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All sourcesNASA NTRSDOE OSTI
NASA NTRS1993

Development of an unstructured solution adaptive method for the quasi-three-dimensional Euler and Navier-Stokes equations

A general solution adaptive scheme-based on a remeshing technique is developed for solving the two-dimensional and quasi-three-dimensional Euler and Favre-averaged Navier-Stokes equations. The numerical scheme is formulated on an unstructured triangular mesh utilizing an edge-based pointer system which defines the edge connectivity of the mesh structure. Jameson's four-stage hybrid Runge-Kutta scheme is used to march the solution in time. The convergence rate is enhanced through the use of local time stepping and implicit residual averaging. As the solution evolves, the mesh is regenerated adaptively using flow field information. Mesh adaptation parameters are evaluated such that an estimated local numerical error is equally distributed over the whole domain. For inviscid flows, the present approach generates a complete unstructured triangular mesh using the advancing front method. For turbulent flows, the approach combines a local highly stretched structured triangular mesh in the boundary layer region with an unstructured mesh in the remaining regions to efficiently resolve the important flow features. One-equation and two-equation turbulence models are incorporated into the present unstructured approach. Results are presented for a wide range of flow problems including two-dimensional multi-element airfoils, two-dimensional cascades, and quasi-three-dimensional cascades. This approach is shown to gain flow resolution in the refined regions while achieving a great reduction in the computational effort and storage requirements since solution points are not wasted in regions where they are not required.

Jiang, Yi-Tsann↗
NASA NTRS2011

A Starshade Petal Error Budget for Exo-Earth Detection and Characterization

We present a starshade error budget with engineering requirements that are well within the current manufacturing and metrology capabilities. The error budget is based on an observational scenario in which the starshade spins about its axis on timescales short relative to the zodi-limited integration time, typically several hours. The scatter from localized petal errors is smoothed into annuli around the center of the image plane, resulting in a large reduction in the background flux variation while reducing thermal gradients caused by structural shadowing. Having identified the performance sensitivity to petal shape errors with spatial periods of 3-4 cycles/petal as the most challenging aspect of the design, we have adopted and modeled a manufacturing approach that mitigates these perturbations with 1-meter-long precision edge segments positioned using commercial metrology that readily meets assembly requirements. We have performed detailed thermal modeling and show that the expected thermal deformations are well within the requirements as well. We compare the requirements for four cases: a 32 meter diameter starshade with a 1.5 meter telescope, analyzed at 75 and 90 milliarcseconds, and a 40 meter diameter starshade with a 4 meter telescope, analyzed at 60 and 75 milliarcseconds.

error budget↗
NASA NTRS1985

The role of a low Earth orbiter in intercontinental time synchronization via GPS satellites

Time synchronization between two sites using differential GPS has been investigated by a number of researchers. When the two sites are widely separated, the common view period of any GPS satellite becomes shorter; low elevation observations are inevitable. This increase the corrupting effects of the atmospheric delay and, at the same time, narrows the window for such time synchronization. This difficulty can be alleviated by synchronization. This difficulty can be alleviated by using a transit site located midway between the two main sites. The main sites can now look at different GPS satellites which are also in view at the transit site. However, a ground transit site may not always be conveniently available, especially across the Pacific Ocean; also, the inclusion of a ground transit site introduce additional errors due to its location error and local atmospheric delay. An alternative is to use a low Earth orbiter (LEO) as the transit site. A LEO is superior to a ground transit site in three ways: (1) It covers a large part of the Earth in a short period of time and, hence, a single LEO provides worldwide transit services; (2) it is above the troposphere and thus its inclusion does not introduce additional tropospheric delay error; and (3) it provides strong dynamics needed to improve GPS satellite positions which are of importance to ultraprecise time synchronization.

