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Results for “second-order accuracy”

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 NTRS1974

The influence of the computational mesh on accuracy for initial value problems with discontinuous or nonunique solutions

Discontinuous, or weak, solutions of the wave equation, the inviscid form of Burgers equation, and the time-dependent, two-dimensional Euler equations are studied. A numerical method of second-order accuracy in two forms, differential and integral, is used to calculate the weak solutions of these equations for several initial value problems, including supersonic flow past a wedge, a double symmetric wedge, and a sphere. The effect of the computational mesh on the accuracy of computed weak solutions including shock waves and expansion phenomena is studied. Modifications to the finite-difference method are presented which aid in obtaining desired solutions for initial value problems in which the solutions are nonunique.

Maccormack, R. W.↗
NASA NTRS1972

Numerical solutions of the Navier-Stokes equations for the supersonic laminar flow over a two-dimensional compression corner

Numerical solutions have been obtained for the supersonic, laminar flow over a two-dimensional compression corner. These solutions were obtained as steady-state solutions to the unsteady Navier-Stokes equations using the finite difference method of Brailovskaya, which has second-order accuracy in the spatial coordinates. Good agreement was obtained between the computed results and wall pressure distributions measured experimentally for Mach numbers of 4 and 6.06, and respective Reynolds numbers, based on free-stream conditions and the distance from the leading edge to the corner. In those calculations, as well as in others, sufficient resolution was obtained to show the streamline pattern in the separation bubble. Upstream boundary conditions to the compression corner flow were provided by numerically solving the unsteady Navier-Stokes equations for the flat plate flow field, beginning at the leading edge. The compression corner flow field was enclosed by a computational boundary with the unknown boundary conditions supplied by extrapolation from internally computed points.

Carter, J. E.↗
NASA NTRS1973

Development and application of a local linearization algorithm for the integration of quaternion rate equations in real-time flight simulation problems

High angular rates encountered in real-time flight simulation problems may require a more stable and accurate integration method than the classical methods normally used. A study was made to develop a general local linearization procedure of integrating dynamic system equations when using a digital computer in real-time. The procedure is specifically applied to the integration of the quaternion rate equations. For this application, results are compared to a classical second-order method. The local linearization approach is shown to have desirable stability characteristics and gives significant improvement in accuracy over the classical second-order integration methods.

Barker, L. E., Jr.↗
NASA NTRS1973

Application of matched asymptotic expansions to lunar and interplanetary trajectories. Volume 1: Technical discussion

Previously published asymptotic solutions for lunar and interplanetary trajectories have been modified and combined to formulate a general analytical solution to the problem on N-bodies. The earlier first-order solutions, derived by the method of matched asymptotic expansions, have been extended to second order for the purpose of obtaining increased accuracy. The derivation of the second-order solution is summarized by showing the essential steps, some in functional form. The general asymptotic solution has been used as a basis for formulating a number of analytical two-point boundary value solutions. These include earth-to-moon, one- and two-impulse moon-to-earth, and interplanetary solutions. The results show that the accuracies of the asymptotic solutions range from an order of magnitude better than conic approximations to that of numerical integration itself. Also, since no iterations are required, the asymptotic boundary value solutions are obtained in a fraction of the time required for comparable numerically integrated solutions. The subject of minimizing the second-order error is discussed, and recommendations made for further work directed toward achieving a uniform accuracy in all applications.

Lancaster, J. E.↗
NASA NTRS1972

Numerical analysis of the asymptotic two-point boundary value solution for N-body trajectories.

Previously published asymptotic solutions for lunar and interplanetary trajectories have been modified and combined to formulate a general analytical boundary value solution applicable to a broad class of trajectory problems. In addition, the earlier first-order solutions have been extended to second-order to determine if improved accuracy is possible. Comparisons between the asymptotic solution and numerical integration for several lunar and interplanetary trajectories show that the asymptotic solution is generally quite accurate. Also, since no iterations are required, a solution to the boundary value problem is obtained in a fraction of the time required for numerically integrated solutions.

