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NASA NTRS2014

On the Computation of High Order Rys Quadrature Weights and Nodes

Since its introduction in 1976, the Rys Quadrature method has proven a very attractive method for evaluating electron repulsion integrals for calculations using Gaussian type orbitals. Since then, there have been considerable refinements of the method, but at it's core, Gaussian weights and nodes are used to exactly evaluate using a numerical approach to the transform integral. One of the powers of the Rys Quadrature method is the relative ease in evaluating integrals involving functions of high angular momentum. In this work we report on the complete resolution of these numerical difficulties, and we have easily computed accurate quadrature weights and nodes up to order 101. All calculations were carried out using 128-bit precision.

Computation↗

DOE OSTI2021

SCEPTRE 2.2 Angular Quadrature Sets

This document includes details of the angular quadrature sets available in SCEPTRE for performing numerical integrations in the angular phase space. The angular dependence of the boundary and fixed-source terms an d initial angular flux are specified by angular index rather than by direction. It is, therefore, necessary to know the mapping from a specific direction to a direction index. This document includes angular quadrature weights and direction cosines for most of the quadrature sets available in SCEPTRE.

38 RADIATION CHEMISTRY, RADIOCHEMISTRY, AND NUCLEA↗

NASA NTRS1971

Effect of angular quadrature on results of two-dimensional space power reactor shield calculations.

Study of the effect of angular quadrature on the results of two-dimensional transport calculations made to evaluate radiation environment in the vicinity of a shielded space power reactor. The application considered is for a manned space-station; crew shielding must be provided within a 25 deg cone angle. The asymmetric shield layout is evaluated using the two-dimensional transport code DOT. Within the shield, an S sub 6 angular quadrature yields satisfactory results. For regions where the ray effect may occur it is not apparent that an S sub 10 quadrature is adequate. Anomalies in results due to transport of particles in a void and necessary geometric approximations were observed.

Connolley, D. J.↗

NASA NTRS1975

Programs for computing abscissas and weights for classical and nonclassical Gaussian quadrature formulas

Computer programs for computing Gaussian quadrature abscissas and weights are described. For the classical case the programs use Laguerre iteration to compute abscissas as zeros of orthogonal polynomials. The polynomials are evaluated from known recursion coefficients. The nonclassical case is handled similarly except that the recursion coefficients are computed by numerical integration. A sample problem, with input and output, is presented to illustrate the use of the programs. It computes the quadrature abscissas and weights associated with the weight function over the interval (0,1) for quadrature orders from 16 to 96 in increments of 8.

Desmarais, R. N.↗

NASA NTRS1975

Multidimensional Hermite-Gaussian quadrature formulae and their application to nonlinear estimation

A simplified technique is proposed for calculating multidimensional Hermite-Gaussian quadratures that involves taking the square root of a matrix by the Cholesky algorithm rather than computation of the eigenvectors of the matrix. Ways of reducing the dimension, number, and order of the quadratures are set forth. If the function f(x) under the integral sign is not well approximated by a low-order algebraic expression, the order of the quadrature may be reduced by factoring f(x) into an expression that is nearly algebraic and one that is Gaussian.

Mcreynolds, S. R.↗

NASA NTRS1981

On the power spectral density of quadrature modulated signals

The conventional (no-offset) quadriphase modulation technique suffers from the fact that hardlimiting will restore the frequency sidelobes removed by proper filtering. Thus, offset keyed quadriphase modulation techniques are often proposed for satellite communication with bandpass hardlimiting. A unified theory is developed which is capable of describing the power spectral density before and after the hardlimiting process. Using the in-phase and the quadrature phase channel with arbitrary pulse shaping, analytical results are established for generalized quadriphase modulation. In particular MSK, OPSK or the recently introduced overlapped raised cosine keying all fall into this general category. It is shown that for a linear communication channel, the power spectral density of the modulated signal remains unchanged regardless of the offset delay. Furthermore, if the in phase and the quadrature phase channel have identical pulse shapes without offset, the spectrum after bandpass hardlimiting will be identical to that of the conventional QPSK modulation. Numerical examples are given for various modulation techniques. A case of different pulse shapes in the in phase and the quadrature phase channel is also considered.

