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NASA NTRS1987

Approximate polynomial preconditioning applied to biharmonic equations on vector supercomputers

Applying a finite difference approximation to a biharmonic equation results in a very ill-conditioned system of equations. This paper examines the conjugate gradient method used in conjunction with the generalized and approximate polynomial preconditionings for solving such linear systems. An approximate polynomial preconditioning is introduced, and is shown to be more efficient than the generalized polynomial preconditionings. This new technique provides a simple but effective preconditioning polynomial, which is based on another coefficient matrix rather than the original matrix operator as commonly used.

Wong, Yau Shu↗

NASA NTRS1983

Structural optimization - Challenges and opportunities

A review of developments in structural optimization techniques and their interface with growing computer capabilities is presented. Structural design steps comprise functional definition of an object, an evaluation phase wherein external influences are quantified, selection of the design concept, material, object geometry, and the internal layout, and quantification of the physical characteristics. Optimization of a fully stressed design is facilitated by use of nonlinear mathematical programming which permits automated definition of the physics of a problem. Design iterations terminate when convergence is acquired between mathematical and physical criteria. A constrained minimum algorithm has been formulated using an Augmented Lagrangian approach and a generalized reduced gradient to obtain fast convergence. Various approximation techniques are mentioned. The synergistic application of all the methods surveyed requires multidisciplinary teamwork during a design effort.

Sobieszczanski-Sobieski, J.↗

NASA NTRS1984

Structural optimization: Challenges and opportunities

Sobieszczanski-Sobieski, J.↗

NASA NTRS1972

Effects of gravity-gradient torque on the rotational motion of a triaxial satellite in a precessing elliptic orbit.

A method of general perturbations, based on the use of Lie series to generate approximate canonical transformations, is applied to study the effects of gravity-gradient torque on the rotational motion of a triaxial, rigid satellite. The center of mass of the satellite is constrained to move in an elliptic orbit about an attracting point mass. The orbit, which has a constant inclination, is free to precess and spin. The method of general perturbations is used to obtain the Hamiltonian for the nonresonant secular and long-period rotational motion of the satellite. The differential equations derivable from the transformed Hamiltonian are integrable, and the solution for the long-term motion may be expressed in terms of Jacobian elliptic functions and elliptic integrals. Geometrical aspects of the long-term rotational motion are discussed, and a comparison of theoretical results with observations is made.

Cochran, J. E.↗

NASA NTRS2003

The Galileo Probe: How it Has Changed Our Understanding of Jupiter

The Galileo Mission to Jupiter, which arrived in December of 1995, provided the first study by an orbiter, and the first in-situ sampling via an entry probe, of an outer planet atmosphere. The rationale for an entry probe is that, even from an orbiter, remote sensing of the jovian atmosphere could not adequately retrieve the information desired. This paper provides a current summary of the most significant aspects of the data returned from the Galileo entry probe. As a result of the probe measurements, there has been a reassessment of our understanding of outer planet formation and evolution of the solar system. The primary scientific objective of the Galileo probe was to determine the composition of the jovian atmosphere, which from remote sensing remained either very uncertain, or completely unknown, with respect to several key elements. The probe found that the global He mass fraction is. significantly above the value reported from the Voyager Jupiter flybys but is slightly below the protosolar value, implying that there has been some settling of He to the deep jovian interior. The probe He measurements have also led to a reevaluation of the Voyager He mass fraction for Saturn, which is now determined to be much closer to that of Jupiter. The elements C, N, S, Ar, Kr, Xe were all found to have global abundances approximately 3 times their respective solar abundances. This result has raised a number of fundamental issues with regard to properties of planetesimals and the solar nebula at the time of giant planet formation. Ne, on the other hand, was found to be highly depleted, probably as the result of it being carried along with helium as helium settles towards the deep interior. The global abundance of O was not obtained by the probe because of the influence of local processes at the probe entry site (PES), processes which depleted condensible species, in this case H2O, well below condensation levels. Other condensible species, namely NH3 and H2S, were similarly affected but attained their deep equilibrium mixing ratios before the maximum depth sampled by the probe. Processes that might be capable of producing such effects on the condensibles are still under investigation. Measured isotopic ratios of noble gases and other heavy elements are solar, and (D + (Sup 3)He)/H is the same to within measurement uncertainties as in the local interstellar medium. No thick clouds were detected, and in particular no significant water cloud, but the PES location clearly affected the probe measurements of clouds. In fact, the probe data must be understood in the context of the location of the PES, which was within what is termed a 5 micron hot spot, a local clearing in the clouds that is bright near the 5 microns spectral region. The thermal structure at the PES was determined from approximately 1000 km above the 1 bar pressure level (10(exp -9 bars)) to 132 km 1 bar (22bars). The probe showed the atmosphere to have a generally sub-adiabatic temperature gradient (static stability) of = 0.1 K/km to as deep as the probe made measurements. In the upper atmosphere the probe derived a maximum positive vertical temperature gradient of approximately 5 K/km, and maximum temperature of = 900 K. The energy sources producing the warm upper atmosphere have yet to be completely identified. At first glance, Doppler tracking of the probe indicates that the long observed cloud level zonal winds extend to levels at least as deep as the probe made measurements. Zonal wind increases from = 80 m/s at pressures less than a bar to about 180 m/s near 5 bars, and remains approximately constant with depth thereafter. However, there is a question as to whether the winds measured from probe tracking are representative of the general wind field, or are considerably influenced by localized winds associated with the PES.

