Engineering Papers⌕ Search

SEARCH · Engineering Papers

Results for “APPROXIMATIONS”

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.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 433 records · Page 24
All sourcesNASA NTRSDOE OSTI
NASA NTRS1983

Heavy ion transport in the straight ahead approximation

An as yet unsolved problem in space radiation protection is the necessary relation between the external cosmic ray heavy ion fluence and the resultant environment within the spacecraft. Such a relation involves the transport of such ions through extended materials. Presented is a derivation of the solution of the transport equation for heavy ions in the straight ahead approximation for directed beam applications. An iterative scheme for the solution of the inhomogeneous integral transport equations is applied to a neon ion beam in water. The iterative scheme requires transport coefficients as input, which are determined from the available data on ion nuclear reactions. The iteration converged for the neon beam in water. Thus this iterative scheme appears to be a suitable approximation of heavy ion transport.

Wilson, J. W.↗
NASA NTRS1982

Spline approximations for nonlinear hereditary control systems

A sline-based approximation scheme is discussed for optimal control problems governed by nonlinear nonautonomous delay differential equations. The approximating framework reduces the original control problem to a sequence of optimization problems governed by ordinary differential equations. Convergence proofs, which appeal directly to dissipative-type estimates for the underlying nonlinear operator, are given and numerical findings are summarized.

Daniel, P. L.↗
NASA NTRS1983

Approximation in control of flexible structures, theory and application

The sense in which the feedback control law based on an approximate finite dimensional model of a continuous structure approximates a control law which is optimal for the distributed, or infinite dimensional, model of the structure is studied. From the analysis of the various control and stability issues associated with this basis question, useful information for designing finite dimensional compensators which produce near-optimal performance in infinite dimensional systems is gained. Some of the important predictions that can be made about large-order finite dimensional control laws, using the theory of infinite dimensional Riccati equations are indicated.

Gibson, J. S.↗
NASA NTRS1983

Approximation of the optimal compensator for a large space structure

This paper considers the approximation of the optimal compensator for a Large Space Structure. The compensator is based upon a solution to the Linear Stochastic Quadratic Regulator problem. Colocation of sensors and actuators is assumed. A small gain analytical solution for the optimal compensator is obtained for a single input/single output system, i.e., certain terms in the compensator can be neglected for sufficiently small gain. The compensator is calculated in terms of the kernel to a Volterra integral operator using a Neumann series. The calculation of the compensator is based upon the C sub 0 semigroup for the infinite dimensional system. A finite dimensional approximation of the compensator is, therefore, obtained through analysis of the infinite dimensional compensator which is a compact operator.

Mackay, M. K.↗
NASA NTRS1982

An approximately 300 Mpc void of rich clusters of galaxies

Kirshner et al. (1981) have reported an apparent absence of relatively bright galaxies over an enormous volume in the direction of Bootes. This result, if statistically significant, has important implications for theories of galaxy clustering and formation. In connection with the study of the correlation function of rich clusters of galaxies, Bahcall and Soneira (1982) have noticed a large area in the sky, extending approximately 100 deg across, which is completely void of nearby, but not distant, rich Abell clusters. This apparent absence of rich clusters may represent a physical void. If this interpretation is correct, the void extends approximately 150 Mpc/h along the line of sight. The sample studied is Abell's (1958) statistical sample of clusters of galaxies. The distribution of rich clusters is discussed, taking into account observed versus expected characteristics. Attention is also given to aspects of absorption and galaxy counts in the considered region.

Bahcall, N. A.↗
NASA NTRS1983

Uniform asymptotic approximations for duct eigenfunctions in a thin boundary layer flow

Analytical approximations for the acoustic modes in a duct carrying a uniform core flow with a thin shear layer at the walls are developed using the Method of Matched Asymptotic Expansions. Both two-dimensional and cylindrical duct propagation are considered. Numerical results for eigenvalues calculated using the theory are presented for the two dimensional problem and compared with results from earlier analyses. It is found that the new approximations yield a significant increase in accuracy.

