Search NASA⌕ Search

SEARCH · Search NASA

Results for “QUADRATURE APPROXIMATION”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 55 records · Page 3
All sourcesNASA NTRSDOE OSTI
NASA NTRS1993

Shear and compression buckling analysis for anisotropic panels with centrally located elliptical cutouts

An approximate analysis for buckling of biaxial- and shear-loaded anisotropic panels with centrally located elliptical cutouts is presented in the present paper. The analysis is composed of two parts, a prebuckling analysis and a buckling analysis. The prebuckling solution is determined using Lekhnitskii's complex variable equations of plane elastostatics combined with a Laurent series approximation and a boundary collocation method. The buckling solution is obtained using the principle of minimum potential energy. A by-product of the minimum potential energy equation is an integral equation which is solved using Gaussian quadrature. Comparisons with documented experimental results and finite element analyses indicate that the approximate analysis accurately predicts the buckling loads of square biaxial- and shear-loaded panels having elliptical cutouts with major axes up to sixty percent of the panel width. Results of a parametric study are presented for shear- and compression-loaded rectangular anisotropic panels with elliptical cutouts. The effects of panel aspect ratio, cutout shape, cutout size, cutout orientation, laminate anisotropy, and combined loading on the buckling load are examined.

Britt, V. O.↗
NASA NTRS1989

Real-time approximate optimal guidance laws for the advanced launch system

An approach to optimal ascent guidance for a launch vehicle is developed using an expansion technique. The problem is to maximize the payload put into orbit subject to the equations of motion of a rocket over a rotating spherical earth. It is assumed that the thrust and gravitational forces dominate over the aerodynamic forces. It is shown that these forces can be separated by a small parameter epsilon, where epsilon is the ratio of the atmospheric scale height to the radius of the earth. The Hamilton-Jacobi-Bellman or dynamic programming equation is expanded in a series where the zeroth-order term (epsilon = 0) can be obtained in closed form. The zeroth-order problem is that of putting maximum payload into orbit subject to the equations of motion of a rocket in a vacuum over a flat earth. The neglected inertial and aerodynamic terms are included in higher order terms of the expansion, which are determined from the solution of first-order linear partial differential equations requiring only quadrature integrations. These quadrature integrations can be performed rapidly, so that real-time approximate optimization can be used to construct the launch guidance law.

Speyer, Jason L.↗
NASA NTRS2005

Performance Analysis of Direct-Sequence Code-Division Multiple-Access Communications with Asymmetric Quadrature Phase-Shift-Keying Modulation

This article considers a quaternary direct-sequence code-division multiple-access (DS-CDMA) communication system with asymmetric quadrature phase-shift-keying (AQPSK) modulation for unequal error protection (UEP) capability. Both time synchronous and asynchronous cases are investigated. An expression for the probability distribution of the multiple-access interference is derived. The exact bit-error performance and the approximate performance using a Gaussian approximation and random signature sequences are evaluated by extending the techniques used for uniform quadrature phase-shift-keying (QPSK) and binary phase-shift-keying (BPSK) DS-CDMA systems. Finally, a general system model with unequal user power and the near-far problem is considered and analyzed. The results show that, for a system with UEP capability, the less protected data bits are more sensitive to the near-far effect that occurs in a multiple-access environment than are the more protected bits.

Wang, C.-W.↗
NASA NTRS1977

Observations on approximate integrations

A class of integration strategies is explored that fall in between the two extremes of symbolic integration and numerical quadrature, which are, respectively, aimed at the computer generation of answers in the form of exact expressions and numerical values. The theoretical advances in symbolic integration are discussed. Three major contexts of applications with attendant case studies, four possible types of strategies for approximate integration are examined. The feasibility and adequacy (or inadequancy) of MACSYMA for implementing these strategies are considered.

Ng, E. W.↗
NASA NTRS1980

The sensitivity of a very long baseline interferometer

The theoretical sensitivity of various methods of acquiring and processing interferometer data are compared. It is shown that for a fixed digital recording capacity one bit quantization of single sideband data filtered with a rectangular bandpass and sampled at the Nyquist rate yields the optimum signal to noise ratio. The losses which result from imperfect bandpass, poor image rejection, approximate methods of fringe rotation, fractional bit correction, and loss of quadrature are discussed. Also discussed is the use of the complex delay function as a maximum likelihood fringe estimator.

