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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.

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At least 109 records · Page 6
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DOE OSTI2024

Collocation methods for nonlinear differential equations on low-rank manifolds

We introduce new methods for integrating nonlinear differential equations on low-rank manifolds. These methods rely on interpolatory projections onto the tangent space, enabling low-rank time integration of vector fields that can be evaluated entry-wise. A key advantage of our approach is that it does not require the vector field to exhibit low-rank structure, thereby overcoming significant limitations of traditional dynamical low-rank methods based on orthogonal projection. To construct the interpolatory projectors, we develop a sparse tensor sampling algorithm based on the discrete empirical interpolation method (DEIM) that parameterizes tensor train manifolds and their tangent spaces with cross interpolation. Using these projectors, we propose two time integration schemes on low-rank tensor train manifolds. The first scheme integrates the solution at selected interpolation indices and constructs the solution with cross interpolation. The second scheme generalizes the well-known orthogonal projector-splitting integrator to interpolatory projectors. We demonstrate the proposed methods with applications to several tensor differential equations arising from the discretization of partial differential equations.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1977

Small deflection of a class of clamped thin plates using collocation

Equations are given for the optimization of a class of two-and three-dimensional structures. The application of existing analytical techniques to the response of thin clamped plates is described. The ratios of deflections to plate thickness are given for uniform transverse loads as well as for uniform plus linearly varying transverse loads. Deflections are presented at angular increments of 5 degrees and at radial increments of 0.1 of the radius.

Worley, W. J.↗
NASA NTRS1978

The determination of gravity anomalies from geoid heights using the inverse Stokes' formula, Fourier transforms, and least squares collocation

A numerical method for the determination of gravity anomalies from geoid heights is described using the inverse Stokes formula. This discrete form of the inverse Stokes formula applies a numerical integration over the azimuth and an integration over a cubic interpolatory spline function which approximates the step function obtained from the numerical integration. The main disadvantage of the procedure is the lack of a reliable error measure. The method was applied on geoid heights derived from GEOS-3 altimeter measurements in the calibration area of the GEOS-3 satellite.

Rummel, R.↗
NASA NTRS1984

A spectral collocation method for the Navier-Stokes equations

A Fourier-Chebyshev spectral method for the incompressible Navier-Stokes equations is described. It is applicable to a variety of problems including some with fluid properties which vary strongly both in the normal direction and in time. In this fully spectral algorithm, a preconditioned iterative technique is used for solving the implicit equations arising from semi-implicit treatment of pressure, mean advection and vertical diffusion terms. The algorithm is tested by applying it to hydrodynamic stability problems in channel flow and in external boundary layers with both constant and variable viscosity.

Malik, M. R.↗
NASA NTRS1985

A spectral collocation method for the Navier-Stokes equations

A Fourier-Chebyshev spectral method for the incompressible Navier-Stokes equations is described. It is applicable to a variety of problems including some with fluid properties which vary strongly both in the normal direction and in time. In this fully spectral algorithm, a preconditioned iterative technique is used for solving the implicit equations arising from semi-implicit treatment of pressure, mean advection and vertical diffusion terms. The algorithm is tested by applying it to hydrodynamic stability problems in channel flow and in external boundary layers with both constant and variable viscosity.

Malik, M. R.↗
NASA NTRS1986

Collocation for an integral equation arising in duct acoustics

A mathematical model is developed to describe the effect of aircraft-engine inlet geometry on the reflected and radiated acoustic field without flow, as studied experimentally using a spinning-mode synthesizer by Silcox (1983). The acoustic pressure in the inlet interior and exterior is modeled by a pure cylindrical azimuthal mode for the Helmholtz equation with hardwall boundary and by the Helmholtz equation and the radiation condition at infinity, respectively. The analytical approach to the solution of the resulting boundary-value problem and the program implementation are explained; numerical results are presented in tables and graphs; and the uniqueness of the problem is demonstrated.

Moss, W. F.↗
NASA NTRS1988

A spectral collocation solution to the compressible stability eigenvalue problem

A newly developed spectral compressible linear stability code (SPECLS) (staggered pressure mesh) is presented for analysis of shear flow stability, and applied to high speed boundary layers and free shear flows. The formulation utilizes the first application of a staggered mesh for a compressible flow analysis by a spectral technique. An order of magnitude less number of points is needed for equivalent accuracy of growth rates compared to those calculated by a finite difference formulation. Supersonic disturbances which are found to have oscillatory structures were resolved by a spectral multi-domain discretization, which requires a factor of three fewer points than the single domain spectral stability code. It is indicated, as expected, that stability of mixing layers is enhanced by viscosity and increasing Mach number. The mean flow involves a jet being injected into a quiescent gas. Higher temperatures of the injected gas is also found to enhance stability characteristics of the free shear layer.

Macaraeg, Michele G.↗
NASA NTRS1988

Intercomparison of satellite laser ranging accuracy of three NASA stations through collocation

The accuracy of laser ranging has been evaluated through comparisons of simultaneous LAGEOS satellite-borne laser ranging data received at three NASA tracking stations in support of the Crustal Dynamics project. Single-shot satellite ranging precisions of 8, 14, and 30 mm have been demonstrated at the three ground stations, with a stability better than 3 mm. The data-processing software used were POLYQUICK and GEODYN; a consistent degree of agreement between the three stations of less than 1 cm is obtained.

