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NASA NTRS1983

Linear approximations of nonlinear systems

The development of a method for designing an automatic flight controller for short and vertical take off aircraft is discussed. This technique involves transformations of nonlinear systems to controllable linear systems and takes into account the nonlinearities of the aircraft. In general, the transformations cannot always be given in closed form. Using partial differential equations, an approximate linear system called the modified tangent model was introduced. A linear transformation of this tangent model to Brunovsky canonical form can be constructed, and from this the linear part (about a state space point x sub 0) of an exact transformation for the nonlinear system can be found. It is shown that a canonical expansion in Lie brackets about the point x sub 0 yields the same modified tangent model.

Hunt, L. R.↗

NASA NTRS1995

Dynamics of Vapour Bubbles in Nucleate Boiling: Basic Equations of Bubble Evolution - 1

We consider the behaviour of a vapour bubble formed at a nucleation site on a heated horizontal wall. There is no forced convection of an ambient liquid, and the bubble is presumably separated from the wall by a thin liquid microlayer. The energy conservation law results in a variational equation for the mechanical energy of the whole system consisting of the bubble and liquid. It leads to a set of two strongly nonlinear equations which govern bubble expansion and motion of its centre of mass. A supplementary equation to find out the vapour temperature follows from consideration of heat transfer to the bubble, both from the bulk of surrounding liquid and through the microlayer. The average thickness of the microlayer is shown to increase monotonously with time as the bubble meniscus spreads along the wall. Bubble expansion is driven by the pressure head between vapour inside and liquid far away from the bubble, with due allowance for surface tension and gravity effects. It is resisted by inertia of liquid being placed into motion as the bubble grows. The inertia originates also a force that presses the bubble to the wall. This force is counteracted by the buoyancy and an effective surface tension force that tends to transform the bubble into a sphere. The analysis brings about quite a new formulation of the familiar problem of bubble growth and detachment under conditions of nucleate pool boiling.

Buyevich, Yu A.↗

NASA NTRS1984

On the use of parametric differentiation to predict weak shock waves

The method of parametric differentiation is shown to be an effective means to transform both the time-linearized and the fully nonlinear transonic small disturbance potential flow equations to a linear system of equations. The globally linearized equations are then solved in phase space by finite difference methods and integral equation methods. The predicted birth, growth, and motion of weak shock waves in two-dimensional transonic flows are compared with existing theory and experiment.

Harris, W. L.↗

NASA NTRS1986

Constrained hierarchical least square nonlinear equation solvers

The current paper develops a constrained hierarchical least square nonlinear equation solver. The procedure can handle the response behavior of systems which possess indefinite tangent stiffness characteristics. Due to the generality of the scheme, this can be achieved at various hierarchical application levels. For instance, in the case of finite element simulations, various combinations of either degree of freedom, nodal, elemental, substructural, and global level iterations are possible. Overall, this enables a solution methodology which is highly stable and storage efficient. To demonstrate the capability of the constrained hierarchical least square methodology, benchmarking examples are presented which treat structure exhibiting highly nonlinear pre- and postbuckling behavior wherein several indefinite stiffness transitions occur.

Padovan, J.↗

NASA NTRS1989

Projection methods for the numerical solution of Markov chain models

Projection methods for computing stationary probability distributions for Markov chain models are presented. A general projection method is a method which seeks an approximation from a subspace of small dimension to the original problem. Thus, the original matrix problem of size N is approximated by one of dimension m, typically much smaller than N. A particularly successful class of methods based on this principle is that of Krylov subspace methods which utilize subspaces of the form span(v,av,...,A(exp m-1)v). These methods are effective in solving linear systems and eigenvalue problems (Lanczos, Arnoldi,...) as well as nonlinear equations. They can be combined with more traditional iterative methods such as successive overrelaxation, symmetric successive overrelaxation, or with incomplete factorization methods to enhance convergence.

Saad, Youcef↗

NASA NTRS1992

Stability of a class of interconnected evolution systems

Stability conditions for a class of interconnected systems modeled by linear abstract evolution equations and a memoryless nonlinearity are derived. These conditions are stated in terms of the passivity of each of the subsystems and can be considered as a partial generalization of the hyperstability theorem. A Liapunov function approach is used in the proof without requiring the positive definiteness of the Liapunov function. Application to the robustness analysis of the infinite-dimensional linear quadratic regulator is also discussed.

