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At least 631 records · Page 35
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NASA NTRS1978

On least squares approximations to indefinite problems of the mixed type

A least squares method is presented for computing approximate solutions of indefinite partial differential equations of the mixed type such as those that arise in connection with transonic flutter analysis. The method retains the advantages of finite difference schemes namely simplicity and sparsity of the resulting matrix system. However, it offers some great advantages over finite difference schemes. First, the method is insensitive to the value of the forcing frequency, i.e., the resulting matrix system is always symmetric and positive definite. As a result, iterative methods may be successfully employed to solve the matrix system, thus taking full advantage of the sparsity. Furthermore, the method is insensitive to the type of the partial differential equation, i.e., the computational algorithm is the same in elliptic and hyperbolic regions. In this work the method is formulated and numerical results for model problems are presented. Some theoretical aspects of least squares approximations are also discussed.

Fix, G. J.↗
NASA NTRS1978

Heating due to the 9.6 micron ozone band in an inhomogeneous atmosphere - A new approximation

The Curtis-Godson (1953) method, when applied to the 9.6-micron ozone band, yields large errors. Kuriyan et al. (1977) suggested a method improving this result and yet retaining the analytic simplicity of the transmission function. In this paper, the analytic form of the derivative of the transmission function is obtained and then the heating rates are calculated with the new approximation. The errors in this case are negligibly small. Algebraic manipulation enables the reduction of this approximation to a form similar to that proposed by Goody (1964). This procedure permits interpretation of the boundary conditions in physical terms.

Kuriyan, J. G.↗
NASA NTRS1978

Compression of ephemerides by discrete Chebyshev approximations

The use of Chebyshev series in representing the ephemerides of satellites and planets in terms of truncated polynomial series is discussed. Emphasis is placed on a FORTRAN package which was developed for fitting satellite orbits. The features desired in any approximation are: (1) the ability to compress a satellite emphemeris; (2) the ability to represent a satellite ephemeris over several orbits; (3) guaranteed accuracy to within prescribed tolerance over the time interval of consideration; and (4) fast processing. These features are imposed with an eye towards adapting the approximation for use on microprocessor applications in which storage is limited and real time processing is required.

Pickard, H. M.↗
NASA NTRS1979

A Study of Linear Approximation Techniques for SAR Azimuth Processing

The application of the step transform subarray processing techniques to synthetic aperture radar (SAR) was studied. The subarray technique permits the application of efficient digital transform computational techniques such as the fast Fourier transform to be applied while offering an effective tool for range migration compensation. Range migration compensation is applied at the subarray level, and with the subarray size based on worst case range migration conditions, a minimum control system is achieved. A baseline processor was designed for a four-look SAR system covering approximately 4096 by 4096 SAR sample field every 2.5 seconds. Implementation of the baseline system was projected using advanced low power technologies. A 20 swath is implemented with approximately 1000 circuits having a power dissipation of from 70 to 195 watts. The baseline batch step transform processor is compared to a continuous strip processor, and variations of the baseline are developed for a wide range of SAR parameters.

Martinson, L. W.↗
NASA NTRS1978

A note on obtaining an approximate value for the period of a periodic solution of x-double prime = -V prime /x/

The first integral of x-double prime = - V prime (x) yields an integral for the period of a periodic solution, if such exists. In general, this integral cannot be evaluated. By means of an approximate solution along with the minimization of a mean-square error, one can obtain an approximate value for the period in terms of the amplitude of the motion. The calculated period agrees very well with the period obtained by means of numerical integration for the case of orbit-orbit resonance involving the motion of two satellites of a planet. The same method is applied to obtain an alternative derivation of the first Krylov-Bogoliuboff averaging method in nonlinear mechanics.

