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All sourcesNASA NTRSDOE OSTI
NASA NTRS1981

Numerical analysis of spectral properties of coupled oscillator Schroedinger operators. I - Single and double well anharmonic oscillators

Several methods for computing many eigenvalues and eigenfunctions of a single anharmonic oscillator Schroedinger operator whose potential may have one or two minima are described. One of the methods requires the solution of an ill-conditioned generalized eigenvalue problem. This method has the virtue of using a bounded amount of work to achieve a given accuracy in both the single and double well regions. Rigorous bounds are given, and it is proved that the approximations converge faster than any inverse power of the size of the matrices needed to compute them. The results of computations for the g:phi(4):1 theory are presented. These results indicate that the methods actually converge exponentially fast.

Isaacson, D.↗
NASA NTRS1972

Adjoint variational methods in nonconservative stability problems.

A general nonself-adjoint eigenvalue problem is examined and it is shown that the commonly employed approximate methods, such as the Galerkin procedure, the method of weighted residuals and the least square technique lack variational descriptions. When used in their previously known forms they do not yield stationary eigenvalues and eigenfunctions. With the help of an adjoint system, however, several analogous variational descriptions may be developed and it is shown in the present study that by properly restating the method of least squares, stationary eigenvalues may be obtained. Several properties of the adjoint eigenvalue problem, known only for a restricted group, are shown to exist for the more general class selected for study.

Prasad, S. N.↗
NASA NTRS1996

Implicity restarted Arnoldi/Lanczos methods for large scale eigenvalue calculations

Eigenvalues and eigenfunctions of linear operators are important to many areas of applied mathematics. The ability to approximate these quantities numerically is becoming increasingly important in a wide variety of applications. This increasing demand has fueled interest in the development of new methods and software for the numerical solution of large-scale algebraic eigenvalue problems. In turn, the existence of these new methods and software, along with the dramatically increased computational capabilities now available, has enabled the solution of problems that would not even have been posed five or ten years ago. Until very recently, software for large-scale nonsymmetric problems was virtually non-existent. Fortunately, the situation is improving rapidly. The purpose of this article is to provide an overview of the numerical solution of large-scale algebraic eigenvalue problems. The focus will be on a class of methods called Krylov subspace projection methods. The well-known Lanczos method is the premier member of this class. The Arnoldi method generalizes the Lanczos method to the nonsymmetric case. A recently developed variant of the Arnoldi/Lanczos scheme called the Implicitly Restarted Arnoldi Method is presented here in some depth. This method is highlighted because of its suitability as a basis for software development.

Sorensen, Danny C.↗
NASA NTRS1991

Dynamical eigenfunction decomposition of turbulent channel flow

The results of an analysis of low-Reynolds-number turbulent channel flow based on the Karhunen-Loeve (K-L) expansion are presented. The turbulent flow field is generated by a direct numerical simulation of the Navier-Stokes equations at a Reynolds number Re(tau) = 80 (based on the wall shear velocity and channel half-width). The K-L procedure is then applied to determine the eigenvalues and eigenfunctions for this flow. The random coefficients of the K-L expansion are subsequently found by projecting the numerical flow field onto these eigenfunctions. The resulting expansion captures 90 percent of the turbulent energy with significantly fewer modes than the original trigonometric expansion. The eigenfunctions, which appear either as rolls or shearing motions, possess viscous boundary layers at the walls and are much richer in harmonics than the original basis functions.

Ball, K. S.↗
NASA NTRS1973

Evaluation of control laws and actuator locations for control systems applicable to deformable astronomical telescope mirrors

Some of the major difficulties associated with large orbiting astronomical telescopes are the cost of manufacturing the primary mirror to precise tolerances and the maintaining of diffraction-limited tolerances while in orbit. One successfully demonstrated approach for minimizing these problem areas is the technique of actively deforming the primary mirror by applying discrete forces to the rear of the mirror. A modal control technique, as applied to active optics, has previously been developed and analyzed. The modal control technique represents the plant to be controlled in terms of its eigenvalues and eigenfunctions which are estimated via numerical approximation techniques. The report includes an extension of previous work using the modal control technique and also describes an optimal feedback controller. The equations for both control laws are developed in state-space differential form and include such considerations as stability, controllability, and observability. These equations are general and allow the incorporation of various mode-analyzer designs; two design approaches are presented. The report also includes a technique for placing actuator and sensor locations at points on the mirror based upon the flexibility matrix of the uncontrolled or unobserved modes of the structure. The locations selected by this technique are used in the computer runs which are described. The results are based upon three different initial error distributions, two mode-analyzer designs, and both the modal and optimal control laws.

