Aeroelastic System Development Using Proper Orthogonal Decomposition and Volterra Theory

Abstract

This research combines Volterra theory and proper orthogonal decomposition (POD) into a hybrid methodology for reduced-order modeling of aeroelastic systems. The outcome of the method is a set of linear ordinary differential equations (ODEs) describing the modal amplitudes associated with both the structural modes and the POD basis functions for the fluid. For this research, the structural modes are sine waves of varying frequency, and the Volterra-POD approach is applied to the fluid dynamics equations. The structural modes are treated as forcing terms which are impulsed as part of the fluid model realization. Using this approach, structural and fluid operators are coupled into a single aeroelastic operator. This coupling converts a free boundary fluid problem into an initial value problem, while preserving the parameter (or parameters) of interest for sensitivity analysis. The approach is applied to an elastic panel in supersonic cross flow. The hybrid Volterra-POD approach provides a low-order fluid model in state-space form. The linear fluid model is tightly coupled with a nonlinear panel model using an implicit integration scheme. The resulting aeroelastic model provides correct limit-cycle oscillation prediction over a wide range of panel dynamic pressure values. Time integration of the reduced-order aeroelastic model is four orders of magnitude faster than the high-order solution procedure developed for this research using traditional fluid and structural solvers.

A reduced order system ID approach to the modelling of nonlinear structural behavior in aeroelasticity

“…Gabbay et al (2000) and Mehner et al (2000) used polynomial functions whose coefficients were identified using strain energy data from a series of finite element runs, to model a microelectromechanical system. Silva et al have used a Volterra series approach to identify linear and nonlinear aerodynamic systems (Silva, 1997(Silva, , 1999Silva and Raveh, 2001;Silva and Bartels, 2002;Lucia et al, 2003). Denegri and Johnson (2001) have conducted a study based upon a neural network model using flight test data which has shown promise.…”

A Reduced Order Cyclic Method for Computation of Limit Cycles

“…Detailed derivations of the POD and its properties are available elsewhere, and references and discussion are provided in recent review articles [3][4][5]16]. In our discussion of POD, N m basis vectors, or "modes" are used to represent deviations of x(t), an N f -dimensional vector, from a base solution, x 0 .…”

“…Much attention is currently being given to the efficient treatment of dynamic systems that develop simple LCO or more complicated dynamic responses. The authors have collaborated with NASA Langley Research Center to review the state-of-the-art in reduced order modeling and system identification techniques for aeroelasticity [3][4][5]. Methods reviewed include those based on the proper orthogonal decomposition (POD) (e.g., [3][4][5]), system identification with Volterra theory (e.g., [6]), harmonic balance (e.g., [7,8]), and a POD/Volterra hybrid strategy (e.g., [3][4][5]).…”

Identification of Nonlinear Aeroelastic Systems Based on the Volterra Theory: Progress and Opportunities

“…A topic of recent interest is the potential development of hybrid POD/Volterra methods. These hybrid techniques would combine the spatial resolution possible with POD methods with the low dimensionality and computational efficiency of Volterra methods [54,55].…”