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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 343 records · Page 19
All sourcesNASA NTRSDOE OSTI
NASA NTRS1992

Mappings and accuracy for Chebyshev pseudo-spectral approximations

The effect of mappings on the approximation, by Chebyshev collocation, of functions which exhibit localized regions of rapid variation is studied. A general strategy is introduced whereby mappings are adaptively constructed which map specified classes of rapidly varying functions into low order polynomials which can be accurately approximated by Chebyshev polynomial expansions. A particular family of mappings constructed in this way is tested on a variety of rapidly varying functions similar to those occurring in approximations. It is shown that the mapped function can be approximated much more accurately by Chebyshev polynomial approximations than in physical space or where mappings constructed from other strategies are employed.

Bayliss, Alvin↗
NASA NTRS1993

Robust algebraic image enhancement for intelligent control systems

Robust vision capability for intelligent control systems has been an elusive goal in image processing. The computationally intensive techniques a necessary for conventional image processing make real-time applications, such as object tracking and collision avoidance difficult. In order to endow an intelligent control system with the needed vision robustness, an adequate image enhancement subsystem capable of compensating for the wide variety of real-world degradations, must exist between the image capturing and the object recognition subsystems. This enhancement stage must be adaptive and must operate with consistency in the presence of both statistical and shape-based noise. To deal with this problem, we have developed an innovative algebraic approach which provides a sound mathematical framework for image representation and manipulation. Our image model provides a natural platform from which to pursue dynamic scene analysis, and its incorporation into a vision system would serve as the front-end to an intelligent control system. We have developed a unique polynomial representation of gray level imagery and applied this representation to develop polynomial operators on complex gray level scenes. This approach is highly advantageous since polynomials can be manipulated very easily, and are readily understood, thus providing a very convenient environment for image processing. Our model presents a highly structured and compact algebraic representation of grey-level images which can be viewed as fuzzy sets.

Lerner, Bao-Ting↗
NASA NTRS1992

Stability margins for multilinear interval systems by way of phase conditions: A unified approach

A simple way of checking the stability with respect to an arbitrary stability region of a family of polynomials containing a vector of parameters varying within prescribed intervals is discussed. It is assumed that the parameters appear affine multilinearly in the characteristic polynomial coefficients. The condition proposed is simply to check the phase difference of the vertex polynomials. This test based on the mapping theorem significantly reduces computational complexity. Mathematical proofs are omitted. The results can be used to determine various stability margins of control systems containing interconnected interval subsystems. These include the gain, phase, time-delay, H(sup infinity), and nonlinear sector bounded stability margins of multilinear interval systems.

Keel, L. H.↗
NASA NTRS1993

Exploiting structure: Introduction and motivation

Research activities performed during the period of 29 June 1993 through 31 Aug. 1993 are summarized. The Robust Stability of Systems where transfer function or characteristic polynomial are multilinear affine functions of parameters of interest in two directions, Algorithmic and Theoretical, was developed. In the algorithmic direction, a new approach that reduces the computational burden of checking the robust stability of the system with multilinear uncertainty is found. This technique is called 'Stability by linear process.' In fact, the 'Stability by linear process' described gives an algorithm. In analysis, we obtained a robustness criterion for the family of polynomials with coefficients of multilinear affine function in the coefficient space and obtained the result for the robust stability of diamond families of polynomials with complex coefficients also. We obtained the limited results for SPR design and we provide a framework for solving ACS. Finally, copies of the outline of our results are provided in the appendix. Also, there is an administration issue in the appendix.

Xu, Zhong Ling↗
NASA NTRS1993

Rational trigonometric approximations using Fourier series partial sums

A class of approximations (S(sub N,M)) to a periodic function f which uses the ideas of Pade, or rational function, approximations based on the Fourier series representation of f, rather than on the Taylor series representation of f, is introduced and studied. Each approximation S(sub N,M) is the quotient of a trigonometric polynomial of degree N and a trigonometric polynomial of degree M. The coefficients in these polynomials are determined by requiring that an appropriate number of the Fourier coefficients of S(sub N,M) agree with those of f. Explicit expressions are derived for these coefficients in terms of the Fourier coefficients of f. It is proven that these 'Fourier-Pade' approximations converge point-wise to (f(x(exp +))+f(x(exp -)))/2 more rapidly (in some cases by a factor of 1/k(exp 2M)) than the Fourier series partial sums on which they are based. The approximations are illustrated by several examples and an application to the solution of an initial, boundary value problem for the simple heat equation is presented.

