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NASA NTRS1998

A Preconditioning Method for Shape Optimization Governed by the Euler Equations

We consider a classical aerodynamic shape optimization problem subject to the compressible Euler flow equations. The gradient of the cost functional with respect to the shape variables is derived with the adjoint method at the continuous level. The Hessian (second order derivative of the cost functional with respect to the shape variables) is approximated also at the continuous level, as first introduced by Arian and Ta'asan (1996). The approximation of the Hessian is used to approximate the Newton step which is essential to accelerate the numerical solution of the optimization problem. The design space is discretized in the maximum dimension, i.e., the location of each point on the intersection of the computational mesh with the airfoil is taken to be an independent design variable. We give numerical examples for 86 design variables in two different flow speeds and achieve an order of magnitude reduction in the cost functional at a computational effort of a full solution of the analysis partial differential equation (PDE).

Arian, Eyal↗

NASA NTRS2005

The Apollo Capsule Optimization for Improved Stability and Computational/Experimental Data Comparisons

Numerical optimization was employed on the Apollo Command Module to modify its external shape. The Apollo Command Module (CM) that was used on all NASA human space flights during the Apollo Space Program is stable and trimmed in an apex forward (alpha of approximately 40 to 80 degrees) position. This poses a safety risk if the CM separates from the launch tower during abort. Optimization was employed on the Apollo CM to remedy the undesirable stability characteristics of the configuration. Geometric shape changes were limited to axisymmetric modifications that altered the radius of the apex (R(sub A)), base radius (R(sub O)), corner radius (R(sub C)), and the cone half angle (theta), while the maximum diameter of the CM was held constant. The results of multipoint optimization on the CM indicated that the cross-range performance can be improved while maintaining robust apex-aft stability with a single trim point. Navier-Stokes computations were performed on the baseline and optimized configurations and confirmed the Euler-based optimization results. Euler Analysis of ten alternative CM vehicles with different values of the above four parameters are compared with the published experimental results of numerous wind tunnel tests during the late 1960's. These comparisons cover a wide Mach number range and a full 180-degree pitch range and show that the Euler methods are capable of fairly accurate force and moment computations and can separate the vehicle characteristics of these ten alternative configurations.

Cliff, Susan E.↗

NASA NTRS1974

Analysis of twisting stiffness for a multifiber composite layer

The twisting stiffness of a rectangular cross section consisting of a single row of solid circular cross-section fibers embedded in a matrix is analyzed. The problem is formulated as a Dirichlet torsion problem of a multielement region and solved by the boundary-point least-squares method. Numerical results for a single-fiber square cross section compare favorably with previous relaxation-method results. New numerical results for three and five-fiber composites suggest that the torsional rigidity of a multifiber composite can be approximated from the torsional rigidities of single and three-fiber models.

Bert, C. W.↗

NASA NTRS1986

Radiation pattern analysis of the tapered slot antenna, appendix 1

A theoretical model for the tapered slot antenna is presented. The model is valid for any smooth taper of the slot. The problem is solved by modeling the slot electric field and using the half plane Green's function to compute the far fields. The aperture field is obtained by affecting a stepped approximation to the continuous taper and utilizing the uniform wide slot line data. The uniform wide slot line is solved by the spectral Galerkin's technique and closed form experssions are developed for the slot wave length and the slot characteristic impedance. Numerous comparisons with measurement are made to demonstrate the versatility of the model in treating an arbitrary slot taper.

Janaswamy, Ramakrishna↗

NASA NTRS1981

Free-Vibration Analysis of Rotating Beams by a Variable-Order Finite-Element Method

The free vibration of rotating beams is analyzed by means of a finite-element method of variable order. This method entails displacement functions that are a complete power series of a variable number of terms. The terms are arranged so that the generalized coordinates are composed of displacements and slopes at the element extremities and, additionally, displacements at certain points within the element. The displacement is assumed to be analytic within an element and thus can be approximated to any degree of accuracy desired by a complete power series. Numerical results are presented for uniform beams with zero and nonzero hub radii, tapered beams, and a nonuniform beam with discontinuities. Since the present method reduces to a conventional beam finite-element method for a cubic displacement function, the results are compared and found to be superior to the conventional results in terms of accuracy for a given number of degrees of freedom. Indeed, essentially exact eigenvalues and eigenvectors are obtained with this technique, which is far more rapidly convergent than other approaches in the literature.

