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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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NASA NTRS2014

Comptonization in Ultra-Strong Magnetic Fields: Numerical Solution to the Radiative Transfer Problem

We consider the radiative transfer problem in a plane-parallel slab of thermal electrons in the presence of an ultra-strong magnetic field (B approximately greater than B(sub c) approx. = 4.4 x 10(exp 13) G). Under these conditions, the magnetic field behaves like a birefringent medium for the propagating photons, and the electromagnetic radiation is split into two polarization modes, ordinary and extraordinary, that have different cross-sections. When the optical depth of the slab is large, the ordinary-mode photons are strongly Comptonized and the photon field is dominated by an isotropic component. Aims. The radiative transfer problem in strong magnetic fields presents many mathematical issues and analytical or numerical solutions can be obtained only under some given approximations. We investigate this problem both from the analytical and numerical point of view, provide a test of the previous analytical estimates, and extend these results with numerical techniques. Methods. We consider here the case of low temperature black-body photons propagating in a sub-relativistic temperature plasma, which allows us to deal with a semi-Fokker-Planck approximation of the radiative transfer equation. The problem can then be treated with the variable separation method, and we use a numerical technique to find solutions to the eigenvalue problem in the case of a singular kernel of the space operator. The singularity of the space kernel is the result of the strong angular dependence of the electron cross-section in the presence of a strong magnetic field. Results. We provide the numerical solution obtained for eigenvalues and eigenfunctions of the space operator, and the emerging Comptonization spectrum of the ordinary-mode photons for any eigenvalue of the space equation and for energies significantly lesser than the cyclotron energy, which is on the order of MeV for the intensity of the magnetic field here considered. Conclusions. We derived the specific intensity of the ordinary photons, under the approximation of large angle and large optical depth. These assumptions allow the equation to be treated using a diffusion-like approximation.

acceleration of particles↗
NASA NTRS1986

Spline-based Rayleigh-Ritz methods for the approximation of the natural modes of vibration for flexible beams with tip bodies

Rayleigh-Ritz methods for the approximation of the natural modes for a class of vibration problems involving flexible beams with tip bodies using subspaces of piecewise polynomial spline functions are developed. An abstract operator-theoretic formulation of the eigenvalue problem is derived and spectral properties investigated. The existing theory for spline-based Rayleigh-Ritz methods applied to elliptic differential operators and the approximation properties of interpolatory splines are used to argue convergence and establish rates of convergence. An example and numerical results are discussed.

Rosen, I. G.↗
NASA NTRS1980

Preliminary demonstration of a robust controller design method

Alternative computational procedures for obtaining a feedback control law which yields a control signal based on measurable quantitites are evaluated. The three methods evaluated are: (1) the standard linear quadratic regulator design model; (2) minimization of the norm of the feedback matrix, k via nonlinear programming subject to the constraint that the closed loop eigenvalues be in a specified domain in the complex plane; and (3) maximize the angles between the closed loop eigenvectors in combination with minimizing the norm of K also via the constrained nonlinear programming. The third or robust design method was chosen to yield a closed loop system whose eigenvalues are insensitive to small changes in the A and B matrices. The relationship between orthogonality of closed loop eigenvectors and the sensitivity of closed loop eigenvalues is described. Computer programs are described.

Anderson, L. R.↗
NASA NTRS1978

On a numerical solution of the plastic buckling problem of structures

An automated digital computer procedure is presented for the accurate and efficient solution of the plastic buckling problem of structures. This is achieved by a Sturm sequence method employing a bisection strategy, which eliminates the need for having to solve the buckling eigenvalue problem at each incremental (decremental) loading stage that is associated with the usual solution techniques. The plastic buckling mode shape is determined by a simple inverse iteration process, once the buckling load has been established. Numerical results are presented for plate problems with various edge conditions. The resulting computer program written in FORTRAN V for the JPL UNIVAC 1108 machine proves to be most economical in comparison with other existing methods of such analysis.

Gupta, K. K.↗
NASA NTRS1980

A method of order reduction for structural dynamics

This paper presents a new method of order reduction based on recent work of a similar nature applicable to system dynamics and control. This method provides a practical computational procedure for producing a condensed model which exactly preserves the slowest n1 modes (where n1 is the number of 'slow' eigenvalues) of the total n modes of the original model for almost any set of n1 degrees of freedom retained in the condensed model. The method can also be used to compute the eigensolutions corresponding to the n1 slowest modes of the original structural dynamics problem. The accuracy of the condensed model and the speed/accuracy performance of the eigensolver are compared with standard methods for a 90 DOF cantilevered plate.

