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NASA NTRS1993

Nonlinear aerodynamic modeling using multivariate orthogonal functions

A technique was developed for global modeling of nonlinear aerodynamic coefficients using multivariate orthogonal functions based on the data. Each orthogonal function retained in the model was decomposed into an expansion of ordinary polynomials in the independent variables, so that the final model could be interpreted as selectively retained terms from a multivariable power series expansion. A predicted squared-error metric was used to determine the orthogonal functions to be retained in the model; analytical derivatives were easily computed. The approach was demonstrated on the Z-body axis aerodynamic force coefficient (Cz) wind tunnel data for an F-18 research vehicle which came from a tabular wind tunnel and covered the entire subsonic flight envelope. For a realistic case, the analytical model predicted experimental values of Cz very well. The modeling technique is shown to be capable of generating a compact, global analytical representation of nonlinear aerodynamics. The polynomial model has good predictive capability, global validity, and analytical differentiability.

Morelli, Eugene A.↗
NASA NTRS1992

Importance of Geosat orbit and tidal errors in the estimation of large-scale Indian Ocean variations

To improve the estimate accuracy of large-scale meridional sea-level variations, Geosat ERM data on the Indian Ocean for a 26-month period were processed using two different techniques of orbit error reduction. The first technique removes an along-track polynomial of degree 1 over about 5000 km and the second technique removes an along-track once-per-revolution sine wave about 40,000 km. Results obtained show that the polynomial technique produces stronger attenuation of both the tidal error and the large-scale oceanic signal. After filtering, the residual difference between the two methods represents 44 percent of the total variance and 23 percent of the annual variance. The sine-wave method yields a larger estimate of annual and interannual meridional variations.

Perigaud, Claire↗
NASA NTRS1955

Generalized indical forces on deforming rectangular wings in supersonic flight

A method is presented for determining the time-dependent flow over a rectangular wing moving with a supersonic forward speed and undergoing small vertical distortions expressible as polynomials involving spanwise and chordwise distances. The solution for the velocity potential is presented in a form analogous to that for steady supersonic flow having the familiar "reflected area" concept discovered by Evvard. Particular attention is paid to indicial-type motions and results are expressed in terms of generalized indicial forces. Numerical results for Mach numbers equal to 1.1 and 1.2 are given for polynomials of the first and fifth degree in the chordwise and spanwise directions, respectively, on a wing having an aspect ratio of 4.

Lomax, Harvard↗
NASA NTRS1990

On well-partial-order theory and its application to combinatorial problems of VLSI design

We nonconstructively prove the existence of decision algorithms with low-degree polynomial running times for a number of well-studied graph layout, placement, and routing problems. Some were not previously known to be in p at all; others were only known to be in p by way of brute force or dynamic programming formulations with unboundedly high-degree polynomial running times. Our methods include the application of the recent Robertson-Seymour theorems on the well-partial-ordering of graphs under both the minor and immersion orders. We also briefly address the complexity of search versions of these problems.

Fellows, M.↗
NASA NTRS1990

Application specific serial arithmetic arrays

High performance systolic arrays of serial-parallel multiplier elements may be rapidly constructed for specific applications by applying hardware description language techniques to a library of full-custom CMOS building blocks. Single clock pre-charged circuits have been implemented for these arrays at clock rates in excess of 100 Mhz using economical 2-micron (minimum feature size) CMOS processes, which may be quickly configured for a variety of applications. A number of application-specific arrays are presented, including a 2-D convolver for image processing, an integer polynomial solver, and a finite-field polynomial solver.

Winters, K.↗
NASA NTRS1993

Flutter analysis using transversality theory

A new method of calculating flutter boundaries of undamped aeronautical structures is presented. The method is an application of the weak transversality theorem used in catastrophe theory. In the first instance, the flutter problem is cast in matrix form using a frequency domain method, leading to an eigenvalue matrix. The characteristic polynomial resulting from this matrix usually has a smooth dependence on the system's parameters. As these parameters change with operating conditions, certain critical values are reached at which flutter sets in. Our approach is to use the transversality theorem in locating such flutter boundaries using this criterion: at a flutter boundary, the characteristic polynomial does not intersect the axis of the abscissa transversally. Formulas for computing the flutter boundaries and flutter frequencies of structures with two degrees of freedom are presented, and extension to multi-degree of freedom systems is indicated. The formulas have obvious applications in, for instance, problems of panel flutter at supersonic Mach numbers.

