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NASA NTRS1989

Numerical methods in Markov chain modeling

Several methods for computing stationary probability distributions of Markov chains are described and compared. The main linear algebra problem consists of computing an eigenvector of a sparse, usually nonsymmetric, matrix associated with a known eigenvalue. It can also be cast as a problem of solving a homogeneous singular linear system. Several methods based on combinations of Krylov subspace techniques are presented. The performance of these methods on some realistic problems are compared.

Philippe, Bernard↗

NASA NTRS1996

Probabilistic Component Mode Synthesis of Nondeterministic Substructures

Standard methods of structural dynamic analysis assume that the structural characteristics are deterministic. Recognizing that these characteristics are actually statistical in nature researchers have recently developed a variety of methods that use this information to determine probabilities of a desired response characteristic, such as natural frequency, without using expensive Monte Carlo simulations. One of the problems in these methods is correctly identifying the statistical properties of primitive variables such as geometry, stiffness, and mass. We present a method where the measured dynamic properties of substructures are used instead as the random variables. The residual flexibility method of component mode synthesis is combined with the probabilistic methods to determine the cumulative distribution function of the system eigenvalues. A simple cantilever beam test problem is presented that illustrates the theory.

Brown, Andrew M.↗

NASA NTRS1995

Development of a Probabilistic Component Mode Synthesis Method for the Analysis of Non-Deterministic Substructures

Standard methods of structural dynamic analysis assume that the structural characteristics are deterministic. Recognizing that these characteristics are actually statistical in nature, researchers have recently developed a variety of methods that use this information to determine probabilities of a desired response characteristic, such as natural frequency, without using expensive Monte Carlo simulations. One of the problems in these methods is correctly identifying the statistical properties of primitive variables such as geometry, stiffness, and mass. This paper presents a method where the measured dynamic properties of substructures are used instead as the random variables. The residual flexibility method of component mode synthesis is combined with the probabilistic methods to determine the cumulative distribution function of the system eigenvalues. A simple cantilever beam test problem is presented that illustrates the theory.

Brown, Andrew M.↗

NASA NTRS1990

Spectral simulation of unsteady compressible flow past a circular cylinder

An unsteady compressible viscous wake flow past a circular cylinder was successfully simulated using spectral methods. A new approach in using the Chebyshev collocation method for periodic problems is introduced. It was further proved that the eigenvalues associated with the differentiation matrix are purely imaginary, reflecting the periodicity of the problem. It was been shown that the solution of a model problem has exponential growth in time if improper boundary conditions are used. A characteristic boundary condition, which is based on the characteristics of the Euler equations of gas dynamics, was derived for the spectral code. The primary vortex shedding frequency computed agrees well with the results in the literature for Mach = 0.4, Re = 80. No secondary frequency is observed in the power spectrum analysis of the pressure data.

Don, Wai-Sun↗

NASA NTRS1972

Localization of the eigenvalues of linear integral equations with applications to linear ordinary differential equations.

The equivalence of a considered integral equation form with an infinite system of linear equations is proved, and the localization of the eigenvalues of the infinite system is expressed. Error estimates are derived, and the problems of finding upper bounds and lower bounds for the eigenvalues are solved simultaneously.

Sloss, J. M.↗

NASA NTRS1985

On the spectrum of some singular equations in magnetohydrodynamics

A class of singular differential equations that often occurs in problems in magnetohydrodynamics, plasma physics, and astrophysics is studied; first by means of the specific and fairly straightforward example of motions in a stratified magnetofluid, and secondly in terms of the nature of the eigenvalue spectrum for the more general class of problems. It is shown that in the absence of discontinuities in the basic profiles, the eigenvalue spectrum is real and continuous, and no discrete spectrum exists. Bounds are obtained on the location of the continuous spectrum.

Adam, J. A.↗

NASA NTRS1976

Gaussian ideal impulsive loading of rigid viscoplastic plates

The response of a thin, rigid, viscoplastic plate subjected to a spatially axisymmetric Gaussian ideal impulse loading was studied analytically. The Gaussian ideal impulse distribution instantaneously imparts a Gaussian initial velocity distribution to the plate, except at the fixed boundary. The plate deforms with monotonically increasing deflections until the initial dynamic energy is completely dissipated in plastic work. The simply supported plate of uniform thickness obeys the von Mises yield criterion and a generalized constitutive equation for rigid, viscoplastic materials. For the small deflection bending response of the plate, neglecting the transverse shear stress in the yield condition and rotary inertia in the equations of dynamic equilibrium, the governing system of equations is essentially nonlinear. A proportional loading technique, known to give excellent approximations of the exact solution for the uniform load case, was used to linearize the problem and obtain analytical solution in the form of eigenvalue expansions. The linearized governing equations required the knowledge of the collapse load of the corresponding static problem.