Lass, H.↗
NASA NTRS1979

Approximations for proton excitation - Erratum and extension

Kastner (1977) proposed relatively simple approximations for proton excitation in coronal ions, which yield both excitation rate coefficients and excitation cross sections. An error in Kastner's expression for one of the partial excitation rate coefficients is corrected. The expression for excitation cross section is extended to higher energies, and the resulting expression for the other partial rate coefficient is given. Results obtained with the corrected and extended total excitation rate coefficient are presented for several representative transitions in Ca XIII, Ca XV, Fe X, Fe XIII, Fe XIV, Fe XVIII, and S X. The corrected and extended approximation is expected to yield excitation rates reliably within 50%.

Kastner, S. O.↗
NASA NTRS1979

Rotational excitation of symmetric top molecules by collisions with atoms. II - Infinite order sudden approximation

The infinite order sudden (IOS) approximation is extended to rotational excitation of symmetric tops by collisions with atoms. After development of a formalism for 'primitive' or 'one-ended' tops, proper parity-adapted linear combinations describing real rotors are considered and modifications needed for asymmetric rigid rotors are noted. The generalized spectroscopic relaxation cross sections are discussed. IOS calculations for NH3-He and H2CO-He are performed and compared with more accurate calculations, and the IOS approximation is found to provide a reasonably accurate description.

Green, S.↗
NASA NTRS1979

Radiative transfer in spherical dust shells using a generalized two-stream Eddington approximation

Application of a generalized two-stream Eddington approximation to the problem of radiative transfer in extended, spherically symmetric dust shells is presented. It is assumed that the radiation field can be characterized by the mean intensity, the flux, and a positionally dependent direction cosine specifying the division into two solid-angle ranges. The direction cosine is not specified a priori and is a function of the geometry, opacity, and emissivity in the dust shell. A multiple-grain-size multiple-temperature-distribution dust shell is postulated in which isotropic and anisotropic scattering as well as absorption and thermal reemission are allowed. A program has been developed that solves for the multiple temperature distributions by applying the constraint of radiative equilibrium to each grain size, and then calculates emergent fluxes. Results of one such calculation are presented for a model dust shell having a maximum optical depth (approximately 41) in the visible, clearly showing large optical extinction and a moderate infrared excess.

Haisch, B. M.↗
NASA NTRS1978

A sequential method for spline approximation with variable knots

In this paper we describe a method for approximating a waveform by a spline. The method is quite efficient, as the data are processed sequentially. The basis of the approach is to view the approximation problem as a question of estimation of a polynomial in noise, with the possibility of abrupt changes in the highest derivative. This allows us to bring several powerful statistical signal processing tools into play. We also present some initial results on the application of our technique to the processing of electrocardiograms, where the knot locations themselves may be some of the most important pieces of diagnostic information.

Mier Muth, A. M.↗
NASA NTRS1978

Computational test of the infinite order sudden approximation for excitation of linear rigid rotors by collisions with atoms

The infinite order sudden approximation for excitation of linear rigid rotors by collisions with atom is tested by comparing integral state-to-state cross sections with accurate close coupling and coupled states results. The systems studied are HCl-Ar, HCl-He, CO-He, HCN-He, CS-H2 and OCS-H2. With the exception of diatomic hydrides (e.g., HCl) which have atypically large rotational constants the method is found to be very accurate to remarkably low collision energies. This approximation should generally be extremely useful for thermal energy collisions.

Green, S.↗
NASA NTRS1979

Approximate convective heating equations for hypersonic flows

Laminar and turbulent heating-rate equations appropriate for engineering predictions of the convective heating rates about blunt reentry spacecraft at hypersonic conditions are developed. The approximate methods are applicable to both nonreacting and reacting gas mixtures for either constant or variable-entropy edge conditions. A procedure which accounts for variable-entropy effects and is not based on mass balancing is presented. Results of the approximate heating methods are in good agreement with existing experimental results as well as boundary-layer and viscous-shock-layer solutions.

Zoby, E. V.↗
NASA NTRS1981

An approximation for inverse Laplace transforms

Programmable calculator runs simple finite-series approximation for Laplace transform inversions. Utilizing family of orthonormal functions, approximation is used for wide range of transforms, including those encountered in feedback control problems. Method works well as long as F(t) decays to zero as it approaches infinity and so is appliable to most physical systems.