Afolabi, D.↗
NASA NTRS1993

Gravity investigation of the Manson impact structure, Iowa

The Manson crater, of probable Cretaceous/Tertiary age, is located in northwestern Iowa (center at 42 deg. 34.44 min N; 94 deg. 33.60 min W). A seismic reflection profile along an east west line across the crater and drill hole data indicate a crater about 35 km in diameter having the classic form for an impact crater, an uplifted central peak composed of uplifted Proterozoic crystalline bedrock, surrounded by a 'moat' filled with impact produced breccia and a ring graben zone composed of tilted fault blocks of the Proterozoic and Paleozoic country rocks. The structure has been significantly eroded. This geologic structure would be expected to produce a significant gravity signature and study of that signature would shed additional light on the details of the crater structure. A gravity study was undertaken to better resolve the crustal structure. The regional Bouguer gravity field is characterized by a southeastward decreasing field. To first order, the Bouguer gravity field can be understood in the context of the geology of the Precambrian basement. The high gravity at the southeast corner is associated with the mid-continent gravity high; the adjacent low to the northwest results from a basin containing low-density clastic sediments shed from the basement high. Modeling of a simple basin and adjacent high predicts much of the observed Bouguer gravity signature. A gravity signature due to structure associated with the Manson impact is not apparent in the Bouguer data. To resolve the gravity signature of the impact, a series of polynomial surfaces were fit to the Bouguer gravity field to isolate the small wavelength residual anomalies. The residual gravity obtained after subtracting a 5th- or 6th-order polynomial seems to remove most of the regional effects and isolate local anomalies. The pattern resolved in the residual gravity is one of a gravity high surrounded by gravity lows and in turn surrounded by isolated gravity highs. The central portion of the crater is characterized by two positive anomalies having amplitudes of about plus 4 mGal separated by a gentle saddle located approximately at the crater center.

Plescia, J. B.↗
NASA NTRS1993

On the Gibbs phenomenon 3: Recovering exponential accuracy in a sub-interval from a spectral partial sum of a piecewise analytic function

The investigation of overcoming Gibbs phenomenon was continued, i.e., obtaining exponential accuracy at all points including at the discontinuities themselves, from the knowledge of a spectral partial sum of a discontinuous but piecewise analytic function. It was shown that if we are given the first N expansion coefficients of an L(sub 2) function f(x) in terms of either the trigonometrical polynomials or the Chebyshev or Legendre polynomials, an exponentially convergent approximation to the point values of f(x) in any sub-interval in which it is analytic can be constructed.

Gottlieb, David↗
NASA NTRS1994

In-flight scale/distortion calibration of the Hubble Space Telescope fixed-head star trackers

This paper describes an in-flight scale and distortion calibration procedure that has been developed for the Ball Aerospace Systems Division Fixed-Head Star Trackers (FHST's) used on the Hubble Space Telescope (HST). The FHST is a magnetically focused and deflected imaging sensor that is designed to track stars as faint as m(sub v) = 5.7 over an 8 degree by 8 degree field of view. Raw FHST position measurements are accurate to approximately 200 arcseconds, but this can be improved to 10-15 arcseconds by processing the raw measurements through calibration polynomials that correct for flat field, temperature intensity, and magnetic field effects. The coefficients for these polynomials were initially determined using ground test data. On HST the use of three FHST's is an integral part of the preliminary attitude update procedures required before the acquisition of guide stars for science observations. To this end, FHST-based attitude determination having single-axis errors no worse than 22 arcseconds (1 sigma) is required. In early 1991 it became evident that one of the HST FHST's was experiencing a significant change in its optical scale. By mid-1993 the size of this error had grown to a point that, if not corrected, it would correspond to a maximum position error on the order of 100 arcseconds. Subsequent investigations demonstrated that substantial, uncompensated cubic distortion effects had also developed, the maximum contribution to position errors from the cubic terms being on the order of 30 arcseconds. To ensure accurate FHST-based attitude updates, procedures have been developed to redetermine the FHST scale and distortion calibration coefficients based on in-flight data gathered during normal HST operations. These scale and distortion calibrations have proven very effective operationally, and procedures are in place to monitor FHST calibration changes on a continuing basis.

