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NASA NTRS1987

Legendre-tau approximation for functional differential equations. II - The linear quadratic optimal control problem

The numerical scheme based on the Legendre-tau approximation is proposed to approximate the feedback solution to the linear quadratic optimal control problem for hereditary differential systems. The convergence property is established using Trotter ideas. The method yields very good approximations at low orders and provides an approximation technique for computing closed-loop eigenvalues of the feedback system. A comparison with existing methods (based on averaging and spline approximations) is made.

Ito, Kazufumi↗

NASA NTRS1983

Computing Graphical Confidence Bounds

Approximation for graphical confidence bounds is simple enough to run on programmable calculator. Approximation is used in lieu of numerical tables not always available, and exact calculations, which often require rather sizable computer resources. Approximation verified for collection of up to 50 data points. Method used to analyze tile-strength data on Space Shuttle thermal-protection system.

Mezzacappa, M. A.↗

NASA NTRS1992

A statistical model for evaluating GOPEX uplink performance

This article describes a statistical model to analyze the signal intensity received at the solid-state imaging (SSI) camera of the Galileo optical communications system from an Earth-based transmitter (GOPEX) demonstration. The analytical model assumes that the optical beam possesses a Gaussian profile and the communication channel has a log-normal scattering characteristic. The atmospheric-induced jitter is modelled as two independent zero mean Gaussian random variables. By modelling the system parameters as a set of independent and identically distributed (iid) random variables, the combined impact of uncertainties due to system parameters and the turbulent atmosphere can be approximated by a log-normal distributed signal intensity at the spacecraft. A Monte-Carlo software simulation package was also developed to compute the confidence interval probabilities for general optical beam profiles. Numerical results show that the approximation is valid for a wide range of operation scenarios.

Kiesaleh, K.↗

NASA NTRS1979

Anomalous growth of single ice crystals in solution

It is shown that major discrepancies exist between experiments and theory for ice crystal growth from solution. Accurate data, taken in a microgravity environment, approximate analytical models, and exact (probably numerical) models all are needed to advance our understanding of ice crystal growth phenomena. A new approximate semi-empirical theory is presented which predicts that a relatively sharp transition from natural convection control to diffusion control for ice growth in pure water occurs at a subcooling of about 10 C (a reduced temperature difference of about 0.125). No reliable data exist to test this prediction. The theory also predicts qualitatively the growth of ice in NaCl solution in which maxima in the growth rates are observed at various levels of subcooling.

Gill, W. N.↗

NASA NTRS1988

Effect of curvature on the backscattering from leaves

Using a model previously developed for the backscattering cross section of a planar leaf at X-band frequencies and above, the effect of leaf curvature is examined. For normal incidence on a rectangular section of a leaf curved in one and two dimensions, an integral expression for the backscattered field is evaluated numerically and by a stationary phase approximation, leading to a simple analytical expression for the cross section reduction produced by the curvature. Numerical results based on the two methods are virtually identical, and in excellent agreement with measured data for rectangular sections of coleus leaves applied to the surfaces of styrofoam cylinders and spheres of different radii.

Sarabandi, K.↗

NASA NTRS1984

Legendre-tau approximation for functional differential equations. Part 2: The linear quadratic optimal control problem

Ito, K.↗

NASA NTRS1973

First-excursion probability in non-stationary random vibration.

The first-excursion probability of a non-stationary Gaussian process with zero mean has been studied. Within the framework of the point process approach, a variety of analytical approximations applicable to stationary random processes is extended herein to non-stationary random processes. The extension is possible owing to a recent definition of non-stationary envelope processes proposed by the author. With the aid of numerical examples, merits of each approximation are examined by comparing with the results of simulation. It is found that under non-stationary excitations with short duration, the Markov approximation is the best among all the approximations discussed in this paper.

Yang, J.-N.↗

NASA NTRS1988

Effect of curvature on the backscattering from a leaf

Using a model previously developed for the backscattering cross section of a planar leaf at X-band frequencies and above, the effect of leaf curvature is examined. For normal incidence on a rectangular section of a leaf curved in one and two dimensions, an integral expression for the backscattered field is evaluated numerically and by a stationary phase approximation, leading to a simple analytical expression for the cross-section reduction produced by the curvature. Numerical results based on the two methods are virtually identical, and in excellent agreement with measured data for rectangular sections of coleus leaves applied to the surfaces of styrofoam cylinders and spheres of different radii.

