# A Study on the Surrogate-Based Optimization of Flexible Wings Considering a Flutter Constraint +
## 3 Results
### Case 3: Cruise Speed of 130 m/s  
For the final case, considering a cruise speed of 130 m/s, the flutter phenomenon emerged for most of the points in the dataset, which made optimization difficult. The performance of the surrogate models on which the optimization was based is shown in Table 7.  
The initial optimization process, despite using an expanded dataset, produced results with high percentage errors compared with SHARPy, especially in terms of \(C_{L}/C_{D}\) and flutter velocity. This was primarily due to the active flutter constraint in a vast majority (82.2%) of the examined design combinations that led to the introduction of a penalty in the objective function. Consequently, in the Pareto front of 180 optimal solutions, 24.4% had active flutter issues. To address this, the optimization was reevaluated by increasing the population size to 280 and the number of generations to 200, as was implemented for the previous cruise speeds.  
The modified approach yielded a substantial reduction in the number of solutions with an active flutter constraint, down to 7.5%. However, this improvement came at a significant computational cost, with a 690% increase in time compared to the initial optimization. Despite the improvement, the percentage errors were still remarkably high, and the Pareto  
\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} & \(C_{L}/C_{D}\) & 36.08 & 36.16 & 0.21\% \\  & Mass & 927.40 kg & 927.06 kg & 0.04\% \\  & Vflutter & 91.94 m/s & 92.23 m/s & 0.32\% \\ \hline \multirow{3}{*}{Mass minimization} & \(C_{L}/C_{D}\) & 21.28 & 21.56 & 1.32\% \\  & Mass & 392.62 kg & 392.32 kg & 0.08\% \\ \cline{1-1}  & Vflutter & 194.92 m/s & 188.92 m/s & 3.08\% \\ \hline \hline \end{tabular}
\end{table}
Table 6: Optimized solutions obtained with the surrogate models compared to the corresponding SHARPy results for Case 2 (cruise speed of 60 m/s).  
\begin{table}
\begin{tabular}{l c c} \hline \hline
**Model** & **RMSE (\%)** & **R\({}^{2}\) (-)** \\ \hline Flutter speed & 7.981 & 0.907 \\ Mass & 0.263 & 0.999 \\ \(C_{D}\) & 0.0017 & 0.997 \\ \(C_{L}\) & 0.011 & 0.999 \\ \hline \hline \end{tabular}
\end{table}
Table 7: Final RMSE and R\({}^{2}\) metrics for Case 3 (cruise speed of 130 m/s).  
Figure 6: Pareto fronts for Case 2 (cruise speed of 60 m/s).  
fronts showed several infeasible solutions due to the active flutter problem and the penalties associated with the results.  
However, a closer examination of the results, focusing particularly on the solutions not affected by flutter, showed commendable accuracy. For the most exhaustive optimization (with 200 generations and a population of 280), errors were consistently less than 5%, as listed in Table 8.  
These results, in line with those obtained for the cruise speeds previously examined, clearly reveal that the high penalties were the main cause of the high percentage errors previously observed.  
Two distinct groups of solutions can be immediately identified from the graph in Figure 7. In both cases, the optimized design space is significantly reduced, but the second offers a more diverse set of solutions.  
Moreover, considering only the feasible solutions, we obtained results that are aligned with the previous cases. This further strengthens the viability of the implemented SBO strategy and shows that, despite the obstacles represented by penalties, the algorithm is able to offer very good solutions.