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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.

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At least 109 records · Page 6
All sourcesNASA NTRSDOE OSTI
DOE OSTI2025

OpenSn: A massively parallel, open-source simulation environment for discrete ordinates radiation transport

OpenSn is an open-source, massively parallel deterministic radiation transport code for solving the discrete-ordinates ( S N ) form of the Boltzmann transport equation on unstructured, arbitrary polyhedral meshes. It supports high-fidelity simulations involving steady-state, eigenvalue, and adjoint problems for neutral particles (e.g., neutrons, photons, multi-particles), using the multigroup approximation in energy. OpenSn combines angular discretization via discrete ordinates with a discontinuous Galerkin finite element method (DGFEM) in space, enabling accurate resolution of transport physics on arbitrary polyhedral cells, included locally refined spatial grids. It includes multiple angular quadrature types, including locally refined angular quadratures. Written in modern C++ with a Python API, OpenSn runs efficiently on platforms ranging from laptops to supercomputers. The transport sweep algorithm is implemented using a task-based, directed-acyclic-graph (DAG) approach for each angle and supports asynchronous parallelism across thousands of MPI ranks. Group-set aggregation improves compute intensity, and synthetic acceleration techniques (e.g., diffusion synthetic acceleration, second-moment method) enhance solver convergence. OpenSn has been verified on reactor physics problems and demonstrated excellent weak and strong scaling performance on more than 32,768 processes, making it a versatile and robust platform for large-scale transport simulations in complex geometries.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
NASA NTRS1989

Numerical solution of large Lyapunov equations

A few methods are proposed for solving large Lyapunov equations that arise in control problems. The common case where the right hand side is a small rank matrix is considered. For the single input case, i.e., when the equation considered is of the form AX + XA(sup T) + bb(sup T) = 0, where b is a column vector, the existence of approximate solutions of the form X = VGV(sup T) where V is N x m and G is m x m, with m small is established. The first class of methods proposed is based on the use of numerical quadrature formulas, such as Gauss-Laguerre formulas, applied to the controllability Grammian. The second is based on a projection process of Galerkin type. Numerical experiments are presented to test the effectiveness of these methods for large problems.

Saad, Youcef↗
NASA NTRS1990

Finite-volume application of high order ENO schemes to multi-dimensional boundary-value problems

The finite volume approach in developing multi-dimensional, high-order accurate essentially non-oscillatory (ENO) schemes is considered. In particular, a two dimensional extension is proposed for the Euler equation of gas dynamics. This requires a spatial reconstruction operator that attains formal high order of accuracy in two dimensions by taking account of cross gradients. Given a set of cell averages in two spatial variables, polynomial interpolation of a two dimensional primitive function is employed in order to extract high-order pointwise values on cell interfaces. These points are appropriately chosen so that correspondingly high-order flux integrals are obtained through each interface by quadrature, at each point having calculated a flux contribution in an upwind fashion. The solution-in-the-small of Riemann's initial value problem (IVP) that is required for this pointwise flux computation is achieved using Roe's approximate Riemann solver. Issues to be considered in this two dimensional extension include the implementation of boundary conditions and application to general curvilinear coordinates. Results of numerical experiments are presented for qualitative and quantitative examination. These results contain the first successful application of ENO schemes to boundary value problems with solid walls.

Casper, Jay↗
DOE OSTI2021

A multi-center quadrature scheme for the molecular continuum

We report a common way to evaluate electronic integrals for polyatomic molecules is to use Becke’s partitioning scheme (Becke and Chem, 1988) in conjunction with overlapping grids centered at each atomic site. The Becke scheme was designed for integrands that fall off rapidly at large distances, such as those approximating bound electronic states. When applied to states in the electronic continuum, however, Becke scheme exhibits slow convergence and it is highly redundant. Here, we present a modified version of Becke scheme that is applicable to functions of the electronic continuum, such as those involved in molecular photoionization and electron–molecule scattering, and which ensures convergence and efficiency comparable to those realized in the calculation of bound states. In this modified scheme, the atomic weights already present in Becke’s partition are smoothly switched off within a range of few bond lengths from their respective nuclei, and complemented by an asymptotically unitary weight. The atomic integrals are evaluated on small spherical grids, centered on each atom, with size commensurate to the support of the corresponding atomic weight. The residual integral of the interstitial and long-range region is evaluated with a central master grid. The accuracy of the method is demonstrated by evaluating integrals involving integrands containing Gaussian Type Orbitals and Yukawa potentials, on the atomic sites, as well as spherical Bessel functions centered on the master grid. These functions are representative of those encountered in realistic electron-scattering and photoionization calculations in polyatomic molecules.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2022

Equatorial Origin of the Observed Tropical Pacific Quasi-Decadal Variability From ENSO Nonlinearity

Quasi-decadal (QD) climate variability is detected in the tropical Pacific based on the recent 70 years of observations. This QD variability is identified in equatorial sea surface temperatures (SSTs), the pattern of which resembles the Central Pacific (CP) El Niño-Southern Oscillation (ENSO) but extends further meridionally to the northeastern subtropical Pacific. Whereas equatorial upper-ocean heat content and SSTs are in quadrature for ENSO, these two quantities are almost in phase on the QD timescale. Further analysis shows that nonlinear dynamical heating, primarily originating from strong El Niño events, tends to lead QD SSTs by a quarter of its dominant period (approximately 30 months) and shapes the dominant QD periodicity in observations. Here, our results suggest that the observed QD variability largely originates from ENSO nonlinearity and thus is fundamentally different from ENSO's oscillatory nature.

