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NASA NTRS1993

Two-dimensional radiative transfer in cloudy atmospheres - The spherical harmonic spatial grid method

A new two-dimensional monochromatic method that computes the transfer of solar or thermal radiation through atmospheres with arbitrary optical properties is described. The model discretizes the radiative transfer equation by expanding the angular part of the radiance field in a spherical harmonic series and representing the spatial part with a discrete grid. The resulting sparse coupled system of equations is solved iteratively with the conjugate gradient method. A Monte Carlo model is used for extensive verification of outgoing flux and radiance values from both smooth and highly variable (multifractal) media. The spherical harmonic expansion naturally allows for different levels of approximation, but tests show that the 2D equivalent of the two-stream approximation is poor at approximating variations in the outgoing flux. The model developed here is shown to be highly efficient so that media with tens of thousands of grid points can be computed in minutes. The large improvement in efficiency will permit quick, accurate radiative transfer calculations of realistic cloud fields and improve our understanding of the effect of inhomogeneity on radiative transfer in cloudy atmospheres.

Evans, K. F.↗
NASA NTRS2017

Introduction: Photons and Ground-Based

A Conference on Measurement Techniques for Solar and Space Physics was held on 20-24 April 2015 in Boulder, Colorado, at the National Center for Atmospheric Research Center Green Campus. The present volume collects together the conference papers for photons and ground-based categories. This gathering of over 200 scientists and instrumentalists was born out of the desire to collect in one place the latest experiment and instrument technologies required for advancement of scientific knowledge in the disciplines of solar and space physics. The two goals for this conference and the subsequent publication of its content are (a) to describe measurement techniques and technology development needed to advance high priority science in the fields of solar and space physics; and (b) to provide a survey or reference of techniques for in situ measurement and remote sensing of space plasmas. Towards this end, our goal has always been inspired by the two 1998 Geophysical Monographs (Nos. 102 and 103) entitled, "Measurement Techniques in Space Plasmas" (particles and fields) [Pfaff et al., 1998a, 1998b], which have served as a reference and resource for advanced students, engineers, and scientists who wish to learn the fundamentals of measurement techniques and technology in this field. Those monographs were the product of an American Geophysical Union Chapman Conference that took place in Santa Fe, NM, in 1995: "Measurement Techniques in Space Plasmas-What Works, What Doesn't." Two decades later, we believe that it is appropriate to revisit this subject, in light of recent advances in technology, research platforms, and analysis techniques. Moreover, we now include direct measurements of neutral gases in the upper atmosphere, optical imaging techniques, and remote observations in space and on the ground. Accordingly, the workshop was organized among four areas of measurement techniques: particles, fields, photons, and ground-based. This two-set volume is largely composed of the content of that workshop. Special attention is given to those techniques and technologies that demonstrate promise of significant advancement in measurements that will enable the highest priority science as described in the 2012 National Research Council Decadal Survey [Baker and Zurbuchen et al., 2013]. Additionally, a broad tutorial survey of the current technologies is provided to serve as reference material and as a basis from which advanced and innovative ideas can be discussed and pursued. Included are instrumentation and techniques to observe the solar environment from its interior to its outer atmosphere, the heliosphere out to the interstellar regions, in geospace, and other planetary magnetospheres and atmospheres. To make significant progress in priority science as expressed in the National Research Council solar and space physics decadal survey and recent NASA Heliophysics roadmaps, identification of enabling new measurement techniques and technologies to be developed is required. Also, it is valuable to the community and future scientists and engineers to have a complete survey of the techniques and technologies used by the practitioners of solar and space physics. As with the 1995 conference and subsequent 1998 publication, it is incumbent on the community to identify those measurements that are particularly challenging and still require new techniques to be identified and tested to enable the necessary accuracy and resolution of certain parameters to be achieved. The following is a partial list of the measurement technique categories that are featured in these special publications: Particles; Thermal plasma to MeV energetic particles, neutral gas properties including winds, density, temperature, and composition, and enhanced neutral atom imaging; Fields; DC electric and magnetic fields, plasma waves, and electron drift instruments from which the plasma velocity information provides a measure of the DC electric field; Photons; Instruments sensitive from the near-infrared to X-rays; Contributions of techniques and technology for optical design, optical components, sensors, material selection for cameras, telescopes, and spectrographs; Ground based; Remote sensing methods for solar and geospace activity and space weather. The focus includes solar observatories, all-sky cameras, lidars, and ionosphere thermosphere mesosphere observatory systems such as radars, ionosondes, GPS receivers, magnetometers, conjugate observations, and airborne campaigns. The present volume collects together the papers for photons and ground-based categories. The companion volume collects together the papers for particles and fields categories. It is recognized that there are measurement techniques that overlap among the four categories. For example, use of microchannel plate detectors is used in photon and particle measurement techniques or the observation of visible photons and magnetic fields in space and on the ground share common technologies. Therefore, the reader should consider the entire collection of papers as they seek to understand particular applications. We hope that these volumes will be as valuable as a reference for our community as the earlier 1998 volumes have been.

