SEARCH · Search NASA

Results for “approximation”

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.

SourceResearch connectionFrom yearThrough year

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

At least 523 records · Page 29

All sources NASA NTRS DOE OSTI

NASA NTRS1986

Approximate Cartesian state transition matrix

Based on a Taylor series expansion, an easily-computed approximation to the Cartesian state transition matrix is presented for a general velocity-independent force field. Suitable for the short time intervals encountered in onboard navigation applications of the extended Kalman filter, it provides approximately five decimal digits of accuracy for earth orbiting spacecraft with update intervals of one minute, and better accuracy for shorter intervals.

Markley, F. L.↗

NASA NTRS1986

Comparison of exact and approximate formulas for the Mott correction to energy loss of relativistic heavy ions

A comparison is conducted between an exact numerical calculation and two approximate analytic expressions for the Mott correction to energy loss of heavy ions. A complete tabulation of the Mott correction is presented together with a comparison of some of the approximate expressions for the Mott correction due to Ahlen (1978) and Morgan and Eby (1973). Comparison is made with results of an experimental calibration of the HEAO-3 Heavy Cosmic Ray experimental chambers at the Bevalac.

Eby, P. B.↗

NASA NTRS1986

Parameter identifiability under approximation

The problem of injectivity of the parameter-to-state map is discussed for Galerkin approximations of a linear parabolic equation. A necessary and sufficient condition is derived and illustrated by means of simple examples. Finally, output least squares identifiability of the Galerkin approximations is discussed.

Kunisch, K.↗

NASA NTRS1987

Elastic interaction of a crack with a microcrack array. I - Formulation of the problem and general form of the solution. II - Elastic solution for two crack configurations (piecewise constant and linear approximations)

The elastic interactions of a two-dimensional configuration consisting of a crack with an array of microcracks located near the tip are studied. The general form of the solution is based on the potential representations and approximations of tractions on the microcracks by polynomials. In the second part, the technique is applied to two simple two-dimensional configurations involving one and two microcracks. The problems of stress shielding and stress amplification (the reduction or increase of the effective stress intensity factor due to the presence of microcracks) are discussed, and the refinements introduced by higher order polynomial approximations are illustrated.

Chudnovsky, A.↗

NASA NTRS1986

Approximate methods for black hole collisions

The head-on collision of two nonrotating black holes, initially falling toward each other with speed of approach equal to escape velocity, is studied. The initial-value problem for the two-black-hole configuration is solved by giving the extrinsic curvature on a Euclidean space slice, with the equal masses of the two holes and their separation as free parameters. An approximating space-time is then constructed by stacking up a whole sequence of initial-value slices with decreasing separation between the black holes. The space-time singularities at the centers of the two black holes collide at some instant of time, and after that time the space-time so constructed is just the Schwarzschild solution with mass equal to the total mass of both black holes. The approximating space-time describes the coalescence of the event horizons, but contains no gravitational waves.

Eardley, D. M.↗

NASA NTRS1987

Convenient stability criteria for difference approximations of hyperbolic initial-boundary value problems. II

The results of Goldberg and Tadmor (1985) are extended to achieve improved stability criteria for a large class of approximations to the initial boundary value problem associated with a particular hyperbolic system in a quarter plane. In a stability analysis, it is shown that the entire approximation is stable if and only if the scalar outflow components of its principal part are stable. Thus, the global stability question is reduced to that of a scalar, homogeneous outflow problem. The stability criteria for the reduced problem, which depend both on the basic scheme and the boundary conditions, but very little on the interaction between the two, are stated and used to establish previous examples and new ones, including a host of dissipative and nondissipative examples. There is no difficulty in extending the stability criteria to two-boundary problems and initial-boundary value problems with variable coefficients.

Goldberg, Moshe↗

NASA NTRS1987

Complex space monofilar approximation of diffraction currents on a conducting half plane

Simple approximation of diffraction surface currents on a conducting half plane, due to an incoming plane wave, is obtained with a line current (monofile) in complex space. When compared to an approximating current at the edge, the diffraction pattern is seen to improve by an order of magnitude for a minimal increase of computation effort. Thus, the inconvient Fresnel integral functions can be avoided for quick calculations of diffracted fields and the accuracy is good in other directions than along the half plane. The method can be applied to general problems involving planar metal edges.