Young, Richard E.↗

NASA NTRS2008

Comparative In Situ Measurements of Plasma Instabilities in the Equatorial and Auroral Electrojets

This presentation provides a comparison of in situ measurements of plasma instabilities gathered by rocket-borne probes in the equatorial and auroral electrojets. Specifically, using detailed measurements of the DC electric fields, current density, and plasma number density within the unstable daytime equatorial electrojet from Brazil (Guara Campaign) and in the auroral electrojet from Sweden (ERRIS Campaign), we present comparative observations and general conclusions regarding the observed physical properties of Farley-Buneman two-stream waves and large scale, gradient drift waves. The two stream observations reveal coherent-like waves propagating near the E x B direction but at reduced speeds (nearer to the presumed acoustic velocity) with wavelengths of approximately 5-10m in both the equatorial and auroral electrojet, as measured using the spaced-receiver technique. The auroral electrojet data generally shows extensions to shorter wavelengths, in concert with the fact that these waves are driven harder. With respect to gradient-drift driven waves, observations of this instability are much more pronounced in the equatorial electrojet, given the more favorable geometry for growth provided by the vertical gradient and horizontal magnetic field lines. We present new analysis of Guara rocket observations of electric field and plasma density data that reveal considerable structuring in the middle and lower portion of the electrojet (90-105 km) where the ambient plasma density gradient is unstable. Although the electric field amplitudes are largest (approximately 10-15 mV/m) in the zonal direction, considerable structure (approximately 5-10 mV/m) is also observed in the vertical electric field component as well, implying that the dominant large scale waves involve significant vertical interaction and coupling within the narrow altitude range where they are observed. Furthermore, a detailed examination of the phase of the waveforms show that on some, but not all occasions, locally enhanced eastward fields are associated with locally enhanced upwards (polarization) electric fields. The measurements are discussed in terms of theories involving the non-linear evolution and structuring of plasma waves.

Pfaff, Robert F.↗

NASA NTRS1976

The numerical evaluation of maximum-likelihood estimates of the parameters for a mixture of normal distributions from partially identified samples

Likelihood equations determined by the two types of samples which are necessary conditions for a maximum-likelihood estimate are considered. These equations, suggest certain successive-approximations iterative procedures for obtaining maximum-likelihood estimates. These are generalized steepest ascent (deflected gradient) procedures. It is shown that, with probability 1 as N sub 0 approaches infinity (regardless of the relative sizes of N sub 0 and N sub 1, i=1,...,m), these procedures converge locally to the strongly consistent maximum-likelihood estimates whenever the step size is between 0 and 2. Furthermore, the value of the step size which yields optimal local convergence rates is bounded from below by a number which always lies between 1 and 2.

Walker, H. F.↗

NASA NTRS1987

Inelastic strain analogy for piecewise linear computation of creep residues in built-up structures

An analogy between inelastic strains caused by temperature and those caused by creep is presented in terms of isotropic elasticity. It is shown how the theoretical aspects can be blended with existing finite-element computer programs to exact a piecewise linear solution. The creep effect is determined by using the thermal stress computational approach, if appropriate alterations are made to the thermal expansion of the individual elements. The overall transient solution is achieved by consecutive piecewise linear iterations. The total residue caused by creep is obtained by accumulating creep residues for each iteration and then resubmitting the total residues for each element as an equivalent input. A typical creep law is tested for incremental time convergence. The results indicate that the approach is practical, with a valid indication of the extent of creep after approximately 20 hr of incremental time. The general analogy between body forces and inelastic strain gradients is discussed with respect to how an inelastic problem can be worked as an elastic problem.

Jenkins, Jerald M.↗

NASA NTRS1976

The numerical evaluation of maximum-likelihood estimates of the parameters for a mixture of normal distributions from partially identified samples

Likelihood equations determined by the two types of samples which are necessary conditions for a maximum-likelihood estimate were considered. These equations suggest certain successive approximations iterative procedures for obtaining maximum likelihood estimates. The procedures, which are generalized steepest ascent (deflected gradient) procedures, contain those of Hosmer as a special case.

