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At least 1,063 records · Page 59
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NASA NTRS1995

SeaWiFS Technical Report Series: The SeaWiFS CZCS-Type Pigment Algorithm - Volume 29

The Sea-viewing Wide Field-of-view Sensor (SeaWiFS) mission will provide operational ocean color that will be superior to the previous Coastal Zone Color Sensor (CZCS) proof-of-concept mission. An algorithm is needed that exploits the full functionality of SeaWiFS whilst remaining compatible in concept with algorithms used for the CZCS. This document describes the theoretical rationale of radiance band-ratio methods for determining chlorophyll-a and other important biogeochemical parameters, and their implementation for the SeaWIFS mission. Pigment interrelationships are examined to explain the success of the CZCS algorithms. In the context where chlorophyll-a absorbs only weakly at 520 nm, the success of the 520 nm to 550 nm CZCS band ratio needs to be explained. This is explained by showing that in pigment data from a range of oceanic provinces chlorophyll-a (absorbing at less than 490 nm), carotenoids (absorbing at greater than 460 nm), and total pigment are highly correlated. Correlations within pigment groups particularly photoprotectant and photosynthetic carotenoids are less robust. The sources of variability in optical data are examined using the NIMBUS Experiment Team (NET) bio-optical data set and bio-optical model. In both the model and NET data, the majority of the variance in the optical data is attributed to variability in pigment (chlorophyll-a), and total particulates, with less than 5% of the variability resulting from pigment assemblage. The relationships between band ratios and chlorophyll is examined analytically, and a new formulation based on a dual hyperbolic model is suggested which gives a better calibration curve than the conventional log-log linear regression fit. The new calibration curve shows the 490:555 ratio is the best single-band ratio and is the recommended CZCS-type pigment algorithm. Using both the model and NET data, a number of multiband algorithms are developed; the best of which is an algorithm based on the 443:555 and 490:555 ratios. From model data, the form of potential algorithms for other products, such as total particulates and dissolved organic matter (DOM), are suggested.

Hooker, Stanford B.↗
NASA NTRS1996

GENIE(++): A Multi-Block Structured Grid System

The computer code GENIE++ is a continuously evolving grid system containing a multitude of proven geometry/grid techniques. The generation process in GENIE++ is based on an earlier version. The process uses several techniques either separately or in combination to quickly and economically generate sculptured geometry descriptions and grids for arbitrary geometries. The computational mesh is formed by using an appropriate algebraic method. Grid clustering is accomplished with either exponential or hyperbolic tangent routines which allow the user to specify a desired point distribution. Grid smoothing can be accomplished by using an elliptic solver with proper forcing functions. B-spline and Non-Uniform Rational B-splines (NURBS) algorithms are used for surface definition and redistribution. The built in sculptured geometry definition with desired distribution of points, automatic Bezier curve/surface generation for interior boundaries/surfaces, and surface redistribution is based on NURBS. Weighted Lagrance/Hermite transfinite interpolation methods, interactive geometry/grid manipulation modules, and on-line graphical visualization of the generation process are salient features of this system which result in a significant time savings for a given geometry/grid application.

Williams, Tonya↗
NASA NTRS1996

Homotopy Solutions of Kepler's Equations

Kepler's Equation is solved using an integrative algorithm developed using homotropy theory. The solution approach is applicable to both elliptic and hyperbolic forms of Kepler's Equation. The results from the proposed algorithm compare quite favorably with those from existing iterative schemes.

Fitz-Coy, Norman↗
NASA NTRS1996

Parallel Newton-Krylov-Schwarz algorithms for the transonic full potential equation

We study parallel two-level overlapping Schwarz algorithms for solving nonlinear finite element problems, in particular, for the full potential equation of aerodynamics discretized in two dimensions with bilinear elements. The overall algorithm, Newton-Krylov-Schwarz (NKS), employs an inexact finite-difference Newton method and a Krylov space iterative method, with a two-level overlapping Schwarz method as a preconditioner. We demonstrate that NKS, combined with a density upwinding continuation strategy for problems with weak shocks, is robust and, economical for this class of mixed elliptic-hyperbolic nonlinear partial differential equations, with proper specification of several parameters. We study upwinding parameters, inner convergence tolerance, coarse grid density, subdomain overlap, and the level of fill-in in the incomplete factorization, and report their effect on numerical convergence rate, overall execution time, and parallel efficiency on a distributed-memory parallel computer.