Ostroff, A. J.↗
NASA NTRS1975

A non-Gaussian model of continuous atmospheric turbulence proposed for use in aircraft design

This paper describes a statistical model proposed for use in forecasting vehicle responses to stationary continuous atmospheric turbulence. The model is suggested by the observed patchy character of turbulence, and differs from models now in use in that it does not assume the gust velocity to be a Gaussian process. For simplicity only the vertical gust component is considered here. The validity of the proposed model is established through comparison with published data. This comparison shows that the model is in better agreement with observed gust velocity probability distributions and exceedance frequencies than is the widely used Gaussian model, especially insofar as high velocity gusts are concerned. A method of applying the proposed model to the determination of vehicle responses is developed. It is shown that response probability distributions as well as exceedance frequencies can be derived from the eigenvalues and eigenfunction of certain unsymmetric kernels.

Reeves, P. M.↗
NASA NTRS1975

Two-electron bond-orbital model. I

The one-electron bond-orbital model of Harrison (1973) for tetrahedrally coordinated solids is modified to a two-electron model using a generalization of Falicov and Harris' (1969) method for treating the hydrogen molecule. Expressions for nuclear-exchange and pseudodipolar interactions are obtained in terms of bond-orbital model parameters. The eigenfunctions and eigenvalues of the Hamiltonian are employed along with standard perturbation theory to calculate the energy shifts resulting from the application of an external electric field and induced by the magnetic interaction between the nuclear spins and the bond-electron spins. The quantitative predictions of the derived formulas are examined for real semiconductors involving elements and compounds from periodic groups III, IV, and V. It is concluded that the present two-electron model makes it possible to assess the quantitative effects of electron correlation in the anion-cation bond for both nonmagnetic properties (electric susceptibility, dielectric constant) and magnetic properties (nuclear-exchange and pseudodipolar interactions) of various materials.

Huang, C.↗
NASA NTRS1977

Solar seismology. I - The stability of the solar p-modes

The stability of the radial p-modes of the sun is investigated by computing nonadiabatic eigenvalues and eigenfunctions for a solar envelope model which extends from an inner radius of about 0.3 solar radius out to an optical depth of about 0.0003. The calculations take into account in a crude fashion the response of the convective flux to the oscillation. The dynamical effect of turbulence in the convection zone is parametrized in terms of a turbulent shear viscosity. The results show that if damping by turbulent viscosity is neglected, all modes with periods longer than 6 minutes are unstable. The familiar kappa-mechanism, which operates in the H ionization-H(-) opacity region, is the dominant source of driving of the oscillations. Modes with periods shorter than 6 minutes are stabilized by radiative damping in the solar atmosphere. When turbulent dissipation of pulsational energy is included, all modes are predicted to be stable. However, the margin of stability is very small. In view of the large uncertainty that must be assigned to the estimate of turbulent damping, it is concluded that theoretical calculations cannot unequivocally resolve the question of the stability of the solar p-modes.

Goldreich, P.↗
NASA NTRS1977

Acoustics-turbulence interaction

An investigation of the instability frequency was undertaken. Measurements revealed that the hot wire probe induces and sustains stable upstream oscillation of the free shear layer. The characteristics of the free shear layer tone are found to be different from the slit jet wedge edgetone phenomenon. The shear tone induced by a plane wedge in a plane free shear layer was then examined in order to further document the phenomenon. The eigenvalues and eigenfunctions of the tone fundamental show agreement with the spatial stability theory. A comprehensive summary of the results is also included.

Hussain, A. K. M. F.↗
NASA NTRS1978

Use of asymptotic methods in vibration analysis

The derivation of dynamic differential equations, suitable for studying the vibrations of rotating, curved, slender structures was examined, and the Hamiltonian procedure was advocated for this purpose. Various reductions of the full system are displayed, which govern the vibrating troposkien when various order of magnitude restrictions are placed on important parameters. Possible advantages of the WKB asymptotic method for solving these classes of problems are discussed. A special case of this method is used illustratively to calculate eigenvalues and eigenfunctions for a flat turbine blade with small flexural stiffness.

Ashley, H.↗
NASA NTRS1978

The free shear layer tone phenomenon and probe interference

The reported investigation shows that the hot-wire probe induces stable upstream oscillations in a free shear layer, similar to the jet edge tone mechanism. This effect can be significant also in measurements involving large-scale organized structure, conditional sampling, space-time correlation, and convection velocity, when a reference or indicator probe may be used near the origin of the free shear layer. It appears that even in a free shear layer without any wedge, an object in the flow sufficiently downstream can also provide feedback to the flow upstream. A description is given of the edge-tone phenomenon which is observed when a thin slit jet impinges on a plane wedge. Attention is given to the free shear layer tone induced by a hot-wire probe, the free shear layer tone phenomenon, and shear layer tone eigenvalues and eigenfunctions.

