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NASA NTRS1972

Calculation of free-fall trajectories using numerical optimization methods.

An important problem in space flight is the calculation of trajectories for nonthrusting vehicles between fixed points in a given time. A new procedure based on Hamilton's principle for solving such two-point boundary-value problems is presented. It employs numerical optimization methods to perform the extremization required by Hamilton's principle. This procedure is applied to the calculation of an Earth-Moon trajectory. The results show that the initial guesses required to obtain an iteration procedure which converges are not critical and that convergence can be obtained to any predetermined degree of accuracy.

Hull, D. G.↗

NASA NTRS1973

Advanced control concepts

The problems of excess control devices and insufficient trim control capability on shuttle ascent vehicles were investigated. The trim problem is solved at all time points of interest using Lagrangian multipliers and a Simplex based iterative algorithm developed as a result of the study. This algorithm has the capability to solve any bounded linear problem with physically realizable constraints, and to minimize any piecewise differentiable cost function. Both solution methods also automatically distribute the command torques to the control devices. It is shown that trim requirements are unrealizable if only the orbiter engines and the aerodynamic surfaces are used.

Sharp, J. B.↗

NASA NTRS1973

A direct procedure for partitioning scanning workload in control display system design

Experimental eye scanning measurements were made from simulated instrument approaches in a flight-like cockpit representing a contemporary jet transport. The procedure for predicting the partition of the pilot's scanning workload required for monitoring and controlling a task with status displays and a flight director was simplified. When there is but a single director control display, iteration in the preliminary design computations is eliminated. The preliminary design computations are based on predictions of closed-loop input-correlated errors in displayed variables with respect to the trimmed flight values. Methods for predicting multiloop error coherence, and for correcting the predicted partition of scanning workload when the pilot's scanning remnant contribution is significant are included.

Clement, W. F.↗

NASA NTRS1975

Prediction and verification of creep behavior in metallic materials and components for the space shuttle thermal protection system. Volume 2: Phase 2 subsize panel cyclic creep predictions

A method for predicting permanent cyclic creep deflections in stiffened panel structures was developed. The resulting computer program may be applied to either the time-hardening or strain-hardening theories of creep accumulation. Iterative techniques were used to determine structural rotations, creep strains, and stresses as a function of time. Deflections were determined by numerical integration of structural rotations along the panel length. The analytical approach was developed for analyzing thin-gage entry vehicle metallic-thermal-protection system panels subjected to cyclic bending loads at high temperatures, but may be applied to any panel subjected to bending loads. Predicted panel creep deflections were compared with results from cyclic tests of subsize corrugation and rib-stiffened panels. Empirical equations were developed for each material based on correlation with tensile cyclic creep data and both the subsize panels and tensile specimens were fabricated from the same sheet material. For Vol. 1, see N75-21431.

Cramer, B. A.↗

NASA NTRS1982

Methodology for multiaircraft minimum noise impact landing trajectories

The problem of optimizing landing trajectories with respect to noise impact index is addressed. In contrast with previous work, multilandings are considered. In order to make the problem tractable the trajectories are specified in functional form with certain parameters left free for selection by the optimization methods. Constraint of aircraft dynamic behavior, trajectory separation, pilot workload, passenger comfort, and maximum noise intensity all enter into the determination of what is an allowable trajectory. A version of the quasi-Newton iterative procedure is used to determine the optimum parameter values. The results show improvement in noise impact to the airport considered and the potential for even greater improvement at many airports.

Cook, G.↗

NASA NTRS1989

The use of Lanczos's method to solve the large generalized symmetric definite eigenvalue problem

The generalized eigenvalue problem, Kx = Lambda Mx, is of significant practical importance, especially in structural enginering where it arises as the vibration and buckling problem. A new algorithm, LANZ, based on Lanczos's method is developed. LANZ uses a technique called dynamic shifting to improve the efficiency and reliability of the Lanczos algorithm. A new algorithm for solving the tridiagonal matrices that arise when using Lanczos's method is described. A modification of Parlett and Scott's selective orthogonalization algorithm is proposed. Results from an implementation of LANZ on a Convex C-220 show it to be superior to a subspace iteration code.

