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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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NASA NTRS1973

Use of three-dimensional photoelasticity in fracture mechanics

The philosophy of fracture mechanics is reviewed and utilized to formulate a simplified approach to the determination of the stress-intensity factor photoelastically for three-dimensional problems. The method involves a Taylor Series correction for the maximum in-plane shear stress (TSCM) and does not involve stress separation. The results are illustrated by applying the TSCM to surface flaws in bending fields. Other three-dimensional problems solved by the TSCM are cited.

Smith, C. W.↗
NASA NTRS1973

Error analysis of multi-conic techniques

A general error analysis of three recently developed multi-conic methods of three-body trajectory integration has been carried out. Single-step error functions for position and velocity have been derived as Taylor series in powers of the time step and also in integral form. These error functions are used to investigate the relative accuracy of the three methods in various regions of the earth-moon space and to provide a method of variable step size control for the trajectory integration procedure. Numerical results are used to compare the multi-step performance of the methods for both large and small step sizes.

D'Amario, L. A.↗
NASA NTRS1974

The modified equation approach to the stability and accuracy analysis of finite-difference methods

The stability and accuracy of finite-difference approximations to simple linear partial differential equations are analyzed by studying the modified partial differential equation. Aside from round-off error, the modified equation represents the actual partial differential equation solved when a numerical solution is computed using a finite-difference equation. The modified equation is derived by first expanding each term of a difference scheme in a Taylor series and then eliminating time derivatives higher than first order by certain algebraic manipulations. The connection between 'heuristic' stability theory based on the modified equation approach and the von Neumann (Fourier) method is established. In addition to the determination of necessary and sufficient conditions for computational stability, a truncated version of the modified equation can be used to gain insight into the nature of both dissipative and dispersive errors.

Warming, R. F.↗
NASA NTRS1973

Digital approximation of continuous-data control systems by point-by-point state comparison

This paper presents a point-by-point state comparison method of approximating a continuous-data system by a sampled-data system. The problem is to attempt the matching of the states of the two systems at the sampling instants. A partial matching has to be conducted if the systems have more states than controls. A weighting matrix is used to regulate the partial matching and weights placed on each state. The digital approximation is affected by use of forward gain E(T) and feedback gain G(T) in the sampled-data system. It is shown that, in general, these gains can be approximated by truncated Taylor series expansions. An illustrative example is given using the one-axis dynamics of the Skylab satellite.

Kuo, B. C.↗
NASA NTRS1975

Electrocardiogram signal analyzer

Algorithm based on Taylor series expansion of Fourier transform has been developed and used for detection of cardiac arrhythmias in real-time electrocardiogram signal.

Portnoy, W. M.↗
NASA NTRS1974

Some approximation concepts for structural synthesis

An efficient automated minimum weight design procedure is presented which is applicable to sizing structural systems that can be idealized by truss, shear panel, and constant strain triangles. Static stress and displacement constraints under alternative loading conditions are considered. The optimization algorithm is an adaptation of the method of inscribed hyperspheres and high efficiency is achieved by using several approximation concepts including temporary deletion of noncritical constraints, design variable linking, and Taylor series expansions for response variables in terms of design variables. Optimum designs for several planar and space truss examples problems are presented. The results reported support the contention that the innovative use of approximation concepts in structural synthesis can produce significant improvements in efficiency.

Schmit, L. A., Jr.↗
NASA NTRS1974

Some self starting integrators for x Prime equals f (x, t)

The integration is discussed of the vector differential equation X = F(x, t) from time t sub i to t sub (i = 1) where only the values of x sub i are available for the the integration. No previous values of x or x prime are used. Using an orbit integration problem, comparisons are made between Taylor series integrators and various types and orders of Runge-Kutta integrators. A fourth order Runge-Kutta type integrator for orbital work is presented, and approximate (there may be no exact) fifth order Runge-Kutta integrators are discussed. Also discussed and compared is a self starting integrator ising delta f/delta x. A numerical method for controlling the accuracy of integration is given, and the special equations for accurately integrating accelerometer data are shown.

Lear, W. M.↗
NASA NTRS1975

Higher order accurate partial implicitization: An unconditionally stable fourth-order-accurate explicit numerical technique

The previously obtained second-order-accurate partial implicitization numerical technique used in the solution of fluid dynamic problems was modified with little complication to achieve fourth-order accuracy. The Von Neumann stability analysis demonstrated the unconditional linear stability of the technique. The order of the truncation error was deduced from the Taylor series expansions of the linearized difference equations and was verified by numerical solutions to Burger's equation. For comparison, results were also obtained for Burger's equation using a second-order-accurate partial-implicitization scheme, as well as the fourth-order scheme of Kreiss.