Spann, James↗
NASA NTRS1988

On conjugate gradient type methods and polynomial preconditioners for a class of complex non-Hermitian matrices

Conjugate gradient type methods are considered for the solution of large linear systems Ax = b with complex coefficient matrices of the type A = T + i(sigma)I where T is Hermitian and sigma, a real scalar. Three different conjugate gradient type approaches with iterates defined by a minimal residual property, a Galerkin type condition, and an Euclidian error minimization, respectively, are investigated. In particular, numerically stable implementations based on the ideas behind Paige and Saunder's SYMMLQ and MINRES for real symmetric matrices are proposed. Error bounds for all three methods are derived. It is shown how the special shift structure of A can be preserved by using polynomial preconditioning. Results on the optimal choice of the polynomial preconditioner are given. Also, some numerical experiments for matrices arising from finite difference approximations to the complex Helmholtz equation are reported.

Freund, Roland↗
NASA NTRS1994

Multivariable Hermite polynomials and phase-space dynamics

The phase-space approach to classical and quantum systems demands for advanced analytical tools. Such an approach characterizes the evolution of a physical system through a set of variables, reducing to the canonically conjugate variables in the classical limit. It often happens that phase-space distributions can be written in terms of quadratic forms involving the above quoted variables. A significant analytical tool to treat these problems may come from the generalized many-variables Hermite polynomials, defined on quadratic forms in R(exp n). They form an orthonormal system in many dimensions and seem the natural tool to treat the harmonic oscillator dynamics in phase-space. In this contribution we discuss the properties of these polynomials and present some applications to physical problems.

Dattoli, G.↗
NASA NTRS2000

Analysis of the DFP/AFCS Systems for Compensating Gravity Distortions on the 70-Meter Antenna

This paper presents the theoretical computations showing the expected performances for both systems. The basic analysis tool is a Physical Optics reflector analysis code that was ported to a parallel computer for faster execution times. There are several steps involved in computing the RF performance of the various systems. 1 . A model of the RF distortions of the main reflector is required. This model is based upon measured holography maps of the 70-meter antenna obtained at 3 elevation angles. The holography maps are then processed (using an appropriate gravity mechanical model of the dish) to provide surface distortion maps at all elevation angles. 2. From the surface distortion maps, ray optics is used to determine the theoretical shape of the DFP that will exactly phase compensate the distortions. 3. From the theoretical shape and a NASTRAN mechanical model of the plate, the actuator positions that generate a surface that provides the best RMS fit to the theoretical model are selected. Using the actuator positions and the NASTRAN model provides an accurate description of the actual mirror shape. 4. Starting from the mechanical drawings of the feed, a computed RF feed pattern is generated. This pattern is expanded into a set of spherical wave modes so that a complete near field analysis of the reflector system can be obtained. 5. For the array feed, the excitation coefficients that provide the maximum gain are computed using a phase conjugate technique. The basic experimental geometry consisted of a dual shaped 70-meter antenna system; a refocusing ellipse, a DFP and an array feed system. To provide physical insight to the systems performance, focal plane field plots are presented at several elevations. Curves of predicted performance are shown for the DFP system, monopulse tracking system, AFCS and combined DFP/AFCS system. The calculated results show that the combined DFP/AFCS system is capable of recovering the majority of the gain lost due to gravity distortion.