Wu, S. C.↗
NASA NTRS1996

Adaptive grid methods for RLV environment assessment and nozzle analysis

Rapid access to highly accurate data about complex configurations is needed for multi-disciplinary optimization and design. In order to efficiently meet these requirements a closer coupling between the analysis algorithms and the discretization process is needed. In some cases, such as free surface, temporally varying geometries, and fluid structure interaction, the need is unavoidable. In other cases the need is to rapidly generate and modify high quality grids. Techniques such as unstructured and/or solution-adaptive methods can be used to speed the grid generation process and to automatically cluster mesh points in regions of interest. Global features of the flow can be significantly affected by isolated regions of inadequately resolved flow. These regions may not exhibit high gradients and can be difficult to detect. Thus excessive resolution in certain regions does not necessarily increase the accuracy of the overall solution. Several approaches have been employed for both structured and unstructured grid adaption. The most widely used involve grid point redistribution, local grid point enrichment/derefinement or local modification of the actual flow solver. However, the success of any one of these methods ultimately depends on the feature detection algorithm used to determine solution domain regions which require a fine mesh for their accurate representation. Typically, weight functions are constructed to mimic the local truncation error and may require substantial user input. Most problems of engineering interest involve multi-block grids and widely disparate length scales. Hence, it is desirable that the adaptive grid feature detection algorithm be developed to recognize flow structures of different type as well as differing intensity, and adequately address scaling and normalization across blocks. These weight functions can then be used to construct blending functions for algebraic redistribution, interpolation functions for unstructured grid generation, forcing functions to attract/repel points in an elliptic system, or to trigger local refinement, based upon application of an equidistribution principle. The popularity of solution-adaptive techniques is growing in tandem with unstructured methods. The difficultly of precisely controlling mesh densities and orientations with current unstructured grid generation systems has driven the use of solution-adaptive meshing. Use of derivatives of density or pressure are widely used for construction of such weight functions, and have been proven very successful for inviscid flows with shocks. However, less success has been realized for flowfields with viscous layers, vortices or shocks of disparate strength. It is difficult to maintain the appropriate mesh point spacing in the various regions which require a fine spacing for adequate resolution. Mesh points often migrate from important regions due to refinement of dominant features. An example of this is the well know tendency of adaptive methods to increase the resolution of shocks in the flowfield around airfoils, but in the incorrect location due to inadequate resolution of the stagnation region. This problem has been the motivation for this research.

Thornburg, Hugh J.↗
NASA NTRS1991

Application of the Global Positioning System to crustal deformation measurement. II - The influence of errors in orbit determination networks

The errors introduced into GPS measurements of a geodetic network by means of adopting different sets of stations in central and southern California as fixed points (termed fiducial stations) were investigated. Solutions for a crustal deformation network spanning distances up to 350 km were computed with a variety of fiducial networks, using fiducial coordinates determined from VLBI. Results showed that three receivers collocated with VLBI sites provide a stable reference system at a few millimeter level in the horizontal components and at 10 mm in the vertical components, indicating that local survey errors between VLBI and GPS markers at fiducial sites do not severely affect the estimates of interstation vectors.

Larson, Kristine M.↗
NASA NTRS1972

Generation and application of the equations of condition for high order Runge-Kutta methods

This thesis develops the equations of condition necessary for determining the coefficients for Runge-Kutta methods used in the solution of ordinary differential equations. The equations of condition are developed for Runge-Kutta methods of order four through order nine. Once developed, these equations are used in a comparison of the local truncation errors for several sets of Runge-Kutta coefficients for methods of order three up through methods of order eight.

Haley, D. C.↗
NASA NTRS1973

Mixed finite-difference scheme for analysis of simply supported thick plates.

A mixed finite-difference scheme is presented for the stress and free vibration analysis of simply supported nonhomogeneous and layered orthotropic thick plates. The analytical formulation is based on the linear, three-dimensional theory of orthotropic elasticity and a Fourier approach is used to reduce the governing equations to six first-order ordinary differential equations in the thickness coordinate. The governing equations possess a symmetric coefficient matrix and are free of derivatives of the elastic characteristics of the plate. In the finite difference discretization two interlacing grids are used for the different fundamental unknowns in such a way as to reduce both the local discretization error and the bandwidth of the resulting finite-difference field equations. Numerical studies are presented for the effects of reducing the interior and boundary discretization errors and of mesh refinement on the accuracy and convergence of solutions. It is shown that the proposed scheme, in addition to a number of other advantages, leads to highly accurate results, even when a small number of finite difference intervals is used.