Myers, M. K.↗
NASA NTRS1983

Three-dimensional transonic nacelle/inlet flowfield computations using an efficient approximate factorization algorithm

A highly efficient computer analysis has been developed for predicting transonic nacelle/inlet flowfields. This algorithm can compute the three-dimensional transonic flowfield about axisymmetric or asymmetric nacelle/inlet configurations at zero or nonzero incidence. The flowfield is determined by solving the full-potential equation in conservative form on a body-fitted curvilinear computational mesh. The difference equations are solved using the AF2 approximate factorization scheme. The effects of boundary layer viscous entrainment are approximated in the inviscid algorithm by applying a surface transpiration velocity which is determined from the calculated boundary layer growth. Computed results and correlations with existing methods and experiment are presented to illustrate application of the analysis.

Vadyak, J.↗
NASA NTRS1983

Stability analysis of intermediate boundary conditions in approximate factorization schemes

In many cases, approximate factorization schemes have provided a significant increase in efficiency over previously used solution methods in certain problems. The present investigation is concerned with the importance of intermediate boundary conditions in approximate factorization schemes, taking into account a specific example regarding a boundary-induced stability restriction in a scheme for the transonic full-potential equation. The considered scheme has been discussed by Holst (1979). Holst's scheme is a variation of the AF2 schemes described by Ballhaus and Steger (1975). The application of the AF2 scheme to the two-dimensional Laplace equation in a rectangle is studied giving attention to the stability of the scheme in connection with various boundary conditions for the intermediate variable.

South, J. C., Jr.↗
NASA NTRS1984

Rational-spline approximation with automatic tension adjustment

An algorithm for weighted least-squares approximation with rational splines is presented. A rational spline is a cubic function containing a distinct tension parameter for each interval defined by two consecutive knots. For zero tension, the rational spline is identical to a cubic spline; for very large tension, the rational spline is a linear function. The approximation algorithm incorporates an algorithm which automatically adjusts the tension on each interval to fulfill a user-specified criterion. Finally, an example is presented comparing results of the rational spline with those of the cubic spline.

Schiess, J. R.↗
NASA NTRS1983

Accuracy of approximations to the Navier-Stokes equations

The systems of truncated differential equations that have been proposed to reduce the complexity and large computational costs of solutions to the full Navier-Stokes equations are considered. These systems are computationally efficient and capture all the physically relevant behavior. The systems follow a certain hierarchy: (1) the classical boundary-layer equations with specified edge properties (usually the streamwise pressure distribution); (2) the coupled boundary-layer/inviscid equations; (3) the so-called thin-layer equations that discard streamwise diffusion; and (4) the Navier-Stokes equations. Consideration is given to each of these approximations applied to an incompressible, laminar-separating flow at low and moderate Reynolds numbers. It is pointed out that for any flow or region of flow for which viscous-inviscid interaction effects are small, classical boundary-layer equations will provide a satisfactory description of the viscous flow at a fraction of the computational cost of any higher approximations.

Murphy, J. D.↗
NASA NTRS1985

Spline-based Rayleigh-Ritz methods for the approximation of the natural modes of vibration for flexible beams with tip bodies

Rayleigh-Ritz methods for the approximation of the natural modes for a class of vibration problems involving flexible beams with tip bodies using subspaces of piecewise polynomial spline functions are developed. An abstract operator theoretic formulation of the eigenvalue problem is derived and spectral properties investigated. The existing theory for spline-based Rayleigh-Ritz methods applied to elliptic differential operators and the approximation properties of interpolatory splines are useed to argue convergence and establish rates of convergence. An example and numerical results are discussed.

Rosen, I. G.↗
NASA NTRS1985

Intermediate boundary conditions for LOD, ADI and approximate factorization methods

A general approach to determining the correct intermediate boundary conditions for dimensional splitting methods is presented. The intermediate solution U is viewed as a second order accurate approximation to a modified equation. Deriving the modified equation and using the relationship between this equation and the original equation allows us to determine the correct boundary conditions for U*. This technique is illustrated by applying it to locally one dimensional (LOD) and alternating direction implicit (ADI) methods for the heat equation in two and three space dimensions. The approximate factorization method is considered in slightly more generality.

