Showing papers on "Nonlinear system published in 1986"

An outer-approximation algorithm for a class of mixed-integer nonlinear programs

Abstract: An outer-approximation algorithm is presented for solving mixed-integer nonlinear programming problems of a particular class. Linearity of the integer (or discrete) variables, and convexity of the nonlinear functions involving continuous variables are the main features in the underlying mathematical structure. Based on principles of decomposition, outer-approximation and relaxation, the proposed algorithm effectively exploits the structure of the problems, and consists of solving an alternating finite sequence of nonlinear programming subproblems and relaxed versions of a mixed-integer linear master program. Convergence and optimality properties of the algorithm are presented, as well as a general discussion on its implementation. Numerical results are reported for several example problems to illustrate the potential of the proposed algorithm for programs in the class addressed in this paper. Finally, a theoretical comparison with generalized Benders decomposition is presented on the lower bounds predicted by the relaxed master programs.

A Theory of Post-Stall Transients in Axial Compression Systems: Part I—Development of Equations

Abstract: An approximate theory is presented for post-stall transients in multistage axial compression systems. The theory leads to a set of three simultaneous nonlinear third-order partial differential equations for pressure rise, and average and disturbed values of flow coefficient, as functions of time and angle around the compressor. By a Galerkin procedure, angular dependence is averaged, and the equations become first order in time. These final equations are capable of describing the growth and possible decay of a rotating-stall cell during a compressor mass-flow transient. It is shown how rotating-stall-like and surgelike motions are coupled through these equations, and also how the instantaneous compressor pumping characteristic changes during the transient stall process.

An analysis of a phase field model of a free boundary

Abstract: A mathematical analysis of a new approach to solidification problems is presented. A free boundary arising from a phase transition is assumed to have finite thickness. The physics leads to a system of nonlinear parabolic differential equations. Existence and regularity of solutions are proved. Invariant regions of the solution space lead to physical interpretations of the interface. A rigorous asymptotic analysis leads to the Gibbs-Thompson condition which relates the temperature at the interface to the surface tension and curvature.

The variational approach to shape from shading

Abstract: We develop a systematic approach to the discovery of parallel iterative schemes for solving the shape-from-shading problem on a grid. A standard procedure for finding such schemes is outlined, and subsequently used to derive several new ones. The shape-from-shading problem is known to be mathematically equivalent to a nonlinear first-order partial differential equation in surface elevation. To avoid the problems inherent in methods used to solve such equations, we follow previous work in reformulating the problem as one of finding a surface orientation field that minimizes the integral of the brightness error. The calculus of variations is then employed to derive the appropriate Euler equations on which iterative schemes can be based. The problem of minimizing the integral of the brightness error term is ill posed, since it has an infinite number of solutions in terms of surface orientation fields. A previous method used a regularization technique to overcome this difficulty. An extra term was added to the integral to obtain an approximation to a solution that was as smooth as possible. We point out here that surface orientation has to obey an integrability constraint if it is to correspond to an underlying smooth surface. Regularization methods do not guarantee that the surface orientation recovered satisfies this constraint. Consequently, we attempt to develop a method that enforces integrability, but fail to find a convergent iterative scheme based on the resulting Euler equations. We show, however, that such a scheme can be derived if, instead of strictly enforcing the constraint, a penalty term derived from the constraint is adopted. This new scheme, while it can be expressed simply and elegantly using the surface gradient, unfortunately cannot deal with constraints imposed by occluding boundaries. These constraints are crucial if ambiguities in the solution of the shape-from-shading problem are to be avoided. Differrent schemes result if one uses different parameters to describe surface orientation. We derive two new schemes, using unit surface normals, that facilitate the incorporation of the occluding boundary information. These schemes, while more complex, have several advantages over previous ones.

Two‐dimensional nonlinear inversion of seismic waveforms: Numerical results

Abstract: The nonlinear problem of inversion of seismic waveforms can be set up using least‐squares methods. The inverse problem is then reduced to the problem of minimizing a lp;nonquadratic) function in a space of many (104to106) variables. Using gradient methods leads to iterative algorithms, each iteration implying a forward propagation generated by the actual sources, a backward propagation generated by the data residuals (acting as if they were sources), and a correlation at each point of the space of the two fields thus obtained, which gives the updated model. The quality of the results of any inverse method depends heavily on the realism of the forward modeling. Finite‐difference schemes are a good choice relative to realism because, although they are time‐consuming, they give excellent results. Numerical tests performed with multioffset synthetic data from a two‐dimensional model prove the feasibility of the approach. If only surface‐recorded reflections are used, the high spatial frequency content of the ...