Yan, T. Y.↗

NASA NTRS1981

Coherent detection of frequency-hopped quadrature modulations in the presence of jamming. II - QPR Class I modulation

This paper considers the performance of quadrature partial response (QPR) in the presence of jamming. Although a QPR system employs a single sample detector in its receiver, while quadrature amplitude shift keying (or quadrature phase shift keying) requires a matched-filter type of receiver, it is shown that the coherent detection performances of the two in the presence of the intentional jammer have definite similarities.

Simon, M. K.↗

NASA NTRS1986

Numerical quadrature methods for integrals of singular periodic functions and their application to singular and weakly singular integral equations

High accuracy numerical quadrature methods for integrals of singular periodic functions are proposed. These methods are based on the appropriate Euler-Maclaurin expansions of trapezoidal rule approximations and their extrapolations. They are used to obtain accurate quadrature methods for the solution of singular and weakly singular Fredholm integral equations. Such periodic equations are used in the solution of planar elliptic boundary value problems, elasticity, potential theory, conformal mapping, boundary element methods, free surface flows, etc. The use of the quadrature methods is demonstrated with numerical examples.

Sidi, A.↗

NASA NTRS1988

Quadrature methods for periodic singular and weakly singular Fredholm integral equations

High-accuracy numerical quadrature methods for integrals of singular periodic functions are proposed. These methods are based on the appropriate Euler-Maclaurin expansions of trapezoidal rule approximations and their extrapolations. They are subsequently used to obtain accurate quadrature methods for the solution of singular and weakly singular Fredholm integral equations. Throughout the development the periodic nature of the problem plays a crucial role. Such periodic equations are used in the solution of planar elliptic boundary value problems such as those that arise in elasticity, potential theory, conformal mapping, and free surface flows. The use of the quadrature methods is demonstrated with numerical examples.

Sidi, Avram↗

NASA NTRS1989

Thin-plate spline quadrature of geodetic integrals

Thin-plate spline functions (known for their flexibility and fidelity in representing experimental data) are especially well-suited for the numerical integration of geodetic integrals in the area where the integration is most sensitive to the data, i.e., in the immediate vicinity of the evaluation point. Spline quadrature rules are derived for the contribution of a circular innermost zone to Stoke's formula, to the formulae of Vening Meinesz, and to the recursively evaluated operator L(n) in the analytical continuation solution of Molodensky's problem. These rules are exact for interpolating thin-plate splines. In cases where the integration data are distributed irregularly, a system of linear equations needs to be solved for the quadrature coefficients. Formulae are given for the terms appearing in these equations. In case the data are regularly distributed, the coefficients may be determined once-and-for-all. Examples are given of some fixed-point rules. With such rules successive evaluation, within a circular disk, of the terms in Molodensky's series becomes relatively easy. The spline quadrature technique presented complements other techniques such as ring integration for intermediate integration zones.

Vangysen, Herman↗

NASA NTRS1990

Optimal displacement in apparent motion and quadrature models of motion sensing

A grating appears to move if it is displaced by some amount between two brief presentations, or between multiple successive presentations. A number of recent experiments have examined the influence of displacement size upon either the sensitivity to motion, or upon the induced motion aftereffect. Several recent motion models are based upon quadrature filters that respond in opposite quadrants in the spatiotemporal frequency plane. Predictions of the quadrature model are derived for both two-frame and multiframe displays. Quadrature models generally predict an optimal displacement of 1/4 cycle for two-frame displays, but in the multiframe case the prediction depends entirely on the frame rate.

Watson, Andrew B.↗

NASA NTRS2000

An Application of the Quadrature-Free Discontinuous Galerkin Method

The process of generating a block-structured mesh with the smoothness required for high-accuracy schemes is still a time-consuming process often measured in weeks or months. Unstructured grids about complex geometries are more easily generated, and for this reason, methods using unstructured grids have gained favor for aerodynamic analyses. The discontinuous Galerkin (DG) method is a compact finite-element projection method that provides a practical framework for the development of a high-order method using unstructured grids. Higher-order accuracy is obtained by representing the solution as a high-degree polynomial whose time evolution is governed by a local Galerkin projection. The traditional implementation of the discontinuous Galerkin uses quadrature for the evaluation of the integral projections and is prohibitively expensive. Atkins and Shu introduced the quadrature-free formulation in which the integrals are evaluated a-priori and exactly for a similarity element. The approach has been demonstrated to possess the accuracy required for acoustics even in cases where the grid is not smooth. Other issues such as boundary conditions and the treatment of non-linear fluxes have also been studied in earlier work This paper describes the application of the quadrature-free discontinuous Galerkin method to a two-dimensional shear layer problem. First, a brief description of the method is given. Next, the problem is described and the solution is presented. Finally, the resources required to perform the calculations are given.