Wen, John T.↗

NASA NTRS1983

Cubic spline approximation techniques for parameter estimation in distributed systems

Approximation schemes employing cubic splines in the context of a linear semigroup framework are developed for both parabolic and hyperbolic second-order partial differential equation parameter estimation problems. Convergence results are established for problems with linear and nonlinear systems, and a summary of numerical experiments with the techniques proposed is given.

Banks, H. T.↗

NASA NTRS2001

Energy Stable Flux Formulas For The Discontinuous Galerkin Discretization Of First Order Nonlinear Conservation Laws

We consider the discontinuous Galerkin (DG) finite element discretization of first order systems of conservation laws derivable as moments of the kinetic Boltzmann equation. This includes well known conservation law systems such as the Euler For the class of first order nonlinear conservation laws equipped with an entropy extension, an energy analysis of the DG method for the Cauchy initial value problem is developed. Using this DG energy analysis, several new variants of existing numerical flux functions are derived and shown to be energy stable.

Barth, Timothy↗

NASA NTRS1993

Reliability enhancement of Navier-Stokes codes through convergence enhancement

Reduction of total computing time required by an iterative algorithm for solving Navier-Stokes equations is an important aspect of making the existing and future analysis codes more cost effective. Several attempts have been made to accelerate the convergence of an explicit Runge-Kutta time-stepping algorithm. These acceleration methods are based on local time stepping, implicit residual smoothing, enthalpy damping, and multigrid techniques. Also, an extrapolation procedure based on the power method and the Minimal Residual Method (MRM) were applied to the Jameson's multigrid algorithm. The MRM uses same values of optimal weights for the corrections to every equation in a system and has not been shown to accelerate the scheme without multigriding. Our Distributed Minimal Residual (DMR) method based on our General Nonlinear Minimal Residual (GNLMR) method allows each component of the solution vector in a system of equations to have its own convergence speed. The DMR method was found capable of reducing the computation time by 10-75 percent depending on the test case and grid used. Recently, we have developed and tested a new method termed Sensitivity Based DMR or SBMR method that is easier to implement in different codes and is even more robust and computationally efficient than our DMR method.

Choi, K.-Y.↗

NASA NTRS1975

Estimation and Analysis of Nonlinear Stochastic Systems

The algebraic and geometric structures of certain classes of nonlinear stochastic systems were exploited in order to obtain useful stability and estimation results. The class of bilinear stochastic systems (or linear systems with multiplicative noise) was discussed. The stochastic stability of bilinear systems driven by colored noise was considered. Approximate methods for obtaining sufficient conditions for the stochastic stability of bilinear systems evolving on general Lie groups were discussed. Two classes of estimation problems involving bilinear systems were considered. It was proved that, for systems described by certain types of Volterra series expansions or by certain bilinear equations evolving on nilpotent or solvable Lie groups, the optimal conditional mean estimator consists of a finite dimensional nonlinear set of equations. The theory of harmonic analysis was used to derive suboptimal estimators for bilinear systems driven by white noise which evolve on compact Lie groups or homogeneous spaces.

Marcus, S. I.↗

NASA NTRS1985

Numerical solution of Euler's equation by perturbed functionals

A perturbed functional iteration has been developed to solve nonlinear systems. It adds at each iteration level, unique perturbation parameters to nonlinear Gauss-Seidel iterates which enhances its convergence properties. As convergence is approached these parameters are damped out. Local linearization along the diagonal has been used to compute these parameters. The method requires no computation of Jacobian or factorization of matrices. Analysis of convergence depends on properties of certain contraction-type mappings, known as D-mappings. In this article, application of this method to solve an implicit finite difference approximation of Euler's equation is studied. Some representative results for the well known shock tube problem and compressible flows in a nozzle are given.

Dey, S. K.↗

NASA NTRS1972

Singular perturbation of absolute stability.

It was previously shown (author, 1969) that the regions of absolute stability in the parameter space can be determined when the parameters appear on the right-hand side of the system equations, i.e., the regular case. Here, the effect on absolute stability of a small parameter attached to higher derivatives in the equations (the singular case) is studied. The Lur'e-Postnikov class of nonlinear systems is considered.