Rogers, A. E. E.↗
NASA NTRS1989

Aeronautical mobile TDMA/MCTDMA system

A multiple carrier Time Division Multiple Access (TDMA) system capable of supporting voice, stream data, and packet data traffic between aircraft and ground terminals is presented. Demand assignment permits efficient resource sharing for voice and stream data. The bandwidth efficiency of uncoded A Quadrature Phase Shift Keying (AQPSK) is 1 bps/Hz. High time efficiency (approximately equals 83 percent) is achieved through the use of symbol synchronous TDMA. Demodulation is achieved without loop hang-up while requiring only a 16 symbol preamble each burst. A concatenated coding system provides reliable transmission under multipath conditions.

Magill, Thomas↗
NASA NTRS2002

Structure Corrections in Modeling VLBI Delays for RDV Data

Since 1997, bimonthly S- and X-band observing sessions have been carried out employing the VLBA (Very Long Baseline Array) and as many as ten additional antennas. Maps of the extended structures have been generated for the 160 sources observed in ten of these experiments (approximately 200,000 observations) taking place during 1997 and 1998. This paper reports the results of the first massive application of such structure maps to correct the modeled VLBI (Very Long Baseline Interferometry) delay in astrometric data analysis. For high-accuracy celestial reference frame work, proper choice of a reference point within each extended source is crucial. Here the reference point is taken at the point of maximum emitted flux. Overall, the weighted delay residuals (approximately equal to 30 ps) are reduced by 8 ps in quadrature upon introducing source maps to model the structure delays of the sources. Residuals of some sources with extended or fast-varying structures improve by as much as 40 ps. Scatter of 'arc positions' about a time-linear model decreases substantially for most sources. Based on our results, it is also concluded that source structure is presently not the dominant error source in astrometric/geodetic VLBI.

Sovers, Ojars J.↗
NASA NTRS1988

Focal region fields of distorted reflectors

The problem of the focal region fields scattered by an arbitrary surface reflector under uniform plane wave illumination is solved. The physical optics (PO) approximation is used to calculate the current induced on the reflector. The surface of the reflector is described by a number of triangular domain-wise 5th degree bivariate polynomials. A 2-dimensional Gaussian quadrature is employed to numerically evaluate the integral expressions of the scattered fields. No Freshnel or Fraunhofer zone approximations are made. The relation of the focal fields problem to surface compensation techniques and other applications are mentioned. Several examples of distorted parabolic reflectors are presented. The computer code developed is included, together with instructions on its usage.

Buris, N. E.↗
DOE OSTI2024

Spectral scheme for atomic structure calculations in density functional theory

In this study, we present a spectral scheme for atomic structure calculations in pseudopotential Kohn-Sham density functional theory. In particular, after applying an exponential transformation of the radial coordinates, we employ global polynomial interpolation on a Chebyshev grid, with derivative operators approximated using the Chebyshev differentiation matrix, and integrations using Clenshaw-Curtis quadrature. We demonstrate the accuracy and efficiency of the scheme through spin-polarized and unpolarized calculations for representative atoms, while considering local, semilocal, and hybrid exchange-correlation functionals. In particular, we find that $\mathcal{O}$(200) grid points are sufficient to achieve an accuracy of 1 microhartree in the eigenvalues for optimized norm conserving Vanderbilt pseudopotentials spanning the periodic table from atomic number Ζ = 1 to 83.

