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NASA NTRS1981

Research on numerical algorithms for large space structures

Numerical algorithms for analysis and design of large space structures are investigated. The sign algorithm and its application to decoupling of differential equations are presented. The generalized sign algorithm is given and its application to several problems discussed. The Laplace transforms of matrix functions and the diagonalization procedure for a finite element equation are discussed. The diagonalization of matrix polynomials is considered. The quadrature method and Laplace transforms is discussed and the identification of linear systems by the quadrature method investigated.

Denman, E. D.↗

NASA NTRS1981

Some aspects of algorithm performance and modeling in transient analysis of structures

The status of an effort to increase the efficiency of calculating transient temperature fields in complex aerospace vehicle structures is described. The advantages and disadvantages of explicit algorithms with variable time steps, known as the GEAR package, is described. Four test problems, used for evaluating and comparing various algorithms, were selected and finite-element models of the configurations are described. These problems include a space shuttle frame component, an insulated cylinder, a metallic panel for a thermal protection system, and a model of the wing of the space shuttle orbiter. Results generally indicate a preference for implicit over explicit algorithms for solution of transient structural heat transfer problems when the governing equations are stiff (typical of many practical problems such as insulated metal structures).

Adelman, H. M.↗

NASA NTRS1981

A multigrid algorithm for steady transonic potential flows around aerofoils using Newton iteration

The application of multigrid relaxation to transonic potential flow calculation was investigated. Conservative potential flows around aerofoils were taken as test problems. The solution algorithm was based on Newton iteration. It was found that the iteration to the circulation has to be kept outside the multigrid algorithm. To obtain meaningful norms of residuals, difference formulas with asymptotic scaling were introduced. Nonlinear instability problems were solved by upwind differencing using mass flux vector splitting instead of artificial viscosity or artificial density. It is suggested that the algorithms efficiency be increased by improving the iteration on the shock positions even though this is a highly nonlinear process.

Boestoel, J. W.↗

NASA NTRS1980

Outline of a fast hardware implementation of Winograd's DFT algorithm

The main characteristics of the discrete Fourier transform (DFT) algorithm considered by Winograd (1976) is a significant reduction in the number of multiplications. Its primary disadvantage is a higher structural complexity. It is, therefore, difficult to translate the reduced number of multiplications into faster execution of the DFT by means of a software implementation of the algorithm. For this reason, a hardware implementation is considered in the current study, taking into account a design based on the algorithm prescription discussed by Zohar (1979). The hardware implementation of a FORTRAN subroutine is proposed, giving attention to a pipelining scheme in which 5 consecutive data batches are being operated on simultaneously, each batch undergoing one of 5 processing phases.

Zohar, S.↗

NASA NTRS1981

Efficient algorithms for single-axis attitude estimation

The computationally efficient algorithms determine attitude from the measurement of art lengths and dihedral angles. The dependence of these algorithms on the solution of trigonometric equations was reduced. Both single time and batch estimators are presented along with the covariance analysis of each algorithm.

Shuster, M. D.↗

NASA NTRS1981

An evaluation of ISOCLS and CLASSY clustering algorithms for forest classification in northern Idaho

Both the iterative self-organizing clustering system (ISOCLS) and the CLASSY algorithms were applied to forest and nonforest classes for one 1:24,000 quadrangle map of northern Idaho and the classification and mapping accuracies were evaluated with 1:30,000 color infrared aerial photography. Confusion matrices for the two clustering algorithms were generated and studied to determine which is most applicable to forest and rangeland inventories in future projects. In an unsupervised mode, ISOCLS requires many trial-and-error runs to find the proper parameters to separate desired information classes. CLASSY tells more in a single run concerning the classes that can be separated, shows more promise for forest stratification than ISOCLS, and shows more promise for consistency. One major drawback to CLASSY is that important forest and range classes that are smaller than a minimum cluster size will be combined with other classes. The algorithm requires so much computer storage that only data sets as small as a quadrangle can be used at one time.

