A method of solving sets of nonlinear algebraic equations Progress report
Methods for solving nonlinear algebraic equations in computer programs for nuclear magnetic resonance spectroscopy
Adler, R. J.↗
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Advanced control concepts
The selection of a trim solution that provides the space shuttle with the highest level of performance and dynamic control in the presense of wind disturbances and bias torques due to misalignment of rocket engines is described. It was determined that engine gimballing is insufficient to provide control to trim the vehicle for headwind and sidewind disturbances, and that it is necessary to use aerodynamic surfaces in conjunction with engine gimballing to achieve trim. The algebraic equations for computing the trim solution were derived from the differential equations describing the motion of the vehicle by substituting the desired trim conditions. The general problem of showing how the trim equations are derived from the equations of motion and the mathematical forms of the performance criterion is discussed in detail, along with the general equations for studying the dynamic response of the trim solution.
Hutton, M. F.↗
Space Mathematics: A Resource for Secondary School Teachers
A collection of mathematical problems related to NASA space science projects is presented. In developing the examples and problems, attention was given to preserving the authenticity and significance of the original setting while keeping the level of mathematics within the secondary school curriculum. Computation and measurement, algebra, geometry, probability and statistics, exponential and logarithmic functions, trigonometry, matrix algebra, conic sections, and calculus are among the areas addressed.
Kastner, Bernice↗
K-epsilon turbulence model assessment with reduced numerical diffusion for coaxial jets
A numerical study has been conducted to evaluate the performance of the k-epsilon turbulence model for axisymmetric, unconfined, swirling, and nonswirling coaxial jet flows. Two lower order schemes, hybrid and power-law, and two higher order schemes, flux-spline and bounded skew upwind differencing, were employed in this investigation. The predicted results indicate that the higher order numerical schemes have greater potential for future model improvements and complex flow calculations in terms of storage, accuracy, and execution time. For the nonswirling flow, computations using an algebraic stress model have also been presented.
Nikjooy, M.↗
Robustness issues in boundary feedback of flexible structures
Transfer function models for basic structural elements with boundary control reveal certain inherent properties relevant to questions of control system realization, design, and analysis of robustness. The transfer functions involved are not strictly proper, may be nonminimum phase (except for the special case of colocated actuation and sensing), and often have large number of poles on the imaginary axis. In this paper these basic questions are considered in terms of modeling and control system design for robustness. The application of algebraic methods for computing stabilizing control, based on certain exact, irrational transfer functions is investigated. Boundary control of the wave equation is considered and extension of the method to more general problems is suggested. Special attention is given to implementation issues associated with a class of infinite dimensional control laws.
Bennett, W. H.↗
Analysis and experimental evaluation of a Stewart platform-based force/torque sensor
The kinematic analysis and experimentation of a force/torque sensor whose design is based on the mechanism of the Stewart Platform are discussed. Besides being used for measurement of forces/torques, the sensor also serves as a compliant platform which provides passive compliance during a robotic assembly task. It consists of two platforms, the upper compliant platform (UCP) and the lower compliant platform (LCP), coupled together through six spring-loaded pistons whose length variations are measured by six linear voltage differential transformers (LVDT) mounted along the pistons. Solutions to the forward and inverse kinematics of the force sensor are derived. Based on the known spring constant and the piston length changes, forces/torques applied to the LCP gripper are computed using vector algebra. Results of experiments conducted to evaluate the sensing capability of the force sensor are reported and discussed.
Nguyen, Charles C.↗
Flight control application of new stability robustness bounds for linear uncertain systems
This paper addresses the issue of obtaining bounds on the real parameter perturbations of a linear state-space model for robust stability. Based on Kronecker algebra, new, easily computable sufficient bounds are derived that are much less conservative than the existing bounds since the technique is meant for only real parameter perturbations (in contrast to specializing complex variation case to real parameter case). The proposed theory is illustrated with application to several flight control examples.
Yedavalli, Rama K.↗
Effects of spatial order of accuracy on the computation of vortical flowfields
The objective of the present investigation is to assess the effect of the spatial order of accuracy used for the evaluation of the inviscid fluxes on the resolution of higher order quantitites, such as velocity gradients. The viscous terms are computed as second-order accurate with central difference formulas, even though for the explicit part of the algorithm higher order approximations may be used. A viscous/inviscid method is used, and the outer part of the flowfield is computed with the inviscid flow equations. The viscous boundary-layer type flow region close to the body surface is computed with an algebraic eddy viscosity model. Results obtained with the conservative and nonconservative formulations and the viscous/inviscid approach are compared with available experimental data. The effect of grid refinement on the accuracy of the solution is also presented.
