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NASA NTRS1974

The Princeton equipment on board

The Princeton telescope-spectrometer on the Copernicus satellite scans stellar spectra between 950 and 1450 A in second order with a resolution of about 0.05 A. The resolution in first order, between 1650 and 3000 A, is twice that in second order. The equipment may be used down to a nominal limit of 5th magnitude with the photometric precision in second order limited only by the statistics of photon counts. At 1100 A, a rate of 1,000 per 14 sec is obtained on an unreddened B 1 star with a visual magnitude of 5.0. In the first order, phototube noise from cosmic rays limits observations to stars brighter than visual magnitude 3.0 in general.

Drake, J. F.↗

NASA NTRS1991

Optimal least-squares finite element method for elliptic problems

An optimal least squares finite element method is proposed for two dimensional and three dimensional elliptic problems and its advantages are discussed over the mixed Galerkin method and the usual least squares finite element method. In the usual least squares finite element method, the second order equation (-Delta x (Delta u) + u = f) is recast as a first order system (-Delta x p + u = f, Delta u - p = 0). The error analysis and numerical experiment show that, in this usual least squares finite element method, the rate of convergence for flux p is one order lower than optimal. In order to get an optimal least squares method, the irrotationality Delta x p = 0 should be included in the first order system.

Jiang, Bo-Nan↗

NASA NTRS2010

Comparison of Node-Centered and Cell-Centered Unstructured Finite-Volume Discretizations: Inviscid Fluxes

Cell-centered and node-centered approaches have been compared for unstructured finite-volume discretization of inviscid fluxes. The grids range from regular grids to irregular grids, including mixed-element grids and grids with random perturbations of nodes. Accuracy, complexity, and convergence rates of defect-correction iterations are studied for eight nominally second-order accurate schemes: two node-centered schemes with weighted and unweighted least-squares (LSQ) methods for gradient reconstruction and six cell-centered schemes two node-averaging with and without clipping and four schemes that employ different stencils for LSQ gradient reconstruction. The cell-centered nearest-neighbor (CC-NN) scheme has the lowest complexity; a version of the scheme that involves smart augmentation of the LSQ stencil (CC-SA) has only marginal complexity increase. All other schemes have larger complexity; complexity of node-centered (NC) schemes are somewhat lower than complexity of cell-centered node-averaging (CC-NA) and full-augmentation (CC-FA) schemes. On highly anisotropic grids typical of those encountered in grid adaptation, discretization errors of five of the six cell-centered schemes converge with second order on all tested grids; the CC-NA scheme with clipping degrades solution accuracy to first order. The NC schemes converge with second order on regular and/or triangular grids and with first order on perturbed quadrilaterals and mixed-element grids. All schemes may produce large relative errors in gradient reconstruction on grids with perturbed nodes. Defect-correction iterations for schemes employing weighted least-square gradient reconstruction diverge on perturbed stretched grids. Overall, the CC-NN and CC-SA schemes offer the best options of the lowest complexity and secondorder discretization errors. On anisotropic grids over a curved body typical of turbulent flow simulations, the discretization errors converge with second order and are small for the CC-NN, CC-SA, and CC-FA schemes on all grids and for NC schemes on triangular grids; the discretization errors of the CC-NA scheme without clipping do not converge on irregular grids. Accurate gradient reconstruction can be achieved by introducing a local approximate mapping; without approximate mapping, only the NC scheme with weighted LSQ method provides accurate gradients. Defect correction iterations for the CC-NA scheme without clipping diverge; for the NC scheme with weighted LSQ method, the iterations either diverge or converge very slowly. The best option in curved geometries is the CC-SA scheme that offers low complexity, second-order discretization errors, and fast convergence.

Diskin, Boris↗

NASA NTRS1972

On optimizing the treatment of exchange perturbations

A method using the zeroth plus first order wave functions, obtained by optimizing the basic equation used in exchange perturbation treatments, is utilized in an attempt to determine the exact energy and wave function in the exchange process. Attempts to determine the first order perturbation solution by optimizing the sum of the first and second order energies were unsuccessful.