Lancaster, J. E.↗
NASA NTRS1957

Comparison of several methods for obtaining the time response of linear systems to either a unit impulse or arbitrary input from frequency-response data

Several methods of obtaining the time response of Linear systems to either a unit impulse or an arbitrary input from frequency-response data are described and compared. Comparisons indicate that all the methods give good accuracy when applied to a second-order system; the main difference is the required computing time. The methods generally classified as inverse Laplace transform methods were found to be most effective in determining the response to a unit impulse from frequency-response data of higher order systems. Some discussion and examples are given of the use of such methods as flight-data-analysis techniques in predicting loads and motions of a flexible aircraft on the basis of simple calculations when the aircraft frequency response is known.

Donegan, James J↗
NASA NTRS1973

Application of matched asymptotic expansions to lunar and interplanetary trajectories. Volume 2: Derivations of second-order asymptotic boundary value solutions

Previously published asymptotic solutions for lunar and interplanetery trajectories have been modified and combined to formulate a general analytical solution to the problem of N-bodies. The earlier first-order solutions, derived by the method of matched asymptotic expansions, have been extended to second order for the purpose of obtaining increased accuracy. The complete derivation of the second-order solution, including the application of a regorous matching principle, is given. It is shown that the outer and inner expansions can be matched in a region of order mu to the alpha power, where 2/5 alpha 1/2, and mu (the moon/earth or planet/sun mass ratio) is much less than one. The second-order asymptotic solution has been used as a basis for formulating a number of analytical two-point boundary value solutions. These include earth-to-moon, one- and two-impulse moon-to-Earth, and interplanetary solutions. Each is presented as an explicit analytical solution which does not require iterative steps to satisfy the boundary conditions. The complete derivation of each solution is shown, as well as instructions for numerical evaluation. For Vol. 1, see N73-27738.

Lancaster, J. E.↗
NASA NTRS1971

On strain energy and constitutive relations for alkali metals.

An expression for the strain energy as a continuous differentiable function of the Green-Cauchy deformation tensor is obtained for the alkali metals at absolute zero temperature. The development is based on well established quantum and classical calculations of the various contributions to the crystal energy. Stress-deformation relations are next obtained. As a check on the accuracy of the strain energy, theoretical calculations of the values of the second-order elastic coefficients are obtained and compared to known experimental data. The predicted values are shown to compare quite well with the experimental values.

Eftis, J.↗
NASA NTRS1974

Fast direct numerical solution of the nonhomogeneous Cauchy-Riemann equations

A fast direct (noniterative) 'Cauchy-Riemann Solver' is developed for solving the finite-difference equations representing systems of first-order elliptic partial differential equations in the form of the nonhomogeneous Cauchy-Riemann equations. The method is second-order accurate and requires approximately the same computer time as a fast cyclic-reduction Poisson solver. The accuracy and efficiency of the direct solver are demonstrated in an application to solving an example problem in aerodynamics: subsonic inviscid flow over a biconvex airfoil. The analytical small-perturbation solution contains singularities, which are captured well by the computational technique. The algorithm is expected to be useful in nonlinear subsonic and transonic aerodynamics.

Lomax, H.↗
NASA NTRS1973

Pogo suppression on space shuttle - early studies

Preliminary studies for pogo prevention on the shuttle vehicle are reported. The importance of the effect of oscillatory outflow from a hydroelastic tank is displayed in terms of excitation of normal modes for a structure containing that tank assuming its outlet is closed. Evaluation of an approximate propulsion frequency response at undamped feedline resonance reveals the conditions for which the contribution of tank outflow is destabilizing and also provides a criterion for identifying those structural modes which are of potential significance for system stability. Various finite-element and normal-mode models for hydraulic feedlines are evaluated relative to accuracy of admittances of a long line. A procedure is recommended for modeling a feed system to minimize the required number of second-order equations. Specific recommendations are made for the analytical estimation of pump cavitation compliance and a first estimate for the shuttle pumps is given. Weakness in past practices of pump testing are identified and a new three-phase program is proposed. Finally results of numerical studies on the early vehicle configuration are presented. It is concluded that an accumulator between the boost and main pump offers promise of higher effectiveness than one at the engine inlet.