Hodges, Dewey H.↗

NASA NTRS1999

Approximation of the Newton Step by a Defect Correction Process

In this paper, an optimal control problem governed by a partial differential equation is considered. The Newton step for this system can be computed by solving a coupled system of equations. To do this efficiently with an iterative defect correction process, a modifying operator is introduced into the system. This operator is motivated by local mode analysis. The operator can be used also for preconditioning in Generalized Minimum Residual (GMRES). We give a detailed convergence analysis for the defect correction process and show the derivation of the modifying operator. Numerical tests are done on the small disturbance shape optimization problem in two dimensions for the defect correction process and for GMRES.

Arian, E.↗

NASA NTRS1998

The Oxidation of AlN in Dry and Wet Oxygen

The oxidation kinetics of AlN containing 3.5 wt% Y2O3 were studied by thermogravimetric analysis in dry oxygen and 10% H2O/balance oxygen at temperatures between 1000 and 1200 C for times between 48 and 100 h. The oxidation kinetics for AlN in dry oxygen were parabolic and of approximately the same magnitude and temperature dependence as other alumina forming materials. In this case, diffusion of oxygen and/or aluminum through the alumina scale is the rate limiting mechanism. The oxidation kinetics for AlN in wet oxygen were nearly linear and much more rapid than rates observed in dry oxygen. Numerous micropores were observed in the alumina formed on AIN in wet oxygen. These pores provide a fast path for oxygen transport. The linear kinetics observed in this case suggest that the interface reaction rate of AlN with wet oxygen is the oxidation rate limiting step.

Opila, Elizabeth↗

NASA NTRS2003

Toroidal wave frequency at L=6-10: AMPTE/CCE Observations and Comparison with Theoretical Model

Magnetospheric standing Alfven waves are guided along the ambient magnetic field and their frequency depends on the mass density of the plasma distributed along the field lines. These properties allow us to use Alfen waves to map time-dependent phenomena between space and ground and to estimate the mass density. In this paper we present a statistical study of the spatial variation of the fundamental frequency f(sub T1) of toroidal-mode standing Alfen waves in the L range from 6 to 10, where L indicates the maximum geocentric distance on the field line. The data used for this analysis are energetic particle flux anisotropy (proxy of transverse electric field) and magnetic field measurements from the Active Magnetospheric Particle Tracer Explorers Charge Composition Explorer (CCE) spacecraft. Using CCE data covering 4 years we obtained approximately 5000 20-min intervals containing a clear signature of toroidal waves. The median f(sub T1), is 6-10 mHz at L=7 and decreases to 4-8 mHz at L = . The frequency tends to be lower at noon than at midnight. The observed frequencies are compared with numerically derived frequencies using an empirical mass density model and a magnetic field model. We found a good agreement for 12-24 MLT but a large discrepancy near MLT= 3. This may indicate that the flux tubes at this local time are more heavily loaded than specified in the Gallagher et al. model.

Takahashi, Kazue↗

NASA NTRS1992

Mars dust storm simulations: Analysis of surface stress

The primary mechanism by which dust is inserted into the Martian atmosphere is the interaction of low-level atmospheric motions with the planet's surface. Near-surface winds exert a shear stress upon dust particles resting on the Martian surface, and at some lower threshold limit of stress magnitude, approximately 0.04 N-m(exp -2), particles are set into motion. Wind tunnel studies indicate that the first particles are too large to remain in suspension in the Martian atmosphere, but their impact back upon the surface can set smaller suspendable particles into motion. This process is termed saltation. Numerical simulations of Martian dust storms were carried out via the interactive coupling of the NASA Ames Mars general circulation Model with an aerosol transport/microphysical model.

Murphy, J. R.↗

NASA NTRS1992

Preliminary analysis of thermal-infrared multispectral scanner data of the Iron Hill, Colorado carbonatite-alkalic rock complex

The Iron Hill carbonatite-alkalic igneous rock complex is in the Powderhorn mining district, approximately 40 km south-southwest of Gunnison, Colorado. The complex, which occupies about 30 sq km, was emplaced in metasedimentay and metavolcanic rocks during the later Precambrian or early Cambrian. The main rock types in the complex, from oldest to youngest, are fenite, pyroxenite, uncompahgrite, ijolite, nepheline syenite, and dolomitic carbonatite. The carbonatite is limonitic and forms an elliptially shaped 4 sq km stock. Calcitic and dolomitic carbonatite dikes are also numerous throughout the complex and in the pre-existing rocks. Pyroxenite is the most widespread rock type within the complex, but pyroxene is extensively altered to biotite, phlogopite, and vermiculite. Fenite, which formed through Na, K-metasomatism of the country rocks, typically contains more feldspar and less quartz than the equivalent unaltered country rocks. The other alkalic rock types are less widespread and less well exposed. Parts of the complex are covered by Oligocene ash-flow tuff and alluvial, colluvial, and glacial deposits. Sagebrush and grass cover is moderately dense to very dense at low to intermediate elevations; coniferous tree cover is dense at high elevations and on some north-facing slopes at lower elevations. A new algorithm was used to compute spectral emissivity ratios, independent of any emissivity assumptions. This algorithm has the advantage that any of the possible emissivity ratios can be computed and, thus, a large variety of composite ratio images can be constructed, which permits examination of various geologic hypotheses based on the spectral properties of the surface materials.

