# A Study on the Surrogate-Based Optimization of Flexible Wings Considering a Flutter Constraint +
## 3 Results
### Comparison and Discussion of the Results  
A final comparison is presented in Table 9, which details the results obtained using consistent parameters and dataset sizes across the three analyzed cruise speeds. The following optimization parameters were set: a population size of 280; 200 generations; a mutation rate of 0.5; and a crossover rate of 0.8. For the last analysis, using a cruise speed of 130 m/s, the results considered are just the feasible ones, i.e., those without the flutter constraint active. \begin{table}
\begin{tabular}{l l c c c} \hline \hline
**Objective** & **Metric** & **SHARPy** & **Optimization** & **Relative Difference** \\ \hline \multirow{3}{*}{\(C_{L}/C_{D}\) maximization} & \(C_{L}/C_{D}\) & 13.12 & 13.71 & 4.47\% \\  & Mass & 1030.99 kg & 1030.06 kg & 0.03\% \\  & V\({}_{\text{flutter}}\) & 189.21 m/s & 194.87 m/s & 2.96\% \\ \hline \multirow{3}{*}{Mass minimization} & \(C_{L}/C_{D}\) & 11.99 & 12.37 & 3.14\% \\  & Mass & 955.82 kg & 956.41 kg & 0.06\% \\ \cline{1-1}  & V\({}_{\text{flutter}}\) & 188.82 m/s & 195.03 m/s & 3.29\% \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Optimized solutions obtained with the surrogate models compared to the corresponding SHARPy results for Case 3 (cruise speed of 130 m/s). Figure 7: Pareto fronts for Case 3 (cruise speed of 130 m/s). Firstly, one can notice that, as expected, when improving one parameter, the other worsens accordingly, typical of the nature of multi-objective optimization. High values of \(C_{L}/C_{D}\) consequently correspond to higher mass values. It is possible to observe that the aspect ratio is consistently higher when maximizing \(C_{L}/C_{D}\) compared to minimizing the mass across all cruise speeds. This observation aligns with established aerodynamic principles: wings with a higher AR have longer spans relative to their chord, which reduces the induced drag and consequently improves the lift-to-drag ratio. However, the flip side of this advantage is that longer, slender wires tend to be more flexible. Indeed, it is possible to observe how the flutter phenomenon in this case occurs at lower speeds. To counterbalance this issue, these wings are heavier and require a higher torsional stiffness to comply with the flutter speed constraint. Deviating from the usual trend is the relatively high aspect ratio observed at the highest cruise speed for the mass minimization objective. This unexpected outcome may be attributed to the narrower feasible design space identified by the surrogate model, and the optimizer that might have been trapped in a local minima region. At the lower cruise speeds (30 m/s and 60 m/s), when the primary objective is to minimize mass, a positive and more pronounced sweep angle is observed. This might be associated with a structural benefit, redistributing the lift toward the root, which can result in a lighter wing structure. However, when maximizing \(C_{L}/C_{D}\), a lower sweep may favor aerodynamic efficiency at these speeds. Variations in torsional stiffness and the angle of attack across the different objectives and velocities are subtler compared to other parameters. However, it is worth noting that higher \(C_{L}/C_{D}\) values are associated with lower angles of attack. This trend aligns with the expectation that a reduced angle of attack would lead to decreased aerodynamic drag. Lowering the angle of attack while still achieving adequate lift is a strategy to enhance the aerodynamic efficiency of the wing. Another important observation about the aspect ratio is that for both 30 m/s and 60 m/s cruise speeds, the optimal solutions approach the boundaries of the defined design space, which was set between 6 and 16. This is significant, as it indicates that the design space might not fully encompass the optimal regions for these objectives at these speeds. The plot presented in Figure 8 delineates the Pareto fronts for the three analyzed cruise speeds, 30 m/s, 60 m/s, and 130 m/s, derived using the same optimization parameters.