Lindell, I. V.↗

NASA NTRS1988

Optimal discrete-time LQR problems for parabolic systems with unbounded input: Approximation and convergence

An abstract approximation and convergence theory for the closed-loop solution of discrete-time linear-quadratic regulator problems for parabolic systems with unbounded input is developed. Under relatively mild stabilizability and detectability assumptions, functional analytic, operator techniques are used to demonstrate the norm convergence of Galerkin-based approximations to the optimal feedback control gains. The application of the general theory to a class of abstract boundary control systems is considered. Two examples, one involving the Neumann boundary control of a one-dimensional heat equation, and the other, the vibration control of a cantilevered viscoelastic beam via shear input at the free end, are discussed.

Rosen, I. G.↗

NASA NTRS1988

Analytical approximation of a distorted reflector surface defined by a discrete set of points

Reflector antennas on Earth orbiting spacecrafts generally cannot be described analytically. The reflector surface is subjected to a large temperature fluctuation and gradients, and is thus warped from its true geometrical shape. Aside from distortion by thermal stresses, reflector surfaces are often purposely shaped to minimize phase aberrations and scanning losses. To analyze distorted reflector antennas defined by discrete surface points, a numerical technique must be applied to compute an interpolatory surface passing through a grid of discrete points. In this paper, the distorted reflector surface points are approximated by two analytical components: an undistorted surface component and a surface error component. The undistorted surface component is a best fit paraboloid polynomial for the given set of points and the surface error component is a Fourier series expansion of the deviation of the actual surface points, from the best fit paraboloid. By applying the numerical technique to approximate the surface normals of the distorted reflector surface, the induced surface current can be obtained using physical optics technique. These surface currents are integrated to find the far field radiation pattern.

Acosta, Roberto J.↗

NASA NTRS1988

On the convergence of difference approximations to scalar conservation laws

A unified treatment is given for time-explicit, two-level, second-order-resolution (SOR), total-variation-diminishing (TVD) approximations to scalar conservation laws. The schemes are assumed only to have conservation form and incremental form. A modified flux and a viscosity coefficient are introduced to obtain results in terms of the latter. 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 TVD-SOR schemes approximating convex or concave conservation laws is shown by enforcing a single discrete entropy inequality.

Osher, Stanley↗

NASA NTRS1988

Numerical approximation for the infinite-dimensional discrete-time optimal linear-quadratic regulator problem

An abstract approximation framework is developed for the finite and infinite time horizon discrete-time linear-quadratic regulator problem for systems whose state dynamics are described by a linear semigroup of operators on an infinite dimensional Hilbert space. The schemes included the framework yield finite dimensional approximations to the linear state feedback gains which determine the optimal control law. Convergence arguments are given. Examples involving hereditary and parabolic systems and the vibration of a flexible beam are considered. Spline-based finite element schemes for these classes of problems, together with numerical results, are presented and discussed.

Gibson, J. S.↗

NASA NTRS1988

Shape optimization of 3D continuum structures via force approximation techniques

The existing need to develop methods whereby the shape design efficiency can be improved through the use of high quality approximation methods is addressed. An efficient approximation method for stress constraints in 3D shape design problems is proposed based on expanding the nodal forces in Taylor series with respect to shape variations. The significance of this new method is shown through elementary beam theory calculations and via numerical computations using 3D solid finite elements. Numerical examples including the classical cantilever beam structure and realistic automotive parts like the engine connecting rod are designed for optimum shape using the proposed method. The numerical results obtained from these methods are compared with other published results, to assess the efficiency and the convergence rate of the proposed method.

Vanderplaats, Garret N.↗

NASA NTRS1988

Approximations for hypervelocity nonequilibrium radiating, reacting, and conducting stagnation regions

An approximate solution which includes the effects of chemical and thermal nonequilibrium and nonequilibrium nongray radiative transfer and which is suitable for rapidly obtaining approximate solutions for flowfield structure and estimates of radiative heat transfer is presented. Comparisons are made with Fire 2 data and example applications to a hypothetical Mars return AOTV are presented. Parametric results indicate that further studies of VUV processes and radiation models are needed.