Graves, R. A., Jr.↗
NASA NTRS1975

Gravitational collapse of a turbulent vortex with application to star formation

The gravitational collapse of a rotating cloud or vortex is analyzed by expanding the dependent variables in the equations of motion in two-dimensional Taylor series in the space variables. It is shown that the gravitation and rotation terms in the equations are of first order in the space variables, the pressure gradient terms are of second order, and the turbulent viscosity term is of third order. The presence of a turbulent viscosity insures that the initial rotation is solid-body-like near the origin. The effect of pressure on the collapse process is found to depend on the shape of the initial density disturbance at the origin. Dimensionless collapse times, as well as the evolution of density and velocity, are calculated by solving numerically the system of nonlinear ordinary differential equations resulting from the series expansions. The axial inflow plays an important role and allows collapse to occur even when the rotation is large. An approximate solution of the governing partial differential equations is also given; the equations are used to study the spacial distributions of the density and velocity.

Deissler, R. G.↗
NASA NTRS1976

On signal design by the R sub 0 criterion for non-white Gaussian noise channels

The use of the R sub 0 criterion for modulation system design is investigated for channels with non-white Gaussian noise. A signal space representation of the waveform channel is developed, and the cut-off rate R sub 0 for vector channels with additive nonwhite Gaussian noise and unquantized demodulation is derived. When the signal unput to the channel is a continuous random vector, maximization of R sub 0 with constrained average signal energy leads to a water-filling interpretation of optimal energy distribution in signal space. The necessary condition for a finite signal set to maximize R sub 0 with constrained energy and an equally likely probability assignment of signal vectors is presented, and an algorithm is outlined for numerically computing the optimum signal set. A necessary condition on a constrained energy, finite signal set is found which maximizes a Taylor series approximation of R sub 0. This signal set is compared with the finite signal set which has the water-filling average energy distribution.

Bordelon, D. L.↗
NASA NTRS1976

On implementation of the extended interior penalty function

The extended interior penalty function formulation is implemented. A rational method for determining the transition between the interior and extended parts is set forth. The formulation includes a straightforward method for avoiding design points with some negative components, which are physically meaningless in structural analysis. The technique, when extended to problems involving parametric constraints, can facilitate closed form integration of the penalty terms over the most important parts of the parameter interval. The method lends itself well to the use of approximation concepts, such as design variable linking, constraint deletion and Taylor series expansions of response quantities in terms of design variables. Examples demonstrating the algorithm, in the context of planar orthogonal frames subjected to ground motion, are included.

Cassis, J. H.↗
NASA NTRS1975

Gravitational collapse of a turbulent vortex with application to star formation

The gravitational collapse of a rotating cloud or vortex is analyzed by expanding the dependent variables in the equations of motion in two-dimensional Taylor series in the space variables. It is shown that the gravitation and rotation terms in the equations are of first order in the space variables, the pressure gradient terms are of second order, and the turbulent viscosity term is of third order. The presence of a turbulent viscosity insures that the initial rotation is solid-body-like near the origin. The effect of pressure on the collapse process is found to depend on the shape of the intial density disturbance at the origin. Dimensionless collapse times, as well as the evolution of density and velocity, are calculated by solving numerically the system of nonlinear ordinary differential equations resulting from the series expansions. The axial inflow plays an important role and allows collapse to occur even when the rotation is large. An approximate solution of the governing partial differential equations is also given, in order to study the spacial distributions of the density and velocity.

Deissler, R. G.↗
NASA NTRS1976

Stress intensity factors for deep cracks emanating from the corner formed by a hole intersecting a plate surface

The stress intensity factors (SIFs) at the end points of flaws emanating from the corner formed by the intersection of a plate with a hole were determined using stress freezing photoelasticity and a numerical technique known as the Taylor series correction method to extract the SIF values from the photoelastic data. The geometries studied were crack depth to thickness ratios of about 0.2, 0.5, and 0.75; crack depth to crack length ratios of about 1.0 to 2.0; and crack length to hole radius ratios of about 0.5 to 2.0. The SIFs were determined at the intersection of the flaw border with the plate surface (KS) and with the edge of the hole (KH). It is shown that extension of a crack emanating from a corner of intersection of a hole with a plate under monotonically increasing load is not self-similar and that as the flaw depth increases, KH decreases and KS increases. Existing theories and design criteria significantly overestimate the SIF at both the hole and the surface except for shallow flaws at the hole and deep flaws at the surface.