Global solutions of nonlinear hyperbolic equations for small initial data

Abstract: On considere des systemes quasilineaires d'equations hyperboliques d'ordre 2 qui sont des deformations non lineaires de l'equation d'onde. On etudie l'existence globale et le comportement asymptotique des solutions du probleme de Cauchy quand la donnee initiale est suffisamment petite

Wave Interactions and Fluid Flows

Abstract: Part I. Introduction: 1. Introduction Part II. Linear Wave Interactions: 2. Flows with piecewise-constant density and velocity 3. Flows with constant density and continuous velocity profile 4. Flows with density stratification and piecewise-constant velocity 5. Flows with continuous profiles of density and velocity 6. Models of mode coupling 7. Eigenvalue spectra and localized disturbances Part III. Introduction to Nonlinear Theory: 8. Introduction to nonlinear theory Part IV. Waves and Mean Flows: 9. Spatially-periodic waves in channel flows 10. Spatially-periodic waves on deformable boundaries 11. Modulated wave-packets 12. Generalized Lagrangian mean (GLM) formulation 13. Spatially-periodic means flows Part V. Three-wave Resonance: 14. Conservative wave interactions 15. Solutions of the conservative interaction equations 16. Linearly damped waves 17. Non-conservative wave interactions Part VI. Evolution of a Nonlinear Wave-Train: 18. Heuristic derivation of the evolution equations 19. Weakly nonlinear waves in inviscid fluids 20. Weakly nonlinear waves in shear flows 21. Properties of the evolution equations 22. Waves of larger amplitude Part VII. Cubic Three- and Four-wave Interactions: 23. Conservative four-wave interactions 24. Mode interactions in Taylor-Couette flow 25. Rayleigh-Benard convection 26. Wave interactions in planar shear flows Part VIII. Strong Interactions, Local Instabilities and Turbulence: A Postscript: 27. Strong interactions, local instabilities and turbulence: A postscript References Index.

Feedback control of nonlinear systems by extended linearization

Abstract: For single-input, multiple-output, nonlinear systems, we consider a design method based on the family of linearizations of the system, parameterized by constant operating points. Nonlinear state feedback control laws and observer/state feedback control laws are designed such that the eigenvalues of the family of linearized closed-loop systems are placed at specified values that are locally invariant with respect to the closed-loop operating point. The method is illustrated by application to the problem of automatically balancing an inverted pendulum.

Simulation of Nonlinear Circuits in the Frequency Domain

Abstract: Simulation in the frequency domain avoids many of the severe problems experienced when trying to use traditional time-domain simulators such as SPICE to find the steady-state behavior of analog and microwave circuits. In particular, frequency-domain simulation eliminates problems from distributed components and high-Q circuits by foregoing a nonlinear differential equation representation of the circuit in favor of a complex algebraic representation. This paper reviews the method of harmonic balance as a general approach to converting a set of differential equations into a nonlinear algebraic system of equations that can be solved for the periodic steady-state solution of the original differential equations. Three different techniques are applied to solve the algebraic system of equations: optimization, relaxation, and Newton's method. The implementation of the algorithm resulting from the combination of Newton's method with harmonic balance is described. Several new ways of exploiting both the structure of the formulation and the characteristics of the circuits that would typically be seen by this type of simulator are presented. These techniques dramatically reduce the time required for a simulation, and allow harmonic balance to be applied to much larger circuits than were previously attempted, making it suitable for use on monolithic microwave integrated circuits (MMIC's).

Blow-up in Nonlinear Schroedinger Equations-I A General Review

Abstract: The general properties of a class of nonlinear Schroedinger equations: iut + p:∇∇u + f(|u|2)u = 0 are reviewed. Conditions for existence, uniqueness, and stability of solitary wave solutions are presented, along with conditions for blow-up and global existence for the Cauchy problem.