Walker, H. F.↗

NASA NTRS1974

First results of a general circulation model applied to the SST-NOx problem

Results of model runs, one using two-dimensional distribution of NO2 in the unperturbed stratosphere and another including an additional localized source of NO2 to approximate the effect of SSTs, are reported. The general circulation model and chemical reactions are described, and corrections in the previous latitudinal and seasonal gradients of total columnar ozone and the diffusion coefficient in the neighborhood of the tropopause are noted. Excellent agreement with previous observations was obtained. Global distributions of ozone and NO2 are described and represented in graphs. Results indicate ozone reduction of approximately 16% in the Northern Hemisphere and approximately 8% in the Southern Hemisphere, with the mid-latitude source of NO2 apparently having a blocking effect on the horizontal transport of ozone, resulting in larger reductions of ozone at high latitudes than at low ones.

Cunnold, D. M.↗

NASA NTRS1993

A Realizable Reynolds Stress Algebraic Equation Model

The invariance theory in continuum mechanics is applied to analyze Reynolds stresses in high Reynolds number turbulent flows. The analysis leads to a turbulent constitutive relation that relates the Reynolds stresses to the mean velocity gradients in a more general form in which the classical isotropic eddy viscosity model is just the linear approximation of the general form. On the basis of realizability analysis, a set of model coefficients are obtained which are functions of the time scale ratios of the turbulence to the mean strain rate and the mean rotation rate. The coefficients will ensure the positivity of each component of the mean rotation rate. These coefficients will ensure the positivity of each component of the turbulent kinetic energy - realizability that most existing turbulence models fail to satisfy. Separated flows over backward-facing step configurations are taken as applications. The calculations are performed with a conservative finite-volume method. Grid-independent and numerical diffusion-free solutions are obtained by using differencing schemes of second-order accuracy on sufficiently fine grids. The calculated results are compared in detail with the experimental data for both mean and turbulent quantities. The comparison shows that the present proposal significantly improves the predictive capability of K-epsilon based two equation models. In addition, the proposed model is able to simulate rotational homogeneous shear flows with large rotation rates which all conventional eddy viscosity models fail to simulate.

Shih, Tsan-Hsing↗

NASA NTRS2012

From the Blazar Sequence to the Blazar Envelope: Revisiting the Relativistic Jet Dichotomy in Radio-Loud AGN

We revisit the concept of a blazar sequence that relates the synchrotron peak frequency (Vpeak) in blazars with synchrotron peak luminosity (Lpeak, in vLv) using a large sample of radio-loud AGN. We present observational evidence that the blazar sequence is formed from two populations in the synchrotron Vpeak - Lpeak plane, each forming an upper edge to an envelope of progressively misaligned blazars, and connecting to an adjacent group of radio galaxies having jets viewed at much larger angles to the line of sight. When binned by jet kinetic power (Lkin; as measured through a scaling relationship with extended radio power), we find that radio core dominance decreases with decreasing synchrotron Lpeak, revealing that sources in the envelope are generally more misaligned. We find population-based evidence of velocity gradients in jets at low kinetic powers (approximately 10(exp 42) - 10(exp 44.5) erg s(exp -1)), corresponding to FR I radio galaxies and most BL Lacs. These low jet power 'weak jet' sources, thought to exhibit radiatively inefficient accretion, are distinguished from the population of non-decelerating, low synchrotron-peaking (LSP) blazars and FR II radio galaxies ('strong' jets) which are thought to exhibit radiatively efficient accretion. The two-population interpretation explains the apparent contradiction of the existence of highly core-dominated, low-power blazars at both low and high synchrotron peak frequencies, and further implies that most intermediate synchrotron peak (ISP) sources are not intermediate in intrinsic jet power between LSP and high synchrotron-peaking (HSP) sources, but are more misaligned versions of HSP sources with similar jet powers.

Meyer, Eileen T.↗

NASA NTRS1980

A simple approximate result for the maximum growth rate of baroclinic instabilities

The Charney problem for baroclinic instability involves the quasi-geostrophic instability of a zonal flow on a beta plane where the zonal flow is characterized by a constant vertical shear. The atmosphere is non-Boussinesq and continuous. The solution of this problem involves confluent hypergeometric functions, and the mathematical difficulty of the problem has precluded extracting simple results of generality. It is shown that there exists very simple, powerful approximate result for the growth rate of the most rapidly growing instability, viz., that this growth rate is linearly proportional to the surface meridional temperature gradient. The coefficient of proportionality is also easily determined. The result extends to substantially more general profiles than those in the Charney problem.

