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NASA NTRS1983

A function space approach to state and model error estimation for elliptic systems

An approach is advanced for the concurrent estimation of the state and of the model errors of a system described by elliptic equations. The estimates are obtained by a deterministic least-squares approach that seeks to minimize a quadratic functional of the model errors, or equivalently, to find the vector of smallest norm subject to linear constraints in a suitably defined function space. The minimum norm solution can be obtained by solving either a Fredholm integral equation of the second kind for the case with continuously distributed data or a related matrix equation for the problem with discretely located measurements. Solution of either one of these equations is obtained in a batch-processing mode in which all of the data is processed simultaneously or, in certain restricted geometries, in a spatially scanning mode in which the data is processed recursively. After the methods for computation of the optimal esimates are developed, an analysis of the second-order statistics of the estimates and of the corresponding estimation error is conducted. Based on this analysis, explicit expressions for the mean-square estimation error associated with both the state and model error estimates are then developed. While this paper focuses on theoretical developments, applications arising in the area of large structure static shape determination are contained in a closely related paper (Rodriguez and Scheid, 1982).

Rodriguez, G.↗

NASA NTRS1987

Conservative finite volume solutions of a linear hyperbolic transport equation in two and three dimensions using multiple grids

The feasibility of the multiple grid technique is investigated by solving linear hyperbolic equations for simple two- and three-dimensional cases. The results are compared with exact solutions and those obtained from the single grid calculations. It is demonstrated that the technique works reasonably well when two grid systems contain grid cells of comparative sizes. The study indicates that use of the multiple grid does not introduce any significant error and that it can be used to attack more complex problems.

Tiwari, Surendra N.↗

NASA NTRS1988

Approximating the linear quadratic optimal control law for hereditary systems with delays in the control

The fundamental control synthesis issue of establishing a priori convergence rates of approximation schemes for feedback controllers for a class of distributed parameter systems is addressed within the context of hereditary schemes. Specifically, a factorization approach is presented for deriving approximations to the optimal feedback gains for the linear regulator-quadratic cost problem associated with time-varying functional differential equations with control delays. The approach is based on a discretization of the state penalty which leads to a simple structure for the feedback control law. General properties of the Volterra factors of Hilbert-Schmidt operators are then used to obtain convergence results for the controls, trajectories and feedback kernels. Two algorithms are derived from the basic approximation scheme, including a fast algorithm, in the time-invariant case. A numerical example is also considered.

Milman, Mark H.↗

NASA NTRS1990

Asymptotic analysis of dissipative waves with applications to their numerical simulation

Various problems involving the interplay of asymptotics and numerics in the analysis of wave propagation in dissipative systems are studied. A general approach to the asymptotic analysis of linear, dissipative waves is developed. It was applied to the derivation of asymptotic boundary conditions for numerical solutions on unbounded domains. Applications include the Navier-Stokes equations. Multidimensional traveling wave solutions to reaction-diffusion equations are also considered. A preliminary numerical investigation of a thermo-diffusive model of flame propagation in a channel with heat loss at the walls is presented.

Hagstrom, Thomas↗

NASA NTRS1985

The modelling of latitudinal gradients in the solar wind in the outer solar system

A steady, axisymmetric, quasi-radial, global model is developed for thermally driven stellar winds with embedded magnetic fields. The asymptotic, linear results are presented for 0(1) latitudinal variations in the radial magnetic field, mass-loss rate, and radial velocity of the wind. The MHD equations are solved for the latitudinal dependence of the rotational velocity and magnetic field. They are driven by the meridional flows that develop naturally from internal magnetic stresses. Most flows open flux tubes in the stars equatorial plane, redistributing mass and magnetic flux as a function of stellar latitude. The plasma spins up to conserve angular momentum in fields and plasma.

Nerney, S.↗

NASA NTRS1989

Estimation on nonlinear damping in second order distributed parameter systems

An approximation and convergence theory for the identification of nonlinear damping in abstract wave equations is developed. It is assumed that the unknown dissipation mechanism to be identified can be described by a maximal monotone operator acting on the generalized velocity. The stiffness is assumed to be linear and symmetric. Functional analytic techniques are used to establish that solutions to a sequence of finite dimensional (Galerkin) approximating identification problems in some sense approximate a solution to the original infinite dimensional inverse problem.

