## 6 Regulator Structure Optimization Analysis
### Analysis of Optimization Results
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Source of** & **Sum of** & **Degree of** & **Mean** & **F** & \(p\)**Value** & **Significance** \\
**Variation** & **Squares** & **Freedom** & **SQUARE** & & & \\ \hline Models & 7243.12 & 9 & 804.79 & 48.33 & \textless{}0.0001 & *** \\ \(x_{1}\) & 201.40 & 1 & 201.40 & 11.35 & 0.0028 & ** \\ \(x_{2}\) & 82.11 & 1 & 82.11 & 178.31 & \textless{}0.0001 & *** \\ \(x_{3}\) & 3168.48 & 1 & 3168.48 & 26.77 & \textless{}0.0001 & *** \\ \(x_{1}\,x_{2}\) & 19.05 & 1 & 19.05 & 93.43 & \textless{}0.0001 & *** \\ \(x_{1}\,x_{3}\) & 223.95 & 1 & 223.95 & 9.12 & 0.0143 & * \\ \(x_{2}\,x_{3}\) & 769.51 & 1 & 769.51 & 112.47 & \textless{}0.0001 & *** \\ \(x_{2}^{2}\) & 480.49 & 1 & 480.49 & 8.44 & 0.011 & * \\ \(x_{2}^{2}\) & 1306.48 & 1 & 1306.48 & 17.91 & 0.0044 & ** \\ \(x_{3}^{2}\) & 714.87 & 1 & 714.87 & 13.29 & 0.0015 & ** \\ Residuals & 2662.31 & 7 & 380.33 & & & \\ Pure Error & 0 & 4 & 0.00 & & & \\ Total & 9905.42 & 16 & & & & \\ \hline \hline \end{tabular} Note. ‘**’ is very significant; **’ is significant, **’ is less significant.  
\end{table} 
Table 5: Analysis of variance. Figure 9 shows the residuals versus the number of trials plot and the predicted value plot, which can be used to check the reasonableness of the model. The plot indicates that all residuals are distributed within the horizontal band between the two red lines, which means that the quadratic polynomial model is reasonable. Table 6 shows the comparison between the predicted and actual values. The maximum relative error is 15.13%, indicating that the predicted and actual values have a good correlation and that the quality of prediction and the accuracy of the model are high. The regression diagram can visualize the trend of each variable, and the change of the equivalent stress can also be visualized. The change in the slope of each curve shows how sensitive the equivalent stress is to each variable. Figure 10 shows that the equivalent stress increases with the sleeve wall thickness and the diameter of the valve bore, and then the value decreases. This result indicates that increasing the above values is beneficial for reducing the equivalent force. Equivalent stress \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Number of Tests** & **Actual Value (MPa)** & **Predicted Value (MPa)** & **Relative Error (\%)** \\ \hline  
\hline \hline \end{tabular}
\end{table} 
Table 6: Summary of predicted and actual values. Figure 9: Residual Curves. (**a**) Residual vs. test times, (**b**) Residual vs. predicted value. with the increase in wall thickness of the outlet seat has a certain upward trend, but the rising trend gradually slows down, negatively affecting the reduction of the equivalent force. Hence, larger is not necessarily better in selecting the outlet seat wall thickness; rather, it should be taken within a reasonable range of values. The three-dimensional response surface plot can more intuitively show the effect of the two variables acting simultaneously on the equivalent force. The two-dimensional contour plot can represent the mutually changing relationship between the two variables. Therefore, the three-dimensional response surface diagram and its two-dimensional contour diagram are shown in Figure 11.  
The gas regulator secondary response surface interaction term surface image and the corresponding contour plot can be analyzed under the simultaneous action of the two factors equivalent effect force change law. As the sleeve wall thickness and valve hole diameter increase, the equivalent effect force first increases and then decreases. When the outlet seat wall thickness is small, the sleeve thickness and valve hole diameter increase, and the equivalent effect force decreases significantly. The above results are consistent with the results of the regenerative analysis.