Geer, James F.↗
NASA NTRS1994

Some rules for polydimensional squeezing

The review of the following results is presented: For mixed state light of N-mode electromagnetic field described by Wigner function which has generic Gaussian form, the photon distribution function is obtained and expressed explicitly in terms of Hermite polynomials of 2N-variables. The momenta of this distribution are calculated and expressed as functions of matrix invariants of the dispersion matrix. The role of new uncertainty relation depending on photon state mixing parameter is elucidated. New sum rules for Hermite polynomials of several variables are found. The photon statistics of polymode even and odd coherent light and squeezed polymode Schroedinger cat light is given explicitly. Photon distribution for polymode squeezed number states expressed in terms of multivariable Hermite polynomials is discussed.

Manko, Vladimir I.↗
NASA NTRS1994

An Efficient Spectral Method for Ordinary Differential Equations with Rational Function Coefficients

We present some relations that allow the efficient approximate inversion of linear differential operators with rational function coefficients. We employ expansions in terms of a large class of orthogonal polynomial families, including all the classical orthogonal polynomials. These families obey a simple three-term recurrence relation for differentiation, which implies that on an appropriately restricted domain the differentiation operator has a unique banded inverse. The inverse is an integration operator for the family, and it is simply the tridiagonal coefficient matrix for the recurrence. Since in these families convolution operators (i.e. matrix representations of multiplication by a function) are banded for polynomials, we are able to obtain a banded representation for linear differential operators with rational coefficients. This leads to a method of solution of initial or boundary value problems that, besides having an operation count that scales linearly with the order of truncation N, is computationally well conditioned. Among the applications considered is the use of rational maps for the resolution of sharp interior layers.

Coutsias, Evangelos A.↗
NASA NTRS1992

A new model for estimating total body water from bioelectrical resistance

Estimation of total body water (T) from bioelectrical resistance (R) is commonly done by stepwise regression models with height squared over R, H(exp 2)/R, age, sex, and weight (W). Polynomials of H(exp 2)/R have not been included in these models. We examined the validity of a model with third order polynomials and W. Methods: T was measured with oxygen-18 labled water in 27 subjects. R at 50 kHz was obtained from electrodes placed on the hand and foot while subjects were in the supine position. A stepwise regression equation was developed with 13 subjects (age 31.5 plus or minus 6.2 years, T 38.2 plus or minus 6.6 L, W 65.2 plus or minus 12.0 kg). Correlations, standard error of estimates and mean differences were computed between T and estimated T's from the new (N) model and other models. Evaluations were completed with the remaining 14 subjects (age 32.4 plus or minus 6.3 years, T 40.3 plus or minus 8 L, W 70.2 plus or minus 12.3 kg) and two of its subgroups (high and low) Results: A regression equation was developed from the model. The only significant mean difference was between T and one of the earlier models. Conclusion: Third order polynomials in regression models may increase the accuracy of estimating total body water. Evaluating the model with a larger population is needed.

Siconolfi, S. F.↗
NASA NTRS1995

Statistically generated weighted curve fit of residual functions for modal analysis of structures

A statistically generated weighting function for a second-order polynomial curve fit of residual functions has been developed. The residual flexibility test method, from which a residual function is generated, is a procedure for modal testing large structures in an external constraint-free environment to measure the effects of higher order modes and interface stiffness. This test method is applicable to structures with distinct degree-of-freedom interfaces to other system components. A theoretical residual function in the displacement/force domain has the characteristics of a relatively flat line in the lower frequencies and a slight upward curvature in the higher frequency range. In the test residual function, the above-mentioned characteristics can be seen in the data, but due to the present limitations in the modal parameter evaluation (natural frequencies and mode shapes) of test data, the residual function has regions of ragged data. A second order polynomial curve fit is required to obtain the residual flexibility term. A weighting function of the data is generated by examining the variances between neighboring data points. From a weighted second-order polynomial curve fit, an accurate residual flexibility value can be obtained. The residual flexibility value and free-free modes from testing are used to improve a mathematical model of the structure. The residual flexibility modal test method is applied to a straight beam with a trunnion appendage and a space shuttle payload pallet simulator.

