equals 90 are compared with filtered DNS fields. Probability density functions (pdfs) of the subgrid energy transfer, total dissipation, and the stretch of the subgrid vorticity by the resolved velocity-gradient tensor show reasonable agreement with the DNS data. The model is also tested in LES of decaying isotropic turbulence where it correctly predicts the decay rate and energy spectra measured by Comte-Bellot & Corrsin (1971).

Misra, Ashish↗

NASA NTRS1997

Spectral Density of Laser Beam Scintillation in Wind Turbulence: Theory - Part 1

The temporal spectral density of the log-amplitude scintillation of a laser beam wave due to a spatially dependent vector-valued crosswind (deterministic as well as random) is evaluated. The path weighting functions for normalized spectral moments are derived, and offer a potential new technique for estimating the wind velocity profile. The Tatarskii-Klyatskin stochastic propagation equation for the Markov turbulence model is used with the solution approximated by the Rytov method. The Taylor 'frozen-in' hypothesis is assumed for the dependence of the refractive index on the wind velocity, and the Kolmogorov spectral density is used for the refractive index field.

Balakrishnan, A. V.↗

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

Rapid iterative reanalysis for automated design

A method for iterative reanalysis in automated structural design is presented for a finite-element analysis using the direct stiffness approach. A basic feature of the method is that the generalized stiffness and inertia matrices are expressed as functions of structural design parameters, and these generalized matrices are expanded in Taylor series about the initial design. Only the linear terms are retained in the expansions. The method is approximate because it uses static condensation, modal reduction, and the linear Taylor series expansions. The exact linear representation of the expansions of the generalized matrices is also described and a basis for the present method is established. Results of applications of the present method to the recalculation of the natural frequencies of two simple platelike structural models are presented and compared with results obtained by using a commonly applied analysis procedure used as a reference. In general, the results are in good agreement. A comparison of the computer times required for the use of the present method and the reference method indicated that the present method required substantially less time for reanalysis. Although the results presented are for relatively small-order problems, the present method will become more efficient relative to the reference method as the problem size increases. An extension of the present method to static reanalysis is described, ana a basis for unifying the static and dynamic reanalysis procedures is presented.

Bhatia, K. G.↗

NASA NTRS2019

Efficient method for approximating nonlinear dynamics: applications to uncertainty propagation and estimation

High-order Taylor series expansions can be used to model nonlinear dynamics at the cost of integrating a large set of variational equations to obtain high-order state-transition tensors (STTs). This paper presents an innovative technique for approximating the high-order STTs that reduces significantly the computational cost by retaining only the dominant secular terms. We propagate the low-order partial derivatives of Kepler’s equation, which only requires the integration of six additional equations to extend an n-th order approximation to order (n + 1). The approximation stems from the Lindstedt-Poincare procedure and exploits the stability properties of orbital motion. Since the method makes no dynamical assumptions, it can accommodate any source of orbital perturbations. We show how the approximation of the second-order STT significantly increases the accuracy of the linear method for uncertainty propagation with only a small computational overhead. Finally, we derive a high-order approximate extended Kalman filter that implements the proposed approximation of the STT and improves the performance of linear filters. Examples of application with different perturbation sources include the heliocentric orbit of an asteroid, an orbiter around Europa, and an Earth-orbiting satellite.

Park, Ryan S↗

NASA NTRS1986

Application of physical parameter identification to finite element models

A time domain technique for matching response predictions of a structural dynamic model to test measurements is developed. Significance is attached to prior estimates of physical model parameters and to experimental data. The Bayesian estimation procedure allows confidence levels in predicted physical and modal parameters to be obtained. Structural optimization procedures are employed to minimize an error functional with physical model parameters describing the finite element model as design variables. The number of complete FEM analyses are reduced using approximation concepts, including the recently developed convoluted Taylor series approach. The error function is represented in closed form by converting free decay test data to a time series model using Prony' method. The technique is demonstrated on simulated response of a simple truss structure.

Bronowicki, Allen J.↗

NASA NTRS1979

Rayleigh-Taylor and wind-driven instabilities of the nighttime equatorial ionosphere

A thorough re-examination has been made of the Rayleigh-Taylor instability in the nighttime equatorial ionosphere from approximately 100 km to the bottomside F region. The following effects have been taken into account explicitly in various combinations: (1) the eastward drift of the ionosphere caused by the nighttime polarization electric field, (2) the eastward nighttime neutral wind, and (3) recombination in the F and E regions. It was found that, well below the bottomside F region, the Rayleigh-Taylor mode can be unstable and is driven by an eastward neutral wind rather than by gravitational drift. Formation of ionospheric bubbles below the bottomside F region is consistent with the observation of lower ionospheric ions in F region ionospheric holes; furthermore, seasonal and shorter term variations in spread-F occurrence may be associated with variations in the neutral wind and polarization electric field.

Chiu, Y. T.↗