Cai, Xiao-Chuan↗
NASA NTRS1995

Dynamical Approach Study of Spurious Steady-State Numerical Solutions of Nonlinear Differential Equations: Global Asymptotic Behavior of Time Discretizations - 2

The global asymptotic nonlinear behavior of 1 1 explicit and implicit time discretizations for four 2 x 2 systems of first-order autonomous nonlinear ordinary differential equations (ODES) is analyzed. The objectives are to gain a basic understanding of the difference in the dynamics of numerics between the scalars and systems of nonlinear autonomous ODEs and to set a baseline global asymptotic solution behavior of these schemes for practical computations in computational fluid dynamics. We show how 'numerical' basins of attraction can complement the bifurcation diagrams in gaining more detailed global asymptotic behavior of time discretizations for nonlinear differential equations (DEs). We show how in the presence of spurious asymptotes the basins of the true stable steady states can be segmented by the basins of the spurious stable and unstable asymptotes. One major consequence of this phenomenon which is not commonly known is that this spurious behavior can result in a dramatic distortion and, in most cases, a dramatic shrinkage and segmentation of the basin of attraction of the true solution for finite time steps. Such distortion, shrinkage and segmentation of the numerical basins of attraction will occur regardless of the stability of the spurious asymptotes, and will occur for unconditionally stable implicit linear multistep methods. In other words, for the same (common) steady-state solution the associated basin of attraction of the DE might be very different from the discretized counterparts and the numerical basin of attraction can be very different from numerical method to numerical method. The results can be used as an explanation for possible causes of error, and slow convergence and nonconvergence of steady-state numerical solutions when using the time-dependent approach for nonlinear hyperbolic or parabolic PDES.

Yee, H. C.↗
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 NTRS1996

The Effects of Dissipation and Coarse Grid Resolution for Multigrid in Flow Problems

The objective of this paper is to investigate the effects of the numerical dissipation and the resolution of the solution on coarser grids for multigrid with the Euler equation approximations. The convergence is accomplished by multi-stage explicit time-stepping to steady state accelerated by FAS multigrid. A theoretical investigation is carried out for linear hyperbolic equations in one and two dimensions. The spectra reveals that for stability and hence robustness of spatial discretizations with a small amount of numerical dissipation the grid transfer operators have to be accurate enough and the smoother of low temporal accuracy. Numerical results give grid independent convergence in one dimension. For two-dimensional problems with a small amount of numerical dissipation, however, only a few grid levels contribute to an increased speed of convergence. This is explained by the small numerical dissipation leading to dispersion. Increasing the mesh density and hence making the problem over resolved increases the number of mesh levels contributing to an increased speed of convergence. If the steady state equations are elliptic, all grid levels contribute to the convergence regardless of the mesh density.

Eliasson, Peter↗
NASA NTRS1996

Time-Dependent Parabolic Finite Difference Formulation for Harmonic Sound Propagation in a Two-Dimensional Duct with Flow

An explicit finite difference real time iteration scheme is developed to study harmonic sound propagation in aircraft engine nacelles. To reduce storage requirements for future large 3D problems, the time dependent potential form of the acoustic wave equation is used. To insure that the finite difference scheme is both explicit and stable for a harmonic monochromatic sound field, a parabolic (in time) approximation is introduced to reduce the order of the governing equation. The analysis begins with a harmonic sound source radiating into a quiescent duct. This fully explicit iteration method then calculates stepwise in time to obtain the 'steady state' harmonic solutions of the acoustic field. For stability, applications of conventional impedance boundary conditions requires coupling to explicit hyperbolic difference equations at the boundary. The introduction of the time parameter eliminates the large matrix storage requirements normally associated with frequency domain solutions, and time marching attains the steady-state quickly enough to make the method favorable when compared to frequency domain methods. For validation, this transient-frequency domain method is applied to sound propagation in a 2D hard wall duct with plug flow.

Kreider, Kevin L.↗
NASA NTRS1996

Compressing an Elliptic Vortex: Transition to Turbulence by Tumble Breakdown

Stability of the elliptic vortex attracted interest in the past decade. Cambon (1982), and Cambon, Teissedre and Jeandel (1985) have studied the stability of such flows with spatially uniform velocity gradient, and have provided RDT solutions for a wide range of the parameter S/Omega (where the strain rate S and the vorticity 2 Omega define the velocity gradient matrix). The range studied included those of hyperbolic streamlines (strain dominated, S/Omega is greater than 1), linear streamlines (simple shear, S/Omega = 1), and elliptical streamlines (vorticity dominated, S/Omega is less than 1). The latter class has more recently attracted interest and several studies appeared (Pierrehumbert 1986, Bayly 1986, Craik and Criminale 1986). These studies will be collectively referred to as PBCC.

