Search NASA⌕ Search

SEARCH · Search NASA

Results for “Elliptic problems”

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 487 records · Page 27
All sourcesNASA NTRSDOE OSTI
NASA NTRS1994

Microwave backscattering and emission model for grass canopies

Microwave radar and radiometer measurements of grasslands indicate a substantial reduction in sensor sensitivity to soil moisture in the presence of a thatch layer. When this layer is wet it masks changes in the underlying soil, making the canopy appear warm in the case of passive sensors (radiometer) and decreasing backscatter in the active case (scatterometer). A model for a grass canopy with thatch will be presented in this paper to explain this behavior and to compare with observations. The canopy model consists of three layers: grass, thatch, and the underlying soil. The grass blades are modeled by elongated elliptical discs and the thatch is modeled as a collection of disk shaped water droplets (i.e., the dry matter is neglected). The ground is homogeneous and flat. The distorted Born approximation is used to compute the radar cross section of this three layer canopy and the emissivity is computed from the radar cross section using the Peake formulation for the passive problem. Results are computed at L-band (1.4 GHz) and C-band (4.75 GHz) using canopy parameters (i.e., plant geometry, soil moisture, plant moisture, etc.) representative of Konza Prairie grasslands. The results are compared to C-band scatterometer measurements and L-band radiometer measurements at these grasslands.

TRT-THEORETICAL↗
NASA NTRS1979

Recent developments in finite element analysis for transonic airfoils

The prediction of aerodynamic forces in the transonic regime generally requires a flow field calculation to solve the governing non-linear mixed elliptic-hyperbolic partial differential equations. Finite difference techniques were developed to the point that design and analysis application are routine, and continual improvements are being made by various research groups. The principal limitation in extending finite difference methods to complex three-dimensional geometries is the construction of a suitable mesh system. Finite element techniques are attractive since their application to other problems have permitted irregular mesh elements to be employed. The purpose of this paper is to review the recent developments in the application of finite element methods to transonic flow problems and to report some recent results.

Hafez, M. M.↗
NASA NTRS1987

Shock structure in jets of arbitrary exit geometry

The shock structure and hydrodynamic stability of single and twin jets with arbitrary exit geometry are studied. Both vortex sheets and realistic mean profiles are considered for the jet shear layer. The boundary element method is used to predict the shock spacing and screech tones in a vortex sheet model of a single jet. Results are given for circular, elliptic and rectangular jets and the results compared with analysis and experiment. A finite difference technique is used to examine the effect of finite mixing layer thickness for a single jet. The hydrodynamic stability of twin circular jets is also considered. The variation of the stability characteristics are presented for the vortex sheet case for several separation distances. The formulation of the problem to account for finite thickness effects is also given.

Morris, P. J.↗
NASA NTRS1994

Structure of merger remnants. 4: Isophotal shapes

This paper examines the shapes of isophotes of galaxy merger remnants. More specifically, we perform a series of numerical experiments to study galaxy mergers. The simulations explore a variety of encounter geometries, types of progenitor galaxies, and particle numbers. We 'observe' each of the remnants from 64 viewpoints to estimate how the isophotal shapes vary with the orientation of the remnant. Also, by comparing the results from the various simulations, we learn how encounter geometry and the structure of the progenitors can affect the shape of the remnants. The encounter geometry, the structure of the progenitors, and the orientation of the remnant play deciding roles in the shape of the 'observed' isophotes, so much so that these simulated merger remnants have both 'boxy' and 'disky' isophotes and ellipticities ranging from E0 to E7. Thus, the shapes seen in our merger simulation remnants span a similar range to observed isophotal shapes of real galaxies. Furthermore, to estimate the errors in this analysis, we introduce the statistical bootstrap. Bootstrapping is used rather widely in observational astronomy when dealing with small samples; however, its use is practically unheard of for tackling theoretical problems. We believe that bootstrapping can be extremely useful for dealing with the small samples found in numerical simulations; therefore, we present the rudiments and basis of the technique with emphasis on its use in N-body calculations. Also, by comparing the errors for the smaller simulations with those of the larger ones, we speculate on the number of particles required to accurately explore isophotal shapes in simulations.