Hirschfelder, J. O.↗

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 NTRS1976

Solution of the minimum time-to-climb problem by matched asymptotic expansions

Application of singular perturbation techniques to trajectory optimization problems of flight mechanics is discussed. The method of matched asymptotic expansions is used to obtain an approximate solution to the aircraft minimum time-to-climb problem. Outer, boundary-layer, and composite solutions are obtained to zeroth and first orders. A stability criterion is derived for the zeroth-order boundary-layer solutions (the theory requires a form of boundary-layer stability). A numerical example is considered for which it is shown that the stability criterion is satisfied and a useful numerical solution is obtained. The zeroth-order solution proves to be a poor approximation, but the first-order solution gives a good approximation for both the trajectory and the minimum time-to-climb. The computational cost of the singular perturbation solution is considerably less than that of a steepest descent solution. Thus singular perturbation methods appear to be promising for the solution of optimal control problems.

Ardema, M. D.↗

NASA NTRS1991

On the Power of Abstract Interpretation

Increasingly sophisticated applications of static analysis place increased burden on the reliability of the analysis techniques. Often, the failure of the analysis technique to detect some information my mean that the time or space complexity of the generated code would be altered. Thus, it is important to precisely characterize the power of static analysis techniques. We follow the approach of Selur et. al. who studied the power of strictness analysis techniques. Their result can be summarized by saying 'strictness analysis is perfect up to variations in constants.' In other words, strictness analysis is as good as it could be, short of actually distinguishing between concrete values. We use this approach to characterize a broad class of analysis techniques based on abstract interpretation including, but not limited to, strictness analysis. For the first-order case, we consider abstract interpretations where the abstract domain for data values is totally ordered. This condition is satisfied by Mycroft's strictness analysis that of Sekar et. al. and Wadler's analysis of list-strictness. For such abstract interpretations, we show that the analysis is complete in the sense that, short of actually distinguishing between concrete values with the same abstraction, it gives the best possible information. We further generalize these results to typed lambda calculus with pairs and higher-order functions. Note that products and function spaces over totally ordered domains are not totally ordered. In fact, the notion of completeness used in the first-order case fails if product domains or function spaces are added. We formulate a weaker notion of completeness based on observability of values. Two values (including pairs and functions) are considered indistinguishable if their observable components are indistinguishable. We show that abstract interpretation of typed lambda calculus programs is complete up to this notion of indistinguishability. We use denotationally-oriented arguments instead of the detailed operational arguments used by Selur et. al.. Hence, our proofs are much simpler. They should be useful for further future improvements.

Reddy, Uday S.↗

NASA NTRS2011

A (201)Hg+ Comagnetometer for (199)Hg+ Trapped Ion Space Atomic Clocks

A method has been developed for unambiguously measuring the exact magnetic field experienced by trapped mercury ions contained within an atomic clock intended for space applications. In general, atomic clocks are insensitive to external perturbations that would change the frequency at which the clocks operate. On a space platform, these perturbative effects can be much larger than they would be on the ground, especially in dealing with the magnetic field environment. The solution is to use a different isotope of mercury held within the same trap as the clock isotope. The magnetic field can be very accurately measured with a magnetic-field-sensitive atomic transition in the added isotope. Further, this measurement can be made simultaneously with normal clock operation, thereby not degrading clock performance. Instead of using a conventional magnetometer to measure ambient fields, which would necessarily be placed some distance away from the clock atoms, first order field-sensitive atomic transition frequency changes in the atoms themselves determine the variations in the magnetic field. As a result, all ambiguity over the exact field value experienced by the atoms is removed. Atoms used in atomic clocks always have an atomic transition (often referred to as the clock transition) that is sensitive to magnetic fields only in second order, and usually have one or more transitions that are first-order field sensitive. For operating parameters used in the (199)Hg(+) clock, the latter can be five orders of magnitude or more sensitive to field fluctuations than the clock transition, thereby providing an unambiguous probe of the magnetic field strength.

Burt, Eric A.↗

NASA NTRS1981

Quantum chemical calculation of the equilibrium structures of small metal atom clusters

A decomposition of the molecular energy is presented that is motivated by the atom superposition and electron delocalization physical model of chemical binding. The energy appears in physically transparent form consisting of a classical electrostatic interaction, a zero order two electron exchange interaction, a relaxation energy, and the atomic energies. Detailed formulae are derived in zero and first order of approximation. The formulation extends beyond first order to any chosen level of approximation leading, in principle, to the exact energy. The structure of this energy decomposition lends itself to the fullest utilization of the solutions to the atomic sub problems to simplify the calculation of the molecular energy. If nonlinear relaxation effects remain minor, the molecular energy calculation requires at most the calculation of two center, two electron integrals. This scheme thus affords the prospects of substantially reducing the computational effort required for the calculation of molecular energies.