Lockard, David P.↗

NASA NTRS2002

Low-Latitude Solar Wind During the Fall 1998 SOHO-Ulysses Quadrature

Solar and Heliospheric Observatory (SOH0)-Ulysses quadratures occur when the SOHO-Sun-Ulysses-included angle is 90 deg. These offer the opportunity to directly compare properties of plasma parcels, observed by SOHO [Dorningo et al.] in the low corona, with properties of the same parcels measured, in due time, in situ, by Ulysses [ Wenzel et al]. We refer the reader to Suess et al. for an extended discussion of SOHO-Ulysses quadrature geometry. Here it suffices to recall that there are two quadratures per year, as SOHO makes its one-year revolution around the Sun. This, because SOHO is at the L1 Lagrangian point, in essentially the same place as the Earth, while Ulysses is in a near-polar -5-year solar orbit with a perihelion of 1.34 AU and aphelion of 5.4 AU.

Poletto, G.↗

NASA NTRS2006

Wide and Narrow CMEs and Their Source Explosions Observed at the Spring 2003 SOHO-Sun-Ulysses Quadrature

At the time of the spring 2003 Ulysses-SOHO-Sun quadrature, Ulysses was off the East limb of the Sun at 14.5 degrees north latitude and 4.91 AU. LASCO/C2 images show small transient events that originated from near the limb on May 25, 26 and 27 in the north-east quadrant, along with a large Coronal Mass Ejection (CME) that originated from an active region near disk center on May 26. Ulysses data bear clear signatures of the large CME, specifically including an enhanced abundance of highly ionized Fe. SOHO/UVCS spectra at 1.75 solar radii, near the radial direction to Ulysses, give no evidence of emission from high temperature lines, even for the large CME: instead, for the small events, occasional transient high emission in cool lines was observed, such as the CIII 977 Angstrom line usually absent at coronal levels. Each of these events lasted ca. 1 hour or less and never affected lines from ions forming above ca. 106K. Compact eruptions in Helium 304 Angstrom EIT images, related to the small UVCS transients, were observed at the limb of the Sun over the same period. At least one of these surge events produced a narrow CME observed in LASCO/C2. Most probably all these events are compact magnetic explosions (surges/jets, from around a small island of included polarity) which ejected cool material from lower levels. Ulysses data have been analyzed to find evidence of the cool, narrow CME events, but none or little was found. This puzzling scenario, where events seen by UVCS have no in situ counterparts and vice versa, can be partially explained once the region where the large CME originated is recognized as being at the center of the solar disk so that the CME material was actually much further from the Sun than the 1.7 Rsun height of the UVCS slit off the limb. Conversely, the narrow events may simply have missed Ulysses or been too brief for reliable signatures in composition and ionization state. A basic feature demonstrated by these observations is that large magnetic explosions produce wide-angle CMEs whereas compact magnetic explosions produce narrow CMEs. The results show that quadrature observations need some luck to be successfull: that is, events must be in the plane of the sky to allow SOHO/UVCS and Ulysses to sample the same plasma. This will most easily occur in winter 2007 and winter 2008, when the quadrature geometry will allow for prolonged observations.

Suess, Steven↗

NASA NTRS2009

Phase measurement device using inphase and quadrature components for phase estimation

A phasemeter for estimating the phase of a signal. For multi-tone signals, multiple phase estimates may be provided. An embodiment includes components operating in the digital domain, where a sampled input signal is multiplied by cosine and sine terms to provide estimates of the inphase and quadrature components. The quadrature component provides an error signal that is provided to a feedback loop, the feedback loop providing a model phase that tends to track the phase of a tone in the input signal. The cosine and sine terms are generated from the model phase. The inphase and quadrature components are used to form a residual phase, which is added to the model phase to provide an estimate of the phase of the input signal. Other embodiments are described and claimed.

