A Novel Filled Function Method for Nonlinear Equations

A novel filled function method is suggested for solving box-constrained systems of nonlinear equations. Firstly, the original problem is converted into an equivalent global optimization problem. Subsequently, a novel filled function with one parameter is proposed for solving the converted global optimization problem. Some properties of the filled function are studied and discussed. Finally, an algorithm based on the proposed novel filled function for solving systems of nonlinear equations is presented. The objective function value can be reduced by quarter in each iteration of our algorithm. The implementation of the algorithm on several test problems is reported with satisfactory numerical results

The Penetration of a Finger into a Viscous Fluid in a Channel and Tube

The steady-state shape of a finger penetrating into a region filled with a viscous fluid is examined. The two-dimensional and axisymmetric problems are solved using Stokes equations for low Reynolds number flow. To solve the equations, an assumption for the shape of the finger is made and the normal-stress boundary condition is dropped. The remaining equations are solved numerically by covering the domain with a composite mesh composed of a curvilinear grid which follows the curved interface, and a rectilinear grid parallel to the straight boundaries. The shape of the finger is then altered to satisfy the normal-stress boundary condition by using a nonlinear least squares iteration method. The results are compared with the singular perturbation solution of Bretherton (J. Fluid Mech., 10 (1961), pp. 166–188). When the axisymmetric finger moves through a tube, a fraction $m$ of the viscous fluid is left behind on the walls of the tube. The fraction $m$ was measured experimentally by Taylor (J. Fluid Mech., 10 (1961), pp. 161–165) as a function of the dimensionless parameter µU/T. The numerical results are compared with the experimental results of Taylor

Optimal design of a linear induction motor applied in transportation

Several optimal design schemes of a single-sided linear induction motor (SLIM) adopted in linear metro are presented in this paper. Firstly, the equivalent circuit of SLIM fully considering the end effects, half-filled slots, back-iron saturation and skin effect is proposed, based on one-dimensional air-gap magnetic equations. In the circuit, several coefficients including longitudinal end effect coefficients Kr(s) and Kx(s), transversal end edge effect coefficients Cr(s) and Cx(s), and skin effect coefficient Kf are achieved by using the dummy electric potential method and complex power equivalence between primary and secondary sides. Furthermore, several optimal design restraint equations of SLIM are provided in order to improve the operational efficiency and reduce the primary weight. These nonlinear equations are solved by using genetic algorithm and mixture penalty function method. The optimal schemes are compared with the original design of one company, where analysis on parameters is made in detail. These results show that the optimal schemes are reasonable for improving the performance of SLIM

Bifurcations in two-dimensional differentially heated cavity

In this work, we propose a numerical analysis of a bidimensional instationary natural convection in a square cavity filled with air and inclined 45 degree versus to horizontal. The vertical walls are subjected to non-uniform temperatures while the horizontal walls are adiabatic. The equations based on the formulation vorticity-stream function are solved using the Alternating Directions Implicit scheme (ADI) and Gauss elimination method. We analyze the influence of Rayleigh number on the roads to chaos borrowed by the natural convection developed in this cavity, and we are looking for stable solutions representing the nonlinear dynamic system. A correlation between the Nusselt number and the Rayleigh number is proposed. We have analyzed the vicinity of the critical point.  The transition of the point attractor to another limit cycle attractor is characterized by the Hopf bifurcation

Reduced-order modeling and dynamics of nonlinear acoustic waves in a combustion chamber

For understanding the fundamental properties of unsteady motions in combustion chambers, and for applications of active feedback control, reduced-order models occupy a uniquely important position. A framework exists for transforming the representation of general behavior by a set of infinite-dimensional partial differential equations to a finite set of nonlinear second-order ordinary differential equations in time. The procedure rests on an expansion of the pressure and velocity fields in modal or basis functions, followed by spatial averaging to give the set of second-order equations in time. Nonlinear gasdynamics is accounted for explicitly, but all other contributing processes require modeling. Reduced-order models of the global behavior of the chamber dynamics, most importantly of the pressure, are obtained simply by truncating the modal expansion to the desired number of terms. Central to the procedures is a criterion for deciding how many modes must be retained to give accurate results. Addressing that problem is the principal purpose of this paper. Our analysis shows that, in case of longitudinal modes, a first mode instability problem requires a minimum of four modes in the modal truncation whereas, for a second mode instability, one needs to retain at least the first eight modes. A second important problem concerns the conditions under which a linearly stable system becomes unstable to sufficiently large disturbances. Previous work has given a partial answer, suggesting that nonlinear gasdynamics alone cannot produce pulsed or 'triggered' true nonlinear instabilities; that suggestion is now theoretically established. Also, a certain form of the nonlinear energy addition by combustion processes is known to lead to stable limit cycles in a linearly stable system. A second form of nonlinear combustion dynamics with a new velocity coupling function that naturally displays a threshold character is shown here also to produce triggered limit cycle behavior

Fast recursive filters for simulating nonlinear dynamic systems

A fast and accurate computational scheme for simulating nonlinear dynamic systems is presented. The scheme assumes that the system can be represented by a combination of components of only two different types: first-order low-pass filters and static nonlinearities. The parameters of these filters and nonlinearities may depend on system variables, and the topology of the system may be complex, including feedback. Several examples taken from neuroscience are given: phototransduction, photopigment bleaching, and spike generation according to the Hodgkin-Huxley equations. The scheme uses two slightly different forms of autoregressive filters, with an implicit delay of zero for feedforward control and an implicit delay of half a sample distance for feedback control. On a fairly complex model of the macaque retinal horizontal cell it computes, for a given level of accuracy, 1-2 orders of magnitude faster than 4th-order Runge-Kutta. The computational scheme has minimal memory requirements, and is also suited for computation on a stream processor, such as a GPU (Graphical Processing Unit).Comment: 20 pages, 8 figures, 1 table. A comparison with 4th-order Runge-Kutta integration shows that the new algorithm is 1-2 orders of magnitude faster. The paper is in press now at Neural Computatio