Bookout, P. S.↗
NASA NTRS1993

Sensitivity Analysis of the Static Aeroelastic Response of a Wing

A technique to obtain the sensitivity of the static aeroelastic response of a three dimensional wing model is designed and implemented. The formulation is quite general and accepts any aerodynamic and structural analysis capability. A program to combine the discipline level, or local, sensitivities into global sensitivity derivatives is developed. A variety of representations of the wing pressure field are developed and tested to determine the most accurate and efficient scheme for representing the field outside of the aerodynamic code. Chebyshev polynomials are used to globally fit the pressure field. This approach had some difficulties in representing local variations in the field, so a variety of local interpolation polynomial pressure representations are also implemented. These panel based representations use a constant pressure value, a bilinearly interpolated value. or a biquadraticallv interpolated value. The interpolation polynomial approaches do an excellent job of reducing the numerical problems of the global approach for comparable computational effort. Regardless of the pressure representation used. sensitivity and response results with excellent accuracy have been produced for large integrated quantities such as wing tip deflection and trim angle of attack. The sensitivities of such things as individual generalized displacements have been found with fair accuracy. In general, accuracy is found to be proportional to the relative size of the derivatives to the quantity itself.

Eldred, Lloyd B.↗
NASA NTRS1998

Optimization of Turbine Blade Design for Reusable Launch Vehicles

To facilitate design optimization of turbine blade shape for reusable launching vehicles, appropriate techniques need to be developed to process and estimate the characteristics of the design variables and the response of the output with respect to the variations of the design variables. The purpose of this report is to offer insight into developing appropriate techniques for supporting such design and optimization needs. Neural network and polynomial-based techniques are applied to process aerodynamic data obtained from computational simulations for flows around a two-dimensional airfoil and a generic three- dimensional wing/blade. For the two-dimensional airfoil, a two-layered radial-basis network is designed and trained. The performances of two different design functions for radial-basis networks, one based on the accuracy requirement, whereas the other one based on the limit on the network size. While the number of neurons needed to satisfactorily reproduce the information depends on the size of the data, the neural network technique is shown to be more accurate for large data set (up to 765 simulations have been used) than the polynomial-based response surface method. For the three-dimensional wing/blade case, smaller aerodynamic data sets (between 9 to 25 simulations) are considered, and both the neural network and the polynomial-based response surface techniques improve their performance as the data size increases. It is found while the relative performance of two different network types, a radial-basis network and a back-propagation network, depends on the number of input data, the number of iterations required for radial-basis network is less than that for the back-propagation network.

Shyy, Wei↗
NASA NTRS1999

Student Support for Research in Hierarchical Control and Trajectory Planning

Generally, classical polynomial splines tend to exhibit unwanted undulations. In this work, we discuss a technique, based on control principles, for eliminating these undulations and increasing the smoothness properties of the spline interpolants. We give a generalization of the classical polynomial splines and show that this generalization is, in fact, a family of splines that covers the broad spectrum of polynomial, trigonometric and exponential splines. A particular element in this family is determined by the appropriate control data. It is shown that this technique is easy to implement. Several numerical and curve-fitting examples are given to illustrate the advantages of this technique over the classical approach. Finally, we discuss the convergence properties of the interpolant.

