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NASA NTRS1977

Routh's algorithm - A centennial survey

One hundred years have passed since the publication of Routh's fundamental work on determining the stability of constant linear systems. The paper presents an outline of the algorithm and considers such aspects of it as the distribution of zeros and applications of it that relate to the greatest common divisor, the abscissa of stability, continued fractions, canonical forms, the nonnegativity of polynomials and polynomial matrices, the absolute stability, optimality and passivity of dynamic systems, and the stability of two-dimensional circuits.

Barnett, S.↗

NASA NTRS1977

Prediction of compliant wall drag reduction, part 1

Computer codes developed to test Bushnell's compliant wall drag reduction model are discussed. One code computes the evolution of mean velocity profiles during the period between bursts as forced by an imposed large-scale pressure pulse due to earlier bursts. Another code computes the local stability characteristics of these computed profiles. The programs use Chebyshev polynomials to resolve the normal boundary layer (y) direction and a staggered grid of mesh points to resolve the x direction. Typically, 257 grid points and 33 Chebyshev polynomials are used in the computations.

Orszag, S. A.↗

NASA NTRS1977

Improved ground truth geoid for the GEOS-3 calibration area

The purpose of this investigation is to develop methods and procedures are reported for computing a detailed geoid to be used as geodetic ground truth for the calibration and verification of GEOS-3 altimeter data. The technique developed is based on rectifying the best available detailed geoid so that the rectified geoid will have correct scale, orientation, shape and position with respect to the geocenter. The approach involved the development of a mathematical model based on a second degree polynomial, in rectangular Cartesian coordinates, describing the geoid undulations at the control stations. A generalized least squares solution was obtained for the polynomial which describes the variation of the undulation differences between the control stations geoid and the gravimetric geoid. Three rectified geoid were determined. These geoids correspond to three sets of tracking station data: (1) WFC/C-band data; (2) GSFC/C-band data; and (3) OSU-275 data. The absolute accuracy of these rectified geoids is linearly correlated with the uncertainties of the tracking station coordinates and, to a certain extent, with those of the detailed geoid being rectified.

Mourad, A. G.↗

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 NTRS1976

Polynominal Interpolation Methods for Viscous Flow Calculations

Higher-order collocation procedures resulting in tridiagonal matrix systems are derived from polynomial spline interpolation and by Hermitian (Taylor series) finite-difference discretization. The similarities and special features of these different developments are discussed. The governing systems apply for both uniform and variable meshes. Hybrid schemes resulting from two different polynomial approximations for the first and second derivatives lead to a nonuniform mesh extension of the so-called compact or Pad? difference technique (Hermite 4). A variety of fourth-order methods are described and the Hermitian approach is extended to sixth-order (Hermite 6). The appropriate spline boundary conditions are derived for all procedures. For central finite differences, this leads to a two-point, second-order accurate generalization of the commonly used three-point end-difference formula. Solutions with several spline and Hermite procedures are presented for the boundary layer equations, with and without mass transfer, and for the incompressible viscous flow in a driven cavity. Divergence and nondivergence equations are considered for the cavity. Among the fourth-order techniques, it is shown that spline 4 has the smallest truncation error. The spline 4 procedure generally requires one-quarter the number of mesh points in a given coordinate direction as a central finite-difference calculation of equal accuracy. The Hermite 6 procedure leads to remarkably accurate boundary layer solutions.

Rubin, S. G.↗

NASA NTRS1979

Nonlinear temperature dependent failure analysis of finite width composite laminates

A quasi-three dimensional, nonlinear elastic finite element stress analysis of finite width composite laminates including curing stresses is presented. Cross-ply, angle-ply, and two quasi-isotropic graphite/epoxy laminates are studied. Curing stresses are calculated using temperature dependent elastic properties that are input as percent retention curves, and stresses due to mechanical loading in the form of an axial strain are calculated using tangent modulii obtained by Ramberg-Osgood parameters. It is shown that curing stresses and stresses due to tensile loading are significant as edge effects in all types of laminate studies. The tensor polynomial failure criterion is used to predict the initiation of failure. The mode of failure is predicted by examining individual stress contributions to the tensor polynomial.

