## 3 Structural Optimization of PIPR's End Face
### The Optimization Results of PIPR's End Face  
Based on the established pressure gradient model, the parameters of PIPR's end face are optimized to minimize the average pressure gradient of the flow field. The fitness function is shown in Equation (12). WOA is a biological heuristic algorithm developed based on the hunting behavior of humpback whales [28]. It updates the position by simulating the way humpback whales search, surround, and capture food to realize the optimization process. Compared to traditional optimization algorithms, WOA has good optimization performance and convergence stability. The search space of WOA is constructed by the four parameters of PIPR's end face, and the optimal parameter combination is the optimal location of the whale population. The search process for the optimal parameter combination can be transformed into a swarm search process for the whales' optimal location. \[ \begin{split} F(X)=\min f\big{(}s,h,r,S_{p}\big{)}=\min\frac{ \overline{\Delta p}}{\overline{\Delta L}}\\ \text{s.t.}\left\{\begin{array}{l}14\leq s\leq 24\\ 2\leq h\leq 12\\ 2\leq r\leq 8\\ 0.01\leq S_{p}\leq 0.99\end{array}\right.\end{split}  \tag{12}\]  
However, for the spiral-hunting mechanism of WOA, the convergence speed in the later stage can be improved, but this method can also cause the population to rapidly gather in the solution space, which leads to a decrease in population diversity and increases the possibility of local optimum. Therefore, the golden sine algorithm (GSA) is introduced to improve the WOA. For this algorithm, the whales capture their prey using a gold sinusoidal spiral motion. At each iteration, the whales communicate information with the optimal individual, and they learn the position difference between themselves and the optimal individual. Meanwhile, the coefficient obtained by the golden section is used to gradually reduce the search space, and the distance and orientation of position updating can be controlled. The optimization method of basic WOA has been improved, which can coordinate global exploration and local development so as to improve the optimization accuracy and convergence speed. The position updating based on GSA can be expressed as follows:  
\[X(t+1)=X(t)\times|\sin(R_{1})|+R_{2}\times\sin(R_{1})\times|x_{1}\times X^{*}( t)-x_{2}\times X(t)| \tag{13}\]  
where \(R_{1}\) is a random number of [0, 2\(\pi\)], which determines the moving distance of the individuals in the next iteration; \(R_{2}\) is a random number of [0, \(\pi\)], which determines the location updating direction of the individuals in the next iteration; \(x_{1}\) and \(x_{2}\) are the coefficients by introducing the golden section, they are expressed as follows [29; 30]:  
\[ \begin{array}{l}x_{1}=-\pi+(1-\tau)\times 2\pi\\ x_{2}=-\pi+\tau\times 2\pi\\ \tau=\frac{\sqrt{5}-1}{2}\end{array}  \tag{14}\]  
The population size of IWOA is 30, and the maximum number of iterations is 200. The values of the fitness function of IWOA and WOA are shown in Figure 22. It can be seen that the original WOA and IWOA both have faster convergence speeds, but the final \begin{table}
\begin{tabular}{c c c c} \hline
**Method** & **MAE** & **MAPE** & **R\({}^{2}\)** \\ \hline BP & 23.514 & 0.001 & 0.999 \\ SVM & 1405.042 & 0.044 & 0.222 \\ GRNN & 331.243 & 0.011 & 0.932 \\ RF & 2049.021 & 0.064 & 1.737 \\ Response surface method & 800.429 & 0.025 & 0.376 \\ \hline \end{tabular}
\end{table} 
Table 6: The evaluation metrics of different methods. results of IWOA are smaller than that of WOA, which indicates the advantages of the WOA algorithm improved by GSA. In order to verify the superiority of IWOA for the optimization of PIPR's end face, the optimization results of WOA, GA, PSO, and response surface method are compared, as shown in Table 7. It can be seen that the average pressure gradient obtained by IWOA is the smallest. Compared with the improved PIPR model before optimization, the average pressure gradient has decreased by 25.11%.