Imbriale, William A.↗
NASA NTRS1996

Multigrid Methods for Fully Implicit Oil Reservoir Simulation

In this paper we consider the simultaneous flow of oil and water in reservoir rock. This displacement process is modeled by two basic equations: the material balance or continuity equations and the equation of motion (Darcy's law). For the numerical solution of this system of nonlinear partial differential equations there are two approaches: the fully implicit or simultaneous solution method and the sequential solution method. In the sequential solution method the system of partial differential equations is manipulated to give an elliptic pressure equation and a hyperbolic (or parabolic) saturation equation. In the IMPES approach the pressure equation is first solved, using values for the saturation from the previous time level. Next the saturations are updated by some explicit time stepping method; this implies that the method is only conditionally stable. For the numerical solution of the linear, elliptic pressure equation multigrid methods have become an accepted technique. On the other hand, the fully implicit method is unconditionally stable, but it has the disadvantage that in every time step a large system of nonlinear algebraic equations has to be solved. The most time-consuming part of any fully implicit reservoir simulator is the solution of this large system of equations. Usually this is done by Newton's method. The resulting systems of linear equations are then either solved by a direct method or by some conjugate gradient type method. In this paper we consider the possibility of applying multigrid methods for the iterative solution of the systems of nonlinear equations. There are two ways of using multigrid for this job: either we use a nonlinear multigrid method or we use a linear multigrid method to deal with the linear systems that arise in Newton's method. So far only a few authors have reported on the use of multigrid methods for fully implicit simulations. Two-level FAS algorithm is presented for the black-oil equations, and linear multigrid for two-phase flow problems with strong heterogeneities and anisotropies is studied. Here we consider both possibilities. Moreover we present a novel way for constructing the coarse grid correction operator in linear multigrid algorithms. This approach has the advantage in that it preserves the sparsity pattern of the fine grid matrix and it can be extended to systems of equations in a straightforward manner. We compare the linear and nonlinear multigrid algorithms by means of a numerical experiment.

Molenaar, J.↗
NASA NTRS1992

On adaptive weighted polynomial preconditioning for Hermitian positive definite matrices

The conjugate gradient algorithm for solving Hermitian positive definite linear systems is usually combined with preconditioning in order to speed up convergence. In recent years, there has been a revival of polynomial preconditioning, motivated by the attractive features of the method on modern architectures. Standard techniques for choosing the preconditioning polynomial are based only on bounds for the extreme eigenvalues. Here a different approach is proposed, which aims at adapting the preconditioner to the eigenvalue distribution of the coefficient matrix. The technique is based on the observation that good estimates for the eigenvalue distribution can be derived after only a few steps of the Lanczos process. This information is then used to construct a weight function for a suitable Chebyshev approximation problem. The solution of this problem yields the polynomial preconditioner. In particular, we investigate the use of Bernstein-Szego weights.

Fischer, Bernd↗
NASA NTRS1985

GMRES acceleration of computational fluid dynamics codes

The generalized minimal residual algorithm (GMRES) is a conjugate-gradient like method that applies directly to nonsymmetric linear systems of equations. In this paper, GMRES is modified to handle nonlinear equations characteristic of computational fluid dynamics. Attention is devoted to the concept of preconditioning and the role it plays in assuring rapid convergence. A formulation is developed that allows GMRES to be preconditioned by the solution procedures already built into existing computer codes. Examples are provided that demonstrate the ability of GMRES to greatly improve the robustness and rate of convergence of current state-of-the-art fluid dynamics codes. Theoretical aspects of GMRES are presented that explain why it works. Finally, the advantage GMRES enjoys over related methods such as conjugate gradients are discussed.

Wigton, L. B.↗
NASA NTRS2011

Numerical Modeling of the Chilldown of Cryogenic Transfer Lines Using a Sinda/GFSSP Integrated Solver

An important first step in cryogenic propellant loading is the chilldown of transfer lines. During the chilldown of the transfer line, the flow is two-phase and unsteady, with solid to fluid heat transfer and therefore a coupled thermo-fluid analysis is necessary to model the system. This paper describes a numerical model of pipe chilldown that utilizes the Sinda/GFSSP Conjugate Integrator (SGCI). SGCI is a new analysis tool developed at NASA's Marshall Space Flight Center (MSFC). SGCI facilitates the solution of thermofluid problems in interconnected solid-fluid systems. The solid component of the system is modeled in MSC Patran and translated into an MSC Sinda thermal network model. The fluid component is modeled in GFSSP, the Generalized Fluid System Simulation Program. GFSSP is a general network flow solver developed at NASA/MSFC. GFSSP uses a finite-volume approach to model fluid systems that can include phase change, multiple species, fluid transients, and heat transfer to simple solid networks. SGCI combines the GFSSP Fortran code with the Sinda input file and compiles the integrated model. Sinda solves for the temperatures of the solid network, while GFSSP simultaneously solves the fluid network for pressure, temperature, and flow rate. The two networks are coupled by convection heat transfer from the solid wall to the cryogenic fluid. The model presented here is based on a series of experiments conducted in 1966 by the National Bureau of Standards (NBS). A vacuum-jacketed, 200 ft copper transfer line was chilled by liquid nitrogen and liquid hydrogen. The predictions of transient temperature profiles and chilldown time of the integrated Sinda/GFSSP model will be compared to the experimental measurements.