Siljak, D. D.↗

NASA NTRS1988

Eigenvalue/Eigenvector derivatives for SAVI Gimbalflex nonlinear transient response analysis

An eigenvector expansion method is utilized to predict eigenvalue and eigenvector derivatives due to geometric reconfiguration of a Gimbalflex fine-pointing/vibration isolation system called SAVI (Space Active Vibration Isolation). The eigenvector expansion method used is a modification of the classical method and allows for rigid body roots. Using the resulting modal derivatives, free-free nonlinear equations of motion are developed with Lagrange's Method. These equations represent a nonlinear plant model to be used in conjunction with a control system transient response simulation.

Orr, M. F., Jr.↗

NASA NTRS1973

Pole and zero placement in multivariable control systems

A method is proposed for designing multivariable systems based on an alternate derivation of Davison's theorem on pole placement and the solution of the nonlinear equations for the feedback gains by the least square error method. Output feedback is used to control a complex dynamical system. The freedom in design, after allocating poles, is used to place zeros and/or satisfy other design objectives. This method results in algorithms which are computationally attractive. However, this is done at a considerable sacrifice in terms of the design freedom available. For a system with m inputs and p outputs only m + p variables are available instead of mp variables.

Sridhar, B.↗

NASA NTRS1991

An efficient solution technique for shockwave-boundary layer interactions with flow separation and slot suction effects

An efficient method for computing two-dimensional compressible Navier-Stokes flow fields is presented. The solution algorithm is a fully-implicit approximate factorization technique based on an unsymmetric line Gauss-Seidel splitting of the equation system Jacobian matrix. Convergence characteristics are improved by the addition of acceleration techniques based on Shamanskii's method for nonlinear equations and Broyden's quasi-Newton update. Characteristic-based differencing of the equations is provided by means of Van Leer's flux vector splitting. In this investigation, emphasis is placed on the fast and accurate computation of shock-wave-boundary layer interactions with and without slot suction effects. In the latter context, a set of numerical boundary conditions for simulating the transpiration flow in an open slot is devised. Both laminar and turbulent cases are considered, with turbulent closure provided by a modified Cebeci-Smith algebraic model. Comparisons with computational and experimental data sets are presented for a variety of interactions, and a fully-coupled simulation of a plenum chamber/inlet flowfield with shock interaction and suction is also shown and discussed.

Edwards, Jack R.↗

NASA NTRS1977

Flutter of articulated pipes at finite amplitude

The plane motion of an articulated pipe made of two segments is examined and the flow velocity at which flutter manifests itself is sought. The pressure in the reservoir feeding the pipe is kept constant. In contrast to previous works, the flow velocity is not taken as a prescribed parameter of the system but is left to follow the laws of motion. This approach requires a nonlinear formulation of the problem and the equations of motion are solved using Krylov-Bogoliubov's method. A graph of the amplitude of the limit cycles, as a function of the fluid-system mass ratio, is presented and conclusions are drawn as to the necessity of considering nonlinearities in the analysis.

Rousselet, J.↗

NASA NTRS1987

On a theory of the evolution of surface cold fronts

The governing vorticity and divergence equations in the surface layer are derived and the roles of the different terms and feedback mechanisms are investigated in semigeostrophic and nongeostrophic cold-frontal systems. A planetary boundary layer model is used to perform sensitivity tests to determine that in a cold front the ageostrophic feedback mechanism as defined by Orlanski and Ross tends to act as a positive feedback mechanism, enhancing vorticity and convergence growth. Therefore, it cannot explain the phase shift between convergence and vorticity as simulated by Orlanski and Ross. An alternative plausible, though tentative, explanation in terms of a gravity wave is offered. It is shown that when the geostrophic deformation increases, nonlinear terms in the divergence equation may become important and further destabilize the system.

Levy, Gad↗

NASA NTRS1980

Engine identification for adaptive control

An attempt to obtain a dynamic model for a turbofan gas turbine engine for the purpose of adaptive control is described. The requirements for adaptive control indicate that a dynamic model should be identified from data sampled during engine operation. The dynamic model identified was of the form of linear differential equations with time varying coefficients. A turbine engine is, however, a highly nonlinear system, so the identified model would be valid only over a small area near the operating point, thus requiring frequent updating of the coefficients in the model. Therefore it is necessary that the identifier use only recent information to perform its function. The identifier selected minimized the square of the equation errors. Known linear systems were used to test the characteristics of the identifier. It was found that the performance was dependent on the number of data points used in the computations and upon the time interval over which the data points were obtained. Preliminary results using an engine deck for the quiet, clean, shorthaul experimental engine indicate that the identified model predicts the engine motion well when there is sufficient dynamic information, that is when the engine is in transient operation.

Leonard, R. G.↗