Anderson, L. R.↗
NASA NTRS1974

Reanalysis information for eigenvalues derived from a differential equation analysis formulation

The investigation reported demonstrates that in the case considered perturbation methods can be used in a straightforward manner to obtain reanalysis information. A perturbation formula for the buckling loads of a general shell of revolution is derived. The accuracy of the obtained relations and their range of application is studied with the aid of a specific example involving a particular stiffened shell of revolution.

Thornton, W. A.↗
NASA NTRS1981

Eigenvector determination by iterative optical methods

Three power algorithms are considered by which an iterative optical processor can be used to compute the eigenvalues and eigenvectors of a matrix. The algorithms are appropriate for three applications: (1) calculating the largest eigenvalue; (2) calculating the eigenvalues in order of decreasingly dominant eigenvalues; and (3) calculating the eigenvector whose eigenvalue is closest to a given value.

Vijaya Kumar, B. V. K.↗
NASA NTRS1991

Eigenvalue perturbation models for flexible structures

It is pointed out that real parametric modal frequency and damping variation for lightly damped systems does not resemble disks on the complex plane. Using complex uncertainty (and therefore disklike) models can introduce conservativeness in the design method. An alternative means of developing suitable complex uncertainty models is presented. It involves treating the uncertainty as perturbations to the system eigenvalues and using a particular linear fractional transformation to cover this uncertainty. This approach is applicable to modes within the bandwidth of control. It can be used in conjunction with the standard perturbation modeling approaches. A simple SISO (single-input single-output) example, motivated by a flexible truss experiment at the Jet Propulsion Laboratory, is discussed in order to illustrate the proposed approach.

Smith, Roy S.↗
NASA NTRS1985

Parameter identification in distributed systems

This paper describes a method for the identification of the parameters entering into the equations of motion of distributed systems. Because the motion of distributed systems is described in terms of partial differential equations, these parameters are in general continuous functions of the spatial variables. For vibrating systems, these parameters ordinarily represent the mass, stiffness and damping distributions. In this paper, these distributions are expanded in terms of finite series of known functions of the spatial variables multiplied by undetermined coefficients. It is assumed that the nature of the equations of motion is known and that a limited number of eigenvalues and eigenfunctions is identified in advance. Use is then made of the least squares method, in conjunction with the eigenfunctions' orthogonality, to compute the undetermined coefficients, thus identifying the system distributed parameters. A method for the identification of the eigensolution is also presented. The procedure for the identification of the eigensolution and of the system parameters is demonstrated via a numerical example.

Baruh, H.↗
NASA NTRS1987

Computational procedures for evaluating the sensitivity derivatives of vibration frequencies and Eigenmodes of framed structures

Computational procedures are presented for evaluating the sensitivity derivatives of the vibration frequencies and eigenmodes of framed structures. Both a displacement and a mixed formulation are used. The two key elements of the computational procedure are: (a) Use of dynamic reduction techniques to substantially reduce the number of degrees of freedom; and (b) Application of iterative techniques to improve the accuracy of the derivatives of the eigenmodes. The two reduction techniques considered are the static condensation and a generalized dynamic reduction technique. Error norms are introduced to assess the accuracy of the eigenvalue and eigenvector derivatives obtained by the reduction techniques. The effectiveness of the methods presented is demonstrated by three numerical examples.

Fetterman, Timothy L.↗
NASA NTRS1989

Quotient-difference type generalizations of the power method and their analysis

The recursion relations that were proposed by W. F. Ford and A. Sidi (Appl. Numer. Math, 4 (1988), pp. 477-489) for implementing vector extrapolation methods are used for devising generalizations of the power method for linear operators. These generalizations are shown to produce approximations to largest eigenvalues of a linear operator under certain conditions. They are similar in form to the quotient-difference algorithm and share similar convergence properties with the latter. These convergence properties also resemble those obtained for the basic LR and QR algorithms. Finally, it is shown that the convergence rate produced by one fo these generalizations is twice as fast for normal operators as it is for nonnormal operators.