Lear, W. M.↗
NASA NTRS1980

Inversion and approximation of Laplace transforms

A method of inverting Laplace transforms by using a set of orthonormal functions is reported. As a byproduct of the inversion, approximation of complicated Laplace transforms by a transform with a series of simple poles along the left half plane real axis is shown. The inversion and approximation process is simple enough to be put on a programmable hand calculator.

Lear, W. M.↗
NASA NTRS1980

An approximate factorization solution of the Navier-Stokes equations for transonic flow using body-fitted coordinates with application to NACA 64A010 airfoils

The implementation of the approximate factorization algorithm and its ability to efficiently and accurately describe transonic flow about an NACA 64A010 airfoil section is examined. The approximate factorization algorithm is developed from the nondimensional, conservative, vectorized Navier-Stokes equations expressed in curvilinear coordinates. Equations of state and transport coefficient relations appropriate to atmospheric air are appended to close the system of partial differential equations. An algebraic turbulence model is also incorporated into the equation set. This algorithm was verified by investigating the flow about an NACA 64A010 airfoil at 0, 2, and 3.5 deg angle of attack for free-stream conditions of 2,000,000 Reynolds number and 0.8 Mach number. Overall results were in good qualitative agreement with wind tunnel data sets. However, while nondimensional times of six were attained, numerical difficulties prevented any case from reaching a true steady state.

Copper, G. K.↗
NASA NTRS1979

Drift boundary approximations in simple magnetospheric convection models

The paper analyzes features of particle behavior in magnetospheric convection using simple analytic forms for the electrostatic potential. It is shown that analytic approximations for the shape and position of surfaces delimiting closed and open particle orbits can be derived. These approximations are good for any pitch angle and at most energies. They break down only for the medium-energy protons that are predicted by such models to penetrate far inside the plasmapause. The derived expressions allow rapid tests of a variety of electric field models when analyzing particle data, and this is illustrated by intercomparing field models with several observations.

Southwood, D. J.↗
NASA NTRS1980

Polynomial approximation of functions in Sobolev spaces

Constructive proofs and several generalizations of approximation results of J. H. Bramble and S. R. Hilbert are presented. Using an averaged Taylor series, we represent a function as a polynomial plus a remainder. The remainder can be manipulated in many ways to give different types of bounds. Approximation of functions in fractional order Sobolev spaces is treated as well as the usual integer order spaces and several nonstandard Sobolev-like spaces.

Dupont, T.↗
NASA NTRS1980

ACCESS 3. Approximation concepts code for efficient structural synthesis: User's guide

A user's guide is presented for ACCESS-3, a research oriented program which combines dual methods and a collection of approximation concepts to achieve excellent efficiency in structural synthesis. The finite element method is used for structural analysis and dual algorithms of mathematical programming are applied in the design optimization procedure. This program retains all of the ACCESS-2 capabilities and the data preparation formats are fully compatible. Four distinct optimizer options were added: interior point penalty function method (NEWSUMT); second order primal projection method (PRIMAL2); second order Newton-type dual method (DUAL2); and first order gradient projection-type dual method (DUAL1). A pure discrete and mixed continuous-discrete design variable capability, and zero order approximation of the stress constraints are also included.

Fleury, C.↗
NASA NTRS1980

Thin-layer approximation for three-dimensional supersonic corner flows

The thin-layer approximation is extended to an axial corner that is formed by the intersection of two perpendicular plates, one of which has an inclination angle with respect to the free stream. A computer code developed by Hung and MacCormack (1978) is modified for the thin-layer approximation, and a case with Mach 5.9 and a wedge angle of 6 deg is computed. In addition, it is shown that it is not necessary to solve the complete Navier-Stokes equations for a three-dimensional high-Reynolds-number corner flow.

Hung, C. M.↗