## 4 Structural Optimization Design
### Constructing the Optimal Model
\begin{table}
\begin{tabular}{c c c c c} \hline  & **Coefficients** & **Standard Error** & \(t\)**Stat** & \(p\)**-Value** \\ \hline Intercept & 4595.752 & 0.051667 & 88,949.98 & 0.001 \\ X Variable 1 & 1,216,390 & 0.577331 & 2,106,921 & 0.001 \\ X Variable 2 & 169,125.3 & 0.769774 & 219,707.7 & 0.001 \\ X Variable 3 & 241,171.9 & 0.659807 & 365,519.1 & 0.001 \\ X Variable 4 & 34,127.85 & 0.461865 & 73,891.46 & 0.001 \\ X Variable 5 & 234,725.8 & 0.384887 & 609,856.2 & 0.001 \\ \hline \end{tabular}
\end{table} 
Table 4: Table of regression analysis results on mass \(M\). \begin{table}
\begin{tabular}{c c c c c c c c} \hline \(R_{1}\) **(m)** & \(R_{3}\) **(m)** & \(R_{4}\) **(m)** & \(R_{5}\) **(m)** & \(R_{6}\) **(m)** & **M (kg)** & **S (Mpa)** & **U (m)** \\ \hline
0.036 & 0.027 & 0.0315 & 0.045 & 0.054 & 74,760.1 & 114 & 0.002402471 \\
0.0206 & 0.0245 & 0.03447 & 0.0446 & 0.0511 & 55,629.1 & 338 & 0.006521 \\
0.04231 & 0.03595 & 0.03296 & 0.02701 & 0.04538 & 81,664 & 87.9 & 0.001984 \\
0.02683 & 0.03912 & 0.02347 & 0.03656 & 0.08548 & 70,820 & 194 & 0.003683 \\
0.03387 & 0.03746 & 0.03854 & 0.05113 & 0.05503 & 76,087 & 27 & 0.002569 \\ \(\vdots\) & \(\vdots\) & \(\vdots\) & \(\vdots\) & \(\vdots\) & \(\vdots\) & \(\vdots\) \\
0.038668 & 0.030999 & 0.0315 & 0.051665 & 0.061998 & 80,786.5 & 96.6 & 0.002107169 \\ \hline \end{tabular}
\end{table} 
Table 3: Table of partial test results. \[S=-6239216457X_{1}-38555965.68X_{2}-44669648.47X_{3}+66759184.62X_{4} \tag{2}\] \[-294959688.1X_{5}+389540494.2\]  
Table of regression analysis results for maximum stress \(S\). \begin{tabular}{c c c c c} \hline \hline  & **Coefficients** & **Standard Error** & **t Stat** & \(p\)**-Value** \\ \hline Intercept & 389,540,494.2 & 18,298,354.5 & 21.28828001 & 1.20376 \(\times\) 10\({}^{-52}\) \\ X Variable 1 & \(-\)6,239,216,457 & 197,832,147 & \(-\)31.53793027 & 2.70762 \(\times\) 10\({}^{-78}\) \\ X Variable 2 & \(-\)38,555,965.68 & 263,724,057.2 & \(-\)0.146198136 & 0.883916779 \\ X Variable 3 & \(-\)44,669,648.47 & 263,764,808.3 & \(-\)0.169354088 & 0.865694592 \\ X Variable 4 & \(66,759,184.62\) & 158,298,029.4 & 0.42173099 & 0.67368832 \\ X Variable 5 & \(-\)294,959,688.1 & 131,903,938.1 & \(-\)2.236170447 & 0.026479707 \\ \hline \hline \end{tabular}   
Finally, a regression analysis was performed on the expression of the maximum deformation function of the large-body structure to obtain a table of the analytical results shown in Table 6.  
Collating the data in Table 6 yields the following expression for the regression function for the largest deformation \(U\):  
\[U=-0.06936X_{1}-0.00191X_{2}-0.00289X_{3}-0.00055X_{4}-0.00603X_{5}+0.005364 \tag{3}\]  
Three objective functions were identified for the optimal design of the large-body structure based on real-world situations: mass, maximum deformation, and maximum stress, which are multi-objective optimization problems. These objectives differ significantly in importance, with the quality objective being the most valuable and having the highest weight, and the maximum deformation and maximum stress objectives being the next most important and being weighted approximately equally. This paper transformed the multi-objective problem into a single-objective optimization problem with optimal solutions. Having identified the three elements of the optimization model in this subsection, the mathematical model for the optimization of large-body structures can be obtained as follows:  
\[\left\{\begin{array}{l} \begin{array}{l}\text{{Minimize }}M(x)\\ x\ =\ [x_{1},x_{2},x_{3},x_{4},x_{5}]\\ 0\ <\ \text{S}\ \leq\ 156.67\\ 0\ <\ \text{U}\ \leq\ 0.006\\ 0.02\ \leq\ x_{1}\ \leq\ 0.06\\ 0.015\ \leq\ x_{2}\ \leq\ 0.045\\ 0.02\ \leq\ x_{3}\ \leq\ 0.05\\ 0.025\ \leq\ x_{4}\ \leq\ 0.075\\ 0.03\ \leq\ x_{5}\ \leq\ 0.09\\ \end{array} \right. \tag{4}\]  
After constructing the mathematical model of the optimized large-body structure, the algorithm needs to be applied to solve the model to find the optimal solution of the model.