Sidi, Avram↗
NASA NTRS1990

On the instabilities of supersonic mixing layers - A high-Mach-number asymptotic theory

The stability of a family of tanh mixing layers is studied at large Mach numbers using perturbation methods. It is found that the eigenfunction develops a multilayered structure, and the eigenvalue is obtained by solving a simplified version of the Rayleigh equation (with homogeneous boundary conditions) in one of these layers which lies in either of the external streams. This analysis leads to a simple hypersonic similarity law which explains how spatial and temporal phase speeds and growth rates scale with Mach number and temperature ratio. Comparisons are made with numerical results, and it is found that this similarity law provides a good qualitative guide for the behavior of the instability at high Mach numbers. In addition to this asymptotic theory, some fully numerical results are also presented (with no limitation on the Mach number) in order to explain the origin of the hypersonic modes (through mode splitting) and to discuss the role of oblique modes over a very wide range of Mach number and temperature ratio.

Balsa, Thomas F.↗
NASA NTRSDate not supplied

SLS Integrated Modal Test Uncertainty Quantification using the Hybrid Parametric Variation Method

Uncertainty in structural loading during launch is a significant concern in the development of spacecraft and launch vehicles. Small variations in launch vehicle and payload mode shapes and their interaction can result in significant variation in system loads. In many cases involving large aerospace systems it is difficult, not economical, or impossible to perform a system modal test. However, it is still vital to obtain test results that can be compared with analytical predictions to validate models. Instead, the “Building Block Approach” is used in which system components are tested individually. Component models are correlated and updated to agree as best they can with test results. The Space Launch System consists of a number of components that are assembled into a launch vehicle. Finite element models of the components are developed, reduced to Hurty/Craig-Bampton models and assembled to represent different phases of flight. The only opportunity to obtain modal test data from an assembled Space Launch System will be during the Integrated Modal Test. There is always uncertainty in every model, which flows into uncertainty in predicted system results. Uncertainty Quantification is used to determine statistical bounds on prediction accuracy based on model uncertainty. For the Space Launch System, model uncertainty is at the Hurty/Craig-Bampton component level. Uncertainty in the Hurty/Craig-Bampton components is quantified using the hybrid parametric variation approach that combines parametric and nonparametric uncertainty. Uncertainty in model form is one of the biggest contributors to uncertainty in complex built-up structures. This type of uncertainty cannot be represented by variations infinite element model input parameters and thus cannot be included in a parametric approach. However, model-form uncertainty can be modeled using a nonparametric approach based on random matrix theory. The hybrid parametric variation method requires the selection of dispersion values for the Hurty/Craig-Bampton fixed-interface eigenvalues, and the Hurty/Craig-Bampton stiffness matrices. Component test/analysis frequency error is used to identify the fixed-interface eigenvalue dispersions, while test/analysis cross-orthogonality is used to identify stiffness dispersion values. The hybrid parametric variation uncertainty quantification approach is applied to the Space Launch System Integrated Modal Test configuration. Monte Carlo analysis is performed, and statistics are determined for modal correlation metrics, frequency response from Integrated Modal Test shakers to selected accelerometers, as well as other metrics for determining how well target modes are excited and identified. If the predicted uncertainty envelopes future Integrated Modal Test results, then there will be increased confidence in the utility of the component-based hybrid parametric variation uncertainty quantification approach.

Uncertainty Quantification↗
NASA NTRS1990

An implementation of the look-ahead Lanczos algorithm for non-Hermitian matrices, part 2

It is shown how the look-ahead Lanczos process (combined with a quasi-minimal residual QMR) approach) can be used to develop a robust black box solver for large sparse non-Hermitian linear systems. Details of an implementation of the resulting QMR algorithm are presented. It is demonstrated that the QMR method is closely related to the biconjugate gradient (BCG) algorithm; however, unlike BCG, the QMR algorithm has smooth convergence curves and good numerical properties. We report numerical experiments with our implementation of the look-ahead Lanczos algorithm, both for eigenvalue problem and linear systems. Also, program listings of FORTRAN implementations of the look-ahead algorithm and the QMR method are included.