Werth, L. F.↗

NASA NTRS1982

Ozone Uncertainties Study Algorithm (OUSA)

An algorithm to carry out sensitivities, uncertainties and overall imprecision studies to a set of input parameters for a one dimensional steady ozone photochemistry model is described. This algorithm can be used to evaluate steady state perturbations due to point source or distributed ejection of H2O, CLX, and NOx, besides, varying the incident solar flux. This algorithm is operational on IBM OS/360-91 computer at NASA/Goddard Space Flight Center's Science and Applications Computer Center (SACC).

Bahethi, O. P.↗

NASA NTRS1981

Solution of the transonic full potential equation in conservative form using an implicit algorithm

The paper presents numerical solutions of the full potential equation in conservative form. The iteration scheme used is a fully implicit approximate factorization technique and provides a significant improvement in convergence speed relative to standard successive line overrelaxation algorithms. The spatial differencing algorithm is centrally differenced in both subsonic and supersonic regions to maintain stability. This effectively approximates rotated differencing, thereby greatly improving the reliability of the algorithm.

Holst, T.↗

NASA NTRS1981

A quadratic weight selection algorithm

A new numerical algorithm is presented which determines a positive semi-definite state weighting matrix in the linear-quadratic optimal control design problem. The algorithm chooses the weighting matrix by placing closed-loop eigenvalues and eigenvectors near desired locations using optimal feedback gains. A simplified flight control design example is used to illustrate the algorithms capabilities.

Broussard, J. R.↗

NASA NTRS1982

A viscous-inviscid interaction algorithm for three-dimensional turbulent subsonic aerodynamic juncture region flow

An order-of magnitude analysis of the subsonic, three-dimensional, steady time-averaged Navier-Stokes equations for semibounded aerodynamic juncture geometries yields the parabolic Navier-Stokes simplification. The numerical solution of the resultant pressure Poisson equation is cast into complementary and particular parts, yielding an iterative interaction algorithm with an exterior three-dimensional potential flow solution. A parabolic transverse momentum equation set is constructed, wherein robust enforcement first-order continuity effects is accomplished using a penalty differential constraint concept within a finite element solution algorithm. A Reynolds stress constitutive equation, with low turbulence Reynolds number wall functions, is employed for closure, using parabolic forms of the two-equation turbulent kinetic energy-dissipation equation system. The algorithm is employed to predict the three-dimensional turbulent flowfield in the juncture region formed by two intersecting parabolic arcs at a free-stream Mach-number of 0.08, and Re/C = 600,000/m.

Baker, A. J.↗

NASA NTRS1981

Piloted simulation of an on-board trajectory optimization algorithm

This paper will describe a real time piloted simulation of algorithms designed for on-board computation of time-optimal intercept trajectories for an F-8 aircraft. The algorithms, which were derived using singular perturbation theory, generate commands that are displayed to the pilot on flight director needles on the 8-ball. By flying the airplane so as to zero the horizontal and vertical needles, the pilot flies an approximation to a time-optimal intercept trajectory. The various display and computation modes that are available will be described and results will be presented illustrating the performance of the algorithms with a pilot in the loop.

Price, D. B.↗

NASA NTRS1981

Fanning - A classification algorithm for mixture landscapes applied to Landsat data of Maine forests

It is pointed out that typical landscapes include a relatively small number of 'pure' land cover types which combine in various proportions to form a myriad of mixture types. Most Landsat classifications algorithms used today require a separate user specification for each category, including mixture categories. Attention is given to a simpler approach, which would require the user to specify only the 'pure' types. Mixture pixels would be classified on the basis of the proportion of the area covered by each pure type within the pixel. The 'fanning' algorithm quantifies varying proportions of two 'pure' land cover types in selected mixture pixels. This algorithm was applied to 200,000 ha of forest land in Maine, taking into account a comparison with standard inventory information. Results compared well with a discrete categories classification of the same area.