Mccutcheon, R.↗
NASA NTRS1994

On the Gibbs phenomenon 5: Recovering exponential accuracy from collocation point values of a piecewise analytic function

The paper presents a method to recover exponential accuracy at all points (including at the discontinuities themselves), from the knowledge of an approximation to the interpolation polynomial (or trigonometrical polynomial). We show that if we are given the collocation point values (or a highly accurate approximation) at the Gauss or Gauss-Lobatto points, we can reconstruct a uniform exponentially convergent approximation to the function f(x) in any sub-interval of analyticity. The proof covers the cases of Fourier, Chebyshev, Legendre, and more general Gegenbauer collocation methods.

Gottlieb, David↗
NASA NTRS1991

Satellite passive-microwave observations and the distribution of net surface accumulation in the interior of Greenland

The rate of net accumulation at the surface and its distribution in ice sheet areas bounded by the dry snow line may be approximated by a determination of either firn external emissivity or various other emissivity terms. A compilation of mean annual brightness temperature for the Greenland ice sheet based on Nimbus-5 Electrically Scanning Microwave Radiometer (19 GHz) data is used to approximate the distribution of the accumulation rate in the area bounded by the dry snow line, applying a simple conversion involving a polynomial function. This is compared with a compilation produced elsewhere using Nimbus-6 SCAMS (31 GHz) data, applying a complex conversion involving the temperature dependence of the absorption coefficient, and with the latest compilation based on surface measurements of the accumulation rate. The findings indicate that in Greenland, the use of polynomial functions to convert for net surface accumulation is a reliable and simple method for estimating both the rate of accumulation and its distribution in areas where there is no melting at the surface.

Giovinetto, M. B.↗
NASA NTRS1994

A uniform geometrical optics and an extended uniform geometrical theory of diffraction for evaluating high frequency EM fields near smooth caustics and composite shadow boundaries

A uniform geometrical optics (UGO) and an extended uniform geometrical theory of diffraction (EUTD) are developed for evaluating high frequency electromagnetic (EM) fields within transition regions associated with a two and three dimensional smooth caustic of reflected rays and a composite shadow boundary formed by the caustic termination or the confluence of the caustic with the reflection shadow boundary (RSB). The UGO is a uniform version of the classic geometrical optics (GO). It retains the simple ray optical expressions of classic GO and employs a new set of uniform reflection coefficients. The UGO also includes a uniform version of the complex GO ray field that exists on the dark side of the smooth caustic. The EUTD is an extension of the classic uniform geometrical theory of diffraction (UTD) and accounts for the non-ray optical behavior of the UGO reflected field near caustics by using a two-variable transition function in the expressions for the edge diffraction coefficients. It also uniformly recovers the classic UTD behavior of the edge diffracted field outside the composite shadow boundary transition region. The approach employed for constructing the UGO/EUTD solution is based on a spatial domain physical optics (PO) radiation integral representation for the fields which is then reduced using uniform asymptotic procedures. The UGO/EUTD analysis is also employed to investigate the far-zone RCS problem of plane wave scattering from two and three dimensional polynomial defined surfaces, and uniform reflection, zero-curvature, and edge diffraction coefficients are derived. Numerical results for the scattering and diffraction from cubic and fourth order polynomial strips are also shown and the UGO/EUTD solution is validated by comparison to an independent moment method (MM) solution. The UGO/EUTD solution is also compared with the classic GO/UTD solution. The failure of the classic techniques near caustics and composite shadow boundaries is clearly demonstrated and it is shown that the UGO/EUTD results remain valid and uniformly reduce to the classic results away from the transition regions. Mathematical details on the asymptotic properties and efficient numerical evaluation of the canonical functions involved in the UGO/EUTD expressions are also provided.

Constantinides, E. D.↗
NASA NTRS1994

Splines and control theory

In this work, the relationship between splines and the control theory has been analyzed. We show that spline functions can be constructed naturally from the control theory. By establishing a framework based on control theory, we provide a simple and systematic way to construct splines. We have constructed the traditional spline functions including the polynomial splines and the classical exponential spline. We have also discovered some new spline functions such as trigonometric splines and the combination of polynomial, exponential and trigonometric splines. The method proposed in this paper is easy to implement. Some numerical experiments are performed to investigate properties of different spline approximations.

