# Metaheuristic Approaches to Solve a Complex Aircraft Performance Optimization Problem
## 5 Case Study
### Optimal Designs by GP and GA  
Metamodels were used to compute the responses of the interests during the MAO process for the representative aircraft conceptual design. The comparison of the optimal result by GP-based and GA-based optimizations and finite element validations is given in Table 3 and the corresponding design variables are shown in Table 4. The genetic parameters used by GA include Population 100, Generation 16, Tournament Selection, Crossover 0.7, and Mutation 0.4. Since the optima obtained by GP metamodel-based and GA-based optimizations satisfy the design constraints, both results are feasible solutions. It is noted that the superiority of the GP method over GA for MAO was demonstrated by the overall performance of the optimal design except for the \begin{table}
\begin{tabular}{c c c} \hline \hline
**Geometric Design Variables** & **GP** & **GA** \\ \hline Inboard taper ratio & 1.74 & 1.60 \\ Outboard taper ratio & 4.25 & 3.52 \\ Inboard aspect ratio & 0.96 & 0.91 \\ Outboard aspect ratio & 2.90 & 2.83 \\ Sweep angle (deg) & 25.63 & 29.90 \\ Inboard dihedral angle (deg) & 5.18 & 4.50 \\ Outboard dihedral angle (deg) & 0.69 & 0.82 \\ Axis position (root) & 0.31 & 0.33 \\ Axis percent (tip) & 0.41 & 0.45 \\ Axis percent (kink) & 0.46 & 0.44 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Geometric parameters and elastic axis position parameters. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Responses** & **GP** & **Validation** & **GA** & **Constraint** \\ \hline Stiffness & 0.533 & 0.535 & 0.806 & - \\ Displacement & 5.61\% & 5.65\% & 6.16\% & \textless{}7\% \\ Torsion (deg) & 1.39 & 1.44 & 2.05 & \textless{}2.14 \\ Efficiency & 0.677 & 0.680 & 0.655 & \textgreater{}0.6 \\ Flutter (m/s) & \textgreater{}320 & \textgreater{}320 & \textgreater{}320 m/s & \textgreater{}320 \\ Drag (N) & 65,158.2 & 63,090.8 & 64,661.2 & \textless{}65,438 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Comparison of responses. Figure 7: Accuracy of the GP model verified by 100 sampling points. drag response shown in Table 3. The stiffness by GP based optimization reduced from 0.806 to 0.535 by 33.6%, while the drag just deteriorated by 0.77% and no constraints were violated. In addition, it is observed that the predictions of responses agree well with finite element validation due to the high-accuracy models. The most remarkable differences among the two cruise shapes shown in Figure 8 are that the optimum by GA-based method had the largest sweep angle of the leading edge, while the optimal design using GP-based method had a largest outboard spanwise length. The increase of the sweep angle contributed to the decrease of the local shock wave drag and the higher aspect ratio led to the induced drag in reduction. The induced drag and the local shock wave drag are two significant types of a large aircraft flight drag under the cruise Mach of 0.785. The increase of the sweep angle also results in the increase of the vertical bending-torsional coupling effect as well as the structural weight. It is the reason why the optimum by GA method had a heavier structural weight and a larger wingtip torsional deformation than the optimal design by GP method. Figure 9 shows the variations of vertical, horizontal and torsional stiffnesses along the wingspan. Since the vertical stiffness is the most important parameter to reflect the aeroelastic character, it is reasonable to identify the slight variation of the vertical bending performance by these two methods. The GP method helped to remarkably reduce the horizontal stiffness in the region near the wing root and decrease the torsional stiffness shown in Figure 9b,c, respectively. The transient behavior of the state variables in Equation (2) for the fourth-order model is shown in Figure 10. It can be found that the disturbance quantities ultimately  
Figure 8: Cruise shapes: the optima by the GP and GA methods. Figure 9: Stiffness comparison between the GA and GP methods. (**a**) Vertical bending; (**b**) Horizontal bending; (**c**) Torsional.