Noor, A. K.↗
NASA NTRS1974

Classical seventh-, sixth-, and fifth-order Runge-Kutta-Nystrom formulas with stepsize control for general second-order differential equations

Runge-Kutta-Nystrom formulas of the seventh, sixth, and fifth order were derived for the general second order (vector) differential equation written as the second derivative of x = f(t, x, the first derivative of x). The formulas include a stepsize control procedure, based on a complete coverage of the leading term of the local truncation error in x, and they require no more evaluations per step than the earlier Runge-Kutta formulas for the first derivative of x = f(t, x). The developed formulas are expected to be time saving in comparison to the Runge-Kutta formulas for first-order differential equations, since it is not necessary to convert the second-order differential equations into twice as many first-order differential equations. The examples shown saved from 25 percent to 60 percent more computer time than the earlier formulas for first-order differential equations, and are comparable in accuracy.

Fehlberg, E.↗
NASA NTRS1973

The Lyapunov stabilization of satellite equations of motion using integrals

A method is introduced that weakens the Lyapunov or in track instability of satellite equations of motion. The method utilizes a linearized energy integral of satellite motion as a constraint on solutions obtained by numerical integration. The procedure prevents local numerical error from altering the frequency associated with the fast angular variable and thereby reduces the Lyapunov instability and the global numerical error. Applications of the method to satellite motion show accuracy improvements of two to three orders of magnitude in position and velocity after 50 revolutions. A modification of the method is presented that allows the use of slowly varying integrals of motion.

Nacozy, P. E.↗
NASA NTRS1974

Time elements

Time elements are introduced for use with Sundman time transformations of the type dt = r(alpha)ds for satellite equations of motion. Two time elements are given, one providing maximum accuracy for alpha = 1, the other for alpha = 2. Time elements and time transformations reduce local truncation error and Liapunov (in track) instability, and provide analytical step size control. Numerical results show accuracy improvements of more than one order of magnitude when time elements are employed with time transformations in the numerical integration of the satellite equations, compared with using time transformations alone.

Nacozy, P. E.↗
NASA NTRS1975

Time elements

Time elements are introduced for use with Sundman time transformations for satellite equations of motion. Two time elements are presented, one providing maximum accuracy when the exponent of the governing equation equals 1, the other when the exponent equals 2. Time elements and time transformations reduce local truncation error and Liapunov (in-track) instability, and provide analytical step size control. Numerical results show accuracy improvements of more than one order of magnitude when time elements are employed with time transformations in the numerical integration of the satellite equations, compared with using time transformations alone.

Nacozy, P. E.↗
NASA NTRS1979

Optimal nodal point distribution for improved accuracy in computational fluid dynamics

In applying finite-difference techniques to flow field problems, the accuracy attained for a fixed number of node points can be improved using unequally-spaced node points. The distribution of these node points is chosen here by minimizing a measure of local truncation error with respect to the parameters which define a transformation between the computational space of equally-spaced node points and the physical space of unequally-spaced node points. The problem then becomes a nonlinear programming problem. Numerical results are presented for two one-dimensional test problems: the Blasius boundary layer problem and the inviscid Burgers' equation.

Pierson, B. L.↗
NASA NTRS1979

Map characteristics of Landsat mosaics

Map characteristics of the Landsat mosaics developed at JPL are considered. Procedures for digital mosaicking of Landsat frames to standard map projections were used to mosaic at full resolution ten scenes over the California desert region and twenty-one scenes over Arizona. The procedures are analyzed for horizontal positioning error (global and local) and the potential for classification error associated with the adjustment of brightness of Z values between frames; the use of this technology for the mapping of extensive features is discussed. Mosaicking facilities, techniques, mapping accuracy, and thematic mapping characteristics are described. A comparative analysis of Landsat mosaicking technology developed at Goddard Space Flight Center, IBM Gaithersburg, and USGS Flagstaff is made, and suggestions are given for algorithm development to improve systems capacity and ability to handle a variety of cases.