Halverson, Peter G.↗

NASA NTRS2013

Testing the Empirical Shock Arrival Model Using Quadrature Observations

The empirical shock arrival (ESA) model was developed based on quadrature data from Helios (in situ) and P-78 (remote sensing) to predict the Sun-Earth travel time of coronal mass ejections (CMEs). The ESA model requires earthward CME speed as input, which is not directly measurable from coronagraphs along the Sun-Earth line. The Solar Terrestrial Relations Observatory (STEREO) and the Solar and Heliospheric Observatory (SOHO) were in quadrature during 20102012, so the speeds of Earth-directed CMEs were observed with minimal projection effects. We identified a set of 20 full halo CMEs in the field of view of SOHO that were also observed in quadrature by STEREO. We used the earthward speed from STEREO measurements as input to the ESA model and compared the resulting travel times with the observed ones from L1 monitors. We find that the model predicts the CME travel time within about 7.3 h, which is similar to the predictions by the ENLIL model. We also find that CME-CME and CME-coronal hole interaction can lead to large deviations from model predictions.

CMEs↗

NASA NTRS2021

Evaluation of Higher-order Quadrature Schemes in Improving Computational Efficiency for Orientation-averaged Single-Scattering Properties of Nonspherical Ice Particles

We evaluate several high-order quadrature schemes for accuracy and efficacy in obtaining orientation-averaged single-scattering properties (SSPs). We use the highly efficient MIDAS to perform electromagnetic scattering calculations to evaluate the gain in efficiency from these schemes. MIDAS is shown to be superior to DDSCAT, a popular discrete dipole approximation (DDA) method. This study is motivated by the fact that quality physical precipitation retrievals rely on using accurate orientation-averaged SSPs derived from realistic hydrometeors as input to radiative transfer Models (RTMs). The DDA has been a popular choice for single-scattering calculations, due to its versatility with respect to target geometry. However, being iterative-solver-based (ISB), the most used DDA codes, e.g. DDSCAT and ADDA, must solve the scattering problem for each orientation of the target separately. As the size parameter and geometric anisotropy of the hydrometeor increase, the number of orientations needed to obtain accurate orientation-averages can increase drastically and so does the computation cost incurred by the ISB-DDA methods. MIDAS is a Direct-Solver-Based (DSB) code, its decomposition of the original large matrix with a high rank into multiple more manageable smaller matrices of lower ranks makes it much more computationally efficient while maintaining excellent accuracy. In addition, direct solvers consider all requested orientations at once, giving MIDAS further advantage over popular ISB-DDA methods. MIDAS, when combined with high-order quadrature for orientation averaging, can be greater than three orders of magnitude more efficient in obtaining RTM-ready SSPs of complex-shaped hydrometeors than existing ISB-DDA methods, with the native quadrature schemes they offer.

Ines Fenni↗

DOE OSTI2022

Spectral quadrature for the first principles study of crystal defects: Application to magnesium

In this work, we present an accurate and efficient finite-difference formulation and parallel implementation of Kohn-Sham Density (Operator) Functional Theory (DFT) for non periodic systems embedded in a bulk environment. Specifically, employing non-local pseudopotentials, local reformulation of electrostatics, and truncation of the spatial Kohn-Sham Hamiltonian, and the Linear Scaling Spectral Quadrature method to solve for the pointwise electronic fields in real-space and the non-local component of the atomic force, we develop a parallel finite difference framework suitable for distributed memory computing architectures to simulate non-periodic systems embedded in a bulk environment. Choosing examples from magnesium-aluminum alloys, we first demonstrate the convergence of energies and forces with respect to spectral quadrature polynomial order, and the width of the spatially truncated Hamiltonian. Next, we demonstrate the parallel scaling of our framework, and show that the computation time and memory scale linearly with respect to the number of atoms. Next, we use the developed framework to simulate isolated point defects and their interactions in magnesium-aluminum alloys. Our findings conclude that the binding energies of divacancies, Al solute-vacancy and two Al solute atoms are anisotropic and are dependent on cell size. Furthermore, the binding is favorable in all three cases.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