When designing aircraft wings shapes, it is important to ensure that the flight envelope does not overlap with regions of flutter or Limit Cycle Oscillation (LCO). A quick assessment of this condition for various design candidates is key to successful design. For this, sensitivity of the flutter velocity needs to be known with the respect of design parameters. The conventional approach of computing this sensitivity has been to differentiate the eigenvalues of the aero-structural system. However, such technique is only applicable for linear or linearized models and cannot be applied to systems undergoing LCO or other nonlinear effects. Though high fidelity CFD modeling can be implemented to overcome this problem for aerodynamic nonlinearities, the method becomes prohibitively expensive, as it requires time accurate calculations. Same goes for computing its sensitivity with respect to design variables. In this work, the time spectral method has been used to compute conditions for both flutter onset, LCOs and their design sensitivities in a computationally efficient way. One of the advantages of the time spectral based formulation is the computational efficiency obtained by eliminating transient flow solutions to reach a periodic steady state, through the solution approximation of a discrete Fourier series. Also, the time spectral form of the governing equation facilitates the application of steady form of adjoint sensitivity analysis for dynamic problems. Large memory and computational cost requirements of unsteady adjoint sensitivity analysis are circumvented by using the periodic steady state formulation. In this paper, steady adjoint formulation for time spectral method based flutter/LCO prediction has been implemented and used to carry out aerodynamic shape optimization for a two DOF flutter model with NACA 64010A airfoil cross section. The objective function has been to maximize with LCO velocity with bounds on the bump function coefficients.
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