Time Dependent Adjoint-based Optimization for Coupled Aeroelastic Problems Asitav Mishra ∗ Karthik Mani † Dimitri Mavriplis ‡ Jay Sitaraman § Department of Mechanical Engineering,University of Wyoming, Laramie, WY 82071-3295. A formulation for sensitivity analysis of fully coupled time-dependent aeroelastic problems is given in this paper. Both forward sensitivity and adjoint sensitivity formulations are derived that correspond to analogues of the non-linear aeroelastic analysis problem. Both sensitivity analysis formulations make use of the same iterative disciplinary solution techniques used for analysis, and make use of an analogous coupling strategy. The information passed between fluid and structural solvers is dimensionally equivalent in all cases, enabling the use of the same data structures for analysis, forward and adjoint problems. Sensitivities from both forward and adjoint formulations for the fully coupled aeroelastic problem are verified using the complex step method and agreement to machine precision is demonstrated. The fully coupled adjoint formulation is then used to perform rotor blade design optimization on a Hart2 rotor in hover while constraining the time-integrated thrust coefficient to the baseline value. The optimized rotor achieves 2% reduced torque with a penalty of 1% reduction of thrust. I. Introduction In the recent past, the use of adjoint equations has become a popular approach for solving aerodynamic design optimization problems based on computational fluid dynamics (CFD). 1–6 Adjoint equations are a very powerful tool in the sense that they allow the computation of sensitivity derivatives of an objective function to a set of given inputs at a cost which is essentially independent of the number of inputs. This is in contrast to the brute-force finite-difference method, where each input or design variable has to be perturbed individually to obtain a corresponding effect on the output. This is a tedious and costly process which is of little use when there are a large number of design variables or inputs. Another major shortcoming of the finite-difference method is that it suffers from step-size limitations which affect the accuracy of the computed gradients. While the use of adjoint equations is now fairly well established in steady-state shape optimization, only recently have inroads been made into extending them to unsteady flow problems. Unsteady discrete adjoint-based shape opti- mization was initially demonstrated in the context of two-dimensional problems by Mani and Mavriplis 7 and also by Rumpfkeil and Zingg. 8 Preliminary demonstration of the method’s feasibility in three-dimensional problems was done by Mavriplis. 9 Full implementation in a general sense and application to large scale problems involving helicopter rotors was then carried out by Nielsen et.al. in the NASA FUN3D code. 10, 11 Since engineering optimization is an inherently multidisciplinary endeavor, the next logical step involves extending adjoint methods to multidisciplinary simulations and using the obtained sensitivities for driving multidisciplinary optimizations. In the context of fixed and especially rotory wing aircraft, aeroelastic coupling effects can be very important and must be considered in the context of a successful optimization strategy. The coupling of computational fluid dynamics (CFD) and computational structural dynamics (CSD) and the use of sensitivity analysis on such a system has been addressed in the past primarily from a steady-state standpoint. Until now, relatively little work has been done addressing unsteady aeroelastic optimization problems, mainly due to complexities in the linearization of coupled time-dependent systems. In previous work, we have derived the fully coupled adjoint ∗ Postdoctoral Research Associate; amishra3@uwyo.edu † Associate Research Scientist; kmani@uwyo.edu ‡ Professor; mavripl@uwyo.edu § Assistant Professor; jsitaram@uwyo.edu 1 of 23 American Institute of Aeronautics and Astronautics