Heyl, Jeremy S.↗
NASA NTRS1950

The spanwise distribution of lift for minimum induced drag of wings having a given lift and a given bending moment

The problem of the minimum induced drag of wings having a given lift and a given span is extended to include cases in which the bending moment to be supported by the wing is also given. The theory is limited to lifting surfaces traveling at subsonic speeds. It is found that the required shape of the downwash distribution can be obtained in an elementary way which is applicable to a variety of such problems. Expressions for the minimum drag and the corresponding spanwise load distributions are also given for the case in which the lift and the bending moment about the wing root are fixed while the span is allowed to vary. The results show a 15-percent reduction of the induced drag with a 15-percent increase in span as compared with results for an elliptically loaded wing having the same total lift and bending moment.

Jones, R. T.↗
NASA NTRS1993

Self-consistent models for triaxial galaxies with flat rotation curves - The disk case

We examine the possibility of constructing scale-free triaxial logarithmic potentials self-consistently, using Schwarzschild's linear programing method. In particular, we explore the limit of nonaxisymmetric disks. In this case it is possible to reduce the problem to the self-consistent reconstruction of the disk surface density on the unit circle, a considerably simpler problem than the usual 2D or 3D one. Models with surface densities of the form Sigma = (x exp n + (y/q) exp n) exp - 1/n with n = 2 or 4 are investigated. We show that the complicated shapes of the 'boxlet' orbit families (which replace the box orbit family found in potentials with smooth cores) limit the possibility of building self-consistent models, though elliptical disks of axis ratio above 0.7 and a restricted range of boxier models can be constructed. This result relies on using sufficiently fine bins, smaller than the 10 deg bins commonly used in 2D or 3D investigations. It also indicates the need for caution in interpreting N-body models of triaxial halos in which the core of the potential is numerically smoothed.

Kuijken, Konrad↗
NASA NTRS1993

Instabilities of two-dimensional inviscid compressible vortices

An investigation of the linear stability and subsequent nonlinear evolution and acoustic radiation of a planar inviscid compressible vortex is presented. The effects of the entropy gradient are investigated, and for the particular entropy profile chosen, the positive average entropy in the vortex core is destabilizing, while the opposite is true for the negative average entropy gradient. Finite-difference methods are used to study the linear initial value problem. These methods are found to be capable of accurately computing the frequencies and weak growth rates of the normal modes. When the initial condition consists of random perturbations, the long-time behavior is found to correspond to the most unstable normal mode in all cases. The numerical solution of the Euler equations is used to study the nonlinear development of an elliptic-mode perturbation.

Chan, W. M.↗
NASA NTRS1998

Mars global surveyor navigation and aerobraking at Mars

The Mars Global Surveyor (MGS) spacecraft was successfully inserted into an elliptical orbit around Mars on 9/12/97, 01:53:49 UTC. This orbit was near polar (inclination = 93.26 deg) with an orbital period of 44.993 hours and apoapsis and periapsis altitudes of 54,025.9 km and 262.9 km respectively. After a short aerobraking (AB) initiation interval (9/12/97 to 10/2/97), the main phase of AB or orbit period reduction was established. However shortly thereafter, a significant problem with the minus-Y axis solar array developed which necessitated a temporary suspension of AB. Ultimately, this forced the Project to abandon the original plan to complete AB on 1/18/98 and establish the mapping orbit on 3/15/98. The revised plan called for a reduced level of AB, thus subjecting the solar array and yoke assembly to less aerodynamic stress. After 201 orbits and 196 days after MOI, the first phase of AB has ended, the orbital period was 11.64 hours with apoapsis and periapsis altitudes of 17,870.3 and 170.7 km respectively. At present, MGS is in a science phasing orbit (SPO) and shall acquire science data from 3/28/98 to 9/11/98. Thereafter the second phase of AB shall begin and is expected to end during Feb 1999 when the orbital period shall be 1.9 hours and the orbit's descending node shall be at the 2:00 am (local mean solar time) orientation.