Banks, H. T.↗

NASA NTRS1995

Phase space analysis in anisotropic optical systems

From the minimal action principle follows the Hamilton equations of evolution for geometric optical rays in anisotropic media. As in classical mechanics of velocity-dependent potentials, the velocity and the canonical momentum are not parallel, but differ by an anisotropy vector potential, similar to that of linear electromagnetism. Descartes' well known diagram for refraction is generalized and a factorization theorem holds for interfaces between two anisotropic media.

Rivera, Ana Leonor↗

NASA NTRS1974

Autonomous orbital navigation using Kepler's equation

A simple method of determining the six elements of elliptic satellite orbits has been developed for use aboard manned and unmanned spacecraft orbiting the earth, moon, or any planet. The system requires the use of a horizon sensor or other device for determining the local vertical, a precision clock or timing device, and Apollo-type navigation equipment including an inertial measurement unit (IMU), a digital computer, and a coupling data unit. The three elements defining the in-plane motion are obtained from simultaneous measurements of central angle traversed around the planet and elapsed flight time using a linearization of Kepler's equation about a reference orbit. It is shown how Kalman filter theory may also be used to determine the in-plane orbital elements. The three elements defining the orbit orientation are obtained from position angles in celestial coordinates derived from the IMU with the spacecraft vertically oriented after alignment of the IMU to a known inertial coordinate frame.

Boltz, F. W.↗

NASA NTRS1968

Design of dissipative linear phase filters

Set of design curves eliminates work involved in designing linear phase filters by being normalized in such a way as to apply to low, band, and high-pass filters of any bandwidth. Similar curves for any number of poles are plotted by solving a system of simultaneous equations.

Phares, R. L.↗

NASA NTRS1973

Decomposition-aggregation stability analysis

This report presents the development and description of the decomposition aggregation approach to stability investigations of high dimension mathematical models of dynamic systems. The high dimension vector differential equation describing a large dynamic system is decomposed into a number of lower dimension vector differential equations which represent interconnected subsystems. Then a method is described by which the stability properties of each subsystem are aggregated into a single vector Liapunov function, representing the aggregate system model, consisting of subsystem Liapunov functions as components. A linear vector differential inequality is then formed in terms of the vector Liapunov function. The matrix of the model, which reflects the stability properties of the subsystems and the nature of their interconnections, is analyzed to conclude over-all system stability characteristics. The technique is applied in detail to investigate the stability characteristics of a dynamic model of a hypothetical spinning Skylab.

Siljak, D. D.↗

NASA NTRS1974

Geometrical geodesy techniques in Goddard earth models

The method for combining geometrical data with satellite dynamical and gravimetry data for the solution of geopotential and station location parameters is discussed. Geometrical tracking data (simultaneous events) from the global network of BC-4 stations are currently being processed in a solution that will greatly enhance of geodetic world system of stations. Previously the stations in Goddard earth models have been derived only from dynamical tracking data. A linear regression model is formulated from combining the data, based upon the statistical technique of weighted least squares. Reduced normal equations, independent of satellite and instrumental parameters, are derived for the solution of the geodetic parameters. Exterior standards for the evaluation of the solution and for the scale of the earth's figure are discussed.

Lerch, F. J.↗

NASA NTRS1976

A theoretical study of the acoustic impedance of orifices in the presence of a steady grazing flow

Oscillatory fluid flow in the vicinity of a circular orifice with a steady grazing flow is analyzed. A simplified system of partial differential equations is obtained by starting with the momentum and continuity equations and by assuming that the flow can be represented as an oscillatory motion superimposed on the (known) steady flow. It is found that the equations are linear in the region where grazing-flow effects dominate; a solution and the resultant orifice impedance are given for this region. It is tentatively concluded that orifice resistance is directly proportional to grazing-flow velocity and that the orifice inductive end correction is not a function of grazing flow.