Chaos and nonlinear dynamics of single‐particle orbits in a magnetotaillike magnetic field

Abstract: The properties of charged-particle motion in Hamiltonian dynamics are studied in a magnetotaillike magnetic field configuration. It is shown by numerical integration of the equation of motion that the system is generally nonintegrable and that the particle motion can be classified into three distinct types of orbits: bounded integrable orbits, unbounded stochastic orbits, and unbounded transient orbits. It is also shown that different regions of the phase space exhibit qualitatively different responses to external influences. The concept of 'differential memory' in single-particle distributions is proposed. Physical implications for the dynamical properties of the magnetotail plasmas and the possible generation of non-Maxwellian features in the distribution functions are discussed.

The nonlinear development of Gortler vortices in growing boundary layers

Abstract: The Growth of Gortler vortices in boundary layers on concave walls is investigated. It is shown that for vortices of wavelength comparable to the boundary-layer thickness the appropriate linear stability equations cannot be reduced to ordinary differential equations. The partial differential equations governing the linear stability of the flow are solved numerically, and neutral stability is defined by the condition that a dimensionless energy function associated with the flow should have a maximum or minimum when plotted as a function of the downstream variable X. The position of neutral stability is found to depend on how and where the boundary layer is perturbed, so that the concept of a unique neutral curve so familiar in hydrodynamic-stability theory is not tenable in the Gortler problem, except for asymptotically small wavelengths. The results obtained are compared with previous parallel-flow theories and the small-wavelength asymptotic results of Hall (1982a, b), which are found to be reasonably accurate even for moderate values of the wavelength. The parallel-flow theories of the growth of Gortler vortices are found to be irrelevant except for the small-wavelength limit. The main deficiency of the parallel-flow theories is shown to arise from the inability of any ordinary differential approximation to the full partial differential stability equations to describe adequately the decay of the vortex at the edge of the boundary layer. This deficiency becomes intensified as the wavelength of the vortices increases and is the cause of the wide spread of the neutral curves predicted by parallel-flow theories. It is found that for a wall of constant radius of curvature a given vortex imposed on the flow can grow for at most a finite range of values of X. This result is entirely consistent with, and is explicable by the asymptotic results of, Hall (1982a).

Navier-Stokes equations for compressible fluids: global existence and qualitative properties of the solutions in the general case

Abstract: We consider the equations which describe the motion of a viscous compressible fluid, taking into consideration the case of inflow and/or outflow through the boundary. By means of some a priori estimates we prove the existence of a global (in time) solution. Moreover, as a consequence of a stability result, we show that there exist a periodic solution and a stationary solution.

Adaptive friction compensation in dc-motor drives

Abstract: This paper proposes a control scheme where the nonlinear effects of friction are compensated adaptively. When the friction is compensated the motor drive can approximately be described by a constant coefficient linear model. Standard methods can be applied to design a regulator for such a model. This results in a control law which is a combination of a fixed linear controller and an adaptive part which compensates for nonlinear friction effects. Experiments have clearly shown that both static and dynamic friction have nonsymmetric characteristic. They depend on the direction of motion. This is considered in the design of the adaptive friction compensation. The proposed scheme has been implemented and tested on a laboratory prototype with good results. The control low is implemented on an IBM-PC. The paper describes the ideas, the algorithm and the experimental results. The results are relevant for many precision drives like those found in industrial robots.

Critical Layers in Shear Flows

Abstract: The normal mode approach to investigating the stability of a parallel shear flow involves the superposition of a small wavelike perturbation on the basic flow. Its evolution in space and/or time is then determined. In the linear inviscid theory, if ū(y) is the basic velocity profile, then a singularity occurs at critical points yc, where ū = c, the perturbation phase speed. This is plausible intuitively because energy can be exchanged most efficiently where the wave and mean flow are travelling at the same speed. The problem is of the singular perturbation type; when viscosity or nonlinearity, for example, are restored to the governing equations, the singularity is removed. In this lecture, the classical viscous theory is first outlined before presenting a newer perturbation approach using a nonlinear critical layer (i.e., nonlinear terms are restored within a thin layer). The application to the case of a density stratified shear flow is discussed and, finally, the results are compared qualitatively with radar observations and also with recent numerical simulations of the full equations. ∗Address for correspondence: Department of Mathematics and Statistics, McGill University, Montreal, QC, H3A 2K6, Canada. e-mail: maslowe@math.mcgill.ca