Esposito, P.↗
NASA NTRS2007

Optically Transparent Split-Ring Antennas for 1 to 10 GHz

Split-ring antennas made from optically transparent, electrically conductive films have been invented for applications in which there are requirements for compact antennas capable of operation over much or all of the frequency band from 1 to 10 GHz. Primary examples of such applications include wireless local-area networks and industrial, scientific, and medical (ISM) applications. These antennas can be conveniently located on such surfaces as those of automobile windows and display screens of diverse hand-held electronic units. They are fabricated by conventional printed-circuit techniques and can easily be integrated with solid-state amplifier circuits to enhance gain. The structure of an antenna of this type includes an antenna/feed layer supported on the top or outer face of a dielectric (e.g., glass) and, optionally, a ground layer on the bottom or inner face of the substrate. The ring can be in the form of either a conductive strip or a slot in the antenna/feed layer. The ring can be of rectangular, square, circular, elliptical, or other suitable shape and can be excited by means of a microstrip, slot line, or coplanar waveguide. For example, the antenna shown in the figure features a square conductive-strip split ring with a microstrip feed. In general, an antenna fed at its external boundary in the manner of this invention presents very high impedance, thereby creating an impedance-matching problem. Splitting the ring . that is, cutting a notch through the ring . offers a solution to the problem in that the notch fixes the location of maximum electric field, which location is directly related to the impedance. Thus, an excellent impedance match can be achieved through proper choice of the location of the notch. In geometric layout, such a ring antenna structure is typically between 1.4 and 1.3 the size of a patch antenna capable of operating in the same frequency range. This miniaturization of the antenna is desirable, not only because it contributes to overall miniaturization of equipment, but also because minimization of the extent of the optically transparent, electrically conductive film helps to minimize the electrical loss associated with the surface resistance ( 5 ohms per square) of the transparent, electrically conductive film material. Incidentally, even at 5 ohms per square, this surface resistance is significantly less than that of indium tin oxide film (typically > 25 ohms per square), which, heretofore has been the transparent, electrically conductive film material of choice. At the time of writing this article, information on the composition of the lower-resistance film used in the antennas of this invention was not available.

Lee, Richard Q.↗
NASA NTRS1975

A numerical scheme to solve unstable boundary value problems

A new iterative scheme for solving boundary value problems is presented. It consists of the introduction of an artificial time dependence into a modified version of the system of equations. Then explicit forward integrations in time are followed by explicit integrations backwards in time. The method converges under much more general conditions than schemes based in forward time integrations (false transient schemes). In particular it can attain a steady state solution of an elliptical system of equations even if the solution is unstable, in which case other iterative schemes fail to converge. The simplicity of its use makes it attractive for solving large systems of nonlinear equations.

Kalnay Derivas, E.↗
NASA NTRS1981

Spectral multigrid methods for elliptic equations

An alternative approach which employs multigrid concepts in the iterative solution of spectral equations was examined. Spectral multigrid methods are described for self adjoint elliptic equations with either periodic or Dirichlet boundary conditions. For realistic fluid calculations the relevant boundary conditions are periodic in at least one (angular) coordinate and Dirichlet (or Neumann) in the remaining coordinates. Spectral methods are always effective for flows in strictly rectangular geometries since corners generally introduce singularities into the solution. If the boundary is smooth, then mapping techniques are used to transform the problem into one with a combination of periodic and Dirichlet boundary conditions. It is suggested that spectral multigrid methods in these geometries can be devised by combining the techniques.

Zang, T. A.↗
NASA NTRS1986

On substructuring algorithms and solution techniques for the numerical approximation of partial differential equations

Substructuring methods are in common use in mechanics problems where typically the associated linear systems of algebraic equations are positive definite. Here these methods are extended to problems which lead to nonpositive definite, nonsymmetric matrices. The extension is based on an algorithm which carries out the block Gauss elimination procedure without the need for interchanges even when a pivot matrix is singular. Examples are provided wherein the method is used in connection with finite element solutions of the stationary Stokes equations and the Helmholtz equation, and dual methods for second-order elliptic equations.

Gunzburger, M. D.↗
NASA NTRS1991

Radiative interactions in chemically reacting supersonic internal flows

The two-dimensional, elliptic Navier-Stokes equations are used to investigate supersonic flows with finite-rate chemistry and radiation for hydrogen-air systems. The chemistry source terms in the species equation is treated implicitly to alleviate the stiffness associated with fast reactions. The explicit, unsplit MacCormack finite-difference scheme is used to advance the governing equations in time, until convergence is achieved. The specific problem considered is the premixed flow in a channel with a ten-degree compression ramp. Three different chemistry models are used, accounting for increasing number of reactions and participating species. Two chemistry models assure nitrogen as inert, while the third model accounts for nitrogen reactions and NO(x) formation. The tangent slab approximation is used in the radiative flux formulation. A pseudo-gray model is used to represent the absorption-emission characteristics of the participating species. Results obtained for specific conditions indicate that the radiative interactions vary substantially, depending on reactions involving HO2 and NO species and that this can have a significant influence on the flowfield.