Jones, Mark T.↗

NASA NTRS1989

Numerical algorithms for steady and unsteady incompressible Navier-Stokes equations

The numerical analysis of the incompressible Navier-Stokes equations are becoming important tools in the understanding of some fluid flow problems which are encountered in research as well as in industry. With the advent of the supercomputers, more realistic problems can be studied with a wider choice of numerical algorithms. An alternative formulation is presented for viscous incompressible flows. The incompressible Navier-Stokes equations are cast in a velocity/vorticity formulation. This formulation consists of solving the Poisson equations for the velocity components and the vorticity transport equation. Two numerical algorithms for the steady two-dimensional laminar flows are presented. The first method is based on the actual partial differential equations. This uses a finite-difference approximation of the governing equations on a staggered grid. The second method uses a finite element discretization with the vorticity transport equation approximated using a Galerkin approximation and the Poisson equations are obtained using a least squares method. The equations are solved efficiently using Newton's method and a banded direct matrix solver (LINPACK). The method is extended to steady three-dimensional laminar flows and applied to a cubic driven cavity using finite difference schemes and a staggered grid arrangement on a Cartesian mesh. The equations are solved iteratively using a plane zebra relaxation scheme. Currently, a two-dimensional, unsteady algorithm is being developed using a generalized coordinate system. The equations are discretized using a finite-volume approach. This work will then be extended to three-dimensional flows.

Hafez, Mohammed↗

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 NTRS1995

Parallel CFD Algorithms for Aerodynamical Flow Solvers on Unstructured Meshes

The Advisory Group for Aerospace Research and Development (AGARD) has requested my participation in the lecture series entitled Parallel Computing in Computational Fluid Dynamics to be held at the von Karman Institute in Brussels, Belgium on May 15-19, 1995. In addition, a request has been made from the US Coordinator for AGARD at the Pentagon for NASA Ames to hold a repetition of the lecture series on October 16-20, 1995. I have been asked to be a local coordinator for the Ames event. All AGARD lecture series events have attendance limited to NATO allied countries. A brief of the lecture series is provided in the attached enclosure. Specifically, I have been asked to give two lectures of approximately 75 minutes each on the subject of parallel solution techniques for the fluid flow equations on unstructured meshes. The title of my lectures is "Parallel CFD Algorithms for Aerodynamical Flow Solvers on Unstructured Meshes" (Parts I-II). The contents of these lectures will be largely review in nature and will draw upon previously published work in this area. Topics of my lectures will include: (1) Mesh partitioning algorithms. Recursive techniques based on coordinate bisection, Cuthill-McKee level structures, and spectral bisection. (2) Newton's method for large scale CFD problems. Size and complexity estimates for Newton's method, modifications for insuring global convergence. (3) Techniques for constructing the Jacobian matrix. Analytic and numerical techniques for Jacobian matrix-vector products, constructing the transposed matrix, extensions to optimization and homotopy theories. (4) Iterative solution algorithms. Practical experience with GIVIRES and BICG-STAB matrix solvers. (5) Parallel matrix preconditioning. Incomplete Lower-Upper (ILU) factorization, domain-decomposed ILU, approximate Schur complement strategies.

Barth, Timothy J.↗

NASA NTRS2011

Combined fast multipole-QR compression technique for solving electrically small to large structures for broadband applications

An approach that efficiently solves for a desired parameter of a system or device that can include both electrically large fast multipole method (FMM) elements, and electrically small QR elements. The system or device is setup as an oct-tree structure that can include regions of both the FMM type and the QR type. An iterative solver is then used to determine a first matrix vector product for any electrically large elements, and a second matrix vector product for any electrically small elements that are included in the structure. These matrix vector products for the electrically large elements and the electrically small elements are combined, and a net delta for a combination of the matrix vector products is determined. The iteration continues until a net delta is obtained that is within predefined limits. The matrix vector products that were last obtained are used to solve for the desired parameter.

Jandhyala, Vikram↗

NASA NTRS2019

Real-time Exponential Curve Fits Using Discrete Calculus

This paper presents an improved solution for curve fitting data to an exponential equation (Y = AeBt + C). This improvement is in four areas ? speed, stability, determinant processing time, and the removal of limits. The solution presented in this paper avoids iterative techniques and their stability errors by using three mathematical ideas ? discrete calculus, a special relationship (between exponential curves and the Mean Value Theorem for Derivatives), and a simple linear curve fit algorithm. This method can also be applied to fitting data to the general power law equation Y = AxB + C and the general geometric growth equation Y = AkBt + C.

Rowe, Geoffrey K.↗

NASA NTRSDate not supplied

Implementation of Machine Learning Methods for Crater-Based Navigation

Terrain Relative Navigation methods require surface feature detectors to gain information from images used to improve on-board state estimates. This paper presents the development of a crater detection method based on Machine Learning that can extract data from optical images with different crater shapes and sizes, under varying lighting conditions. This work includes an automated capability for generating labeled training data and iterative testing of the neural network-based crater detector. Preliminary results are included to quantify the detector’s accuracy compared to a known crater catalog, given a set of real lunar images from the Lunar Reconnaissance Orbiter.

