# Shape Optimization of the Streamlined Train Head for Reducing Aerodynamic Resistance and Noise
## 4 Results  
The OLHS was used to obtain experimental design points of each input variable, which is shown in Figure 4. As plotted in Figure 4, each design variable has a relatively uniform distribution in its design space. Figure 5 shows the objective variables corresponding to the experimental design points, which are the aerodynamic resistance and the dipole noise source of a high-speed train. The RBF neural network was used to establish the approximate calculation model between input variables and output variables. Figure 6 presents the differences between the predicted and actual values of the aerodynamic resistance and dipole noise source, where Figure 6 indicates the percentage error of aerodynamic resistance, and Figure 6 indicates that of the dipole noise source. As shown in Figure 6, the percentage error between the predicted values and actual ones of aerodynamic resistance is less than 1%, and that of the dipole noise source is less than 3 dB. Thus, the approximate calculation model established by the RBF neural network has a good approximation effect. In the optimization calculation, the initial sampling points of the NSGA-II algorithm are set to be 20, and 50 generations of genetic calculation are performed. Thus, the optimization is completed after completion of 1000 designs. Figure 7 depicts histories of design variables during optimization, and the pattern \"\(\bigstar\)\" in the figure indicates the Pareto optimal solution. As plotted in Figure 7, each optimization design variable shows a convergence trend through the sampling of the optimization algorithm. The NSGA-II algorithm will find the Pareto optimal design point that minimizes the objective value through iterative computation. The variable \(x_{5}\) does not seem to converge. Nevertheless, this variable is basically irrelevant to the aerodynamic resistance and dipole noise source according to the correlation analysis. \begin{table}
\begin{tabular}{c c c c} \hline  & **Maximum Surface Grid** & **Total Cells (Million)** & **Aerodynamic Resistance(N)** & **Dipole Noise Source(dB)** \\ \hline mesh1 & 250 & 4.18 & 3456.2 & 110.7 \\ mesh2 & 200 & 5.92 & 3298.9 & 111.2 \\ mesh 3 & 150 & 8.56 & 3327.3 & 111.3 \\ \hline \end{tabular}
\end{table} 
Table 2: Grid-independence tests. Figure 4: Experimental design points for each input variable: (**a**) \(x_{1}\), (**b**) \(x_{2}\), (**c**) \(x_{3}\), (**d**) \(x_{4}\), (**e**) \(x_{5}\). Figure 5: Output variables at each design point: (**a**) Aerodynamic resistance, (**b**) Dipole noise source. Figure 6: Error between predicted value and actual value of output variables: (**a**) Percentage of aerodynamic resistance, (**b**) Difference of dipole noise source. Figure 7: Histories of optimization design variables: (**a**) \(x_{1}\), (**b**) \(x_{2}\), (**c**) \(x_{3}\), (**d**) \(x_{4}\), (**e**) \(x_{5}\). Figure 8 shows histories of objective variables. The pattern \"\(\bigstar\)\" indicates the value of objective variable on the Pareto front. As plotted in Figure 8, through the sampling of the optimization algorithm in the design space, the time histories of aerodynamic resistance and dipole noise source show a decreasing trend. The streamlined head shape is gradually improved towards the optimization of aerodynamic resistance and dipole noise source. The Pareto front is shown in Figure 9. The dot \"\(\bullet\)\" in the figure denotes the aerodynamic resistance and dipole noise source of the initial streamlined head shape. As depicted in Figure 9, both the aerodynamic resistance and dipole noise source have been improved through the optimization. In addition, the Pareto front obtained by the approximate calculation model is not much different from that obtained by the direct optimization, which demonstrates that the approximate calculation model can achieve good optimization results. The optimal aerodynamic resistances for the optimization based on the approximate calculation model and direct optimization are 3150.7 N and 3149.1 N, respectively, which are very close to each other.