Rubin, S.↗
NASA NTRS1967

Subsonic and Supersonic Flutter Analysis of a Highly Tapered Swept-Wing Planform, Including Effects of Density Variation and Finite Wing Thickness, and Comparison with Experiments

The flutter characteristics of several wings with an aspect-ratio of 4.0, a taper ratio of 0.2, and a quarter-chord sweepback of 45 deg. have been investigated analytically for Mach numbers up to 2.0. The calculations were based on the modified-strip-analysis method, the subsonic-kernel-function method, piston theory, and quasi-steady second-order theory. Results of t h e analysis and comparisons with experiment indicated that: (1) Flutter speeds were accurately predicted by the modified strip analysis, although accuracy at t h e highest Mach numbers required the use of nonlinear aerodynamic theory (which accounts for effects of wing thickness) for the calculation of the aerodynamic parameters. (2) An abrupt increase of flutter-speed coefficient with increasing Mach number, observed experimentally in the transonic range, was also indicated by the modified strip analysis. (3) In the low supersonic range for some densities, a discontinuous variation of flutter frequency with Mach number was indicated by the modified strip analysis. An abrupt change of frequency appeared experimentally in the transonic range. (4) Differences in flutter-speed-coefficient levels obtained from tests at low supersonic Mach numbers in two wind tunnels were also predicted by the modified strip analysis and were shown to be caused primarily by differences in mass ratio. (5) Flutter speeds calculated by the subsonic-kernel-function method were in good agreement with experiment and with the results of the modified strip analysis. (6) Flutter speed obtained from piston theory and from quasi-steady second-order theory were higher than experimental values by at least 38 percent.

Yates, Carson, Jr.↗
NASA NTRS1973

The analysis of nonequilibrium, chemically reacting, supersonic flow in three dimensions using a bicharacteristics method.

A second-order numerical method of characteristics based on a bicharacteristics scheme was developed for the evaluation of steady, supersonic, nonequilibrium, chemically reacting flows. The absolute accuracy and order of accuracy of the method was shown by comparisons with spherical source flow and axisymmetric, nonequilibrium flows. Numerical results were obtained for elliptical and super-elliptical nozzles. These results illustrate the complex nature of three-dimensional flows and the inadequacy of quasi-three-dimensional methods which neglect cross flows.

Cline, M. C.↗
NASA NTRS1952

Practical Calculation of Second-order Supersonic Flow past Nonlifting Bodies of Revolution

Calculation of second-order supersonic flow past bodies of revolution at zero angle of attack is described in detail, and reduced to routine computation. Use of an approximate tangency condition is shown to increase the accuracy for bodies with corners. Tables of basic functions and standard computing forms are presented. The procedure is summarized so that one can apply it without necessarily understanding the details of the theory. A sample calculation is given, and several examples are compared with solutions calculated by the method of characteristics.

Van Dyke, Milton D↗
NASA NTRS1972

Classical eight- and lower-order Runge-Kutta-Nystroem formulas with stepsize control for special second-order differential equations

The formulas include a stepsize control procedure, based on a complete coverage of the leading term of the truncation error in x. The formulas require fewer evaluations per stop than other Runge-Kutta-Nystrom formulas if the latter are operated by using the standard procedure for stepsize control. An example is presented. With results being of the same accuracy, Runge-Kutta-Nystrom formulas discussed save 50 percent or more computer time compared with other Runge-Kutta-Nystrom formulas.