54 ENVIRONMENTAL SCIENCES↗
NASA NTRS2002

Calculation of Scattering Amplitude Without Partial Analysis: Inclusion of Exchange - II

There was a method for calculating the whole scattering amplitude, f(Omega(sub k)), directly. The idea was to calculate the complete wave function Psi numerically, and use it in an integral expression for f, which can be reduced to a 2 dimensional quadrature. The original application was for e-H scattering without exchange. There the Schrodinger reduces a 2-d partial differential equation (pde), which was solved using the finite element method (FEM). Here we extend the method to the exchange approximation. The S.E. can be reduced to a pair of coupled pde's, which are again solved by the FEM. The formal expression for f(Omega(sub k)) consists two integrals, f+/- = f(sub d) +/- f(sub e); f(sub d) is formally the same integral as the no-exchange f. We have also succeeded in reducing f(sub e) to a 2-d integral. Results will be presented at the meeting.

Temkin, Aaron↗
NASA NTRS1995

A conservative staggered-grid Chebyshev multidomain method for compressible flows

We present a new multidomain spectral collocation method that uses staggered grids for the solution of compressible flow problems. The solution unknowns are defined at the nodes of a Gauss quadrature rule. The fluxes are evaluated at the nodes of a Gauss-Lobatto rule. The method is conservative, free-stream preserving, and exponentially accurate. A significant advantage of the method is that subdomain corners are not included in the approximation, making solutions in complex geometries easier to compute.

Kopriva, David A.↗
DOE OSTI2025

Selective mass scaling for single-layer thick shell elements in DYNA3D

Hexahedral elements can be adapted to model thin and moderately thick structures by neglecting the coupling of through-thickness stress, resulting in a fully three-dimensional, but simplified, state of stress. These specialized elements, often referred to as “thick” or “solid” shells, are generally employed to model thin-walled structures using continuum mechanics-based material models. In explicit dynamics simulations, where computational speed is important, these elements are integrated with a single quadrature point and a set of anti-hourglassing (stabilizing) forces. Thick shells, by definition, have a thickness dimension smaller than their in-plane dimensions, and this small thickness often determines the stable time step size in simulations, despite the mechanics being approximated. To alleviate this limitation while retaining the relevant dynamics of thin-walled structures, selective mass scaling (SMS), or selective mass “augmentation,” has been proposed in the literature. In this technical report, we explore the application of SMS to single-layer thick shells in the simulation software DYNA3D.

42 ENGINEERING↗
NASA NTRS1995

Effect of various approximations on predicted progressive failure in plain weave composites

Three-dimensional finite element analysis was used to simulate progressive failure of a plain weave composite subjected to in-plane extension. The loading was parallel to one of the tow directions. The effects of various characteristics of the finite element model on predicted behavior were examined. The predicted behavior was found to be sensitive to quadrature order, mesh refinement, and the material degradation model. Also the sensitivity of the predictions to the tow waviness was studied. The predicted strength decreased considerably with increased waviness. More numerical studies and comparisons with experimental data are needed to establish reliable guidelines for accurate progressive failure prediction.

Whitcomb, John↗
NASA NTRS1981

Spherical-earth Gravity and Magnetic Anomaly Modeling by Gauss-legendre Quadrature Integration

The usefulness of long-wavelength potential field anomalies in lithospheric interpretation is greatly increased with spherical Earth modeling techniques. Gauss-Legendre quadrature integration is used to calculate the anomalous potential of gravity and magnetic fields and their spatial derivatives on a spherical Earth for an arbitrary body represented by an equivalent point source distribution of gravity poles or magnetic dipoles. The distribution of equivalent point sources is determined directly from the coordinate limits of the source volume. Variable integration limits for an arbitrarily shaped body are derived from interpolation of points which approximate the body's surface envelope. The versatility of the method is enhanced by the ability to treat physical property variations within the source volume and to consider variable magnetic fields over the source and observation surface. A number of examples verify and illustrate the capabilities of the technique, including preliminary modeling of potential field signatures for Mississippi embayment crustal structure at satellite elevations.

Spherical coordinate gravity modeling↗
NASA NTRS1975

Finite element computation of a viscous compressible free shear flow governed by the time dependent Navier-Stokes equations

A finite element algorithm for solution of fluid flow problems characterized by the two-dimensional compressible Navier-Stokes equations was developed. The program is intended for viscous compressible high speed flow; hence, primitive variables are utilized. The physical solution was approximated by trial functions which at a fixed time are piecewise cubic on triangular elements. The Galerkin technique was employed to determine the finite-element model equations. A leapfrog time integration is used for marching asymptotically from initial to steady state, with iterated integrals evaluated by numerical quadratures. The nonsymmetric linear systems of equations governing time transition from step-to-step are solved using a rather economical block iterative triangular decomposition scheme. The concept was applied to the numerical computation of a free shear flow. Numerical results of the finite-element method are in excellent agreement with those obtained from a finite difference solution of the same problem.