74 ATOMIC AND MOLECULAR PHYSICS↗
NASA NTRSDate not supplied

Discontinuous Galerkin and Related Methods for ODE

A defining feature of the discontinuous Galerkin (DG) method for ODE is that the piecewise polynomial solution can have a jump discontinuity at the beginning of each step. Starting from the standard integral formulation, the DG method is derived here in differential form. The key ingredient is a polynomial called the correction function, which helps ‘correct’ the discontinuous solution by approximating the jump and yields a continuous one. Under the right Radau quadrature, this continuous solution is identical to the solutions by the right Radau collocation and the continuous Galerkin (CG) methods. Next, the correction function facilitates the construction of the associated implicit Runge-Kutta schemes (IRK-DG). Different quadratures for DG result in different IRK-DG methods: left Radau quadrature in Radau IA, right Radau quadrature in Radau IIA or right Radau collocation, and Gauss quadrature in a method called DG-Gauss. The construction of IRK-DG clarifies the meaning and facilitates the proofs of various 𝐵(𝑝), 𝐶(𝜂), and 𝐷(𝜁) conditions for accuracy. The two consequences of these conditions are that all 𝑠-stage IRK-DG methods are accurate to order 2𝑠 − 1, and the IRK-DG methods of Radau type are unique. Numerical examples showing the behavior of the DG solutions are provided. In all, the correction function plays a key role and helps establish the relations among the DG, IRK-DG, collocation, and CG schemes.

numerical methods↗
NASA NTRSDate not supplied

Discontinuous Galerkin and Related Methods for ODE

Starting from the standard integral formulation, the DG method is derived here in differential form. The key ingredient is a polynomial called the correction function, which helps ‘correct’ the discontinuous solution by approximating the jump and yields a continuous one. Under the right Radau quadrature, this continuous solution is identical to the solutions by the right Radau collocation and the continuous Galerkin (CG) methods. Next, the correction function facilitates the construction of the associated implicit Runge-Kutta schemes (IRK-DG). Different quadratures for DG result in different IRK-DG methods: left Radau quadrature in Radau IA, right Radau quadrature in Radau IIA or right Radau collocation, and Gauss quadrature in a method called DG-Gauss. The construction of IRK-DG clarifies the meaning and facilitates the proofs of various 𝐵(𝑝), 𝐶(𝜂), and 𝐷(𝜁) conditions for accuracy. The two consequences of these conditions are that all 𝑠-stage IRK-DG methods are accurate to order 2𝑠− 1, and the IRK-DG methods of Radau type are unique. Numerical examples showing the behavior of the DG solutions are provided. In all, the correction function plays a key role and helps establish the relations among the DG, IRK DG, collocation, and CG methods.

Numerical Methods for Ordinary Differential Equati↗
NASA NTRS1990

A real-time approximate optimal guidance law for flight in a plane

A real-time guidance scheme is presented for the problem of maximizing the payload into orbit subject to the equations of motion of a rocket over a nonrotating spherical earth. The flight is constrained to a path in the equatorial plane while reaching an orbital altitude at orbital injection speeds. The dynamics of the problem can be separated into primary and perturbation effects by a small parameter, epsilon, which is the ratio of the atmospheric scale height to the radius of the earth. The Hamilton-Jacobi-Bellman or dynamic programming equation is expanded in an asymptotic series where the zeroth-order term (epsilon = 0) can be obtained in closed form. The neglected perturbation terms are included in the higher-order terms of the expansion, which are determined from the solution of first-order linear partial differential equations requiring only integrations which are quadratures. The quadratures can be performed rapidly with emerging computer capability, so that real-time approximate optimization can be used to construct the launch guidance law. The application of this technique to flight in three-dimensions is made apparent from the solution presented.

Feeley, Timothy S.↗
NASA NTRS1982

Conversion and matched filter approximations for serial minimum-shift keyed modulation

Serial minimum-shift keyed (MSK) modulation, a technique for generating and detecting MSK using series filtering, is ideally suited for high data rate applications provided the required conversion and matched filters can be closely approximated. Low-pass implementations of these filters as parallel inphase- and quadrature-mixer structures are characterized in this paper in terms of signal-to-noise ratio (SNR) degradation from ideal and envelope deviation. Several hardware implementation techniques utilizing microwave devices or lumped elements are presented. Optimization of parameter values results in realizations whose SNR degradation is less than 0.5 dB at error probabilities of .000001.