Ungar, S. G.↗

NASA NTRS1982

A vectorized, finite-volume, adaptive grid algorithm applied to planetary entry problems

An adaptive grid, finite-volume method has been applied to problems in planetary entry for computing complete flowfields. The adaption algorithm is implicit in nature and is keyed to resolve user specified gradients. The finite-volume algorithm is explicit, utilizing a maximum time step advancement at each grid point to accelerate convergence to the steady state. The present version of the code is for the laminar flow of a perfect gas. The role of the adaption algorithm in resolving various features of blunt body/wake flow for planetary entry conditions is emphasized.

Gnoffo, P. A.↗

NASA NTRS1982

An accurate radiative heating and cooling algorithm for use in a dynamical model of the middle atmosphere

The circulation of the middle atmosphere of the earth (15-90 km) is driven by the unequal distribution of net radiative heating. Calculations have shown that local radiative heating is nearly balanced by radiative cooling throughout parts of the stratosphere and mesosphere. The 15 micrometer band of CO2 is the dominant component of the infrared cooling. The present investigation is concerned with an algorithm regarding the involved cooling process. The algorithm was designed for the semispectral primitive equation model of the stratosphere and mesosphere described by Holton and Wehrbein (1980). The model consists of 16 layers, each nominally 5 km thick, between the base of the stratosphere at 100 mb (approximately 16 km) and the base of the thermosphere (approximately 96 km). The considered algorithm provides a convenient means of incorporating cooling due to CO2 into dynamical models of the middle atmosphere.

Wehrbein, W. M.↗

NASA NTRS1984

Algorithm for Constructing Contour Plots

General computer algorithm developed for construction of contour plots. algorithm accepts as input data values at set of points irregularly distributed over plane. Algorithm based on interpolation scheme: points in plane connected by straight-line segments to form set of triangles. Program written in FORTRAN IV.

Johnson, W.↗

NASA NTRS1982

Application of the Hughes-LIU algorithm to the 2-dimensional heat equation

An implicit explicit algorithm for the solution of transient problems in structural dynamics is described. The method involved dividing the finite elements into implicit and explicit groups while automatically satisfying the conditions. This algorithm is applied to the solution of the linear, transient, two dimensional heat equation subject to an initial condition derived from the soluton of a steady state problem over an L-shaped region made up of a good conductor and an insulating material. Using the IIT/PRIME computer with virtual memory, a FORTRAN computer program code was developed to make accuracy, stability, and cost comparisons among the fully explicit Euler, the Hughes-Liu, and the fully implicit Crank-Nicholson algorithms. The Hughes-Liu claim that the explicit group governs the stability of the entire region while maintaining the unconditional stability of the implicit group is illustrated.

Malkus, D. S.↗

NASA NTRS1982

Mixture densities, maximum likelihood, and the EM algorithm

The problem of estimating the parameters which determine a mixture density is reviewed as well as maximum likelihood estimation for it. A particular iterative procedure for numerically approximating maximum likelihood estimates for mixture density problems is considered. This EM algorithm, is a specialization to the mixture density context of a general algorithm of the same name used to approximate maximum likelihood estimates for incomplete data problems. The formulation and theoretical and practical properties of the EM algorithm for mixture densities are discussed focussing in particular on mixtures of densities from exponential families.

Redner, R. A.↗

NASA NTRS1982

A recursive algorithm for Zernike polynomials

The analysis of a function defined on a rotationally symmetric system, with either a circular or annular pupil is discussed. In order to numerically analyze such systems it is typical to expand the given function in terms of a class of orthogonal polynomials. Because of their particular properties, the Zernike polynomials are especially suited for numerical calculations. Developed is a recursive algorithm that can be used to generate the Zernike polynomials up to a given order. The algorithm is recursively defined over J where R(J,N) is the Zernike polynomial of degree N obtained by orthogonalizing the sequence R(J), R(J+2), ..., R(J+2N) over (epsilon, 1). The terms in the preceding row - the (J-1) row - up to the N+1 term is needed for generating the (J,N)th term. Thus, the algorith generates an upper left-triangular table. This algorithm was placed in the computer with the necessary support program also included.

Davenport, J. W.↗