A computational study of the error in the vortex method associated with discontinuous vorticity

We investigate computationally the error computed by the vortex method for a discontinuous patch of vorticity. Specifically, the computed velocity and vorticity of an elliptical path of constant vorticity, known as the Kirchhoff ellipse, are compared to the analytic velocity and vorticity. The error in the velocity and the vorticity for the Kirchhoff ellipse as computed by the vortex method is presented. This error is studied as a function of the aspect ratio of the ellipse, the blob function, the spacing between the centers of the computational elements, and the blob radius. Both the error at the initial time and the error after three revolutions of the ellipse are discussed. © 1996 John Wiley & Sons, Inc.     

Fast numerical method for the solution of the quasigeostrophic vorticity equation

Particle methods are typically O(N²), where N is the number of computational elements. We present an O(N) particle method for the equations for the conservation of potential vorticity. This method is based on the idea of grouping the particles. The necessary expansions and truncation errors are given. The accuracy and speed of the method are presented for both scalar and vector machines. © 1993 John Wiley & Sons, Inc.     

Numreical solutoin of a class of parabolic partial differential equation arising in optimal control problems with uncertainty

In this paper the optimal control of uncertain parabolic systems of partial differential equations is investigated. In order to search for controllers that are insensitive to uncertainties in these systems, an iterative optimization procedure is proposed. This procedure involves the solution of a set of operator valued parabolic partial differential equations. The existence and uniqueness of solutions to these operator equations is proved, and a stable numerical algorithm to approximate the uncertain optimal control problem is proposed. The viability of the proposed algorithm is demonstrated by applying it to the control of parabolic systems having two different types of uncertainty.     

Melnikov-Based Dynamical Analysis of Microcantilevers in Scanning Probe Microscopy

We study the dynamical behavior of a microcantilever-sample system that forms the basis for the operation of atomic force microscopes (AFM). We model the microcantilever by a single mode approximation. The interaction between the sample and the cantilever is modeled by a Lennard--Jones potential which consists of a short-range repulsive potential and a long-range van der Waals (vdW) attractive potential. We analyze the dynamics of the cantilever sample system when the cantilever is subjected to a sinusoidal forcing. Using the Melnikov method, the region in the space of physical parameters where chaotic motion is present is determined. In addition, using a proportional and derivative controller, we compute the Melnikov function in terms of the parameters of the controller. Using this relation, controllers can be designed to selectively change the regime of dynamical interaction.     

ℓ1-Minimization with Magnitude Constraints in the Frequency Domain

In this paper, we study the -optimal control problem with additional constraints on the magnitude of the closed-loop frequency response. In particular, we study the case of magnitude constraints at fixed frequency points (a finite number of such constraints can be used to approximate an -norm constraint). In previous work, we have shown that the primal-dual formulation for this problem has no duality gap and both primal and dual problems are equivalent to convex, possibly infinite-dimensional, optimization problems with LMI constraints. Here, we study the effect of approximating the convex magnitude constraints with a finite number of linear constraints and provide a bound on the accuracy of the approximation. The resulting problems are linear programs. In the one-block case, both primal and dual programs are semi-infinite dimensional. The optimal cost can be approximated, arbitrarily well from above and within any predefined accuracy from below, by the solutions of finite-dimensional linear programs. In the multiblock case, the approximate LP problem (as well as the exact LMI problem) is infinite-dimensional in both the variables and the constraints. We show that the standard finite-dimensional approximation method, based on approximating the dual linear programming problem by sequences of finite-support problems, may fail to converge to the optimal cost of the infinite-dimensional problem.     

Controller design to optimize a composite performance measure

This paper studies a mixed objective problem of minimizing a composite measure of thel ₁, ₂, andl -norms together with thel -norm of the step response of the closed loop. This performance index can be used to generate Pareto-optimal solutions with respect to the individual measures. The problem is analyzed for discrete-time, single-input single-output (SISO), linear time-invariant systems. It is shown via Lagrange duality theory that the problem can be reduced to a convex optimization problem with a priori known dimension. In addition, continuity of the unique optimal solution with respect to changes in the coefficients of the linear combination that defines the performance measure is estabilished.This research was supported by the National Science Foundation under Grants No. ECS-92-04309, ECS-92-16690 and ECS-93-08481.     

On the Dynamics of a Harmonic Oscillator Undergoing Impacts with a Vibrating Platform

This paper deals with a model for repeated impacts of a massattached to a spring with a massive, sinusoidally vibrating table. Thismodel has been studied in an attempt to understand the cantilever-sampledynamics in atomic force microscopy. In this work, we have shown thatfor some values of the frequency of the vibrating table, there arecountably many orbits of arbitrarily long periods and the system issensitive to the initial conditions with which the experiments areconducted. We have also shown existence of complex dynamics in the caseswhen the natural frequency of the spring-mass system is very low; andwhen it is the same as the oscillating frequency of the table.     

Analysis of a circulant based preconditioner for a class of lower rank extracted systems

This paper proposes and studies the performance of a preconditioner suitable for solving a class of symmetric positive definite systems, A_px=b, which we call lower rank extracted systems (LRES), by the preconditioned conjugate gradient method. These systems correspond to integral equations with convolution kernels defined on a union of many line segments in contrast to only one line segment in the case of Toeplitz systems. The p × p matrix, A_p, is shown to be a principal submatrix of a larger N × N Toeplitz matrix, A_N. The preconditioner is provided in terms of the inverse of a 2N × 2N circulant matrix constructed from the elements of A_N. The preconditioner is shown to yield clustering in the spectrum of the preconditioned matrix similar to the clustering results for iterative algorithms used to solve Toeplitz systems. The analysis also demonstrates that the computational expense to solve LRE systems is reduced to O(N log N). Copyright © 2004 John Wiley & Sons, Ltd.