Leveque, R. J.↗
NASA NTRS1985

On the convergence of difference approximations to scalar conservation laws

A unified treatment of explicit in time, two level, second order resolution, total variation diminishing, approximations to scalar conservation laws are presented. The schemes are assumed only to have conservation form and incremental form. A modified flux and a viscosity coefficient are introduced and results in terms of the latter are obtained. The existence of a cell entropy inequality is discussed and such an equality for all entropies is shown to imply that the scheme is an E scheme on monotone (actually more general) data, hence at most only first order accurate in general. Convergence for total variation diminishing-second order resolution schemes approximating convex or concave conservation laws is shown by enforcing a single discrete entropy inequality.

Osher, S.↗
NASA NTRS1984

Higher-order parabolic approximations for sound propagation in stratified moving media

Asymptotic solutions of order k-n are developed for the equations governing the propagation of sound through a stratified moving medium. Here k is a dimensionless wave number and n is the arbitrary order of the approximation. These approximations are an extension of geometric acoustics theory and provide corrections to that theory in the form of multiplicative functions which satisfy parabolic partial differential equations. These corrections account for the diffraction effects caused by variation of the field normal to the ray path and the interaction of these transverse variations with the variation of the field along the ray. The theory is illustrated by application to simple examples.

Mcaninch, G. L.↗
NASA NTRS1984

Approximate analytic solutions for singular non-linear oscillators

Mickens (1981, 1984) has considered analytic techniques for obtaining approximate solutions to one-dimensional nonlinear oscillatory systems x(double-dot) + x = lambda f(x, x/dot/, lambda) where lambda is a small positive parameter and f is a nonlinear polynomial function of its arguments. However, in certain cases there is interest in the analysis of physical systems for which the nonlinear function f(x, x/dot/, lambda) is singular for finite values of x or x(dot). The present investigation is concerned with the use of existing approximate analytic schemes to obtain solutions to singular nonlinear oscillatory differential equations.

Bota, K. B.↗
NASA NTRS1985

Bounds and approximations for rapid evaluation of coherent MPSK error probabilities

The error probability for MPSK detection is expressed here as a sum of a familiar upper-bound approximation and a correction term which is well suited for numerical integration. An asymptotic expansion of the correction term is derived, from which simple and extremely tight upper and lower bounds are obtained. Relative errors in using these bounds and approximations of them are graphed for cases of practical interest.

Chie, C. M.↗
NASA NTRS1983

Linear approximations of nonlinear systems

A method for designing an automatic flight controller for short and vertical takeoff aircraft is presently being developed at NASA Ames Research Center. This technique involves transformations of nonlinear systems to controllable linear systems and takes into account the nonlinearities of the aircraft. In general, the transformations cannot always be given in closed form. Using partial differential equations, an approximate linear system, called the modified tangent model, was recently introduced. A linear transformation of this tangent model to Brunovsky canonical form can be constructed, and from this an approximation of an exact transformation for the nonlinear system can be found. It is shown that a canonical expansion in Lie brackets about the point x(0) yields the same modified tangent model.

Hunt, L. R.↗
NASA NTRS1985

Ladder and cross terms in second-order distorted Born approximation

In the strong fluctuation theory for a bounded layer of random discrete scatterers, the second moments of the fields in the second-order distorted Born approximation are obtained for copolarized and cross-polarized fields. The backscattering cross sections per unit area are calculated by including the mutual coherence of the fields due to the coincidental ray paths, and that due to the opposite ray paths, corresponding to the ladder and cross terms in the Feynman diagramatic representation. It is proved that the contributions from ladder and cross terms for the copolarized backscattering cross sections are the same, while the contributions for the cross-polarized backscattering cross sections are of the same order. The bistatic scattering coefficients in the second-order approximation for both the ladder and cross terms are also obtained. The contributions from the cross terms explain the enhancement in the backscattering direction.

Jin, Y. Q.↗