Rice, E. J.↗

NASA NTRS1977

Effect of chordwise forces and deformations and deformations due to steady lift on wing flutter

This investigation explores the effects of chordwise forces and deformations and steady-state deformation due to lift on the static and dynamic aeroelastic stability of a uniform cantilever wing. Results of this analysis are believed to have practical applications for high-performance sailplanes and certain RPV's. The airfoil cross section is assumed to be symmetric and camber bending is neglected. Motions in vertical bending, fore-and-aft bending, and torsion are considered. A differential equation model is developed, which included the nonlinear elastic bending-torsion coupling that accompanies even moderate deflections. A linearized expansion in small time-dependent deflections is made about a steady flight condition. The stability determinant of the linearized system then contains coefficients that depend on steady displacements. Loads derived from two-dimensional incompressible aerodynamic theory are used to obtain the majority of the results, but cases using three-dimensional subsonic compressible theory are also studied. The stability analysis is carried out in terms of the dynamically uncoupled natural modes of vibration of the uniform cantilever.

Boyd, W. N.↗

NASA NTRS1978

A computer program to automatically generate state equations and macro-models

A computer program, PROSE, that produces nonlinear state equations from a simple topological description of an electrical or mechanical network is described. Unnecessary states are also automatically eliminated, so that a simplified terminal circuit model is obtained. The program also prints out the eigenvalues of a linearized system and the sensitivities of the eigenvalue of largest magnitude.

Garrett, S. J.↗

NASA NTRS1979

Viscoelastic properties of entangled polymers - Ternary blends of monodisperse homopolymers

In a previous publication from this laboratory, the Rouse-Bueche-Zimm molecular theory of viscoelasticity has been extended by using a transient network model to apply to binary blends of monodisperse polymers with chain entanglements. The dynamics of the entanglements were modeled both by the enhanced frictional coefficients and by the additional elastic couplings. It was recognized that entanglements not only may form between chains of the same lengths (intracomponent entanglements) but also between those of different lengths (intercomponent entanglements). At a given intercomponent entanglement, the longer chain was assumed to have the frictional coefficient of the shorter chain. Similarly, for blends consisting of several monodisperse components with different molecular weights, such modifications are also required to predict their linear viscoelastic behavior. The frequency of these interactions is assumed to be proportional to the weight ratio of the respective component chains in the blend. Equations of motion are formulated for each component and solved numerically for the relaxation time spectra. Linear viscoelastic properties such as the dynamic mechanical moduli, stress relaxation moduli, and zero-shear viscosity can then be computed for these systems by linear summation of those of the components.

Soong, D.↗

NASA NTRS1985

Direct computation of optimal control of forced linear system

It is known that the optimal control of a forced linear system may be reduced to that of tracking the system without forces. The solution of the tracking problem is available via the costate variables method. This procedure is computationally expensive for large order systems. It requires solution of matrix Riccati equation and two final value problems. An alternate approach is outlined for the direct computation of the optimal control. Instead of Riccati equation, a matrix Volterra integral must be solved. For this purpose two computational schemes are described, and an illustrative example is given. The results compare favorably with the classical solution. This alternative approach may be especially useful for the control of large space structure where large order models are required.

Utku, S.↗

NASA NTRS1985

Waveguide PIN junction electrooptic phase modulators Theoretical analysis and design criteria

Basic design equations for waveguide PIN heterojunction electrooptic phase modulators utilizing the linear electrooptic effect are derived, and criteria for optimum operation (in terms of obtainable phase shifts) are discussed. Although the numerical examples pertain to devices fabricated of the AlGaAs system on (100) substrates, the same method can be directly applied to modulators fabricated of other materials.

Marshall, W. K.↗

NASA NTRS1987

A generalized procedure for constructing an upwind based TVD scheme

A generalized formulation for constructing second- and higher-order accurate TVD (total variation diminishing) schemes is presented. A given scheme is made TVD by limiting antidiffusive flux differences with some linear functions, so-called limiters. The general idea of the formulation and its mathematical proof of Harten's TVD conditions is shown by applying the Lax-Wendroff method to scalar nonlinear equations and a constant-coefficient system of conservation laws. For the system of equations, several definitions are derived for the argument used in the limiter function and present their performance in numerical experiments. The formulation is extended to the nonlinear system. It is demonstrated that the present procedure can easily convert existing central or upwind, and second- or higher-order differencing schemes to preserve monotonicity and yield physically admissible solutions. The formulation is simple mathematically as well as numerically; both matrix-vector multiplication and Riemann solver are avoided. Although the notion of TVD is based on the initial value problem, application to the steady Euler equations of the formulation is also made.

Liou, Meng-Sing↗