Sofia G Catalan↗

NASA NTRS1969

Determination of permissible applied load stress in structural elements

Graphic method is used to select allowable stresses in thermally loaded structures. Equations are used for determining the mode of failure for specific materials in order to plot a range of stress curves. Linear assumption and iterative calculations are eliminated resulting in comparatively high accuracy.

Loy, R. E.↗

NASA NTRS1983

Continuation in a parameter - Experience with viscous and free surface flows

The results of modifications in continuation methods applied to obtain solutions to the Navier-Stokes systems of equations for incompressible, two-dimensional, steady flows are reported. It is shown that parameter continuation permits prediction of accurate, initial estimates for iterative processing of nonlinear finite difference and finite element equations of motions. The new parameter steps are derived from values of the preceding parameter steps. The accuracy of the estimates is ensured with appropriate choices of the step size. The continuation predictor/iterative corrector is demonstrated to trace the branches of parameter space along which steady flow states are found, and techniques are available for tracing multiply branching paths. The techniques are applied to solving the Navier-Stokes equations for flow through a rotating square channel, the formation of a falling liquid curtain, and gyrostatic equilibria of rotating cylindrical drops.

Kheshgi, H. S.↗

NASA NTRS1985

Steady and unsteady incompressible free-wake analysis

The flows around highly sweptback wings and bodies of revolution at high angle of attack are described, and inviscid model approximations and mathematical formulation of the problem are given to steady and unsteady incompressible flows. A general presentation of the methods of solution is given, with emphasis on current computational techniques. Detailed descriptions of the nonlinear vortex-lattice and vortex-panel techniques are presented to show how the boundary conditions are enforced using iteration. Typical numerical results are compared with the available experimental data.

Kandil, O. A.↗

NASA NTRS1986

Efficient inverse solution of Kepler's equation

A bicubic polynomial approximation to Kepler's equation for elliptic orbits is shown to provide accurate starting values for efficient numerical solution of this equation for eccentric anomaly. In the approximate equation, a cubic in mean anomaly is set equal to a cubic in eccentric anomaly. The coefficients in the two cubics are obtained as functions of eccentricity by specifying values of function and slope at the midpoint and the endpoint of the complete interval (0 to pi). The initial estimate of eccentric anomaly to use in an iteration formula is obtained by evaluating the cubic in mean anomaly and finding the single real root of the cubic in eccentric anomaly. Numerical results are presented which indicate that the estimate accuracy of this method is roughly an order of magnitude better than that of other recently-reported formulas.

Boltz, Frederick W.↗

NASA NTRS1988

Evaluation of a three-dimensional empirically derived wing at supersonic speeds

A novel wing design concept is introduced which takes advantage of the existence of conical flow at supersonic speeds. The present wing design concept is to create a near conical wing geometry by redistributing airfoils in a spanwise direction. In addition, a set of graphs which review the supersonic aerodynamics of delta wings have been employed to select a design wing sweep and Mach number. An iteration through the wing design logic resulted in the selection of a 65 deg swept delta wing and a design Mach number of 1.62. Theoretical analysis was performed with a nonlinear full-potential analysis method to assess the merits of the wing design approach. The analysis showed large reductions in drag due to lift compared to delta wings configured with traditional thickness and airfoil distributions.

Wood, Richard M.↗

NASA NTRS1992

Analysis and design of planar and non-planar wings for induced drag minimization

The goal of the work reported herein is to develop and validate computational tools to be used for the design of planar and non-planar wing geometries for minimum induced drag. Because of the iterative nature of the design problem, it is important that, in addition to being sufficiently accurate for the problem at hand, these tools need to be reasonably fast and computationally efficient. Toward this end, a method of predicting induced drag in the presence of a free wake has been coupled with a panel method. The induced drag prediction technique is based on the application of the Kutta-Joukowski law at the trailing edge. Until now, the use of this method has not been fully explored and pressure integration and Trefftz-plane calculations favored. As is shown in this report, however, the Kutta-Joukowski method is able to give better results for a given amount of effort than the more commonly used techniques, particularly when relaxed wakes and non-planar wing geometries are considered. Using these methods, it is demonstrated that a reduction in induced drag can be achieved through non-planar wing geometries. It remains to determine what overall drag reductions are possible when the induced drag reduction is traded-off against increased wetted area. With the design methodology that is described herein, such trade studies can be performed in which the non-linear effects of the free wake are taken into account.

Mortara, Karl W.↗