Rowan, Lawrence C.↗

NASA NTRS1977

An approximate method for design and analysis of an ALOHA system

An approximate method for the design and performance prediction of a multiaccess communication system which employs the ALOHA packet-switching technique is developed, based on the use of a diffusion process approximation of an ALOHA-like system (with or without time-slotting). A simple closed-form solution for the variable Q(t), a variant of the number of backlog messages at time t, is given in terms of a few system and user parameters. Final results are expressed in terms of ordinary performance measures such as throughput and average delay. Several numerical examples are given to demonstrate the usefulness of the approximation technique developed.

Kobayashi, H.↗

NASA NTRS1976

Numerical solution of nonlinear partial differential equations of mixed type

A review is presented of some recently developed numerical methods for the solution of nonlinear equations of mixed type. The methods considered use finite difference approximations to the differential equation. Central difference formulas are employed in the subsonic zone and upwind difference formulas are used in the supersonic zone. The relaxation method for the small disturbance equation is discussed and a description is given of difference schemes for the potential flow equation in quasi-linear form. Attention is also given to difference schemes for the potential flow equation in conservation form, the analysis of relaxation schemes by the time dependent analogy, the accelerated iterative method, and three-dimensional calculations.

Jameson, A.↗

NASA NTRS1977

Chemical characterization of lunar core 60010

Twenty-eight major, minor, and trace-element abundances have been determined in nine fines fractions of core 60010 by instrumental thermal and 14 MeV neutron activation analysis. With the exception of the elements Fe, Co, and Ni, core section 60010 exhibits a much more uniform composition than the lower section (60009) of the double drive tube collected at the Apollo 16 LM site. The siderophilic meteorite indicator elements Fe, Co, and Ni are significantly enriched in subsample 40, which is located 3.0-4.5 cm below the lunar surface, and are highly variable throughout the 60010 core section. Microscopic examination of subsample 40 revealed the presence of numerous metallic fragments including an approximately 1-mm metal fragment exhibiting only Fe, Co, Ni, and Ir gamma-ray peaks and having a composition typical of a coarse octahedrite. Composition profiles of 60010 exhibit small variations in lithophilic element abundances which can be related to most of the seven stratigraphic units of Duke and Nagle (1976). Although the basal portion of 60010 has been described as texturally similar to the upper unit of 60009, the present chemical studies show a marked compositional difference at this interface.

Ali, M. Z.↗

NASA NTRS2010

Fast Solution in Sparse LDA for Binary Classification

An algorithm that performs sparse linear discriminant analysis (Sparse-LDA) finds near-optimal solutions in far less time than the prior art when specialized to binary classification (of 2 classes). Sparse-LDA is a type of feature- or variable- selection problem with numerous applications in statistics, machine learning, computer vision, computational finance, operations research, and bio-informatics. Because of its combinatorial nature, feature- or variable-selection problems are NP-hard or computationally intractable in cases involving more than 30 variables or features. Therefore, one typically seeks approximate solutions by means of greedy search algorithms. The prior Sparse-LDA algorithm was a greedy algorithm that considered the best variable or feature to add/ delete to/ from its subsets in order to maximally discriminate between multiple classes of data. The present algorithm is designed for the special but prevalent case of 2-class or binary classification (e.g. 1 vs. 0, functioning vs. malfunctioning, or change versus no change). The present algorithm provides near-optimal solutions on large real-world datasets having hundreds or even thousands of variables or features (e.g. selecting the fewest wavelength bands in a hyperspectral sensor to do terrain classification) and does so in typical computation times of minutes as compared to days or weeks as taken by the prior art. Sparse LDA requires solving generalized eigenvalue problems for a large number of variable subsets (represented by the submatrices of the input within-class and between-class covariance matrices). In the general (fullrank) case, the amount of computation scales at least cubically with the number of variables and thus the size of the problems that can be solved is limited accordingly. However, in binary classification, the principal eigenvalues can be found using a special analytic formula, without resorting to costly iterative techniques. The present algorithm exploits this analytic form along with the inherent sequential nature of greedy search itself. Together this enables the use of highly-efficient partitioned-matrix-inverse techniques that result in large speedups of computation in both the forward-selection and backward-elimination stages of greedy algorithms in general.