Hussain, A. K. M. F.↗
NASA NTRS1978

The free shear layer edgetone and instability measurements

The hot-wire probe is known to be capable of triggering and sustaining upstream instability modes like the slit jet-wedge edgetone termed the free shear layer tone. In this paper, the hot-wire probe is used to investigate the shear tone phenomenon in axisymmetric and plane free shear layers. For all the measurements reported, the initial boundary layers producing the free shear layers are documented (at 0.2 cm upstream from the lip) and classified as laminar. It is shown that the free shear layer, inherently different from the fully vortical slit jet, produces a different tone behavior. The data on the free shear layer tone eigenfunctions and eigenvalues suggest that the feedback is hydrodynamic rather than acoustic.

Zaman, K. B. M. Q.↗
NASA NTRS1980

Numerical methods for studying anharmonic oscillator approximations to the phi super 4 sub 2 quantum field theory

This paper is an expanded version of a talk given at the 1979 T.I.C.O.M. conference. It is a self-contained introduction, for applied mathematicians and numerical analysts, to quantum mechanics and quantum field theory. It also contains a brief description of the authors' numerical approach to the problems of quantum field theory, which may best be summarized by the question; Can we compute the eigenvalues and eigenfunctions of Schrodinger operators in infinitely many variables.

Isaacson, D.↗
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

Linear baroclinic instability in the presence of large scale topography

The effect of a planetary-scale, wavenumber 2 topography on baroclinically active disturbances is investigated for a channel domain in a two-layer, quasi-geostrophic context. When the lower-layer zonal velocity is nonzero, the topography influences the disturbances by forcing a stationary wave, and the topography and the forced wave influence the growth rates and the spatial structures of the time-dependent solutions. The case of zero zonal velocity in the lower layer was also investigated, for which no forced wave exists. Asymptotic forms of the equations, valid when the topographic effect (governed by the ratio of the nondimensional topographic height to the rotational Froude number) is small, are used to obtain both the stationary and time-dependent solutions. The latter are also obtained using a numerical approach, in which is determined the eigenvalues and eigenfunctions of a matrix representing the dynamical equations. Agreement is good between the two approaches. Recent laboratory experiments with a baroclinic annulus in which there is a false bottom with wavenumber 2 topography, are used to select governing parameters. The simultaneous presence of a stationary forced wave of wavenumber 2 and a time-dependent baroclinic wave of wavenumber 4, which has wavenumber 2 and 6 sidebands due to the topography, yields a flow field that exhibits some principal features of the laboratory experiments. The position of the forced wave and the location of an excursion in latitude of the storm track show qualitative resemblance to those features observed in the atmosphere.

Reynolds, Nathaniel Dunton↗
NASA NTRS1989

Nonaxisymmetric instabilities in thin self-gravitating rings and disks

The stability of geometrically thin self-gravitating rings and disks is studied by computing the eigenvalues and eigenfunctions of linearized normal mode oscillations in these systems. For the vertically averaged incompressible models and models of gaseous and stellar disks with softened gravity, analytic treatment is possible. The existence of oscillatory normal modes is established using a variational principle and infer instability using perturbation theory. Effects due to the Lindblad and corotation resonances are analyzed in detail. The distribution of the ratio of vorticity to surface density is important for determining the stability of the systems.

Papaloizou, John C. B.↗
NASA NTRS1990

Invariant criteria for bound states, degree of ionization, and plasma phase transition

Basis invariant characterizations of bound states and bound fraction of a partially ionized hydrogen plasma are given in terms of properties of the spectrum of eigenvalues and eigenfunctions of the equilibrium quantum statistical one-proton-one-electron reduced density matrix. It is suggested that these can be used to place theories of a proposed plasma-ionization phase transition on a firm foundation. This general approach may be relevant to cosmological questions such as the quark deconfinement-confinement transition.

Girardeau, M. D.↗
NASA NTRS1993

Quantized discrete space oscillators

A quasi-canonical sequence of finite dimensional quantizations was found which has canonical quantization as its limit. In order to demonstrate its practical utility and its numerical convergence, this formalism is applied to the eigenvalue and 'eigenfunction' problem of several harmonic and anharmonic oscillators.

Uzes, C. A.↗