Kahn, L. R.↗

NASA NTRS1995

Analytic MHD Theory for Earth's Bow Shock at Low Mach Numbers

A previous MHD theory for the density jump at the Earth's bow shock, which assumed the Alfven M(A) and sonic M(s) Mach numbers are both much greater than 1, is reanalyzed and generalized. It is shown that the MHD jump equation can be analytically solved much more directly using perturbation theory, with the ordering determined by M(A) and M(s), and that the first-order perturbation solution is identical to the solution found in the earlier theory. The second-order perturbation solution is calculated, whereas the earlier approach cannot be used to obtain it. The second-order terms generally are important over most of the range of M(A) and M(s) in the solar wind when the angle theta between the normal to the bow shock and magnetic field is not close to 0 deg or 180 deg (the solutions are symmetric about 90 deg). This new perturbation solution is generally accurate under most solar wind conditions at 1 AU, with the exception of low Mach numbers when theta is close to 90 deg. In this exceptional case the new solution does not improve on the first-order solutions obtained earlier, and the predicted density ratio can vary by 10-20% from the exact numerical MHD solutions. For theta approx. = 90 deg another perturbation solution is derived that predicts the density ratio much more accurately. This second solution is typically accurate for quasi-perpendicular conditions. Taken together, these two analytical solutions are generally accurate for the Earth's bow shock, except in the rare circumstance that M(A) is less than or = 2. MHD and gasdynamic simulations have produced empirical models in which the shock's standoff distance a(s) is linearly related to the density jump ratio X at the subsolar point. Using an empirical relationship between a(s) and X obtained from MHD simulations, a(s) values predicted using the MHD solutions for X are compared with the predictions of phenomenological models commonly used for modeling observational data, and with the predictions of a modified phenomenological model proposed recently. The similarities and differences between these results are illustrated using plots of X and a(s) predicted for the Earth's bow shock. The plots show that the new analytic solutions agree very well with the exact numerical MHD solutions and that these MHD solutions should replace the corresponding phenomenological relations in comparisons with data. Furthermore, significant differences exist between the standoff distances predicted at low M(A) using the MHD models versus those predicted by the new modified phenomenological model. These differences should be amenable to observational testing.

Grabbe, Crockett L.↗

NASA NTRS1969

Fine-line sensitivity for holographic interferograms

Improvement in sensitivity of holography, the technique of lensless interferometry, is obtained by enhancing the higher-order structure in the interferogram. By using the light diffracted into higher orders than the first, phase sensitivity is increased over the first order sensitivity by a factor equal to the order number used.

Heflinger, L. O.↗

NASA NTRS1976

Vibration of a flexible spacecraft with momentum exchange controllers

Floating reference frames were investigated in order to allow first order vibration analysis in the presence of large system rotations. When the deformations of an elastic continuum are expanded in terms of the free-free modes of an unconstrained system, the rigid body modes are found to be fixed relative to the Tisserand frame, with respect to which the relative momentum is zero. The proof presented for this is based on the orthogonality condition for modes with distinct natural frequencies. This result also guarantees the independence of coordinates for all modes with nonzero natural frequencies. A Modified Tisserand Constraint is introduced in order to define a floating reference frame with similar properties for an elastic body which contains a spinning rotor. Finite element equations of motion are derived for a completely flexible spacecraft with momentum exchange controllers, using a Modified Tisserand Frame. The deformable systems covered in this application are assumed to undergo only small rotations, and therefore the rotor torques must formally be small, although in engineering applications it may be possible to relax this constraint. A modal analysis is performed for the system and the resulting set of equations is reduced in number by a truncation procedure for more efficient system simulation.

Canavin, J. R.↗

NASA NTRS1978

Temperature dependence of the rate constants for reactions of O/1D/ atoms with a number of halocarbons

Results are reported for an experimental study of the reactions of O(1D) with CCl4, CFCl3, CF2Cl2, CHFCl2, CHF2Cl, and CF2ClCFCl2 over the temperature range from 173 to 343 K, based on the time-resolved emission of O(1D) at 630 nm. The experiments involved photolysis of O3 by Nd-YAG laser pulses and measurements of 630-nm emission intensity in mixtures with He and the various halocarbons. Pseudo-first-order rate constants are derived from plots of the logarithms of the 630-nm emission intensities as a function of time for particular He/O3/halocarbon mixtures, and second-order rate constants are then obtained from plots of these pseudo-first-order rate constants as a function of halocarbon concentration at constant He and O3 concentrations. No temperature dependence of the second-order rate constants is observed over the range studied, within the precision of the measurements. It is found that the rate constants decrease as H is substituted for Cl in a halocarbon molecule and as F is substituted for H or Cl.