Freund, Roland W.↗
NASA NTRS1985

Probabilistic finite elements for transient analysis in nonlinear continua

The probabilistic finite element method (PFEM), which is a combination of finite element methods and second-moment analysis, is formulated for linear and nonlinear continua with inhomogeneous random fields. Analogous to the discretization of the displacement field in finite element methods, the random field is also discretized. The formulation is simplified by transforming the correlated variables to a set of uncorrelated variables through an eigenvalue orthogonalization. Furthermore, it is shown that a reduced set of the uncorrelated variables is sufficient for the second-moment analysis. Based on the linear formulation of the PFEM, the method is then extended to transient analysis in nonlinear continua. The accuracy and efficiency of the method is demonstrated by application to a one-dimensional, elastic/plastic wave propagation problem. The moments calculated compare favorably with those obtained by Monte Carlo simulation. Also, the procedure is amenable to implementation in deterministic FEM based computer programs.

Liu, W. K.↗
NASA NTRS1992

Rational approximations from power series of vector-valued meromorphic functions

Let F(z) be a vector-valued function, F: C yields C(sup N), which is analytic at z = 0 and meromorphic in a neighborhood of z = 0, and let its Maclaurin series be given. In this work we developed vector-valued rational approximation procedures for F(z) by applying vector extrapolation methods to the sequence of partial sums of its Maclaurin series. We analyzed some of the algebraic and analytic properties of the rational approximations thus obtained, and showed that they were akin to Pade approximations. In particular, we proved a Koenig type theorem concerning their poles and a de Montessus type theorem concerning their uniform convergence. We showed how optical approximations to multiple poles and to Laurent expansions about these poles can be constructed. Extensions of the procedures above and the accompanying theoretical results to functions defined in arbitrary linear spaces was also considered. One of the most interesting and immediate applications of the results of this work is to the matrix eigenvalue problem. In a forthcoming paper we exploited the developments of the present work to devise bona fide generalizations of the classical power method that are especially suitable for very large and sparse matrices. These generalizations can be used to approximate simultaneously several of the largest distinct eigenvalues and corresponding eigenvectors and invariant subspaces of arbitrary matrices which may or may not be diagonalizable, and are very closely related with known Krylov subspace methods.

Sidi, Avram↗
NASA NTRS1999

How to Estimate Attitude from Vector Observations

The most robust estimators minimizing Wahba's loss function are Davenport's q method and the Singular Value Decomposition (SVD) method. The q method is faster than the SVD method with three or more measurements. The other algorithms are less robust since they solve the characteristic polynomial equation to find the maximum eigenvalue of Davenport's K matrix. They are only preferable when speed or processor power is an important consideration. Of these, Fast Optimal Attitude Matrix (FOAM) is the most robust and faster than the q method. Robustness is only an issue for measurements with widely differing accuracies, so the fastest algorithms, Quaternion ESTimator (QUEST), EStimator of the Optimal Quaternion (ESOQ), and ESOQ2, are well suited to star sensor applications.

Markley, F. Landis↗
NASA NTRS1986

Landsat D Thematic Mapper image dimensionality reduction and geometric correction accuracy

To characterize and quantify the performance of the Landsat thematic mapper (TM), techniques for dimensionality reduction by linear transformation have been studied and evaluated and the accuracy of the correction of geometric errors in TM images analyzed. Theoretical evaluations and comparisons for existing methods for the design of linear transformation for dimensionality reduction are presented. These methods include the discrete Karhunen Loeve (KL) expansion, Multiple Discriminant Analysis (MDA), Thematic Mapper (TM)-Tasseled Cap Linear Transformation and Singular Value Decomposition (SVD). A unified approach to these design problems is presented in which each method involves optimizing an objective function with respect to the linear transformation matrix. From these studies, four modified methods are proposed. They are referred to as the Space Variant Linear Transformation, the KL Transform-MDA hybrid method, and the First and Second Version of the Weighted MDA method. The modifications involve the assignment of weights to classes to achieve improvements in the class conditional probability of error for classes with high weights. Experimental evaluations of the existing and proposed methods have been performed using the six reflective bands of the TM data. It is shown that in terms of probability of classification error and the percentage of the cumulative eigenvalues, the six reflective bands of the TM data require only a three dimensional feature space. It is shown experimentally as well that for the proposed methods, the classes with high weights have improvements in class conditional probability of error estimates as expected.

Ford, G. E.↗