Ekaterinaris, J. A.↗
Computation of turbulent boundary layer flows with an algebraic stress turbulence model
An algebraic stress turbulence model is presented, characterized by the following: (1) the eddy viscosity expression is derived from the Reynolds stress turbulence model; (2) the turbulent kinetic energy dissipation rate equation is improved by including a production range time scale; and (3) the diffusion coefficients for turbulence equations are adjusted so that the kinetic energy profile extends further into the free stream region found in most experimental data. The turbulent flow equations were solved using a finite element method. Examples include: fully developed channel flow, fully developed pipe flow, flat plate boundary layer flow, plane jet exhausting into a moving stream, circular jet exhausting into a moving stream, and wall jet flow. Computational results compare favorably with experimental data for most of the examples considered. Significantly improved results were obtained for the plane jet flow, the circular jet flow, and the wall jet flow; whereas the remainder are comparable to those obtained by finite difference methods using the standard kappa-epsilon turbulence model. The latter seems to be promising with further improvement of the expression for the eddy viscosity coefficient.
Kim, Sang-Wook↗
Numerical methods in Markov chain modeling
Several methods for computing stationary probability distributions of Markov chains are described and compared. The main linear algebra problem consists of computing an eigenvector of a sparse, usually nonsymmetric, matrix associated with a known eigenvalue. It can also be cast as a problem of solving a homogeneous singular linear system. Several methods based on combinations of Krylov subspace techniques are presented. The performance of these methods on some realistic problems are compared.
Philippe, Bernard↗
Solution of nonlinear algebraic equations characteristic of filter circuits Summary technical report
Digital computer program developed for solving nonlinear algebraic equations characteristic of filter circuits
COMPUTER PROGRAM↗
Computation of turbulent rotating channel flow with an algebraic Reynolds stress model
An Algebraic Reynolds Stress Model has been implemented to modify the Kolmogorov-Prandtl eddy viscosity relation to produce an anisotropic turbulence model. The eddy viscosity relation becomes a function of the local turbulent production to dissipation ratio and local turbulence/rotation parameters. The model is used to predict fully-developed rotating channel flow over a diverse range of rotation numbers. In addition, predictions are obtained for a developing channel flow with high rotation. The predictions are compared with the experimental data available. Good predictions are achieved for mean velocity and wall shear stress over most of the rotation speeds tested. There is some prediction breakdown at high rotation (rotation number greater than .10) where the effects of the rotation on turbulence become quite complex. At high rotation and low Reynolds number, the laminarization on the trailing side represents a complex effect of rotation which is difficult to predict with the described models.
Warfield, M. J.↗
A method of revitalizing Sampson's theory of the Galilean satellites
A method is developed by which Sampson's theory of motion of Jupiter's Galilean satellites can be revitalized by use of algebraic manipulation software on a digital computer. The technique seeks (1) to remove algebraic errors existing in the current Sampson theory, (2) to introduce some neglected effects due to solar interactions and the 3-7 commensurability between the outer two satellites, (3) to allow for nonzero amplitude and phase of the libration, (4) to allow future revision of the arbitrary constants of integration, (5) to express the final results as analytic functions of variations in the numerous arbitrary constants of integration and arbitrary parameters, and (6) to provide analytic partial derivatives by means of which the numerical values of coefficients in the expressions for the coordinates can be adjusted. The level of precision desired is one arc second (Jovicentric) for the coordinates (2, 3.3, 5.2, and 9.1 km, respectively, for satellites I through IV).