Davidson, J. A.↗

NASA NTRS1982

Higher-order flow angle corrections for three-dimensional wind tunnel wall interference

A second-order theory including camber effects in wind tunnel wall interference corrections is described. Changes in the geometrical configuration of the model tested are avoided by introducing the camber correction as an equivalent angle-of-attack correction. Tabular and graphic data are presented which indicate improved accuracy for second-order over first-order theory.

Rizk, M. H.↗

NASA NTRS1984

Preliminary concept of a 100-kilowatt miniaturized Cassegrainian concentrator solar array

A miniaturized Cassegrainian concentrator (MCC) solar array system is examined to assess the practicality of assembling the basic MCC element into a total array system capable of producing multi-hundred kilowatts of power for Space Platform/Space Station or other low Earth orbit long lifetime missions. Preliminary mechanical and electrical subsystems are developed in order to determine first order performance characteristics. Results of the study support the feasibility of a 100-kilowatt MCC array system with beginning-of-life (BOL) performance of 160 watts per square meter and 28 W/kg. The performance numbers are based on 20 percent efficient (at operating temperature) solar cells and 0.25-millimeter thick electroformed nickel optics. These performance numbers can be improved upon significantly with the development of higher efficiency solar cells and/or lighter weight optics.

Patterson, R. E.↗

NASA NTRS1981

On the relation between upwind-differencing schemes of Godunov, Engquist-Osher and Roe

The upwind differencing first order schemes of Godunov, Engquist-Osher and Roe are discussed on the basis of the inviscid Burgers equations. The differences between the schemes are interpreted as differences between the approximate Riemann solutions on which their numerical flux functions are based. Special attention is given to the proper formulation of these schemes when a source term is present. Second order two step schemes, based on the numerical flux functions of the first order schemes are also described. The schemes are compared in a numerical experiment, and recommendations on their use are included.

Vanleer, B.↗

NASA NTRS1985

Tone excited jets. IV - Acoustic measurements

Attention is given to acoustic results from an experiment designed to deepen insight into the noise generated by a tone-excited jet, with emphasis on the mechanism of broadband jet noise amplification. Results are presented for both heated and unheated jets, with and without the effect of forward flight simulation over a range of excitation frequencies and levels, and for zero-order and first-order spinning modes. Broadband jet noise amplifications of up to 5 dB have been observed in this study. Relative velocity effects are the same for both excited and unexcited jets. It is concluded that small scale turbulence generates the additional broadband noise upon excitation of the jet by an upstream-injected discrete tone acoustic wave.

Ahuja, K. K.↗

NASA NTRS1992

Studies on spectral analysis of randomly sampled signals: Application to laser velocimetry data

Spectral analysis is very useful in determining the frequency characteristics of many turbulent flows, for example, vortex flows, tail buffeting, and other pulsating flows. It is also used for obtaining turbulence spectra from which the time and length scales associated with the turbulence structure can be estimated. These estimates, in turn, can be helpful for validation of theoretical/numerical flow turbulence models. Laser velocimetry (LV) is being extensively used in the experimental investigation of different types of flows, because of its inherent advantages; nonintrusive probing, high frequency response, no calibration requirements, etc. Typically, the output of an individual realization laser velocimeter is a set of randomly sampled velocity data. Spectral analysis of such data requires special techniques to obtain reliable estimates of correlation and power spectral density functions that describe the flow characteristics. FORTRAN codes for obtaining the autocorrelation and power spectral density estimates using the correlation-based slotting technique were developed. Extensive studies have been conducted on simulated first-order spectrum and sine signals to improve the spectral estimates. A first-order spectrum was chosen because it represents the characteristics of a typical one-dimensional turbulence spectrum. Digital prefiltering techniques, to improve the spectral estimates from randomly sampled data were applied. Studies show that the spectral estimates can be increased up to about five times the mean sampling rate.

Sree, David↗