## 3 Numerical Examples
### Influence of Node Deviations on the Reliability Performance of Optimized Structures  
In order to study the influence of node deviations on the structure, the optimization results in Section 3.1 are taken as the research object, and the _probability density evolution theory_ is used to calculate the PDF of the maximum displacement response of the structure when the structure is subjected to a vertical uniformly distributed load at each typical iteration step. Two groups of basic random variables are used in calculating the PDF of the structural response, one of which only contains the _material yield strength_ and _elastic modulus_ (Table 3), and the other group adds node deviations on the basis of Table 3 (Table 4). The other parameters of the two groups are the same. When the range is divided, the two groups of random variables are divided into 365 regions based on the GF discrepancy-based technique [18]. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Random Variable**} & \multicolumn{2}{c}{**Probability Distribution Model**} & \multirow{2}{*}{**Source**} & \multirow{2}{*}{**Mean**} & \multicolumn{1}{c}{**Coefficient of Variation**} \\ \hline Yield Strength of & \multicolumn{1}{c}{} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{} & \\ Material (MPa) & \multicolumn{1}{c}{} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{} & \\ Elastic Modulus & \multicolumn{1}{c}{} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{} & \\ (\(\times 10^{5}\) Mpa) & \multicolumn{1}{c}{} & \multicolumn{1}{c}{} & & & \\ Node deviations (m) & \multicolumn{1}{c}{} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{} & & \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Parameters and sources of random variables (considering node deviations). Figure 6: [PERSON] stress of the optimized structure. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Random Variable**} & \multicolumn{2}{c}{**Probability Distribution Model**} & \multirow{2}{*}{**Source**} & \multirow{2}{*}{**Mean**} & \multicolumn{1}{c}{**Coefficient of Variation**} \\  & \multicolumn{1}{c}{} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{} & \\ Material (MPa) & \multicolumn{1}{c}{} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{} & \\ Elastic Modulus & \multicolumn{1}{c}{} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{} & \\ (\(\times 10^{5}\) Mpa) & \multicolumn{1}{c}{} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{} & & \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Parameters and sources of random variables. After trial calculation, the bearing capacity of the initial structure is about 30 \(\rm{kN/m^{2}}\). The PDF of the maximum node displacement response of the structure is calculated using this load. The calculation results are shown in Figure 7.  
Figure 7: Comparison of the displacement PDF at each representative iteration step. As shown in Figure 7, during the initial iteration stages--particularly for the initial structure--the PDF of the structural response calculated with nodal deviations exhibits significant discrepancies compared to those without nodal deviations. As optimization progresses, the gap between the two curves narrows, indicating that the influence of nodal deviations on the PDF of the structural response diminishes. In order to better compare the effect of nodal deviations on the structure, we calculated the expected value and variance of the displacement response of the initial structure and the optimized structure based on the data in Figure 7 as follows:  
* Initial structure (without nodal deviations): Expected displacement: \(-2.73\times 10^{-2}\) m (upward direction defined as positive); Variance: \(6.68\times 10^{-3}\) m\({}^{2}\). Initial structure (with nodal deviations): Expected displacement: \(-2.84\times 10^{-2}\) m; Variance: \(1.42\times 10^{-3}\) m\({}^{2}\); Relative differences: \(4.05\%\) in expectation and \(78.69\%\) in variance.