LeClair, Andre↗
NASA NTRS1983

M-step preconditioned conjugate gradient methods

Preconditioned conjugate gradient methods for solving sparse symmetric and positive finite systems of linear equations are described. Necessary and sufficient conditions are given for when these preconditioners can be used and an analysis of their effectiveness is given. Efficient computer implementations of these methods are discussed and results on the CYBER 203 and the Finite Element Machine under construction at NASA Langley Research Center are included.

Adams, L.↗
NASA NTRS1985

Objectives and design of the Echo 6 electron beam experiment - Large ionospheric perturbations and energetic particle patterns

The data from the March 30, 1983 launching of Echo 6 are presented. Echo 6 contained duplex accelerators and a system of four throw away detectors to detect and analyze conjugate electron echoes. The model used to calculate the proper trajectory of the rocket and the reason for not detecting echoes are discussed. The components of the plasma diagnostic package which was used to study the electron beam interactions with the ionospheric plasma, and the functions of the two controllable electron guns and the two sets of orthogonal electric probes are described. The signal-mode potentials between each probe and the plasma diagnostic payload are investigated. The signals produced by the gun pulses are analyzed and the interpretation of the electric field probe response is examined.

Winckler, J. R.↗
NASA NTRS1989

Efficient ICCG on a shared memory multiprocessor

Different approaches are discussed for exploiting parallelism in the ICCG (Incomplete Cholesky Conjugate Gradient) method for solving large sparse symmetric positive definite systems of equations on a shared memory parallel computer. Techniques for efficiently solving triangular systems and computing sparse matrix-vector products are explored. Three methods for scheduling the tasks in solving triangular systems are implemented on the Sequent Balance 21000. Sample problems that are representative of a large class of problems solved using iterative methods are used. We show that a static analysis to determine data dependences in the triangular solve can greatly improve its parallel efficiency. We also show that ignoring symmetry and storing the whole matrix can reduce solution time substantially.

Hammond, Steven W.↗
NASA NTRS1993

Large-scale computation of incompressible viscous flow by least-squares finite element method

The least-squares finite element method (LSFEM) based on the velocity-pressure-vorticity formulation is applied to large-scale/three-dimensional steady incompressible Navier-Stokes problems. This method can accommodate equal-order interpolations and results in symmetric, positive definite algebraic system which can be solved effectively by simple iterative methods. The first-order velocity-Bernoulli function-vorticity formulation for incompressible viscous flows is also tested. For three-dimensional cases, an additional compatibility equation, i.e., the divergence of the vorticity vector should be zero, is included to make the first-order system elliptic. The simple substitution of the Newton's method is employed to linearize the partial differential equations, the LSFEM is used to obtain discretized equations, and the system of algebraic equations is solved using the Jacobi preconditioned conjugate gradient method which avoids formation of either element or global matrices (matrix-free) to achieve high efficiency. To show the validity of this scheme for large-scale computation, we give numerical results for 2D driven cavity problem at Re = 10000 with 408 x 400 bilinear elements. The flow in a 3D cavity is calculated at Re = 100, 400, and 1,000 with 50 x 50 x 50 trilinear elements. The Taylor-Goertler-like vortices are observed for Re = 1,000.

Jiang, Bo-Nan↗
NASA NTRS1993

A least-squares finite element method for 3D incompressible Navier-Stokes equations

The least-squares finite element method (LSFEM) based on the velocity-pressure-vorticity formulation is applied to three-dimensional steady incompressible Navier-Stokes problems. This method can accommodate equal-order interpolations, and results in symmetric, positive definite algebraic system. An additional compatibility equation, i.e., the divergence of vorticity vector should be zero, is included to make the first-order system elliptic. The Newton's method is employed to linearize the partial differential equations, the LSFEM is used to obtain discretized equations, and the system of algebraic equations is solved using the Jacobi preconditioned conjugate gradient method which avoids formation of either element or global matrices (matrix-free) to achieve high efficiency. The flow in a half of 3D cubic cavity is calculated at Re = 100, 400, and 1,000 with 50 x 52 x 25 trilinear elements. The Taylor-Gortler-like vortices are observed at Re = 1,000.