Nagarkar, A. P.↗

NASA NTRS1979

Recurrence relations for computing with modified divided differences

Modified divided differences (MDD) provide a good way of representing a polynomial passing through points with unequally spaced abscissas. This note gives recurrence relations for computing coefficients in either the monomial or Chebyshev basis from the MDD coefficients, and for computing the MDD coefficients for either the differentiated or the integrated polynomial. The latter operation is likely to be useful if MDD are used in a method for solving stiff differential equations.

Krogh, F. T.↗

NASA NTRS1980

An analytical technique for approximating unsteady aerodynamics in the time domain

An analytical technique is presented for approximating unsteady aerodynamic forces in the time domain. The order of elements of a matrix Pade approximation was postulated, and the resulting polynomial coefficients were determined through a combination of least squares estimates for the numerator coefficients and a constrained gradient search for the denominator coefficients which insures stable approximating functions. The number of differential equations required to represent the aerodynamic forces to a given accuracy tends to be smaller than that employed in certain existing techniques where the denominator coefficients are chosen a priori. Results are shown for an aeroelastic, cantilevered, semispan wing which indicate a good fit to the aerodynamic forces for oscillatory motion can be achieved with a matrix Pade approximation having fourth order numerator and second order denominator polynomials.

Dunn, H. J.↗

NASA NTRS1979

The generalized pole assignment problem

For some linear, strictly proper system given by its transfer function, two dynamic output feedback problems can be posed. The first one is that of using dynamic-output feedback to assign the closed-loop characteristic polynomial and the second that of assigning the closed-loop invariant factors. These problems and their interrelationships are discussed. The formulation is done in the frequency domain and the investigation carried out from an algebraic point of view, in terms of linear equations over rings of polynomials. Using the notion of genericity, several necessary and sufficient conditions are expressed.

Djaferis, T. E.↗

NASA NTRS1981

Heat transfer of phase-change materials in two-dimensional cylindrical coordinates

Two-dimensional phase-change problem is numerically solved in cylindrical coordinates (r and z) by utilizing two Taylor series expansions for the temperature distributions in the neighborhood of the interface location. These two expansions form two polynomials in r and z directions. For the regions sufficiently away from the interface the temperature field equations are numerically solved in the usual way and the results are coupled with the polynomials. The main advantages of this efficient approach include ability to accept arbitrarily time dependent boundary conditions of all types and arbitrarily specified initial temperature distributions. A modified approach using a single Taylor series expansion in two variables is also suggested.

Labdon, M. B.↗

NASA NTRS1980

Geometrical rectification of spin-scan images from Pioneer 11

Images of Saturn received from Pioneer 11 suffer from geometrical distortions due to the curvilinear scan lines and the unequal sampling intervals in orthogonal directions, which are inherent in spin-scan imaging. In this paper geometrical image rectification by polynomial transformation based on control points is discussed. Factors that affect the accuracy of reconstruction are shown to include the spatial distribution and spatial density of control points, and the order of the polynomial distortion model. A computer implementation of the technique is described.

Strickland, R. N.↗

NASA NTRS1980

Alignment and evaluation of the cryogenic corrected infrared astronomical satellite /IRAS/ telescope

Room temperature alignment and evaluation techniques for the Infrared Astronomical Satellite (IRAS) telescope, which has a primary mirror figured to correct for surface distortions and the 2 K operating temperature are discussed. Interferometric cryogenic testing of the 0.6 m, f/1.5 lightweighted beryllium primary mirror at its intended operating temperature reveals surface distortions that can be modeled with Zernike polynomials. With this model, it becomes possible to derive the 'inverse' of the cryowavefront error (ideal cryo mirror) and to figure the cryo correction into the primary mirror using Perkin-Elmer's Computer Controlled Polisher. It is recognized that during room temperature assembly of the system, misalignment of the secondary mirror can introduce additional unwanted aberrations that may cancel or distort the wavefront errors purposely introduced by the cryo figuring. To avoid this possible degradation and to ensure optimum telescope performance, the system Zernike polynomial coefficients and wavefront maps generated from the in-process alignment interferograms are monitored and compared to Zernike coefficients and wavefront maps for the cryo corrected primary mirror.

