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NASA NTRS1982

Effect of particle size distribution and chlorophyll content on beam attenuation spectra

The relationships between beam attenuation spectra, chlorophyll and pheophytin pigment concentrations, and particle size distributions are examined for a coastal region believed to have negligible concentrations of terrestrially derived dissolved organic compounds but large quantities of phytoplankton and resuspended sediments. It was found that the slope of the beam attenuation spectra increases when the hyperbolic slope of the size distribution increases. The magnitude of this increase in slope was consistent with calculations based on a range of particle diameters from 0.5 to 30 microns so that it would be possible to predict the slope of the particle size distribution if the slope of the beam attenuation spectra is known. The ratio of chlorophyll and pheophytin pigments to suspended volume concentrations affected the beam attenuation spectra to a lesser degree and in a more complex manner. Because of the strong effect of slope, it was concluded that the chlorophyll and pheophytin pigment content of suspended particles could not be efficiently predicted by means of beam attenuation measurements.

Kitchen, J. C.↗

NASA NTRS1982

An approximation theory for nonlinear partial differential equations with applications to identification and control

Approximation results from linear semigroup theory are used to develop a general framework for convergence of approximation schemes in parameter estimation and optimal control problems for nonlinear partial differential equations. These ideas are used to establish theoretical convergence results for parameter identification using modal (eigenfunction) approximation techniques. Results from numerical investigations of these schemes for both hyperbolic and parabolic systems are given.

Banks, H. T.↗

NASA NTRS1983

Computations of two-phase supersonic nozzle flows by a space-marching method

A practical method for efficiently analyzing the two-phase flow field in the supersonic region of an axisymmetric rocket nozzle is presented. The governing hyperbolic equations for the gas- and particle-phases are simultaneously integrated downstream using a space-marching procedure. Both one- and two-phase flow computations are performed for the JPL axisymmetric nozzle. The results show the nonequilibrium effects which are due to the existence of particles in the flow. The comparisons with the experimental data and other numerical solutions are made to examine the accuracy of the present method.

Fujii, K.↗

NASA NTRS1982

Cassegrainian concentrator solar array exploratory development module

A miniaturized Cassegrainian concentrator solar array concept is under development to reduce the cost of multi-kW spacecraft solar arrays. A primary parabolic reflector directs incoming solar energy to a secondary, centrally mounted inverted hyperbolic reflector and down onto a solar cell mounted on an Mo heat spreader on a 0.25 mm thick Al heat fin. Each unit is 12.7 mm thick, which makes the concentrator assembly roughly as thick as a conventional panel. The output is 100 W/sq and 20 W/kg, considering 20% efficient Si cells at 100 suns. A tertiary light catcher is mounted around the cell to ameliorate optic errors. The primary reflector is electroformed Ni with protective and reflective coatings. The cells have back surface reflectors and a SiO antireflective coating. An optical efficiency of 80% is projected, and GaAs cells are being considered in an attempt to raise cell efficiencies to over 30%.

Patterson, R. E.↗

NASA NTRS1982

Boundary-fitted coordinate systems for numerical solution of partial differential equations - A review

A comprehensive review of methods of numerically generating curvilinear coordinate systems with coordinate lines coincident with all boundary segments is given. Some general mathematical framework and error analysis common to such coordinate systems is also included. The general categories of generating systems are those based on conformal mapping, orthogonal systems, nearly orthogonal systems, systems produced as the solution of elliptic and hyperbolic partial differential equations, and systems generated algebraically by interpolation among the boundaries. Also covered are the control of coordinate line spacing by functions embedded in the partial differential operators of the generating system and by subsequent stretching transformation. Dynamically adaptive coordinate systems, coupled with the physical solution, and time-dependent systems that follow moving boundaries are treated. References reporting experience using such coordinate systems are reviewed as well as those covering the system development.

Thompson, J. F.↗

NASA NTRS1982

Numerical boundary condition procedures and multigrid methods; Proceedings of the Symposium, NASA Ames Research Center, Moffett Field, CA, October 19-22, 1981

Papers presented in this volume provide an overview of recent work on numerical boundary condition procedures and multigrid methods. The topics discussed include implicit boundary conditions for the solution of the parabolized Navier-Stokes equations for supersonic flows; far field boundary conditions for compressible flows; and influence of boundary approximations and conditions on finite-difference solutions. Papers are also presented on fully implicit shock tracking and on the stability of two-dimensional hyperbolic initial boundary value problems for explicit and implicit schemes.