Carlson, Leland A.↗

NASA NTRS1988

On Hilbert-Schmidt norm convergence of Galerkin approximation for operator Riccati equations

An abstract approximation framework for the solution of operator algebraic Riccati equations is developed. The approach taken is based on a formulation of the Riccati equation as an abstract nonlinear operator equation on the space of Hilbert-Schmidt operators. Hilbert-Schmidt norm convergence of solutions to generic finite dimensional Galerkin approximations to the Riccati equation to the solution of the original infinite dimensional problem is argued. The application of the general theory is illustrated via an operator Riccati equation arising in the linear-quadratic design of an optimal feedback control law for a 1-D heat/diffusion equation. Numerical results demonstrating the convergence of the associated Hilbert-Schmidt kernels are included.

Rosen, I. G.↗

NASA NTRS1989

Optimal feedback control infinite dimensional parabolic evolution systems: Approximation techniques

A general approximation framework is discussed for computation of optimal feedback controls in linear quadratic regular problems for nonautonomous parabolic distributed parameter systems. This is done in the context of a theoretical framework using general evolution systems in infinite dimensional Hilbert spaces. Conditions are discussed for preservation under approximation of stabilizability and detectability hypotheses on the infinite dimensional system. The special case of periodic systems is also treated.

Banks, H. T.↗

NASA NTRS1989

Charge exchange transition probability for collisions between unlike ions and atoms within the adiabatic approximation

A simple formula for the transition probability for electron exchange between unlike ions and atoms is established within the adiabatic approximation by employing the Linear Combination of Atomic Orbitals (LCAO) method. The formula also involves an adiabatic parameter, introduced by Massey, and thus the difficulties arising from the internal energy defect and the adiabatic approximation are avoided. Specific reactions Li(+++) + H to Li(++) + H(+) and Be(4+) + H to Be(3+) + H(+) are considered as examples. The calculated capture cross section results of the present work are compared with the experimental data and with the calculation of other authors over the velocity range of 10(7) cm/sec to 10(8) cm/sec.

Xu, Y. J.↗

NASA NTRS1989

Low-frequency sound propagation modeling over a locally-reacting boundary using the parabolic approximation

There is substantial interest in the analytical and numerical modeling of low-frequency, long-range atmospheric acoustic propagation. Ray-based models, because of frequency limitations, do not always give an adequate prediction of quantities such as sound pressure or intensity levels. However, the parabolic approximation method, widely used in ocean acoustics, and often more accurate than ray models for lower frequencies of interest, can be applied to acoustic propagation in the atmosphere. Modifications of an existing implicit finite-difference implementation for computing solutions to the parabolic approximation are discussed. A locally-reacting boundary is used together with a one-parameter impedance model. Intensity calculations are performed for a number of flow resistivity values in both quiescent and windy atmospheres. Variations in the value of this parameter are shown to have substantial effects on the spatial variation of the acoustic signal.

Robertson, J. S.↗

NASA NTRS1988

Gaussian approximation versus nearly exact performance analysis of optical communication systems with PPM signaling and APD receivers

A 25-Mb/s direct-detection optical communication system that used Q = 4 PPM (pulse-position modulation) signaling was constructed and its performance measured under laboratory conditions. The system used a single-mode AlGaAs laser diode (lambda = 834 nm) and low-noise silicon avalanche photodiode (APD). A procedure for numerically computing system performance is given that uses the nearly exact Webb approximation of the true Conradi distribution for the APD output and does not require excessive amounts of computer time. Comparison revealed that modeling the APD ouptut as a Gaussian process under conditions of negligible background radiation and low (less than 10 to the -12 A) APD bulk leakage currents leads to substantial underestimates of optimal APD gain and overestimates of system bit error probability. Examples illustrate the breakdown of the Gaussian approximation in assessing system performance. The measured performance was in excellent agreement with that predicted by the computational procedure. The system achieved a bit error probability of 10 to the -6 at a received signal energy, corresponding to an average of 60 absorbed photons/bit and optimal APD gain of 700.

Davidson, Frederic M.↗