Varghese, T.↗
NASA NTRS1990

Conforming Chebyshev spectral collocation methods for the solution of laminar flow in a constricted channel

The numerical simulation of steady planar two-dimensional, laminar flow of an incompressible fluid through an abruptly contracting channel using spectral domain decomposition methods is described. The key features of the method are the decomposition of the flow region into a number of rectangular subregions and spectral approximations which are pointwise C(1) continuous across subregion interfaces. Spectral approximations to the solution are obtained for Reynolds numbers in the range 0 to 500. The size of the salient corner vortex decreases as the Reynolds number increases from 0 to around 45. As the Reynolds number is increased further the vortex grows slowly. A vortex is detected downstream of the contraction at a Reynolds number of around 175 that continues to grow as the Reynolds number is increased further.

Karageorghis, Andreas↗
NASA NTRS1991

Compensator design for stability enhancement with collocated controllers

A continuum model rather than a finite element model is used. The optimal compensator design is formulated as a stochastic regulator problem and is shown to be solvable by the general infinite-dimensional theory developed by the author despite the lack of exponential stabilizability. Infinite-dimensional steady-state Riccati equations characterizing the feedback control gain and the Kalman filter gain operators can be solved explicitly. The associated performance indexes including the mean-square control-effort are calculated in closed form. As a first approximation, the compensator transfer function can be realized as a bank of bandpass filters in parallel centered at the undamped mode frequencies. Numerical calculations for the gain and bandwidths for a typical configuration are presented. The performance of the compensator is evaluated when in fact in the true model there is no actuator noise. The theoretical problem involved is to show that the infinite-dimensional stochastic process is asymptotically stationary. It is possible to calculate the steady-state covariance in closed form and thereby calculate performance indexes of interest explicitly, facilitating the choice of optimal design parameters.

Balakrishnan, A. V.↗
NASA NTRS1996

Time Varying Compensator Design for Reconfigurable Structures Using Non-Collocated Feedback

Analysis and synthesis tools are developed to improved the dynamic performance of reconfigurable nonminimum phase, nonstrictly positive real-time variant systems. A novel Spline Varying Optimal (SVO) controller is developed for the kinematic nonlinear system. There are several advantages to using the SVO controller, in which the spline function approximates the system model, observer, and controller gain. They are: The spline function approximation is simply connected, thus the SVO controller is more continuous than traditional gain scheduled controllers when implemented on a time varying plant; ft is easier for real-time implementations in storage and computational effort; where system identification is required, the spline function requires fewer experiments, namely four experiments; and initial startup estimator transients are eliminated. The SVO compensator was evaluated on a high fidelity simulation of the Shuttle Remote Manipulator System. The SVO controller demonstrated significant improvement over the present arm performance: (1) Damping level was improved by a factor of 3; and (2) Peak joint torque was reduced by a factor of 2 following Shuttle thruster firings.

Scott, Michael A.↗
NASA NTRS1997

The Potential for Collocated AGLP and ERBE data for Fire, Smoke, and Radiation Budget Studies

One month of the Advanced Very High Resolution Radiometer (AVHRR) Global Area Coverage (GAC) Land Pathfinder (AGLP) data from September 1985 are used to examine the spatial and temporal distribution of fires over four major ecosystems in South America. The Earth Radiation Budget Experiment (ERBE) scanner data are used to examine the top of atmosphere (TOA) shortwave and longwave fluxes over smoke generated from biomass burn- ing. The relationship between the AGLP-derived Normalized Difference Vegetation Index (NDVI) and the ERBE-estimated clear sky albedos are also examined as a function of the four ecosystems. This study shows that the grassland areas in South America have the highest number of fires for September 1985, and their corresponding NDVI values are smaller than the tropical rainforest region where the number of fires were comparatively small. Clear sky statistics accumulated during the days when smoke was not present show that clear sky albedos derived from ERBE are higher for grassland areas when compared to the tropical rainforest. The results show that the AGLP can be used to determine the spatial and temporal distribution of fires along with vegetation characteristics, while ERBE data can provide necessary information on broadband albedos and regional top of atmosphere radiative impacts of biomass burning aerosols. Since the AGLP data are available from 1981 to the present day, several climate-related issues can be addressed.

Christropher, S. A.↗
NASA NTRS1997

The Potential for Collocated AGLP and ERBE Data for Fire, Smoke, and Radiation Budget Studies

One month of the Advanced Very High Resolution Radiometer (AVHRR) Global Area Coverage (GAC) Land Pathfinder (AGLP) data from September 1985 are used to examine the spatial and temporal distribution of fires over four major ecosystems in South America. The Earth Radiation Budget Experiment (ERBE) scanner data are used to examine the top of atmosphere (TOA) shortwave and longwave fluxes over smoke generated from biomass burning. The relationship between the AGLP-derived Normalized Difference Vegetation Index (NDVI) and the ERBE-estimated clear sky albedos are also examined as a function of the four ecosystems. This study shows that the grassland areas in South America have the highest number of fires for September 1985. and their corresponding NDVI values are smaller than the tropical rainforest region where the number of fires were comparatively small. Clear sky statistics accumulated during the days when smoke was not present show that clear sky albedos derived from ERBE are higher for grassland areas when compared to the tropical rainforest. The results show that the AGLP can be used to determine the spatial and temporal distribution of fires along with vegetation characteristics, while ERBE data can provide necessary information on broadband albedos and regional top of atmosphere radiative impacts of biomass burning aerosols. Since the AGLP data are available from 1981 to the present day, several climate-related issues can be addressed,

Christopher, S. A.↗