Bogdan I. Epureanu

Fractal Basins of Attraction Associated with a Damped Newton's Method

An intriguing and unexpected result for students learning numerical analysis is that Newtons method, applied to the simple polynomial z 3 - 1 = 0 in the complex plane, leads to intricately interwoven basins of attraction of the roots. As an example of an interesting open question that may help to stimulate student interest in numerical analysis, we investigate the question of whether a damping method, which is designed to increase the likelihood of convergence for Newtons method, modifies the fractal structure of the basin boundaries. The overlap of the frontiers of numerical analysis and nonlinear dynamics provides many other problems that can help to make numerical analysis courses interesting.

On the optimality of the Ott-Grebogi-Yorke control scheme

Abstract Some of the characteristics of the Ott-Grebogi-Yorke (OGY) control technique are presented as applied to nonlinear flows, as distinct from nonlinear maps. Specifically, we consider the case where the magnitude of the control parameter varies in time within each control cycle in proportion to a given function referred to as a basis function . The choice of the basis function is shown to influence the basin of convergence for a given level of parameter variation in the OGY controller. An algorithm for designing the optimal basis function is presented. The optimal basis function is shown to be defined by a step function with potentially several jumps, thus revealing the intrinsic power of a standard OGY technique that uses a single step function as a basis function. Two numerical applications of the optimal design technique to a Duffing oscillator are also presented to show that the standard OGY technique may be significantly improved by making an optimal choice of a basis function. Copyright

A Parametric Analysis of Reduced Order Models of Potential Flows in Turbomachinery Using Proper Orthogonal Decomposition

An unsteady inviscid flow through a cascade of oscillating airfoils is investigated. An inviscid nonlinear subsonic and transonic model is used to compute the steady flow solution. Then a small amplitude motion of the airfoils about their steady flow configuration is considered. The unsteady flow is linearized about the nonlinear steady response based on the observation that in many practical cases the unsteadiness in the flow has a substantially smaller magnitude than the steady component. Several reduced order modal models are constructed in the frequency domain using the proper orthogonal decomposition technique. The dependency of the required number of aerodynamic modes in a reduced order model on the far-field upstream Mach number is investigated. It is shown that the transonic reduced order models require a larger number of modes than the subsonic models for a similar geometry, range of reduced frequencies and interblade phase angles. The increased number of modes may be due to the increased Mach number per se, or the presence of the strong spatial gradients in the region of the shock. These two possible causes are investigated. Also, the geometry of the cascade is shown to influence strongly the shape of the aerodynamic modes, but only weakly the required dimension of the reduced order models.Copyright

Study of warping torsion of thin-walled beams with open cross-section using macro-elements

Thin-walled beams with open cross-section under torsion or complex load are studied based on the hypotheses of the classical theory (Vlasov). Different from previous techniques presented in the literature, the concept of a strip-plate is introduced. This concept is used to accurately model the effect of bending induced by torsion and to define an alternate finite element called macro-element. The macro-elements are shown to model more accurately the thin-walled beams under warping torsion or complex load therefore giving better results than the classical theory. Copyright

Pattern formation and linear stability analysis in centreless grinding

Abstract Quite often a centreless ground surface has an undesired wavy shape instead of a circular shape owing to a geometric instability. In this paper the equations describing the kinematics of the grinding process obtained by Rowe et al.(1) are used to build a linearized model that describes the formation and evolution of the pattern on the manufactured surface. The profile of the workpiece is discretized and linearly interpolated. Stable or unstable patterns or eigenvectors are obtained for the configurations of the centreless grinding system. Circular profiles are shown to appear for configurations that have no unstable patterns while wavy profiles appear for configurations that have at least one unstable pattern. Some of the most important non-linear effects are considered as limits on the exponential growth of the unstable patterns predicted by the linear analysis. Specifically, the non-linear analysis is based on the limited allowed curvature of the profile of the workpiece. Although it is an approximation of the real phenomena, the non-linear analysis is able to predict patterns that appear on the ground surface that are very close to the experimental observations even if unstable patterns are present.

Time-Filtered Limit Cycle Computation for Aeroelastic Systems

A technique for computing limit cycle oscillations of aeroelastic systems is presented. This technique may be used also for constructing and identifying reduced order models which characterize accurately limit cycle oscillations. The models presented use a spatial model reduction combined with a nonlinear identification technique. The time-filtered method, a methodology similar to finite element in time, is discussed. The specific novel aspects of the proposed method are presented. Numerical examples analyzing the dynamics of a a panel under a buffeting aerodynamic load are presented to demonstrate the performance of the proposed methods.