Hayduk, R. J.↗

NASA NTRS1972

A fast numerical method for determining the optimum SNR of an array subject to a Q factor constraint.

The problem of finding the optimum current excitations for a point-detector array which extracts a signal from background noise using the criterion of maximizing the SNR subject to a constraint on the supergain ratio (or Q factor) has recently been solved. The formulation reduced the problem to one of solving for the roots of a polynomial whose coefficients are, in general, complex. We demonstrate that by making a certain transformation, it becomes possible to employ a state variable technique which changes the problem into one of finding the eigenvalues of a real matrix which is considerably faster to do.

Winkler, L. P.↗

NASA NTRS1976

Determination of poles and zeros of transfer functions for flexible spacecraft attitude control

The transfer function matrix is obtained for a three-input and three-output model of minimum sensors and actuators for the attitude control system of flexible spacecraft, and a method is described for determining the poles and zeros of this transfer function. Three cases are considered: (1) the actuators and the sensors are all attached to the primary body, (2) the actuators are on the primary body and the sensors are on the sub-body, and (3) the actuators are on the sub-body and the sensors are on the primary body. The zero-determination problem is shown to reduce to eigenvalue calculations of a matrix which is constructed from the inertial and modal matrices in a simple fashion.

Ohkami, Y.↗

NASA NTRS1992

Negative-energy states in the Dirac-Hartree-Fock problem - The effect of omission of two-electron integrals involving the small component

The effect of omission of two-electron integrals involving basis functions for the small component of the wavefunction on the eigenvalue spectrum in the Dirac-Hartree-Fock problem is studied. From an analysis of the Fock matrix it is shown that omission of these integrals moves the negative-energy states down, not up. Their complete omission does not give rise to intruder states. The appearance of intruder states occurs when only some of the core integrals are omitted, due to the nature of particular contraction schemes used for the core basis functions. Use of radially localized functions rather than atomic functions alleviates the intruder state problem.

Dyall, Kenneth G.↗

NASA NTRS1988

Shape sensitivity analysis of flutter response of a laminated wing

A method is presented for calculating the shape sensitivity of a wing aeroelastic response with respect to changes in geometric shape. Yates' modified strip method is used in conjunction with Giles' equivalent plate analysis to predict the flutter speed, frequency, and reduced frequency of the wing. Three methods are used to calculate the sensitivity of the eigenvalue. The first method is purely a finite difference calculation of the eigenvalue derivative directly from the solution of the flutter problem corresponding to the two different values of the shape parameters. The second method uses an analytic expression for the eigenvalue sensitivities of a general complex matrix, where the derivatives of the aerodynamic, mass, and stiffness matrices are computed using a finite difference approximation. The third method also uses an analytic expression for the eigenvalue sensitivities, but the aerodynamic matrix is computed analytically. All three methods are found to be in good agreement with each other. The sensitivities of the eigenvalues were used to predict the flutter speed, frequency, and reduced frequency. These approximations were found to be in good agreement with those obtained using a complete reanalysis.

Bergen, Fred D.↗

NASA NTRS1989

Shape sensitivity analysis of flutter response of a laminated wing

Kapania, Rakesh K.↗

NASA NTRS1990

Spurious frequencies as a result of numerical boundary treatments

The stability theory for finite difference Initial Boundary-Value approximations to systems of hyperbolic partial differential equations states that the exclusion of eigenvalues and generalized eigenvalues is a sufficient condition for stability. The theory, however, does not discuss the nature of numerical approximations in the presence of such eigenvalues. In fact, as was shown previously, for the problem of vortex shedding by a 2-D cylinder in subsonic flow, stating boundary conditions in terms of the primitive (non-characteristic) variables may lead to such eigenvalues, causing perturbations that decay slowly in space and remain periodic time. Characteristic formulation of the boundary conditions avoided this problem. A more systematic study of the behavior of the (linearized) one-dimensional gas dynamic equations under various sets of oscillation-inducing legal boundary conditions is reported.

Abarbanel, Saul↗

NASA NTRS1991

Sequential design of linear quadratic state regulators with prescribed eigenvalues and specified relative stability

This paper considers the problem of optimal regulator design of linear multivariable systems with prescribed pole locations and/or poles corresponding to specified relative stability. A sequential method based on the frequency-domain optimality condition is proposed for achieving the desired pole assignment and determination of the corresponding quadratic performance index. This design method enables the retention of some stable open-loop poles and the associated eigenvectors in the closed-loop system. An illustrative example is provided to demonstrate the effectiveness of the proposed method.