Sarabandi, K.↗

NASA NTRS1991

Discretized Newton-relaxation solution of high resolution flux-difference split schemes

A method is presented for the implicit solution of high resolution flux-difference split schemes for which it is impractical to obtain the Jacobian matrix analytically. The method is used frequently in numerical schemes for solving nonlinear systems of equations, but apparently has not been used to date for approximating the Jacobian of the Roe numerical flux vectors. The method consists of obtaining a discretized Jacobian by using simple finite-differences of the flux vectors, and then using this discretized Jacobian in a Newton-relaxation solution method. The method is applied to both compressible and incompressible flows. Numerical results are presented that demonstrate the method works for both inviscid and viscous flows.

Whitfield, David L.↗

NASA NTRS1972

Reflection and transmission of acoustic waves from a moving layer

The refraction of acoustic waves by a moving medium layer is theoretically treated and the expressions for reflection and transmission coefficients are determined. The moving medium layer velocity is assumed to have a space dependence in one direction. A partitioning of the moving medium layer into constant-velocity sublayers is introduced and the number of sublayers is allowed to increase until the reflection and transmission coefficients converage to their respective values. Numerical results for several sublayer approximations of Poiseuille's flow are presented as functions of the moving layer velocity for several angles of incidence of the acoustic wave. The degenerate case of single constant-velocity layer is also treated, both theoretically and by a numerical analysis.

Steinmetz, G. G.↗

NASA NTRS1989

Cross correlation and length scales in turbulent flows near surfaces

Two kinds of length scales are used in turbulent flows; 'functional length scales' such as mixing length, dissipation length L(sub epsilon), etc., and 'flow-field length scales' derived from cross correlations of velocity, pressure, etc. in the flow. Some connection between these scales are derived here. We first consider the cross correlation R(sub vv)(y,y(sub 1)) of the normal components u at two heights y, y(sub 1) above a rigid surface, normalized by the velocity y(sub 1) (greater than y). For shear-free boundary layers it is found theoretically, and in field and numerical experiments that R(sub vv) approximately equals y/y(sub 1). For shear layers it is also found that R(sub vv) approximately equals f(y/y(sub 1)) less than or equal to y,y(sub 1). This function f differs slightly between low Reynolds number numerical simulations and field experiments. The lateral structure defined by R(sub vv)(y,r(sub 3); y(sub 1),0) is also self similar and shows that the eddies centered at about y(sub 1) appear to have constant lateral width a(sub 3) above and below y(sub 1), where a(sub 3, sup +) approximately equals 7+1/(1.4dU(sup +)/dy(sup +)), when normalized on u(sub *) and v, where U is the mean velocity. Results for L(sub epsilon, sup -1) from direct numerical simulation are found to compare well with the formula L(sub epsilon, sup -1) = A(sub B)/y + A(sub S)dU/dy/v, for unidirectional and reversing turbulent boundary layers and channel flow, except near where dU/dy approximately equals 0. The conclusion is that the large-scale eddy structure and length scales in these flows are determined by a combination of shear and blocking, and that the vertical component of turbulence has a self-similar structure in both kinds of boundary layer.

Hunt, J. C. R.↗

NASA NTRS1993

The chaotic obliquity of Mars

The discovery (by Laskar, 1989, 1990) that the evolution of the solar system is chaotic, made in a numerical integration of the averaged secular approximation of the equations of motions for the planets, was confirmed by Sussman and Wisdom (1992) by direct numerical integration of the whole solar system. This paper presents results of direct integrations of the rotation of Mars in the chaotically evolved planetary system, made using the same model as that used by Sussman and Wisdom. The numerical integration shows that the obliquity of Mars undergoes large chaotic variations, which occur as the system evolves in the chaotic zone associated with a secular spin-orbit resonance.