Source record↗

NASA NTRS1983

On one-dimensional stretching functions for finite-difference calculations

The class of one-dimensional stretching functions used in finite-difference calculations is studied. For solutions containing a highly localized region of rapid variation, simple criteria for a stretching function are derived using a truncation error analysis. These criteria are used to investigate two types of stretching functions. One an interior stretching function, for which the location and slope of an interior clustering region are specified. The simplest such function satisfying the criteria is found to be one based on the inverse hyperbolic sine. The other type of function is a two-sided stretching function, for which the arbitrary slopes at the two ends of the one-dimensional interval are specified. The simplest such general function is found to be one based on the inverse tangent. Previously announced in STAR as N80-25055

Vinokur, M.↗

NASA NTRS1983

Implicit total variation diminishing (TVD) schemes for steady-state calculations

The application of a new implicit unconditionally stable high resolution total variation diminishing (TVD) scheme to steady state calculations. It is a member of a one parameter family of explicit and implicit second order accurate schemes developed by Harten for the computation of weak solutions of hyperbolic conservation laws. This scheme is guaranteed not to generate spurious oscillations for a nonlinear scalar equation and a constant coefficient system. Numerical experiments show that this scheme not only has a rapid convergence rate, but also generates a highly resolved approximation to the steady state solution. A detailed implementation of the implicit scheme for the one and two dimensional compressible inviscid equations of gas dynamics is presented. Some numerical computations of one and two dimensional fluid flows containing shocks demonstrate the efficiency and accuracy of this new scheme. Previously announced in STAR as N83-23085

Yee, H. C.↗

NASA NTRS1983

High resolution applications of the Osher upwind scheme for the Euler equations

The 'Osher' scheme was introduced by Osher (1981). It represents an upwind finite-difference method for hyperbolic systems of conservation laws, including the Euler equations. In studies conducted by Osher (1981) and Osher and Solomon (1982), the method was applied to the nonisentropic form of the Euler equations in one dimension and the isentropically restricted form in two spatial dimensions, both in Cartesian coordinates. Chakravarthy and Osher (1982) have shown an approach for extending the Osher scheme to the Euler equations written for general geometries, taking into account the use of mappings to arbitrary curvilinear coordinate systems. The present investigation is concerned with the high resolution extension of the Osher scheme to second-order accuracy. Results are presented for several example problems, giving attention to quasi-one-dimensional Laval nozzle flow, a one-dimensional shock tube problem, and supersonic flow over a cylinder.

Chakravarthy, S. R.↗

NASA NTRS1983

Vibrations of cantilevered doubly-curved shallow shells

Vibrational characteristics are determined for a previously unsolved class of problems, that of doubly-curved shallow shells having rectangular planforms, clamped along one edge and free on the other three. The solution procedure uses the Ritz method with algebraic polynomial trial functions. Convergence studies are made, and accurate frequencies and contour plots of mode shapes are presented for various curvature ratios, including spherical, circular cylindrical and hyperbolic paraboloidal shells. Particular emphasis is given to the effect of adding spanwise curvature to shells having chordwise curvature; numerous published references already exist for the case of zero spanwise curvature. The effects of changing aspect ratio, thickness ratio and Poisson's ratio are also studied.

Leissa, A. W.↗

NASA NTRS1983

Aerocapture and aeroassisted orbital transfer - A high performance leverage space technology

Single pass aerocapture and aeroassisted orbital transfer designs are discussed for modifying spacecraft orbits using planetary atmospheres. Aerodynamic drag can be used to shift a hyperbolic flyby trajectory to a planetary orbit by having the spacecraft altering its in-plane L/D ratio to maintain constant drag until the desired velocity is attained. The spacecraft would then move out of the atmosphere, sometimes with a small propulsive assist. Sample missions to Uranus, Jupiter, and Titan are mentioned, with mission scenarios commencing with a Shuttle launch. Biconic-shaped spacecraft are cited as having the best L/D for insertion accuracy, a low hypersonic ballistic coefficient, and volumeric efficiency. Aeroassisted orbital transfer involves atmospheric braking in order to shed velocity on a return trip from higher orbits. No decision has yet been made as to the spacecraft shape, but calculations indicate that aeroassist braking before return to LEO for rendezvous with the Shuttle has the potential for the lowest cost/mass to GEO of any delivery system currently under consideration.