Mcgowan, J. J.↗
NASA NTRS1976

Gravitational collapse of a turbulent vortex with application to star formation

The gravitational collapse of a rotating cloud or vortex is analyzed by expanding the dependent variables in the equations of motion in two-dimensional Taylor series in the space variables. It is shown that the gravitational and rotational terms in the equations are of first order in the space variables, the pressure-gradient terms are of second order, and the turbulent-viscosity term is of third order. The presence of turbulent viscosity ensures that the initial rotation is solid-body-like near the origin. The effect of pressure on the collapse process is found to depend on the shape of the initial density disturbance at the origin. Dimensionless collapse times, as well as the evolution of density and velocity, are calculated by solving numerically the system of nonlinear ordinary differential equations resulting from the series expansions. The axial flow is always inward and allows collapse to occur (axially) even when the rotation is large. An approximate solution of the governing partial differential equations is also given in order to study the spatial distributions of the density and velocity.

Deissler, R. G.↗
NASA NTRS1977

Some MACSYMA program for solving difference equations

A set of MACSYMA programs are described for finding closed form solutions to linear recurrence relations in equations having either constant or variable coefficients. In the homogenous case, a polymonial equation is obtained and the solution to the recurrence relation can be written as a linear combination of the roots of the polynomial. Exponential generating functions are used to solve variable coefficient relations. Taking successive derivatives and using the recurrence relation, an ordinary differential equation is obtained. Expanding the solution to the differential equation in a Taylor series, shows that the nth term of the series is the solution to the recurrence relation. For second order recurrences, a check is made for those that can be solved in terms of Bessel functions.

Ivie, J.↗
NASA NTRS1976

Numerical computation of transient coaxial entry tube flows

A numerical program was developed to compute transient laminar flows in two dimensions including multicomponent mixing and chemical reaction. The program can compute both incompressible flows and compressible flows at all speeds, and it is applied to describe transient and steady state solutions for low subsonic, coaxial entry, tue flows. Single component, nonreacting flows comprise most of the solutions, but one steady state solution is presented for trace concentration constituents engaging in a second order reaction. Numerical stability was obtained by adding at each calculation point a correction for numerical diffusion errors caused by truncation of the Taylor series used to finite difference the conservation equations. Transient computations were made for fluids initially at rest, then subjected to step velocity inputs that were uniform across each region of the entry plane and were held constant throughout the computation period. For center tube to annulus velocity ratios of 0.5 and 2.0, the bulk fluid in the tube initially moved in plug flow, but strong radial flows developed near the injection plane which moved the fluid into the high shear region between the jets and away from the tube wall.

Wieber, P. R.↗
NASA NTRS1976

Stress intensity factors for long, deep surface flaws in plates under extensional fields

Using a singular solution for a part circular crack by Smith, a Taylor Series Correction Method (TSCM) was verified for extracting stress intensity factors (SIFs) from photoelastic data. Photoelastic experiments were then conducted on plates with part circular and flat bottomed cracks for flaw depth to thickness ratios of 0.25, 0.50, and 0.75 and for equivalent flaw depth to equivalent ellipse length values ranging from 0.066 to 0.319. Experimental results agreed well with the Smith theory but indicated that the use of the 'equivalent' semi-elliptical flaw for correlating the part circular flaw results with semi-elliptical flaw results was not valid for a/2c much less than 0.20. Best overall agreement for the moderate (a/t about 0.5) to deep flaws (a/t about 0.75) and a/2c greater than 0.15 was found with a semi-empirical theory due to Newman when compared on the basis of equivalent flaw depth and area. The Smith theory, when correlated on the basis of flaw depth and area, appears to yield reasonable estimates (within 10%) of the SIF for flat bottomed flaws for the geometries studied here.

Harms, A. E.↗
NASA NTRS1977

Polynomial interpolation methods for viscous flow calculations

Higher-order collocation procedures which result in block-tridiagonal matrix systems are derived from (1) Taylor series expansions and from (2) polynomial interpolation, and the relationships between the two formulations, called respectively Hermite and spline collocation, are investigated. A Hermite block-tridiagonal system for a nonuniform mesh is derived, and the Hermite approach is extended in order to develop a variable-mesh sixth-order block-tridiagonal procedure. It is shown that all results obtained by Hermite development can be recovered by appropriate spline polynomial interpolation. The additional boundary conditions required for these higher-order procedures are also given. Comparative solutions using second-order accurate finite difference and spline and Hermite formulations are presented for the boundary layer on a flat plate, boundary layers with uniform and variable mass transfer, and the viscous incompressible Navier-Stokes equations describing flow in a driven cavity.

Rubin, S. G.↗