Flutter onset characteristic are an important consideration in commercial airliner design. Previous work with high-fidelity aerostructural optimization has shown a tendency for optimization algorithms to produce unrealistically high span designs, particularly when maximizing range or minimizing fuel burn, in an effort to maximize aerodynamic efficiency. In order to constraint this tendency, we propose to include a flutter constraint to these optimization problems. To be meaningful for this class of optimization problem, the flutter model should be able to predict the strong nonlinearity in the transonic regime, as commercial airliners largely operate in this regime. In addition, these problems typically include a large number of design variables, therefore we use gradient-based optimization algorithms, requiring the constraint formulation to be continuous, differentiable and have an efficient method for computing the gradients of the constraint. In this paper, we apply Euler time-spectral computational fluid dynamics methods to model the flutter constraint and we propose a coupled adjoint method to calculate the constraint sensitivity with respect to the design variables. The coupled adjoint method has the advantage that the gradient evaluation time is independent of the number of design variables. In the literature, the harmonic balance method has been used to model flutter constraints. A two-level adjoint formulation has been proposed to evaluate the sensitivity of flutter onset velocity with respect to the design variables. Such method requires solving the harmonic balance aerodynamic adjoint $\mathcal{O}(N_{CSD})$ times where $N_{CSD}$ is the structural degree of freedom multiplied by number of time instances. On the contrary, we propose the coupled adjoint method which directly deals with the whole aeroelastic system and solves only $1$ adjoint equation. We verify the adjoint sensitivity computation with the finite difference method. Finally, we present a MDO problem for the classic Isogai case in which we maximize the flutter velocity index with respect to aerodynamic shape design variables. We gain a $10.9\%$ increase of the flutter velocity index through this optimization.

}, keywords = {flutter, time-spectral}, doi = {10.2514/6.2019-0697 }, author = {Sicheng He and Eirikur Jonsson and Charles A. Mader and Joaquim R.R.A. Martins} }