## 5 Multiobjective Optimization of Main Beam Structure Based on RSM
### Multiobjective Optimization of Main Beam Structure
#### 5.2.4 Fitting Response Value Solution and Goodness-Of-Fit Analysis  
We calculated the values of Formulas (35)-(39) to obtain the fitting response values of the corresponding objective functions such as the maximum deformation, maximum equivalent stress, and minimum fatigue life of the main beam at each sample point, as shown in Table 11.  
Because there was a deviation between the predicted value of the response surface model and the true response value, only the approximate solution of the objective function could be obtained. To ensure the accuracy of the predicted data of the response surface model, its goodness of fit was verified through a significance analysis. The prediction ability of the response surface model could be determined according to the decision coefficient \(R^{2}\), the modified decision coefficient \(R^{2}_{a}\), and the normalized root-mean-square difference \(NRMSD\). The closer the decision coefficient \(R^{2}\) and the modified decision coefficient \(R^{2}_{a}\) were to 1, and the closer the normalized root-mean-square difference \(NRMSD\) was to 0, the better the significance of the response surface model was, and the more accurate the predictive value was.  
The determination coefficient \(R^{2}\) is defined as follows  
\[R^{2}=1-\frac{SSE}{SST} \tag{40}\]  
where \(SSE\) is the sum of residual squares, \(SST\) is the sum of total squares, and they are defined as follows, respectively,  
\[SST=\sum_{i=1}^{n}(y_{i}-\overline{y})^{2} \tag{41}\]  
\[SSE=\sum_{i=1}^{N}(y_{i}-\hat{y_{i}})^{2} \tag{42}\]  
where \(y_{i}\) is the true response value of sample point, \(\overline{y}\) is the average value of the real point response value of the sample, \(\hat{y_{i}}\) is the sample point fitting response value, and \(n\) is the number of sample point response values.  
The corrected decision coefficient \(R^{2}_{a}\) is defined as follows  
\[R^{2}_{a}=1-\frac{SSE(n-1)}{SST(n-n_{s}-1)} \tag{43}\]  
where \(n_{s}\) is the number of design variables.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**No.** & **Max. Deformation (mm)** & **Max. Equivalent Stress (MPa)** & **Min. Fatigue Life (Times)** & **First-Order Natural Frequency (Hz)** & **Mass (kg)** \\ \hline
1 & 5.803 & 205.732 & 82,113.190 & 26.426 & 8436.186 \\
2 & 5.822 & 212.233 & 77,230.004 & 26.639 & 8347.852 \\
3 & 5.801 & 210.420 & 79,294.534 & 26.181 & 8524.519 \\
4 & 5.821 & 212.238 & 77,093.582 & 26.634 & 8347.852 \\
5 & 5.811 & 211.445 & 78,022.916 & 26.177 & 8524.519 \\
6 & 6.122 & 225.696 & 64,658.508 & 25.970 & 8204.226 \\
7 & 5.580 & 203.672 & 84,816.411 & 26.750 & 8668.145 \\
8 & 5.989 & 257.428 & 33,054.988 & 26.215 & 8204.220 \\
9 & 5.673 & 211.320 & 75,044.701 & 26.516 & 8668.151 \\
10 & 6.178 & 257.849 & 377.7253 & 26.280 & 7981.767 \\..  
The normalized root-mean-square difference \(NRMSD\) is defined as follows  
\[NRMSD=\frac{\sqrt{\frac{1}{n}\sum_{i=1}^{n}(y_{i}-\hat{y_{i}})^{2}}}{\frac{1}{n} \sum_{i=1}^{n}y_{i}} \tag{44}\]  
The significance analysis parameters of the response surface model calculated according to Formulas (40)-(44) are shown in Table 12. \(R^{2}\) and \(R_{a}^{2}\) of each objective function were close to 1, and \(NRMSD\) was close to 0, which indicated that the data had good significance, that is, the goodness of fit of the response surface model was good.  
The goodness of fit of the response surface model obtained through ANSYS Workbench is shown in Figure 27. The test points of the objective function in the figure were all distributed on the fitting curve, which further showed that the response surface model had a good fit.