Zobrist, A. L.↗
NASA NTRS1979

An accurate method for two-point boundary value problems

A second-order method for solving two-point boundary value problems on a uniform mesh is presented where the local truncation error is obtained for use with the deferred correction process. In this simple finite difference method the tridiagonal nature of the classical method is preserved but the magnitude of each term in the truncation error is reduced by a factor of two. The method is applied to a number of linear and nonlinear problems and it is shown to produce more accurate results than either the classical method or the technique proposed by Keller (1969).

Walker, J. D. A.↗
NASA NTRS1980

Grid generation for time dependent problems: Criteria and methods

The problem of generating local mesh refinements when solving time dependent partial differential equations was examined. The problem of creating an appropriate grid, given a mesh function h defined over the spatial domain is discussed. A data structure which permits efficient use of the resulting grid is described. A good choice for h is an estimate of the local truncation error, and several ways to estimate it are discussed. The efficiency and implementation problems of these error estimates were compared.

Berger, M.↗
NASA NTRS1981

Influence of boundary approximations and conditions on finite difference solutions

Numerical representations of boundary approximations and conditions for three problems are investigated to determine the resulting global accuracy of the steady state solution. Numerical accuracy with various boundary approximations is determined for quasi-one-dimensional inviscid flow in a duct with the interior grid points evaluated using the MacCormack scheme. When an extrapolation approximation with first order local truncation error is used, the global second order accuracy of the difference scheme can be destroyed. For one dimensional flow in a porous medium, an implicit midpoint difference scheme which is consistent with the boundary conditions is developed without the need of boundary approximations. A dissipative model problem is solved with the boundary conditions discretized with first and second order accuracy. The overall second order accuracy of the difference scheme is destroyed if first order numerical representation of one of the boundary conditions is used. With a boundary approximation, the second order global accuracy of the model problem is retained if either second order extrapolation or first order representation of the governing equation is used.

Blottner, F. G.↗
NASA NTRS1984

Recent advances in methods for numerical solution of O.D.E. initial value problems

In the mathematical modeling of physical systems, it is often necessary to solve an initial value problem (IVP), consisting of a system of ordinary differential equations (ODE). A typical program produces approximate solutions at certain mesh points. Almost all existing codes try to control the local truncation error, while the user is really interested in controlling the true or global error. The present investigation provides a review of recent advances regarding the solution of the IVP, giving particular attention to stiff systems. Stiff phenomena are customarily defined in terms of the eigenvalues of the Jacobian. There are, however, some difficulties connected with this approach. It is pointed out that an estimate of the Lipschitz constant proves to be a very practical way to determine the stiffness of a problem.

Bui, T. D.↗
NASA NTRS1987

Loran-C approach guidance project current status

There are four areas of work in the Loran-C flight test project. Current results provide performance data on the effects of Signal Noise Ratio (SNR) on the dynamic performance of the receiver filters for Loran-C data, and data on Loran-C grid deformation at a microscale of 100 meters. The Loran-C receiver provides a line of position (LOP) Master and Slave transmitter at an angle 0 to magnetic north. No transformation to latitude-longitude reference frame is required since this is the major source of Loran-C navigation errors. A local coordinate frame is established centered at touchdown point on the runway with directions along and across the runway. A Loran-C data collection system was set up. The Loran-C data are sent directly to an Apple II computer with a 12 inch monitor. The effect of SNR on Loran-C precision is shown for two receiver filters of different frequency response. A set of ground level static readings of touchdown was taken around Hanscom Field and transferred to an accurate detailed layout drawing; this showed local distortions of the average touchdown values.

Elias, A. L.↗