# Aerodynamic Optimization and Characterization of a Ducted Tail for a Box-Launched Aircraft
## 4 Aerodynamic Optimization
### Approximate Model Selection  
The SSA algorithm, proposed by [PERSON] [40] in 2020, is a bionic intelligent algorithm modeled after the predatory and anti-predatory behaviors of sparrows, which has a higher performance compared with the Particle Swarm Algorithm (PSO) [41; 42; 43]. The BP neural network is a commonly used approximation model, but in a BP neural network each neuron transfer parameter needs to rely on weights and thresholds, so the assignment of the weights and thresholds directly determines the robustness of the BP neural network. Therefore, in order to improve the accuracy of the BP neural network approximate model, this paper fuses the SSA algorithm with the BP neural network approximate model. The SSA-BP approximate model is proposed as shown in Figure 5, and the PSO algorithm module is also introduced for comparison. The predator, joiner, and vigilante position update functions in the SSA algorithm are as follows:  
\[X_{ij}^{t+1}=\left\{ \begin{matrix}X_{ij}^{t}.\exp\left(\frac{-i}{a.N}\right),R_ {2}L<S_{T}\\ X_{ij}^{t}+Q.L,R_{2}>S_{T}\end{matrix} \right. \tag{1}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Number** & **Parameters** & **Lift Coefficient** & **Drag Coefficient** \\ \hline \multirow{4}{*}{3} & H1 = 80 mm & \\  & H2 = 50 mm & \\  & L3 = 420 mm & \\  & L4 = 216 mm & \\  & H1 = 100 mm & \\
4 & H2 = 00 mm & \\  & L3 = 465 mm & \\  & L4 = 00 mm & \\  & H1 = 110 mm & \\
5 & H2 = 00 mm & \\  & L3 = 460 mm & \\  & L4 = 243 mm & \\  & H1 = 110 mm & \\
6 & H2 = 00 mm & \\  & L3 = 400 mm & \\  & L4 = 270 mm & \\  & H1 = 120 mm & \\
7 & H2 = 00 mm & \\  & L3 = 380 mm & \\  & L4 = 135 mm & \\  & H1 = 140 mm & \\
8 & H2 = 00 mm & \\  & L3 = 410 mm & \\  & L4 = 270 mm & \\ \hline \hline \end{tabular}
\end{table} 
Table 4: _Cont._\[X_{ij}^{t+1}= \begin{cases}Q.\exp(\frac{X_{burst}^{t}-X_{ij}^{t}}{F}),i>n/2\\ X_{P}^{t+1}+\left|X_{ij}^{t}-X_{P}^{t+1}\right|.A^{+}.L,i\leq n/2\end{cases}  \tag{2}\]  
\[X_{ij}^{t+1}= \begin{cases}X_{best}^{t}+\beta.\left|X_{ij}^{t}-X_{best}^{t} \right|,f_{i}>f_{best}\\ X_{ij}^{t}+K\left(\frac{|X_{ij}^{t}-X_{burst}^{t}|}{f_{i}-f_{burst}+\epsilon} \right),f_{i}=f_{best}\end{cases}  \tag{3}\]  
In the above equation, \(Q\) is a normally distributed random number; \(L\) is a unit row vector; \(a\) is a random number between \([0,\,1]\); \(\epsilon=0\), \(\beta\) are step parameters; \(A^{+}\) is a row vector consisting of \(1\) and \(-1\); \(K\) is a random number between \([-1,\,1]\); \(X_{ij}^{t+1}\) is the position information of the \(i\) sparrow in the \(j\) dimension after the \(t\) iteration; \(X_{p}\) is the current discoverer position; \(X_{burst}\) is the global worst position; \(X_{best}\) is the current global optimal position; \(N\) and \(t\) are the maximum number of iterations and the current number of iterations; \(R_{2}\) and \(S_{T}\) are the warning value and safety value; \(f_{i}.f_{best},f_{worst}\) are the individual fitness, the global optimal fitness, and the global worst fitness, respectively. In order to verify the superiority of the SSA-BP neural network approximate model, and to comprehensively compare the prediction errors of BP, SSA-BP, and PSO-BP approximate models on the lift-drag characteristic parameters of the ducted tail, the following tests were performed. Among the 30 sample models, 20 sample models are selected as the training set and 10 sample models are selected as the test set, and the sets of lift-drag coefficients of the sample models at different angles of attack are calculated for sample training and testing. The BP neural network adopts a typical 3-layer topology. The logsig function is used for the hidden layer activation function, the pureline function is used for the output layer activation function, and the learning rate is set to be 0.015. The number of iterations is 1500, and the training expectation error is 0.00001. Based on this, the initial parameter settings of the SSA-BP and PSO-BP are shown in Table 5.  
Figure 5: Flowchart of SSA-BP approximate model.