Martin, Clyde F.↗
NASA NTRS1999

Modeling the High Speed Research Cycle 2B Longitudinal Aerodynamic Database Using Multivariate Orthogonal Functions

The data for longitudinal non-dimensional, aerodynamic coefficients in the High Speed Research Cycle 2B aerodynamic database were modeled using polynomial expressions identified with an orthogonal function modeling technique. The discrepancy between the tabular aerodynamic data and the polynomial models was tested and shown to be less than 15 percent for drag, lift, and pitching moment coefficients over the entire flight envelope. Most of this discrepancy was traced to smoothing local measurement noise and to the omission of mass case 5 data in the modeling process. A simulation check case showed that the polynomial models provided a compact and accurate representation of the nonlinear aerodynamic dependencies contained in the HSR Cycle 2B tabular aerodynamic database.

Morelli, E. A.↗
NASA NTRS2002

Spectral (Finite) Volume Method for One Dimensional Euler Equations

Consider a mesh of unstructured triangular cells. Each cell is called a Spectral Volume (SV), denoted by Si, which is further partitioned into subcells named Control Volumes (CVs), indicated by C(sub i,j). To represent the solution as a polynomial of degree m in two dimensions (2D) we need N = (m+1)(m+2)/2 pieces of independent information, or degrees of freedom (DOFs). The DOFs in a SV method are the volume-averaged mean variables at the N CVs. For example, to build a quadratic reconstruction in 2D, we need at least (2+1)(3+1)/2 = 6 DOFs. There are numerous ways of partitioning a SV, and not every partition is admissible in the sense that the partition may not be capable of producing a degree m polynomial. Once N mean solutions in the CVs of a SV are given, a unique polynomial reconstruction can be obtained.

Wang, Z. J.↗
NASA NTRS2005

Radiometer Calibrations: Saving Time by Automating the Gathering and Analysis Procedures

Mr. Abtahi custom-designs radiometers for Mr. Hook's research group. Inherently, when the radiometers report the temperature of arbitrary surfaces, the results are affected by errors in accuracy. This problem can be reduced if the errors can be accounted for in a polynomial. This is achieved by pointing the radiometer at a constant-temperature surface. We have been using a Hartford Scientific WaterBath. The measurements from the radiometer are collected at many different temperatures and compared to the measurements made by a Hartford Chubb thermometer with a four-decimal point resolution. The data is analyzed and fit to a fifth-order polynomial. This formula is then uploaded into the radiometer software, enabling accurate data gathering. Traditionally, Mr. Abtahi has done this by hand, spending several hours of his time setting the temperature, waiting for stabilization, taking measurements, and then repeating for other temperatures. My program, written in the Python language, has enabled the data gathering and analysis process to be handed off to a less-senior member of the team. Simply by entering several initial settings, the program will simultaneously control all three instruments and organize the data suitable for computer analyses, thus giving the desired fifth-order polynomial. This will save time, allow for a more complete calibration data set, and allow for base calibrations to be developed. The program is expandable to simultaneously take any type of measurement from up to nine distinct instruments.

Sadino, Jeffrey L.↗
NASA NTRS2007

Zernike-like Orthogonal Basis Functions for Wavefront Characterization over Sampled, Irregular Apertures

For optical systems with circular apertures, wavefronts are often analyzed using Zernike polynomials, and individual Zernike functions are associated with familiar optical aberrations. For systems with noncircular apertures, or in practical situations in which wavefronts are measured at a limited number of points in the aperture, the Zernike polynomials are no longer an orthogonal basis for the measured data. Although there are an endless number of ways to create a basis for such measured data, a "Zernike-like" basis is useful to connect with our experience with the usual optical aberrations. In this paper, the steps required to identify a Zernike-like basis for wavefronts over sampled, irregular apertures are presented, based on the Gram-Schmidt orthogonalization technique. The benefits of analyzing optical wavefronts using an orthogonal basis specific to an optical system's aperture shape and wavefront sampling, instead of using the traditional Zernike polynomials, are detailed in two examples, from image-based wavefront sensing on a segmented-aperture telescope (the James Webb Space Telescope Testbed Telescope at Ball Aerospace) and from interferometer characterization for surface metrology of a hexagonal mirror segment.