Ganesan, Sekar↗

NASA NTRS1976

Real eigenvalue analysis in NASTRAN by the tridiagonal reduction (FEER) method

Implementation of the tridiagonal reduction method for real eigenvalue extraction in structural vibration and buckling problems is described. The basic concepts underlying the method are summarized and special features, such as the computation of error bounds and default modes of operation are discussed. In addition, the new user information and error messages and optional diagnostic output relating to the tridiagonal reduction method are presented. Some numerical results and initial experiences relating to usage in the NASTRAN environment are provided, including comparisons with other existing NASTRAN eigenvalue methods.

Newman, M.↗

NASA NTRS1988

On the eigenvalue and eigenvector derivatives of a non-defective matrix

A novel approach is introduced to address the problem of existence of differentiable eigenvectors for a nondefective matrix which may have repeated eigenvalues. The existence of eigenvector derivatives for a unique set of continuous eigenvectors corresponding to a repeated eigenvalue is rigorously established for nondefective and analytic matrices. A numerically implementable method is then developed to compute the differentiable eigenvectors associated with repeated eigenvalues. The solutions of eigenvalue and eigenvector derivatives for repeated eigenvalues are then derived. An example is given to illustrate the validity of formulations developed in this paper.

Juang, Jer-Nan↗

NASA NTRS2013

Comparative Study on High-Order Positivity-preserving WENO Schemes

In gas dynamics and magnetohydrodynamics flows, physically, the density and the pressure p should both be positive. In a standard conservative numerical scheme, however, the computed internal energy is obtained by subtracting the kinetic energy from the total energy, resulting in a computed p that may be negative. Examples are problems in which the dominant energy is kinetic. Negative may often emerge in computing blast waves. In such situations the computed eigenvalues of the Jacobian will become imaginary. Consequently, the initial value problem for the linearized system will be ill posed. This explains why failure of preserving positivity of density or pressure may cause blow-ups of the numerical algorithm. The adhoc methods in numerical strategy which modify the computed negative density and/or the computed negative pressure to be positive are neither a conservative cure nor a stable solution. Conservative positivity-preserving schemes are more appropriate for such flow problems. The ideas of Zhang & Shu (2012) and Hu et al. (2012) precisely address the aforementioned issue. Zhang & Shu constructed a new conservative positivity-preserving procedure to preserve positive density and pressure for high-order WENO schemes by the Lax-Friedrichs flux (WENO/LLF). In general, WENO/LLF is too dissipative for flows such as turbulence with strong shocks computed in direct numerical simulations (DNS) and large eddy simulations (LES). The new conservative positivity-preserving procedure proposed in Hu et al. (2012) can be used with any high-order shock-capturing scheme, including high-order WENO schemes using the Roe's flux (WENO/Roe). The goal of this study is to compare the results obtained by non-positivity-preserving methods with the recently developed positivity-preserving schemes for representative test cases. In particular the more difficult 3D Noh and Sedov problems are considered. These test cases are chosen because of the negative pressure/density most often exhibited by standard high-order shock-capturing schemes. The simulation of a hypersonic nonequilibrium viscous shock tube that is related to the NASA Electric Arc Shock Tube (EAST) is also included. EAST is a high-temperature and high Mach number viscous nonequilibrium flow consisting of 13 species. In addition, as most common shock-capturing schemes have been developed for problems without source terms, when applied to problems with nonlinear and/or sti source terms these methods can result in spurious solutions, even when solving a conservative system of equations with a conservative scheme. This kind of behavior can be observed even for a scalar case (LeVeque & Yee 1990) as well as for the case consisting of two species and one reaction (Wang et al. 2012). For further information concerning this issue see (LeVeque & Yee 1990; Griffiths et al. 1992; Lafon & Yee 1996; Yee et al. 2012). This EAST example indicated that standard high-order shock-capturing methods exhibit instability of density/pressure in addition to grid-dependent discontinuity locations with insufficient grid points. The evaluation of these test cases is based on the stability of the numerical schemes together with the accuracy of the obtained solutions.

WENO↗

NASA NTRS1976

Nonlinear periodic waves

Systematic perturbation procedures for the analysis of nonlinear problems are reviewed. The cases when the multiplicity of an eigenvalue is finite or infinite are treated for self-sustained and forced oscillations. The possibility of the formation of shock waves is discussed. Applications to acoustic problems are presented.

Ting, L.↗