Tiwari, S. N.↗
NASA NTRS1991

Radiative interactions in a hydrogen-fueled supersonic combustor

The two-dimensional, elliptic Navier-Stokes equations are used to investigate supersonic flows with finite-rate chemistry and radiation, for hydrogen-air systems. The chemistry source term in the species equation is treated implicitly to alleviate the stiffness associated with fast reactions. The explicit, unsplit MacCormack finite-difference scheme is used to advance the governing equations in time, until convergence is achieved. The specific problem considered is the premixed flow in a channel with a 10-deg compression ramp. Three different chemistry models are used, accounting for increasing number of reactions and participating species. Two chemistry models assume nitrogen as inert, while the third model accounts for nitrogen reactions and NO(x) formation. The tangent slab approximation is used in the radiative flux formulation. A pseudogray model is used to represent the absorption-emission characteristics of the participating species. Results otained for specific conditions indicate that the radiative interactions vary substantially, depending on reactions involving HO2 and NO species, and that this can have a significant influence on the flowfield.

Chandrasekhar, R.↗
NASA NTRS1976

Motions of a liquid in a pulsating bulb with application to problems of blood flow

Potential flows may be utilized to represent motions produced in pulsating bulbs. While the initial bulb shape may be arbitrary, sequential shapes are related by affine transformations. Two components appear in the distribution of pressure, one dependent on the instantaneous velocity and the other on the acceleration. For flows with stationary streamlines the inertial impedance is that of a simple mass, and is proportional to the first moment of the actual mass of fluid contained within the bulb. Examples treated are: (1) Expanding and collapsing circular cylinders, and (2) elliptical cylinders in which the perimeter is held constant. The thickness of the pulsatile laminar boundary layer is found to be approximately on millimeter for conditions in the vicinity of the heart. Conditions for separation and turbulence differ from those in steady flow.

Jones, R. T.↗
NASA NTRS1992

Thermochemical nonequilibrium and radiative interactions in supersonic hydrogen-air combustion

The two-dimensional, elliptic Navier-Stokes equations are used to investigate supersonic flows with nonequilibrium chemistry and thermodynamics, coupled with radiation, for hydrogen-air systems. The chemistry source term in the species equation is treated implicitly to alleviate the stiffness associated with fast reactions. The explicit, unsplit MacCormack finite-difference scheme is used to advance the governing equations in time, until convergence is achieved. The specific problem considered is the premixed, expanding flow in a supersonic nozzle. The reacting flow consists of seven species, one of which is the inert N2 molecule. The thermal state of the gas is modeled with one translational-rotational temperature and five vibrational temperatures. The harmonic oscillator model is used in the formulation for vibrational relaxation. The tangent slab approximation is used in the radiative flux formulation. A pseudo-gray model is used to represent the absorption-emission characteristics of the participating species. Results obtained for specific conditions indicate the presence of nonequilibrium in the expansion region. This reduces the radiative interactions and can have a significant influence on the flowfield.

Chandrasekhar, R.↗
NASA NTRS1991

A Rayleigh-Ritz analysis methodology for cutouts in composite structures

A new Rayleigh-Ritz stress analysis methodology that was developed for composite panels containing cutouts is described. The procedure, which makes use of a general assumed displacement field, accommodates circular and elliptical cutouts in biaxially loaded rectangular composite panels. Symmetric integral padups around the cutout can be included in the analysis. Benchmark results are presented to demonstrate the accuracy of the technique. Strength predictions based on the average stress criterion are generated and compared with experimental data. Finally, the stress analysis methodology is integrated into a design procedure for sizing integral padups around circular cutouts, and a sample problem is solved to illustrate its use.

Russell, Steven G.↗
NASA NTRS2020

Bearing Anomaly for the Sentinel 6 Supplemental Calibration System

The Supplemental Calibration System (SCS) was designed to rotate an elliptical reflector for the Sentinel 6 Advanced Microwave Radiometer (AMR). By rotating the reflector, the system can perform calibrations by comparing the reflection from space to the reflection from an onboard calibration target. During the qualification testing of the SCS, a flaw was discovered that resulted in apparent brinelling of bearings that support the rotating reflector. An extensive investigation ensued to determine the root cause of the problem. Though the most likely root causes of the presumed brinelling ended up being the result of undersized bearing clamp rings and thermal mismatch, there were requirements that drove these design choices. This paper describes the design of the bearing assembly, requirements that drove the design, and documents a challenging investigation filled with ambiguous test results.

Paulsen, Gale↗