## 4 Optimization Design Process and Results
### Design of Experiments  
The experimental design method is a mathematical method about how to arrange experiments scientifically and reasonably [32]. It is a technology of arranging experiments economically and scientifically on the basis of probability theory and mathematical statistics [33]. In the present study, according to surrogate method and the computational cost, 19 samples of the three design variables are generated by Latin hypercube method. The distribution of sample points is shown in Table 6 and Figure 13. The design variables of sample points are uniformly distributed in the design space.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Test Number** & **x\_Flap/mm** & **y\_Flap/mm** & **Angle\_Flap/** \\ \hline  
\hline \hline \end{tabular}
\end{table}
Table 6: Sampling points.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Variable** & **x\_Flap/mm** & **y\_Flap/mm** & **Angle\_Flap/** \\ \hline Lower limit & 10 & 10 & 0 \\ \hline Upper limit & 10 & 6 & 7 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Design space.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Objective** & **Weight Coefficient** \\ \hline OASPL & Minimize & 1 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Optimization objective settings.  
The relationship between design variable and objective function can be clarified by the scatter diagram between design variable and objective function [34, 35]. Related tables and correlation maps can reflect the relationship between the two variables [36], but it is difficult to accurately show the degree of correlation between the two variables. The correlation coefficient is used to reflect the degree of linear correlation between two variables, which takes the form as  
\[r(X,Y)=\frac{Cov(X,Y)}{\sqrt{Var[X]Var[Y]}}\]  
In which, _Cov_(X,Y) is the covariance of X and Y, \(Var[X]\) and \(Var[Y]\) are the variances of X and Y, respectively.  
The scatter plots and correlation coefficients of three design variables (x_flap, y_flap, angle_flap) and lift-drag ratio (L/D) are given in Figure 14. In the scatter plots, the values of x-axis and y-axis represent the variation range of the two variables to be analyzed. The value of the correlation coefficient indicates the degree of linear correlation between the two variables, the closer the value of the correlation coefficient is to 1, the positive correlation between the two variables is shown. On the contrary, the closer the value of the correlation coefficient is to 1, the more the negative correlation between the two variables is shown. If the correlation coefficient is close to 0, the two variables have no linear correlation. In Figure 14a,b, the correlation coefficients are 0.2564 and 0.3190. The two design variables of x_flap and y_flap have positive correlation with lift-drag ratio (L/D). In Figure 14c, the correlation coefficient is 0.9862, indicating that the correlation between angle_flap and constraints is very strong.  
Figure 13: Distribution of sample points.  
The scatter plots and correlation coefficients of three design variables (x_flap, y_flap, angle_flap) and the objectives (OASPL) are shown in Figure 15. The correlation coefficients are 0.2073, 0.3190, and 0.1832, respectively. The design variables of x_flap, y_flap, and angle_flap have positive correlation with the objectives. The scatter diagram shows that there is no linear correlation between variables.