Ziemer, R. E.↗
NASA NTRS1979

Precession of the epsilon ring of Uranus

It is noted that the outer and inner boundaries of the epsilon ring of Uranus can be fitted by aligned Keplerian ellipses. Four possible mechanisms for maintaining uniform precession in the epsilon ring are considered: the ring's self-gravity, precession due to a satellite, smooth pressure gradients, and shocklike phenomena. It is proposed that apse alignment is maintained by the self-gravity of the ring. In this case, a ring mass of approximately 5 x 10 to the 18th g and a mean surface density at quadrature of about 25 g/sq cm are estimated.

Goldreich, P.↗
NASA NTRS1995

Solution of acoustic workshop problems by a spectral multidomain method

We use a new staggered grid Chebyshev spectral multidomain method to solve three of the Workshop benchmark problems. The method defines solution unknowns at the nodes of the Chebyshev Gauss quadrature, and the fluxes at the nodes of the Chebyshev Gauss-Lobatto quadrature. The Chebyshev spectral method gives exponentially convergent phase and dissipation errors. The multidomain approximation gives the method flexibility. Using the method, we solve problems in Categories 1 and 5 of the benchmark problems.

Kopriva, Davis A.↗
NASA NTRS1999

Front end for GPS receivers

The front end in GPS receivers has the functions of amplifying, down-converting, filtering and sampling the received signals. In the preferred embodiment, only two operations, A/D conversion and a sum, bring the signal from RF to filtered quadrature baseband samples. After amplification and filtering at RF, the L1 and L2 signals are each sampled at RF at a high selected subharmonic rate. The subharmonic sample rates are approximately 900 MHz for L1 and 982 MHz for L2. With the selected subharmonic sampling, the A/D conversion effectively down-converts the signal from RF to quadrature components at baseband. The resulting sample streams for L1 and L2 are each reduced to a lower rate with a digital filter, which becomes a straight sum in the simplest embodiment. The frequency subsystem can be very simple, only requiring the generation of a single reference frequency (e.g. 20.46 MHz minus a small offset) and the simple multiplication of this reference up to the subharmonic sample rates for L1 and L2. The small offset in the reference frequency serves the dual purpose of providing an advantageous offset in the down-converted carrier frequency and in the final baseband sample rate.

Thomas, Jr., Jess Brooks↗
NASA NTRS1984

The uniform asymptotic swallowtail approximation - Practical methods for oscillating integrals with four coalescing saddle points

Methods that can be used in the numerical implementation of the uniform swallowtail approximation are described. An explicit expression for that approximation is presented to the lowest order, showing that there are three problems which must be overcome in practice before the approximation can be applied to any given problem. It is shown that a recently developed quadrature method can be used for the accurate numerical evaluation of the swallowtail canonical integral and its partial derivatives. Isometric plots of these are presented to illustrate some of their properties. The problem of obtaining the arguments of the swallowtail integral from an analytical function of its argument is considered, describing two methods of solving this problem. The asymptotic evaluation of the butterfly canonical integral is addressed.

Connor, J. N. L.↗
NASA NTRS1992

Design of an essentially non-oscillatory reconstruction procedure in finite-element type meshes

An essentially non oscillatory reconstruction for functions defined on finite element type meshes is designed. Two related problems are studied: the interpolation of possibly unsmooth multivariate functions on arbitary meshes and the reconstruction of a function from its averages in the control volumes surrounding the nodes of the mesh. Concerning the first problem, the behavior of the highest coefficients of two polynomial interpolations of a function that may admit discontinuities of locally regular curves is studied: the Lagrange interpolation and an approximation such that the mean of the polynomial on any control volume is equal to that of the function to be approximated. This enables the best stencil for the approximation to be chosen. The choice of the smallest possible number of stencils is addressed. Concerning the reconstruction problem, two methods were studied: one based on an adaptation of the so called reconstruction via deconvolution method to irregular meshes and one that lies on the approximation on the mean as defined above. The first method is conservative up to a quadrature formula and the second one is exactly conservative. The two methods have the expected order of accuracy, but the second one is much less expensive than the first one. Some numerical examples are given which demonstrate the efficiency of the reconstruction.

Abgrall, Remi↗