Godeferd, Fabien S.↗
NASA NTRS1994

Numerical Simulations of High-Speed Chemically Reacting Flow

The Essentially NonOscillatory (ENO) shock-capturing scheme for the solution of hyperbolic equations is extended to solve a system of coupled conservation equations governing two-dimensional, time-dependent, compressible chemically reacting flow with full chemistry. The thermodynamic properties of the mixture are modeled accurately, and stiff kinetic terms are separated from the fluid motion by a fractional step algorithm. The methodology is used to study the concept of shock-induced mixing and combustion, a process by which the interaction of a shock wave with a jet of low-density hydrogen fuel enhances mixing through streamwise vorticity generation. Test cases with and without chemical reaction are explored here. Our results indicate that, in the temperature range examined, vorticity generation as well as the distribution of atomic species do not change significantly with the introduction of a chemical reaction and subsequent heat release. The actual diffusion of hydrogen is also relatively unaffected by the reaction process. This suggests that the fluid mechanics of this problem may be successfully decoupled from the combustion processes, and that computation of the mixing problem (without combustion chemistry) can elucidate much of the important physical features of the flow.

Ton, V. T.↗
NASA NTRS1994

Numerical Simulations of High-Speed Chemically Reacting Flow

The essentially nonoscillatory (ENO) shock-capturing scheme for the solution of hyperbolic equations is extended to solve a system of coupled conservation equations governing two-dimensional, time-dependent, compressible chemically reacting flow with full chemistry. The thermodynamic properties of the mixture are modeled accurately, and stiff kinetic terms are separated from the fluid motion by a fractional step algorithm. The methodology is used to study the concept of shock-induced mixing and combustion, a process by which the interaction of a shock wave with a jet of low-density hydrogen fuel enhances mixing through streamwise vorticity generation. Test cases with and without chemical reaction are explored here. Our results indicate that, in the temperature range examined, vorticity generation as well as the distribution of atomic species do not change significantly with the introduction of a chemical reaction and subsequent heat release. The actual diffusion of hydrogen is also relatively unaffected by the reaction process. This suggests that the fluid mechanics of this problem may be successfully decoupled from the combustion processes, and that computation of the mixing problem (without combustion chemistry) can elucidate much of the important physical features of the flow.

Ton, V. T.↗
NASA NTRS1995

Dynamical Approach Study of Spurious Steady-State Numerical Solutions of Nonlinear Differential Equations: Global Asymptotic Behavior of Time Discretizations - Part 2

The global asymptotic nonlinear behavior of 11 explicit and implicit time discretizations for four 2 x 2 systems of first-order autonomous nonlinear ordinary differential equations (ODEs) is analyzed. The objectives are to gain a basic understanding of the difference in the dynamics of numerics between the scalars and systems of nonlinear autonomous ODEs and to set a baseline global asymptotic solution behavior of these schemes for practical computations in computational fluid dynamics. We show how 'numerical' basins of attraction can complement the bifurcation diagrams in gaining more detailed global asymptotic behavior of time discretizations for nonlinear differential equations (DEs). We show how in the presence of spurious asymptotes the basins of the true stable steady states can be segmented by the basins of the spurious stable and unstable asymptotes. One major consequence of this phenomenon which is not commonly known is that this spurious behavior can result in a dramatic distortion and, in most cases, a dramatic shrinkage and segmentation of the basin of attraction of the true solution for finite time steps. Such distortion, shrinkage and segmentation of the numerical basins of attraction will occur regardless of the stability of the spurious asymptotes, and will occur for unconditionally stable implicit linear multistep methods. In other words, for the same (common) steady-state solution the associated basin of attraction of the DE might be very different from the discretized counterparts and the numerical basin of attraction can be very different from numerical method to numerical method. The results can be used as an explanation for possible causes of error, and slow convergence and nonconvergence of steady-state numerical solutions when using the time-dependent approach for nonlinear hyperbolic or parabolic PDEs.

Yee, H. C.↗
NASA NTRS1997

The Local Discontinuous Galerkin Method for Time-Dependent Convection-Diffusion Systems

In this paper, we study the Local Discontinuous Galerkin methods for nonlinear, time-dependent convection-diffusion systems. These methods are an extension of the Runge-Kutta Discontinuous Galerkin methods for purely hyperbolic systems to convection-diffusion systems and share with those methods their high parallelizability, their high-order formal accuracy, and their easy handling of complicated geometries, for convection dominated problems. It is proven that for scalar equations, the Local Discontinuous Galerkin methods are L(sup 2)-stable in the nonlinear case. Moreover, in the linear case, it is shown that if polynomials of degree k are used, the methods are k-th order accurate for general triangulations; although this order of convergence is suboptimal, it is sharp for the LDG methods. Preliminary numerical examples displaying the performance of the method are shown.