Jiang, Bo-Nan↗
NASA NTRS1996

A viscoelastic higher-order beam finite element

A viscoelastic internal variable constitutive theory is applied to a higher-order elastic beam theory and finite element formulation. The behavior of the viscous material in the beam is approximately modeled as a Maxwell solid. The finite element formulation requires additional sets of nodal variables for each relaxation time constant needed by the Maxwell solid. Recent developments in modeling viscoelastic material behavior with strain variables that are conjugate to the elastic strain measures are combined with advances in modeling through-the-thickness stresses and strains in thick beams. The result is a viscous thick-beam finite element that possesses superior characteristics for transient analysis since its nodal viscous forces are not linearly dependent an the nodal velocities, which is the case when damping matrices are used. Instead, the nodal viscous forces are directly dependent on the material's relaxation spectrum and the history of the nodal variables through a differential form of the constitutive law for a Maxwell solid. The thick beam quasistatic analysis is explored herein as a first step towards developing more complex viscoelastic models for thick plates and shells, and for dynamic analyses. The internal variable constitutive theory is derived directly from the Boltzmann superposition theorem. The mechanical strains and the conjugate internal strains are shown to be related through a system of first-order, ordinary differential equations. The total time-dependent stress is the superposition of its elastic and viscous components. Equations of motion for the solid are derived from the virtual work principle using the total time-dependent stress. Numerical examples for the problems of relaxation, creep, and cyclic creep are carried out for a beam made from an orthotropic Maxwell solid.

Johnson, Arthur R.↗
NASA NTRS2001

Investigation of Conjugate Heat Transfer in Turbine Blades and Vanes

We report on work carried out to develop a 3-D coupled Finite Volume/BEM-based temperature forward/flux back (TFFB) coupling algorithm to solve the conjugate heat transfer (CHT) which arises naturally in analysis of systems exposed to a convective environment. Here, heat conduction within a structure is coupled to heat transfer to the external fluid which is convecting heat into or out of the solid structure. There are two basic approaches to solving coupled fluid structural systems. The first is a direct coupling where the solution of the different fields is solved simultaneously in one large set of equations. The second approach is a loose coupling strategy where each set of field equations is solved to provide boundary conditions for the other. The equations are solved in turn until an iterated convergence criterion is met at the fluid-solid interface. The loose coupling strategy is particularly attractive when coupling auxiliary field equations to computational fluid dynamics codes. We adopt the latter method in which the BEM is used to solve heat conduction inside a structure which is exposed to a convective field which in turn is resolved by solving the NASA Glenn compressible Navier-Stokes finite volume code Glenn-HT. The BEM code features constant and bi-linear discontinuous elements and an ILU-preconditioned GMRES iterative solver for the resulting non-symmetric algebraic set arising in the conduction solution. Interface of flux and temperature is enforced at the solid/fluid interface, and a radial-basis function scheme is used to interpolated information between the CFD and BEM surface grids. Additionally, relaxation is implemented in passing the fluxes from the conduction solution to the fluid solution. Results from a simple test example are reported.

Kassab, A. J.↗
NASA NTRS1996

Design Techniques for Beam Waveguide Systems

Various design techniques used for beam waveguide (BWG) feed systems are characterized. These include the use of gaussian beam, geometrical optics, and physical optics systems. Also introduced is a new technique based upon a conjugate phase matching focal plane technique. Advantages, disadvantages, and range of applicability of each technique is given, In addition, comparisons of computed and measured results from each type of design are presented.

beam↗
NASA NTRS1998

Comparison of Conjugate Gradient Density Matrix Search and Chebyshev Expansion Methods for Avoiding Diagonalization in Large-Scale Electronic Structure Calculations

We report a comparison of two linear-scaling methods which avoid the diagonalization bottleneck of traditional electronic structure algorithms. The Chebyshev expansion method (CEM) is implemented for carbon tight-binding calculations of large systems and its memory and timing requirements compared to those of our previously implemented conjugate gradient density matrix search (CG-DMS). Benchmark calculations are carried out on icosahedral fullerenes from C60 to C8640 and the linear scaling memory and CPU requirements of the CEM demonstrated. We show that the CPU requisites of the CEM and CG-DMS are similar for calculations with comparable accuracy.

Bates, Kevin R.↗