Moghaddam, Baback↗

NASA NTRS1990

Factorization and the synthesis of optimal feedback gains for distributed parameter systems

An approach based on Volterra factorization leads to a new methodology for the analysis and synthesis of the optimal feedback gain in the finite-time linear quadratic control problem for distributed parameter systems. The approach circumvents the need for solving and analyzing Riccati equations and provides a more transparent connection between the system dynamics and the optimal gain. The general results are further extended and specialized for the case where the underlying state is characterized by autonomous differential-delay dynamics. Numerical examples are given to illustrate the second-order convergence rate that is derived for an approximation scheme for the optimal feedback gain in the differential-delay problem.

Milman, Mark H.↗

NASA NTRS1992

Integrated multidisciplinary analysis of segmented reflector telescopes

The present multidisciplinary telescope-analysis approach, which encompasses thermal, structural, control and optical considerations, is illustrated for the case of an IR telescope in LEO; attention is given to end-to-end evaluations of the effects of mechanical disturbances and thermal gradients in measures of optical performance. Both geometric ray-tracing and surface-to-surface diffraction approximations are used in the telescope's optical model. Also noted is the role played by NASA-JPL's Integrated Modeling of Advanced Optical Systems computation tool, in view of numerical samples.

Briggs, Hugh C.↗

NASA NTRS2008

A Verification-Driven Approach to Control Analysis and Tuning

This paper proposes a methodology for the analysis and tuning of controllers using control verification metrics. These metrics, which are introduced in a companion paper, measure the size of the largest uncertainty set of a given class for which the closed-loop specifications are satisfied. This framework integrates deterministic and probabilistic uncertainty models into a setting that enables the deformation of sets in the parameter space, the control design space, and in the union of these two spaces. In regard to control analysis, we propose strategies that enable bounding regions of the design space where the specifications are satisfied by all the closed-loop systems associated with a prescribed uncertainty set. When this is unfeasible, we bound regions where the probability of satisfying the requirements exceeds a prescribed value. In regard to control tuning, we propose strategies for the improvement of the robust characteristics of a baseline controller. Some of these strategies use multi-point approximations to the control verification metrics in order to alleviate the numerical burden of solving a min-max problem. Since this methodology targets non-linear systems having an arbitrary, possibly implicit, functional dependency on the uncertain parameters and for which high-fidelity simulations are available, they are applicable to realistic engineering problems..

Crespo, Luis G.↗

NASA NTRS1994

A numerical analysis of shipboard and coastal zone color scanner time series of new production within Gulf Stream cyclonic eddies in the South Atlantic Bight

Eddy-induced upwelling occurs along the western edge of the Gulf Stream between Cape Canaveral, Florida, and Cape Hatteras, North Carolina, in the South Atlantic Bight (SAB). Coastal zone color scanner images of 1-km resolution spanning the period April 13-21, 1979, were processed to examine these eddy features in relation to concurrent shipboard and current/temperature measurements at moored arrays. A quasi-one-dimensional (z), time dependent biological model, using only nitrate as a nutrient source, has been combined with a three-dimensional physical model in an attempt to replicate the observed phytoplankton field at the northward edge of an eddy. The model is applicable only to the SAB south of the Charleston Bump, at approximately 31.5 deg N, since no feature analogous to the bump exists in the model bathymetry. The modeled chlorophyll, nitrate, and primary production fields of the euphotic zone are very similar to those obtained from the satellite and shipboard data at the leading edges of the observed eddies south of the Charleston Bump. The horizontal and vertical simulated fluxes of nitrate and chlorophyll show that only approximately 10% of the upwelled nitrate is utilized by the phytoplankton of the modeled grid box on the northern edge of the cyclone, while approximately 75% is lost horizontally, with the remainder still in the euphotic zone after the 10-day period of the model. Loss of chlorophyll due to sinking is very small in this strong upwelling region of the cyclone. The model is relatively insensitive to variations in the sinking parameterization and the external nitrate and chlorophyll fields but is very sensitive to a reduction of the maximum potential growth rate to half that measured. Given the success of this model in simulating the new production of the selcted upwelling region, other upwelling regions for which measurements or successful models of physical and biological quantities and rates exist could be modeled similarly.

Pribble, J. Raymond↗