Fehlberg, E.↗
NASA NTRS1972

Navigation strategy and filter design for solar electric missions

Methods which have been proposed to improve the navigation accuracy for the low-thrust space vehicle include modifications to the standard Sequential- and Batch-type orbit determination procedures and the use of inertial measuring units (IMU) which measures directly the acceleration applied to the vehicle. The navigation accuracy obtained using one of the more promising modifications to the orbit determination procedures is compared with a combined IMU-Standard. The unknown accelerations are approximated as both first-order and second-order Gauss-Markov processes. The comparison is based on numerical results obtained in a study of the navigation requirements of a numerically simulated 152-day low-thrust mission to the asteroid Eros. The results obtained in the simulation indicate that the DMC algorithm will yield a significant improvement over the navigation accuracies achieved with previous estimation algorithms. In addition, the DMC algorithms will yield better navigation accuracies than the IMU-Standard Orbit Determination algorithm, except for extremely precise IMU measurements, i.e., gyroplatform alignment .01 deg and accelerometer signal-to-noise ratio .07. Unless these accuracies are achieved, the IMU navigation accuracies are generally unacceptable.

Tapley, B. D.↗
NASA NTRS1973

Classical eighth- and lower-order Runge-Kutta-Nystroem formulas with a new stepsize control procedure for special second-order differential equations

New Runge-Kutta-Nystrom formulas of the eighth, seventh, sixth, and fifth order are derived for the special second-order (vector) differential equation x = f (t,x). In contrast to Runge-Kutta-Nystrom formulas of an earlier NASA report, these formulas provide a stepsize control procedure based on the leading term of the local truncation error in x. This new procedure is more accurate than the earlier Runge-Kutta-Nystrom procedure (with stepsize control based on the leading term of the local truncation error in x) when integrating close to singularities. Two central orbits are presented as examples. For these orbits, the accuracy and speed of the formulas of this report are compared with those of Runge-Kutta-Nystrom and Runge-Kutta formulas of earlier NASA reports.

Fehlberg, E.↗
NASA NTRS1973

Analysis of axisymmetrical vibration of a partially liquid-filled elastic sphere by the method of Green's function

The longitudinal dynamic behavior of launch vehicles is largely determined by their huge liquid propellant masses which are spring supported by the elastic tank walls. Throughout much of the powered flight time, these masses constitute a high percentage of the entire vehicle mass and, therefore, may dominate the fundamental modes of the vehicle. In this report, a spherical container is considered. The analysis is based on a Galerkin approach, in the course of which a second-order differential equation must be solved. The solution has been obtained by the method of Green's function. This method is favorable because it displays the manner in which the analysis can be extended to partially liquid-filled general shells of revolution. The computer programs currently available for partially liquid-filled propellant tanks are based on the finite element methods and result in analytical models having as many as several hundred degrees of freedom. The method applied in this report results in a model having less than 10 degrees of freedom as can be shown by numerical evaluation. Therefore, it will be possible to analyze propellant tanks using much less computer time with comparable accuracy.

Glaser, R. F.↗
NASA NTRS1965

Human Tracking Performance in Uncoupled and Coupled Two-Axis Systems

This report presents tile results of an experimental and analytical study of human performance in uncoupled and coupled control systems. Human pilot performance in single and two-axis systems was mathematically modeled by linear second-order describing functions. Model parameters were determined using model matching techniques. Analysis of the models showed that the amplitude ratio and phase lead of the describing function increased with training indicating an increase in open loop bandwidth. The phase margin also decreased with training. Increasing the plant lag time constant resulted in an increase in the model lead time constant and a decrease in the zero frequency gain. No significant difference was found to exist in the normalized tracking error per axis between the two-axis tasks and the single-axis tasks. However tile model lead time constant was significantly greater in two-axis tracking. Manual tracking of two-axis systems with cross-coupling was studied experimentally and analytically. Approximate methods for modeling two-axis performance were developed and checked using a precise spectral analysis approach. Coupled and uncoupled, symmetrical and asymmetrical two-axis performance was compared. The results show that modeling of cross-coupled systems is feasible and that trained subjects are capable of decoupling the axes of some systems. A methodology study compared the identification performance of continuous, iterative, and extrapolation model matching techniques. An iterative technique employing sensitivity equations for the generation of influence coefficients was found to be the best technique due to its excellent identification accuracy and ease of implementation. Convergence in iterative techniques can be improved substantially by equalizing the parameter adjustment rates and limiting the maximum correction per iteration.

Todosiev, E. P.↗