The following paper presents a reduced-order-modeling approach for nonlinear aerodynamic sys- tems utilizing a pruned Volterra series. The method is applied ...
50th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference
17th 4 - 7 May 2009, Palm Springs, California
AIAA 2009-2319
Reduced Order Modeling of Nonlinear Transonic Aerodynamics Using a Pruned Volterra Series Maciej Balajewicz∗ Duke University, Durham, NC, 27708-0300
Fred Nitzsche† and Daniel Feszty‡ Carleton University, Ottawa, ON, K1S 5B6, Canada
The following paper presents a reduced-order-modeling approach for nonlinear aerodynamic systems utilizing a pruned Volterra series. The method is applied to a two-dimensional transonic airfoil undergoing forced pitch oscillations. Pruned Volterra series reduced-order-models up to fourth-order are identified and compared against computational fluid dynamics models. Very favorable accuracies are attained over a wide range of Mach number, reduced frequency and oscillation amplitude. The computational resources associated with the pruned Volterra series are demonstrated to be several ordersof-magnitude lower compared to the standard Volterra series.
I. Introduction
T
HE Volterra series forms a popular Reduced Order Modeling (ROM) approach to nonlinear aerodynamic systems.1 However, the applicability of the method has been limited due to Volterra series convergence issues. In his recent review paper, Silva states ”...potential disadvantages of the Volterra theory include input amplitude limitations related to convergence issues and the need for higher-order kernels”.1 Identification of Volterra kernels is a resource intensive endeavor; limiting most aerodynamic applications to second-order truncations of the Volterra series. Unfortunately, such low-ordered series are applicable to weakly nonlinear systems only. For transonic aerodynamic applications, this requirement translates to small structural perturbations in low Mach number flow regimes. In this paper, an alternate formulation of the Volterra series aimed at addressing the problem of Volterra series convergence is presented. This new formulation of the series, called the pruned Volterra series contains several favorable properties. Most importantly, the pruned Volterra series is capable of efficiently modeling stronger nonlinearities. This paper illustrates the applicability of the pruned Volterra series to nonlinear aerodynamic systems by modeling the unsteady, two-dimensional, transonic flow of a pitching airfoil.
II. Volterra Series The Volterra theory of nonlinear systems is quite mature and several texts are available.2, 3 It was first applied to nonlinear engineering problems by Wiener4 and first applied to subsonic and transonic aerodynamic systems by Tromp and Jenkins5 and Silva,6 respectively. The output y(t), of a continuous-time, causal, time-invariant, fading memory, nonlinear system Ψ, due to an input x(t) y(t) = Ψ{x(t)} (1) ∗ Research Assistant, Department of Mechanical Engineering and Materials Science; formally Research Assistant, Department of Mechanical and Aerospace Engineering, Carleton University, Student Member AIAA. † Professor, Department of Mechanical and Aerospace Engineering, Senior Member AIAA. ‡ Assistant Professor, Department of Mechanical and Aerospace Engineering, Member AIAA.
1 of 8 American Institute of Aeronautics and Astronautics Copyright © 2009 by Maciej Balajewicz. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.