Touma, Jihad↗

NASA NTRS1992

A reliable algorithm for optimal control synthesis

In recent years, powerful design tools for linear time-invariant multivariable control systems have been developed based on direct parameter optimization. In this report, an algorithm for reliable optimal control synthesis using parameter optimization is presented. Specifically, a robust numerical algorithm is developed for the evaluation of the H(sup 2)-like cost functional and its gradients with respect to the controller design parameters. The method is specifically designed to handle defective degenerate systems and is based on the well-known Pade series approximation of the matrix exponential. Numerical test problems in control synthesis for simple mechanical systems and for a flexible structure with densely packed modes illustrate positively the reliability of this method when compared to a method based on diagonalization. Several types of cost functions have been considered: a cost function for robust control consisting of a linear combination of quadratic objectives for deterministic and random disturbances, and one representing an upper bound on the quadratic objective for worst case initial conditions. Finally, a framework for multivariable control synthesis has been developed combining the concept of closed-loop transfer recovery with numerical parameter optimization. The procedure enables designers to synthesize not only observer-based controllers but also controllers of arbitrary order and structure. Numerical design solutions rely heavily on the robust algorithm due to the high order of the synthesis model and the presence of near-overlapping modes. The design approach is successfully applied to the design of a high-bandwidth control system for a rotorcraft.

Vansteenwyk, Brett↗

NASA NTRS1974

Magnetic field dependence of plasma relaxation times

A previously derived Fokker-Planck collision integral for an electron plasma in a dc magnetic field is examined in the limit in which the Debye length is greater than the thermal gyroradius, which is in turn greater than the mean distance of closest approach. It is demonstrated that the collision integral can be satisfactorily approximated by the classical Landau value (which ignores the presence of a dc magnetic field) if the following replacement is made: In the Coulomb logarithm, the Debye length is replaced by the gyroradius. This induces a fundamental logarithmic dependence on magnetic field in the relaxation times. Numerical comparison of the asymptotic approximations with the previously derived exact results is made, and good agreement is found. The simplification this introduces into the description of collision processes in magnetized plasma is considerable.

Montgomery, D.↗

NASA NTRS1989

Transonic Navier-Stokes solutions of three-dimensional afterbody flows

The performance of a three-dimensional Navier-Stokes solution technique in predicting the transonic flow past a nonaxisymmetric nozzle was investigated. The investigation was conducted at free-stream Mach numbers ranging from 0.60 to 0.94 and an angle of attack of 0 degrees. The numerical solution procedure employs the three-dimensional, unsteady, Reynolds-averaged Navier-Stokes equations written in strong conservation form, a thin layer assumption, and the Baldwin-Lomax turbulence model. The equations are solved by using the finite-volume principle in conjunction with an approximately factored upwind-biased numerical algorithm. In the numerical procedure, the jet exhaust is represented by a solid sting. Wind-tunnel data with the jet exhaust simulated by high pressure air were also obtained to compare with the numerical calculations.

Compton, William B., III↗

NASA NTRS1993

A test of the adhesion approximation for gravitational clustering

We quantitatively compare a particle implementation of the adhesion approximation to fully non-linear, numerical 'N-body' simulations. Our primary tool, cross-correlation of N-body simulations with the adhesion approximation, indicates good agreement, better than that found by the same test performed with the Zel-dovich approximation (hereafter ZA). However, the cross-correlation is not as good as that of the truncated Zel-dovich approximation (TZA), obtained by applying the Zel'dovich approximation after smoothing the initial density field with a Gaussian filter. We confirm that the adhesion approximation produces an excessively filamentary distribution. Relative to the N-body results, we also find that: (a) the power spectrum obtained from the adhesion approximation is more accurate than that from ZA or TZA, (b) the error in the phase angle of Fourier components is worse than that from TZA, and (c) the mass distribution function is more accurate than that from ZA or TZA. It appears that adhesion performs well statistically, but that TZA is more accurate dynamically, in the sense of moving mass to the right place.

Melott, Adrian L.↗

NASA NTRS1972

An approximation to midcourse correction direction errors.

A new approximation to the components of midcourse correction direction errors is described that is more accurate than a previously used approximation. The calculation effort involved is much less than that for numerical integration. A comparison of numerical integration results with those of the new approximation is presented in a diagram.

Kibler, J. F.↗

NASA NTRS1973

Measurement of the lunar photoelectron layer in the geomagnetic tail

Stable photoelectron fluxes (with energies between 40 and 200 eV) were observed by the Apollo 15 Charged Particle Lunar Environment Experiment (CPLEE). These observations, made in the magnetotail under near vacuum conditions, are compared with numerically calculated photoemission spectra to determine the approximate potential difference between ground and CPLEE'S apertures (a distance of 26 cm). Numerically calculated density and potential distributions are compared with the measured values to provide an estimate of the photoelectron yield function of the dust layer covering the moon.

Reasoner, D. L.↗