Lieske, J. H.↗
Summary of the functions and capabilities of the structural analysis and matrix interpretive system computer program
Functions and capabilities of large capacity structural analysis and matrix interpretive system digital computer program to analyze frame and shell structures
STRUCTURAL DYNAMICS↗
Computations of Internal and External Axisymmetric Nozzle Aerodynamics at Transonic Speeds
Computational Fluid Dynamics (CFD) analyses of axisymmetric circular-arc boattail nozzles have been completed in support of NASA's Next Generation Launch Technology Program to investigate the effects of high-speed nozzle geometries on the nozzle internal flow and the surrounding boattail regions. These computations span the very difficult transonic flight regime, with shock-induced separations and strong adverse pressure gradients. External afterbody and internal nozzle pressure distributions computed with the Wind code are compared with experimental data. A range of turbulence models were examined in Wind, including an Explicit Algebraic Stress model (EASM). Computations on two nozzle geometries have been completed at freestream Mach numbers ranging from 0.6 to 0.9, driven by nozzle pressure ratios (NPR) ranging from 2.9 to 5. Results obtained on converging-only geometry indicate reasonable agreement to experimental data, with the EASM and Shear Stress Transport (SST) turbulence models providing the best agreement. Calculations completed on a converging-diverging geometry involving large-scale internal flow separation did not converge to a true steady-state solution when run with variable timestepping (steady-state). Calculations obtained using constant timestepping (time-accurate) indicate less variations in flow properties compared with steady-state solutions. This failure to converge to a steady-state solution was found to be the result of difficulties in using variable time-stepping with large-scale separations present in the flow. Nevertheless, time-averaged boattail surface pressure coefficient and internal nozzle pressures show fairly good agreement with experimental data. The SST turbulence model demonstrates the best over-all agreement with experimental data.
Dalbello, Teryn↗
Hypersonic turbulent wall boundary layer computations
The Baldwin-Lomax algebraic turbulence model was modified for hypersonic flow conditions. Two coefficients in the outer layer eddy viscosity model were determined as functions of Mach number and temperature ratio. By matching the solutions from the Baldwin-Lomax model to those from the Cebeci-Smith model for a flat plate at hypersonic speed, the new values of the coefficient were obtained. The results show that the values of C sub cp and C sub kleb are functions of both Mach number and wall temperature ratio. The C sub cp and C sub kleb variations with Mach number and wall temperature were used for the calculations of both a 4 deg wedge flow at Mach 18 and an axisymmetric Mach 20 nozzle flow. The Navier-Stokes equations with thin layer approximation were solved for the above hypersonic flow conditions and the results were compared with existing experimental data. The agreement between the numerical solutions and the existing experimental data were good. The modified Baldwin-Lomax model thus is useful in the computations of hypersonic flows.
Kim, S. C.↗
Using Multiplicative Algebraic Reconstruction Techniques (MART) to Derive Instrument Requirements for Computed Tomography Imaging Spectrographs
Computed Tomography Imaging Spectrographs (CTISs), which are generally slitless or large-aperture spectrographs that observe a wide, dispersed field of view in multiple dispersion angles or diffraction orders, provide a unique opportunity to capture spectral information over an extended source, such as the Sun, but require software reconstruction techniques to be fully utilized. A useful data product that can be recovered from CTIS observations is intensity maps of solar features in single spectral lines. These intensity maps can then be used to determine temperature, density, abundance, and equilibrium properties of the emitting plasma. We apply the multiplicative algebraic reconstruction technique (MART) to example data from a variety of CTIS configurations to determine the capability and limitations of the method to return spectrally-pure maps. By completing this study, we aim to establish a path to derive requirements for CTIS instruments that rely on reconstruction techniques to meet their science objectives.
Amy Winebarger↗
Intelligent machines in the twenty-first century: foundations of inference and inquiry
The last century saw the application of Boolean algebra to the construction of computing machines, which work by applying logical transformations to information contained in their memory. The development of information theory and the generalization of Boolean algebra to Bayesian inference have enabled these computing machines, in the last quarter of the twentieth century, to be endowed with the ability to learn by making inferences from data. This revolution is just beginning as new computational techniques continue to make difficult problems more accessible. Recent advances in our understanding of the foundations of probability theory have revealed implications for areas other than logic. Of relevance to intelligent machines, we recently identified the algebra of questions as the free distributive algebra, which will now allow us to work with questions in a way analogous to that which Boolean algebra enables us to work with logical statements. In this paper, we examine the foundations of inference and inquiry. We begin with a history of inferential reasoning, highlighting key concepts that have led to the automation of inference in modern machine-learning systems. We then discuss the foundations of inference in more detail using a modern viewpoint that relies on the mathematics of partially ordered sets and the scaffolding of lattice theory. This new viewpoint allows us to develop the logic of inquiry and introduce a measure describing the relevance of a proposed question to an unresolved issue. Last, we will demonstrate the automation of inference, and discuss how this new logic of inquiry will enable intelligent machines to ask questions. Automation of both inference and inquiry promises to allow robots to perform science in the far reaches of our solar system and in other star systems by enabling them not only to make inferences from data, but also to decide which question to ask, which experiment to perform, or which measurement to take given what they have learned and what they are designed to understand.
Review↗