Cruz, M. I.↗

NASA NTRS1983

Cubic spline approximation techniques for parameter estimation in distributed systems

Approximation schemes employing cubic splines in the context of a linear semigroup framework are developed for both parabolic and hyperbolic second-order partial differential equation parameter estimation problems. Convergence results are established for problems with linear and nonlinear systems, and a summary of numerical experiments with the techniques proposed is given.

Banks, H. T.↗

NASA NTRS1983

A least squares finite element scheme for transonic flow around harmonically oscillating airfoils

The present investigation shows that a finite element scheme with a weighted least squares variational principle is applicable to the problem of transonic flow around a harmonically oscillating airfoil. For the flat plate case, numerical results compare favorably with the exact solution. The obtained numerical results for the transonic problem, for which an exact solution is not known, have the characteristics of known experimental results. It is demonstrated that the performance of the employed numerical method is independent of equation type (elliptic or hyperbolic) and frequency. The weighted least squares principle allows the appropriate modeling of singularities, which such a modeling of singularities is not possible with normal least squares.

Cox, C. L.↗

NASA NTRS1983

High resolution schemes and the entropy condition

A systematic procedure for constructing semidiscrete, second order accurate, variation diminishing, five point band width, approximations to scalar conservation laws, is presented. These schemes are constructed to also satisfy a single discrete entropy inequality. Thus, in the convex flux case, convergence is proven to be the unique physically correct solution. For hyperbolic systems of conservation laws, this construction is used formally to extend the first author's first order accurate scheme, and show (under some minor technical hypotheses) that limit solutions satisfy an entropy inequality. Results concerning discrete shocks, a maximum principle, and maximal order of accuracy are obtained. Numerical applications are also presented.

Osher, S.↗

NASA NTRS1983

Spectral methods for time dependent partial differential equations

The theory of spectral methods for time dependent partial differential equations is reviewed. When the domain is periodic Fourier methods are presented while for nonperiodic problems both Chebyshev and Legendre methods are discussed. The theory is presented for both hyperbolic and parabolic systems using both Galerkin and collocation procedures. While most of the review considers problems with constant coefficients the extension to nonlinear problems is also discussed. Some results for problems with shocks are presented.

Gottlieb, D.↗

NASA NTRS1983

Enhanced character sizes for the VDM-1 video display board

Use of a microprocessor for providing navigation information from the inherent hyperbolic geometries of Loran-C was discussed. The advantages Loran-C offers over VOR/DME as a primary navigation aid are identified. These advantages include long range coverage by several stations, coverage at low altitudes, and the capability to have nonprecision approaches at airports not already served by landing aids. Modern digital electronic technology is used to produce a device to convert Loran-C data to useful pilot information using simple software algorithms and low cost microprocessor devices. The cost and lack of availability of suitable processors to execute these algorithms have prevented a Loran-C navigator from being developed. Results indicate that the microprocessor based Loran-C navigator has an accuracy of 1.0 nm or less over an area typically covered by a triad of Loran-C stations and can execute a position update in less than 0.2 seconds.

Novacki, S. M., III↗

NASA NTRS1983

Application of the Lienard-Wiechert solution to a lightning return stroke model

The electric and magnetic fields associated with the lightning return stroke are expressed as a convolution of the current waveform shape and the fields generated by a moving charge of amplitude one (i.e., the Lienard-Wiechert solution for a unit charge). The representation can be used to compute the fields produced by a current waveform of non-uniform velocity that propagates along a filament of arbitrary, but finite, curvature. To study numerically the effects of linear charge acceleration and channel curvature two simple channel models are used: the linear and the hyperbolic.

Meneghini, R.↗

NASA NTRS1983

Spectral methods for partial differential equations

Origins of spectral methods, especially their relation to the Method of Weighted Residuals, are surveyed. Basic Fourier, Chebyshev, and Legendre spectral concepts are reviewed, and demonstrated through application to simple model problems. Both collocation and tau methods are considered. These techniques are then applied to a number of difficult, nonlinear problems of hyperbolic, parabolic, elliptic, and mixed type. Fluid dynamical applications are emphasized.

Hussaini, M. Y.↗