Cockburn, Bernardo↗
NASA NTRS1997

Guidance of Nonlinear Nonminimum-Phase Dynamic Systems

The first two years research work has advanced the inversion-based guidance theory for: (1) systems with non-hyperbolic internal dynamics; (2) systems with parameter jumps; (3) systems where a redesign of the output trajectory is desired; and (4) the generation of recovery guidance maneuvers.

Devasia, Santosh↗
NASA NTRS1998

Preconditioning the Helmholtz Equation for Rigid Ducts

An innovative hyperbolic preconditioning technique is developed for the numerical solution of the Helmholtz equation which governs acoustic propagation in ducts. Two pseudo-time parameters are used to produce an explicit iterative finite difference scheme. This scheme eliminates the large matrix storage requirements normally associated with numerical solutions to the Helmholtz equation. The solution procedure is very fast when compared to other transient and steady methods. Optimization and an error analysis of the preconditioning factors are present. For validation, the method is applied to sound propagation in a 2D semi-infinite hard wall duct.

Baumeister, Kenneth J.↗
NASA NTRS1998

A Formulation of Asymptotic and Exact Boundary Conditions Using Local Operators

In this paper we describe a systematic approach for constructing asymptotic boundary conditions for isotropic wave-like equations using local operators. The conditions take a recursive form with increasing order of accuracy. In three dimensions the recursion terminates and the resulting conditions are exact for solutions which are described by finite combinations of angular spherical harmonics. First, we develop the expansion for the two-dimensional wave equation and construct a sequence of easily implementable boundary conditions. We show that in three dimensions and analogous conditions are again easily implementable in addition to being exact. Also, we provide extensions of these ideas to hyperbolic systems. Namely, Maxwell's equations for TM waves are used to demonstrate the construction. Finally, we provide numerical examples to demonstrate the effectiveness of these conditions for a model problem governed by the wave equation.

Hagstrom, T.↗
NASA NTRS1996

A Flexible Alignment Fixture for the Fabrication of Replication Mandrels

NASA uses precision diamond turning technology to fabricate replication mandrels for its X-ray Calibration Facility (XRCF) optics. The XRCF optics are tubular, and the internal surface contains a parabolic profile over the first section and a hyperbolic profile over the last. The optic is fabricated by depositing layers of gold and nickel on to the replication mandrel and then separating it from the mandrel. Since the mandrel serves as a replication form, it must contain the inverse image of the surface. The difficulty in aligning the mandrel comes from the fabrication steps which it undergoes. The mandrel is rough machined and heat treated prior to diamond turning. After diamond turning, silicon rubber separators which are undercut in radius by 3 mm (0.12 in.) are inserted between the two end caps of the mandrel to allow the plating to wrap around the ends (to prevent flaking). The mandrel is then plated with a nickel-phosphor alloy using an electroless nickel process. At this point, the separators are removed and the mandrel is reassembled for the final cut on the DTM. The mandrel is measured for profile and finish, and polished to achieve an acceptable surface finish. Wrapping the plating around the edges helps to prevent flaking, but it also destroys the alignment surfaces between the parts of the mandrel that insure that the axes of the parts are coincident. Several mandrels have been realigned by trial-and-error methods, consuming significant amounts of setup time. When the mandrel studied in this paper was reassembled, multiple efforts resulted in a minimum radial error motion of 100 microns. Since 50 microns of nickel plating was to be removed, and a minimum plating thickness of 25 microns was to remain on the part, the radial error motion had to be reduced to less than 25 microns. The mandrel was therefore not usable in its current state.

Cuttino, James F.↗
NASA NTRS1998

Gas-Kinetic Theory Based Flux Splitting Method for Ideal Magnetohydrodynamics

A gas-kinetic solver is developed for the ideal magnetohydrodynamics (MHD) equations. The new scheme is based on the direct splitting of the flux function of the MHD equations with the inclusion of "particle" collisions in the transport process. Consequently, the artificial dissipation in the new scheme is much reduced in comparison with the MHD Flux Vector Splitting Scheme. At the same time, the new scheme is compared with the well-developed Roe-type MHD solver. It is concluded that the kinetic MHD scheme is more robust and efficient than the Roe- type method, and the accuracy is competitive. In this paper the general principle of splitting the macroscopic flux function based on the gas-kinetic theory is presented. The flux construction strategy may shed some light on the possible modification of AUSM- and CUSP-type schemes for the compressible Euler equations, as well as to the development of new schemes for a non-strictly hyperbolic system.

Xu, Kun↗