Harned, N.↗

NASA NTRS1982

Tolerance analysis of optical telescopes using coherent addition of wavefront errors

A near diffraction-limited telescope requires that tolerance analysis be done on the basis of system wavefront error. One method of analyzing the wavefront error is to represent the wavefront error function in terms of its Zernike polynomial expansion. A Ramsey-Korsch ray trace package, a computer program that simulates the tracing of rays through an optical telescope system, was expanded to include the Zernike polynomial expansion up through the fifth-order spherical term. An option to determine a 3 dimensional plot of the wavefront error function was also included in the Ramsey-Korsch package. Several assimulation runs were analyzed to determine the particular set of coefficients in the Zernike expansion that are effected by various errors such as tilt, decenter and despace. A 3 dimensional plot of each error up through the fifth-order spherical term was also included in the study. Tolerance analysis data are presented.

Davenport, J. W.↗

NASA NTRS1983

VLSI Unit for Two-Dimensional Convolutions

Universal logic structure allows same VLSI chip to be used for variety of computational functions required for two dimensional convolutions. Fast polynomial transform technique is extended into tree computational structure composed of two units: fast polynomial transform (FPT) unit and Chinese remainder theorem (CRT) computational unit.

Liu, K. Y.↗

NASA NTRS1982

Investigation of Antarctic crust and upper mantle using MAGSAT and other geophysical data

Antarctica is the perfect proving ground for testing data reduction procedures which are alternatives to the standard procedure employing polynomial fitting. Unreduced data (observed values minus corefield model theo each pass) were averaged in 3 degree bins. The resulting map was then high pass filtered using a finite Fourier transform filter so that spectral peaks between 4200 km and 5280 km were diminished by 1/3, peaks corresponding to wavelengths greater than or equal to 5280 km were diminished by 1/2, and the d.c. component was set equal to zero. The surface subtracted in this way differs from the polynomials used in the standard method in that it is static over the 5 north data window. The map produced is presented and the effects of the static model are compared to those of the dynamic mode.

Bentley, C. R.↗

NASA NTRS1983

Simplified Syndrome Decoding of (n, 1) Convolutional Codes

A new syndrome decoding algorithm for the (n, 1) convolutional codes (CC) that is different and simpler than the previous syndrome decoding algorithm of Schalkwijk and Vinck is presented. The new algorithm uses the general solution of the polynomial linear Diophantine equation for the error polynomial vector E(D). This set of Diophantine solutions is a coset of the CC space. A recursive or Viterbi-like algorithm is developed to find the minimum weight error vector cirumflex E(D) in this error coset. An example illustrating the new decoding algorithm is given for the binary nonsymmetric (2,1)CC.

I. S. Reed↗

NASA NTRS1983

New syndrome decoder for (n, 1) convolutional codes

The letter presents a new syndrome decoding algorithm for the (n, 1) convolutional codes (CC) that is different and simpler than the previous syndrome decoding algorithm of Schalkwijk and Vinck. The new technique uses the general solution of the polynomial linear Diophantine equation for the error polynomial vector E(D). A recursive, Viterbi-like, algorithm is developed to find the minimum weight error vector E(D). An example is given for the binary nonsystematic (2, 1) CC.

Reed, I. S.↗

NASA NTRS1982

Modelling of TGS growth in space

Attention is called to the necessity of programming the temperature downward in order to maintain a constant growth rate in the absence of convection. Trial and error numerical computations are performed to find the first approximation for the required isothermal dissolution period, linear ramp, and polynomial temperature variation period. The linear ramp rate is limited by the specified maximum temperature gradient. The isothermal dissolution turns out to be unnecessary. The linear ramp must be stopped before the polynomial period is begun in order to avoid overshooting the specified maximum growth rate. After a few hours, the temperature profile approaches steady-state behavior.

Liu, L. C.↗