Lindzen, R. S.↗

NASA NTRS1973

Long-term motion in a restricted problem of rotational motion

A method of general perturbations, based on the use of Lie series to generate approximate canonical transformations, is applied to study the long-term effects of gravity-gradient torque and orbital evolution on the rotational motion of a triaxial, rigid satellite. The center of mass of the satellite is constrained to move in an elliptic orbit about an attracting point mass. The orbit, which has a constant inclination, is constrained to precess and spin with constant rates. The method of general perturbations is used to obtain the Hamiltonian for the nonresonant secular and long-period rotational motion of the satellite to second order in n/omega sub 0, where n is the orbital mean motion of the center of mass and omega sub 0 is a reference value of the magnitude of the satellite's rotational angular velocity. The differential equations derivable from the transformed Hamiltonian are integrable, and the solution for the long-term motion may be expressed in terms of Jacobian elliptic functions and elliptic integrals. Geometrical aspects of the long-term rotational motion are discussed, and a comparison of theoretical results with observations is made.

Cochran, J. E.↗

NASA NTRS1995

The anomalous nuclear component in the three-dimensional heliosphere

The anomalous nuclear component is neither of solar nor galactic cosmic ray origin. Its presence in the heliosphere is an independent probe for both interplanetary electrodynamical investigations--especially solar modulation--and probably the most direct means for determining the elemental and isotopic composition of those neutral atoms in the local interstellar medium that have high first ionization potentials (e.g., He, N, O, Ne, Ar, etc.). This report is a brief account of the evolution of our knowledge of this component for readers not specializing in the subject. Included are the initial discoveries of the component, its composition, spectra, heliospheric radial and latitudinal intensity gradients, modulation over the approximately 22 year solar magnetic cycle, trapping in the magnetosphere and its use for estimating the location of a heliospheric termination shock. Recent measurements from the ULYSSES spacecraft have provided conclusive evidence that incoming neutral atoms, after photoionization by solar uv, are picked up by the solar wind, thus lending support for the concept that after their acceleration--probably at a termination shock--they return to the inner heliosphere as pseudo-cosmic rays. ULYSSES spacecraft investigations extending to approximately 56 deg south latitude reveal, for both the anomalous nuclear component and the galactic cosmic rays, that there is a surprisingly small latitudinal intensity gradient. Thus, for the current phase of the solar cycle, modulation is much more spherically symmetric in the inner solar system than had generally been believed. A further surprise is the continual presence of approximately 26 day recurrent modulation at hight latitudes, without corresponding magnetic field compressions. These results are changing our ideas and requiring modification of our models for solar modulation in three dimensions.

Simpson, J. A.↗

NASA NTRS1985

The response of turbulent boundary layers to sudden pertubations

The processes present in a turbulent boundary layer (BL) which experiences a sudden change in the surface roughness are discussed in terms of numerical models in comparison with empirical data. Consideration is limited to flow perturbations which preserve the BL approximation. Generalized continuity, momentum and turbulent energy transport equations are reviewed. The perturbed BL has two regions: one with large velocity gradients and an outer zone with smaller gradients. Expressions are defined for the velocity and temperature profiles and the roughness length after a step change in surface roughness, which causes the disturbance. A pertrurbation could also arise from a heat flux input of a pressure gradient. The effects of the shape of the perturbing step and the presence of several disturbing edges are explored. Finally, it is shown that closure forms and constants for self-preserving flows are worthy of further development because of their demonstrable abilities to describe simple perturbed flows.

Smits, A. J.↗

NASA NTRS2008

Accuracy of Gradient Reconstruction on Grids with High Aspect Ratio

Gradient approximation methods commonly used in unstructured-grid finite-volume schemes intended for solutions of high Reynolds number flow equations are studied comprehensively. The accuracy of gradients within cells and within faces is evaluated systematically for both node-centered and cell-centered formulations. Computational and analytical evaluations are made on a series of high-aspect-ratio grids with different primal elements, including quadrilateral, triangular, and mixed element grids, with and without random perturbations to the mesh. Both rectangular and cylindrical geometries are considered; the latter serves to study the effects of geometric curvature. The study shows that the accuracy of gradient reconstruction on high-aspect-ratio grids is determined by a combination of the grid and the solution. The contributors to the error are identified and approaches to reduce errors are given, including the addition of higher-order terms in the direction of larger mesh spacing. A parameter GAMMA characterizing accuracy on curved high-aspect-ratio grids is discussed and an approximate-mapped-least-square method using a commonly-available distance function is presented; the method provides accurate gradient reconstruction on general grids. The study is intended to be a reference guide accompanying the construction of accurate and efficient methods for high Reynolds number applications

Thomas, James↗

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.↗