Fluctuating nonlinear hydrodynamics and the liquid-glass transition

Abstract: We study the fluctuating nonlinear hydrodynamics of compressible fluids. Development of the appropriate field-theoretical description for this problem requires treatment of nonlinearities which arise through the relationship g=\ensuremath{\rho}V, where g is the momentum density, \ensuremath{\rho} is the mass density, and V is the velocity field. We show how this constraint can be naturally included in a field theory of the Martin-Siggia-Rose type. We analyze the structure of the resulting field theory using the available fluctuation-dissipation theorem. We also develop the perturbation-theory expansion in powers of the temperature and evaluate the contributions from the nonlinearities to one-loop order. We show that the theory is renormalizable in the hydrodynamic limit. This field-theoretical model is used to systematically investigate the origins and viability of the nonlinear density feedback mechanism first identified by Leutheusser as a source of the liquid-glass transition. While we find that the nonlinear couplings driving this mechanism are present, we also find contributions, arising from the nonlinear constraint relating g, \ensuremath{\rho}, and V, which cut off the mechanism. The cutoff arises from a nonhydrodynamic correction not treated in previous work. While we find that there is no sharp transition, we do find evidence for a rounded version of the transition.

The identification of nonlinear biological systems: LNL cascade models

Abstract: Systems that can be represented by a cascade of a dynamic linear (L), a static nonlinear (N) and a dynamic linear (L) subsystem are considered. Various identification schemes that have been proposed for these LNL systems are critically reviewed with reference to the special problems that arise in the identification of nonlinear biological systems. A simulated LNL system is identified from limited duration input-output data using an iterative identification scheme.

Numerical analysis of parametrized nonlinear equations

Abstract: Some Sample Problems Some Background Material Solution Manifolds and Their Parameterizations Discretization Errors One-Distributions and Augmented Equations A Continuation Method Some Numerical Examples The Computation of Limit Points Differential Equations on Manifolds Error Estimates and Related Topics References Index.

Introduction to the Theory of Nonlinear Elliptic Equations

Abstract: Preface The Topic of the Lecture Notes and Something on Modelling by Partial Differential Equations Sobolev and Morrey-Campanato Spaces Existence of Weak Solutions to Boundary Value Problems for Nonlinear Second Order Elliptic Systems An Excursion to Approximate Methods Intermediary Regularity Regularity of Weak Solutions to Second Order Elliptic Systems References Index.

Neural networks with nonlinear synapses and a static noise

Abstract: The theory of neural networks is extended to include a static noise as well as nonlinear updating of synapses by learning. The noise appears either in the form of spin-glass interactions, which are independent of the learning process, or as a random decaying of synapses. In an unsaturated network, the nonlinear learning algorithms may modify the energy surface and lead to interesting new computational capabilities. Close to saturation, they act as an additional source of a static noise. The effect of the noise on memory storage is calculated.

Self-trapping on a dimer: Time-dependent solutions of a discrete nonlinear Schrödinger equation

Abstract: From the discrete nonlinear Schr\"odinger equation describing transport on a dimer we derive and solve a closed nonlinear equation for the site-occupation probability difference. Our results, which are directly relevant to specific experiments such as neutron scattering in physically realizable dimers, exhibit a transition from "free" to "self-trapped" behavior and illustrate features expected in extended systems, including soliton/polaron bandwidth reduction and the dependence of energy-transfer efficiency on initial conditions.

Nonlinear theory for plates and shells including the effects of transverse shearing

Abstract: Nonlinear strain displacement relations for three-dimensional elasticity are determined in orthogonal curvilinear coordinates. To develop a two-dimensiona l theory, the displacements are expressed by trigonometric series representation through the thickness. The nonlinear strain-displacement relations are expanded into a series that contains all first- and second-degree terms. In the series for the displacements only the first few terms are retained. Insertion of the expansions into the three-dimensional virtual work expression leads to nonlinear equations of equilibrium for laminated and thick plates and shells that include the effects of transverse shearing. Equations of equilibrium and buckling equations are derived for flat plates and cylindrical shells. The shell equations reduce to conventional transverse shearing shell equations when the effects of the trigonometric terms are omitted and to classical shell equations when the trigonometric terms are omitted and the shell is assumed to be thin. Numerical results are presented for the buckling of a thick simply supported flat rectangular plate in longitudinal compression.