Aronstein, David L.↗
NASA NTRS2011

Improvements to a Response Surface Thermal Model for Orion Mated to the International Space Station

This study is an extension of previous work to evaluate the applicability of Design of Experiments (DOE)/Response Surface Methodology to on-orbit thermal analysis. The goal was to determine if the methodology could produce a Response Surface Equation (RSE) that predicted the thermal model temperature results within +/-10 F. An RSE is a polynomial expression that can then be used to predict temperatures for a defined range of factor combinations. Based on suggestions received from the previous work, this study used a model with simpler geometry, considered polynomials up to fifth order, and evaluated orbital temperature variations to establish a minimum and maximum temperature for each component. A simplified Outer Mold Line (OML) thermal model of the Orion spacecraft was used in this study. The factors chosen were the vehicle's Yaw, Pitch, and Roll (defining the on-orbit attitude), the Beta angle (restricted to positive beta angles from 0 to 75), and the environmental constants (varying from cold to hot). All factors were normalized from their native ranges to a non-dimensional range from -1.0 to 1.0. Twenty-three components from the OML were chosen and the minimum and maximum orbital temperatures were calculated for each to produce forty-six responses for the DOE model. A customized DOE case matrix of 145 analysis cases was developed which used analysis points at the factor corners, mid-points, and center. From this data set, RSE s were developed which consisted of cubic, quartic, and fifth order polynomials. The results presented are for the fifth order RSE. The RSE results were then evaluated for agreement with the analytical model predictions to produce a +/-3(sigma) error band. Forty of the 46 responses had a +/-3(sigma) value of 10 F or less. Encouraged by this initial success, two additional sets of verification cases were selected. One contained 20 cases, the other 50 cases. These cases were evaluated both with the fifth order RSE and with the analytical model. For the maximum temperature predictions, 12 of the 23 components had all predictions within +/-10 F and 17 were within +/-20 F. For the minimum temperature predictions, only 4 of the 23 components (the four radiator temperatures), were within the 10 F goal. The maximum temperature RSEs were then run through 59,049 screening cases. The RSE predictions were then filtered to find 55 cases that produced the hottest temperatures. These 55 cases were then analyzed using the thermal model and the results compared against the RSE predictions. As noted earlier, 12 of the 23 responses were within +/-10 F at 17 within +/-20 F. These results demonstrate that if properly formulated, an RSE can provide a reliable, fast temperature prediction. Despite this progress, additional work is needed to determine why the minimum temperatures responses and 6 of the hot temperature responses did not produce reliable RSEs. Recommend focus areas are the model itself (arithmetic vs. diffusion nodes) and seeking consultations with statistical application experts.

Miller, StephenW.↗
NASA NTRS2011

A Maximum Likelihood Approach to Determine Sensor Radiometric Response Coefficients for NPP VIIRS Reflective Solar Bands

Optical sensors aboard Earth orbiting satellites such as the next generation Visible/Infrared Imager/Radiometer Suite (VIIRS) assume that the sensors radiometric response in the Reflective Solar Bands (RSB) is described by a quadratic polynomial, in relating the aperture spectral radiance to the sensor Digital Number (DN) readout. For VIIRS Flight Unit 1, the coefficients are to be determined before launch by an attenuation method, although the linear coefficient will be further determined on-orbit through observing the Solar Diffuser. In determining the quadratic polynomial coefficients by the attenuation method, a Maximum Likelihood approach is applied in carrying out the least-squares procedure. Crucial to the Maximum Likelihood least-squares procedure is the computation of the weight. The weight not only has a contribution from the noise of the sensor s digital count, with an important contribution from digitization error, but also is affected heavily by the mathematical expression used to predict the value of the dependent variable, because both the independent and the dependent variables contain random noise. In addition, model errors have a major impact on the uncertainties of the coefficients. The Maximum Likelihood approach demonstrates the inadequacy of the attenuation method model with a quadratic polynomial for the retrieved spectral radiance. We show that using the inadequate model dramatically increases the uncertainties of the coefficients. We compute the coefficient values and their uncertainties, considering both measurement and model errors.

Lei, Ning↗