Cooke, C. H.↗
NASA NTRS1988

Leapfrog variants of iterative methods for linear algebra equations

Two iterative methods are considered, Richardson's method and a general second order method. For both methods, a variant of the method is derived for which only even numbered iterates are computed. The variant is called a leapfrog method. Comparisons between the conventional form of the methods and the leapfrog form are made under the assumption that the number of unknowns is large. In the case of Richardson's method, it is possible to express the final iterate in terms of only the initial approximation, a variant of the iteration called the grand-leap method. In the case of the grand-leap variant, a set of parameters is required. An algorithm is presented to compute these parameters that is related to algorithms to compute the weights and abscissas for Gaussian quadrature. General algorithms to implement the leapfrog and grand-leap methods are presented. Algorithms for the important special case of the Chebyshev method are also given.

Saylor, Paul E.↗
NASA NTRS2001

Comparison of Implicit Collocation Methods for the Heat Equation

We combine a high-order compact finite difference scheme to approximate spatial derivatives arid collocation techniques for the time component to numerically solve the two dimensional heat equation. We use two approaches to implement the collocation methods. The first one is based on an explicit computation of the coefficients of polynomials and the second one relies on differential quadrature. We compare them by studying their merits and analyzing their numerical performance. All our computations, based on parallel algorithms, are carried out on the CRAY SV1.

Kouatchou, Jules↗
NASA NTRS2000

P-code enhanced method for processing encrypted GPS signals without knowledge of the encryption code

In the preferred embodiment, an encrypted GPS signal is down-converted from RF to baseband to generate two quadrature components for each RF signal (L1 and L2). Separately and independently for each RF signal and each quadrature component, the four down-converted signals are counter-rotated with a respective model phase, correlated with a respective model P code, and then successively summed and dumped over presum intervals substantially coincident with chips of the respective encryption code. Without knowledge of the encryption-code signs, the effect of encryption-code sign flips is then substantially reduced by selected combinations of the resulting presums between associated quadrature components for each RF signal, separately and independently for the L1 and L2 signals. The resulting combined presums are then summed and dumped over longer intervals and further processed to extract amplitude, phase and delay for each RF signal. Precision of the resulting phase and delay values is approximately four times better than that obtained from straight cross-correlation of L1 and L2. This improved method provides the following options: separate and independent tracking of the L1-Y and L2-Y channels; separate and independent measurement of amplitude, phase and delay L1-Y channel; and removal of the half-cycle ambiguity in L1-Y and L2-Y carrier phase.

Meehan, Thomas K.↗
DOE OSTI2021

Approximation of periodic Green's operator in real space using numerical integration and its use in fast Fourier transform-based micromechanical models

In this work, we propose an expression for the periodic first derivative of Green's function in real space. The proposed expression allows an alternative way of computing the periodic Green's operator based on periodically summing the free-space Green's operator in terms of an appropriate quadrature rule. We provide computational examples, which show the accuracy of the proposed approach, together with reduced spurious oscillations in the solution fields.

42 ENGINEERING↗
NASA NTRS1974

A method for the measurement and the statistical analysis of atmospheric turbulence

The instantaneous values of output voltages representing the wind velocity vector and the temperature at different elevations of the 250-foot meteorological tower located at NASA Wallops Flight Center are provided with the three dimensional split-film TSI Model 1080 anemometer system. The output voltages are sampled at a rate of one every 5 milliseconds, digitized and stored on digital magnetic tapes for a time period of approximately 40 minutes, with the use of a specially designed data acqusition system. A new calibration procedure permits the conversion of the digital voltages to the respective values of the temperature and the velocity components in a Cartesian coordinate system connected with the TSI probe with considerable accuracy. Power, cross, coincidence and quadrature spectra of the wind components and the temperature are obtained with the use of the fast Fourier transform. The cosine taper data window and ensemble and frequency smoothing techniques are used to provide smooth estimates of the spectral functions.

Tieleman, H. W.↗
NASA NTRS1993

A Fourier-Riccati approach to radiative transfer. I - Foundations

The three-dimensional equation of radiative transfer is formally solved using a Fourier-Riccati approach while calculations are performed on cloudy media embedded in a two-dimensional space. An extension to Stephens' work this study addresses the coupling between space and angle asserted by the equation of transfer. In particular, the accuracy of the computed radiation field as it is influenced by the angular resolution of the phase function and spatial discretization of the cloudy medium is discussed. The necessity of using a large number of quadrature points to calculate fluxes even when the phase function is isotropic for media exhibiting vertical and horizontal inhomogeneities is demonstrated. Effects of incorrect spatial sampling on both radiance and flux fields are also quantified by example. Radiance and flux comparisons obtained by the Fourier-Riccati model and the independent pixel approximation for inhomogeneous cloudy media illustrate the inadequacy of the latter even for tenuous clouds.

Gabriel, P. M.↗