Linearization of the Coupled Unsteady Fluid-Structure Equations: Application to Flutter Control

Abstract

A method to compute sensitivities for use in gradient-based shape optimization applied to unsteady aeroelasticity problems is presented. The method takes into account the coupling between all three fundamental aspects of computational aeroelasticity, namely, unsteady flow equations, time varying structural response to aerodynamic loads, and dynamic meshes that accommodate geometric deformations. The devised method provides discretely exact sensitivities of time-integrated and non-time-integrated objective functions with respect to design variables that control the shape of geometry. The algorithm is formulated in a general manner and can be readily extended to coupled multidisciplinary problems involving any number of disciplines. The algorithm is applied to a simple two-dimensional flutter model in order to demonstrate the proof-of-concept. I. Introduction With advances in computational power, large scale aeroelastic simulations in CFD are becoming more and more commonplace. This paper is concerned with advancing the role of unsteady aeroelastic simulations in aircraft design by complementing them with the strength of adjoint equations for obtaining functional sensitivities. The strength of adjoint equations lie in the fact that they permit the computation of functional sensitivity derivatives at a cost that is essentially independent of the number of design variables. Adjoint equations have become very popular in solving aerodynamic shape optimization problems, particularly for steady-state conditions. 1‐8 While only recently adjoint methods have begun gaining ground in the aerodynamics community especially for time-dependent problems, they have been well established in the field of structural optimization for some time now. 9 The coupling between the fluid and structure equations and the use of sensitivity analysis on such a system has been addressed but primarily from a steady-state standpoint. 10,11 Until now relatively little work has been done toward addressing unsteady aeroelastic optimization problems mainly due to prohibitive cost and complexities in the linearization of coupled time-dependent systems. Recently, efficient linearization techniques for the unsteady governing flow equations both viscous and inviscid have been developed and applied to unsteady shape optimization problems. 12‐16 It is only natural that such methodology be extended to include coupling effects brought on by the introduction of structural equations or any other set of governing equations. This paper presents work done on developing a generalized modular framework for obtaining sensitivities for the coupled unsteady fluid-structure equations. The method is modular in the sense that multiple sets of governing equations from various disciplines that are coupled may be addressed in a unified manner independent of the number of disciplines. Also, the choice of solution technique employed for the individual disciplines has little impact on the overall framework. This is particularly important from a cost standpoint since efficient solution techniques such as multigrid methods and high-order time integration schemes exploited for the purpose of reducing overall costs inherent in unsteady simulations should carry over to the sensitivity computations. It should also be noted that no approximations in the process of linearization have to be made and all components contributing to the solution of the multidisciplinary analysis problem are taken into account. The work presented here utilizes the time-dependent inviscid Euler equations in an unstructured finite-volume framework for the flow solver and a simple two-dimensional flutter model with pitch and plunge degrees-of-freedom to represent the structural response. While we use a direct solution procedure for the structures discipline by transforming the governing equations into linear differential equations, comprehensive problems based on modal analysis or nonlinear structural models may also be treated by this method.