Zhang, Zhimin↗
NASA NTRS1995

Grid generation and surface modeling for CFD

When computing the flow around complex three dimensional configurations, the generation of the mesh is the most time consuming part of any calculation. With some meshing technologies this can take of the order of a man month or more. The requirement for a number of design iterations coupled with ever decreasing time allocated for design leads to the need for a significant acceleration of this process. Of the two competing approaches, block-structured and unstructured, only the unstructured approach will allow fully automatic mesh generation directly from a CAD model. Using this approach coupled with the techniques described in this paper, it is possible to reduce the mesh generation time from man months to a few hours on a workstation. The desire to closely couple a CFD code with a design or optimization algorithm requires that the changes to the geometry be performed quickly and in a smooth manner. This need for smoothness necessitates the use of Bezier polynomials in place of the more usual NURBS or cubic splines. A two dimensional Bezier polynomial based design system is described.

Connell, Stuart D.↗
NASA NTRS1995

The construction of high-accuracy schemes for acoustic equations

An accuracy analysis of various high order schemes is performed from an interpolation point of view. The analysis indicates that classical high order finite difference schemes, which use polynomial interpolation, hold high accuracy only at nodes and are therefore not suitable for time-dependent problems. Thus, some schemes improve their numerical accuracy within grid cells by the near-minimax approximation method, but their practical significance is degraded by maintaining the same stencil as classical schemes. One-step methods in space discretization, which use piecewise polynomial interpolation and involve data at only two points, can generate a uniform accuracy over the whole grid cell and avoid spurious roots. As a result, they are more accurate and efficient than multistep methods. In particular, the Cubic-Interpolated Psuedoparticle (CIP) scheme is recommended for computational acoustics.

Tang, Lei↗
NASA NTRS1994

Rovibrational intensities of the minor isotopes of the CO X (sup 1)Sigma (sup +) state for v less than or equal to 20 and J less than or equal to 150

Electric dipole transition matrix elements for rovibrational transitions in the X (sup 1)Sigma(sup +) state of the CO minor isotopes (14)C(16)O and (13)C(17)O are calculated for the first time for all the delta v = +1, +2, and +3 transitions for which v less than or equal to 20 and J less than or equal to 150. Improved electric dipole transition matrix elements are also calculated for the minor isotopes (12)C(17)O, (12)C(18)O, (13)C(18)O. We have fitted polynomials to these matrix elements as a function of the parameter m which is defined in terms of the lower state angular momentum quantum number J; the convenient to use polynomial representations are given in tabular form. These results for the minor species of CO complement those previously reported by us for (12)C(16)O and (13)C(16)O.

Goorvitch, D.↗
NASA NTRS1994

The meteorological parameterization of specific attenuation in rain viewed at Nadir

Polynomial expressions are presented for the parameterization of the specific attenuation in rain from 9 to 38 GHz that are applicable to a wide range of naturally occurring drop size distributions and for viewing angles close to nadir. Because the temperature T affects the specific attenuation at some frequencies, expressions for the polynomial coefficients as functions of T are also provided for -10C less than or equal to T less than or equal to 30C. The advantage of this parameterization is that even without a detailed specification of the drop size distribution, useful estimates of the specific attenuation are often possible given only two out of three parameters, namely, the rainfall rate R, the rainwater content W, or D(sub m) (the mass-weighted mean drop diameter), particularly if the temperature is also specified.

Jameson, A. R.↗
NASA NTRS1994

Calculation of (12)C(16)O and (13)C(16)O X(1)Sigma(+) rovibrational intensities for v less than or equal to 20 and J less than or equal to 150

Improved electric dipole transition matrix elements for rovibrational transitions in the ground state X(1)Sigma(+) of (12)C(6)O and (13)C(16)O are calculated for all the delta v = +1, +2, and +3 transitions for which v less than or equal to 20 and J less than or equal to 150. We have fitted polynomials to these matrix elements as a function of the parameter m which is defined in terms of the lower state angular momentum quantum number J. These convenient to use polynomial representations are given in Tables 1-4 for (12)C(16)O and in Tables 5-8 for (13)C(16)O. We observe that there is intensity enhancement due to vibration-rotation interaction for the P-branch transitions at the expense of the R-branch transitions for delta v = +1. This enhancement can be as large as 40% at the highest J. For the delta v = +2 and +3 transitions, the R-branch transitions are enhanced by as much as a factor of 2.75 and 10 at the highest J, respectively. The P-branch transitions exhibit only minor decreases. Comparisons with previous calculations show good agreement for the delta v = +1 transitions. The comparison for delta v = +2 and +3 transitions show differences as large as a factor of 5.

Goorvitch, D.↗