# Aerodynamic Optimization and Characterization of a Ducted Tail for a Box-Launched Aircraft
## 4 Aerodynamic Optimization
### Approximate Model Selection
Based on the above approximate models and parameter settings, 10 sets of sample models are selected as test samples. The lift-drag characteristics of the ducted tail at an incoming flow velocity of 0.35 Ma and the attack angle of 0\(\sim\)14\({}^{\circ}\) are simulated and predicted by BP, SSA-BP, and PSO-BP under given constraints such as H1/L3/L4. The prediction data of one test model is randomly selected, as shown in Figure 6. For the prediction of lift coefficients, the predicted values generated by the SSA-BP approximation model have the same trend of change compared to the sample values. However, the predicted values generated by the BP and PSO-BP approximation models are locally distorted due to the fact that the weights and thresholds on which the neuron transmission parameters depend are not optimal. The predicted values generated by the BP approximation model gradually deviated from the sample values as the attack angle increased, and the predicted values generated by the PSO-BP approximation model had a larger local distortion at an attack angle of 12\({}^{\circ}\). Regarding the prediction of the drag coefficients, the same phenomenon exists, and the predicted values generated by the BP and PSO-BP approximation models have a larger error compared to the sample values due to the higher degree of discretization of the drag coefficients. Initially, the SSA-BP approximation model is judged to have a higher accuracy in approximating the lift and drag coefficients. The above results only qualitatively analyze the parameter approximation results and lack quantitative assessment guidelines. To comprehensively assess the accuracy of the approximation model, the absolute error (MAE), the mean percentage error (MAPE), and the root mean square error (RMSE) were found for the parameter approximation results \begin{table}
\begin{tabular}{c c c} \hline \hline
**Number** & **Parameters** & **Value** \\ \hline \multirow{4}{*}{SSA-BP} & Initial sparrow population & 60 \\  & Number of iterations & 100 \\  & Warning value & 0.8 \\  & Percentage of discoverers & 0.4 \\  & Percentage of anti-predatory sparrows & 0.3 \\ \hline \multirow{4}{*}{PSO-BP} & Initial population size & 60 \\  & Number of iterations & 100 \\  & Acceleration constant C1 & 1.5 \\  & Inertial constant C2 & 1.5 \\  & Inertial weighting & 0.8 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Initial parameter settings of the SSA-BP and PSO-BP. Figure 6: Parameter approximation comparison. (**a**) Lift coefficient approximation; (**b**) drag coefficient approximation. according to the following formulas. Here, n represents the number of test samples, and \(y,\hat{y}\) are the actual and predicted values, respectively. \[E_{MAE}=\frac{1}{n}{\sum\limits_{i=1}^{n}}{\left|{y-\hat{y}}\right|} \tag{4}\]  
\[E_{MAPE}=\frac{1}{n}{\sum\limits_{i=1}^{n}}{\left|{\frac{y-\hat{y}}{y}}\right|} \tag{5}\]  
\[E_{RMSE}=\sqrt{\frac{1}{n}{\sum\limits_{i=1}^{n}}{\left|{y-\hat{y}}\right|^{2}}} \tag{6}\]  
Figure 7 shows the approximate values of SSA-BP compared with the sample values. The approximate values of the lift coefficient corresponded to MAE = 0.0467, MAPE = 0.0310, RMSE = 0.0559; the approximate values of the drag coefficient corresponded to MAE = 0.0049, MAPE = 0.0098, RMSE = 0.0064. Comparing the approximate values of PSO-BP with the sample values, the approximate values of the lift coefficient corresponded to MAE = 0.0896, MAPE = 0.1380, RMSE = 0.1174; the approximate values of the drag coefficient corresponded to MAE = 0.0119, MAPE = 0.0336, RMSE = 0.0183. Comparing the approximate values of BP with the sample values, the approximate values of the lift coefficient corresponded to MAE = 0.1446, MAPE = 0.3140, RMSE = 0.1773; the approximate values of the drag coefficient corresponded to MAE = 0.0230, MAPE = 0.0817, RMSE = 0.0286.