## 3 Case Study
### Update Model Parameters through the PSO Algorithm  
In some actual large structures, it is pretty challenging to determine the dimensions and geometrical features of the structure so that the geometric features will be considered an uncertain parameter. However, in this case, the parameters of the geometrical dimensions can be completely accurate based on detailed measurements or design documents, so this can be considered a specific parameter to reduce the amount of computation. Usually, the material parameters should be determined through direct tests (destructive tests). Thereby, the exact parameters will be selected. However, this is relatively expensive. For existing structures, sampling interference will affect the structure. In the case of the plate under consideration, UHPC was applied for the first time at a construction site in Vietnam. Therefore, the parameters of UHPC are entirely unknown. In addition, uncertain parameters related to steel materials and bearing stiffness are only empirical estimates. Moreover, the fact that the two materials, UHPC and steel, are combined by working together through the rivet system also influences the structural stiffness calculation results, and this part will be adjusted and updated through the parameters of UHPC and steel. The model's uncertain parameters are described in detail in Table 4. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**No.** & **Uncertainty Parameters** & **Initial Value** & **Upper Bound** & **Lower Bound** \\ \hline \hline \multirow{4}{*}{1} & \multicolumn{3}{c}{Young’s modulus} & \multirow{4}{*}{33.91} & \multirow{2}{*}{33.91} & \multirow{2}{*}{29.91} \\  & - & _UHPC_: \(E_{\mathrm{c}}\) (GPa) & & & \\  & - & _Steel_\(E_{\mathrm{s}}\) (GPa) & & & \\ \hline \hline \multirow{4}{*}{2} & \multicolumn{3}{c}{Weight density} & \multirow{4}{*}{2} & \multirow{4}{*}{10} \\  & - & _UHPC_: \(\rho_{\mathrm{c}}\) (kg/m\({}^{3}\)) & & & \\ \cline{1-1}  & - & _Steel_\(p_{\mathrm{s}}\) (kg/m\({}^{3}\)) & & & \\ \hline \hline \multirow{4}{*}{3} & \multicolumn{3}{c}{Stiffness of bearing} & \multirow{4}{*}{3} & \multirow{4}{*}{10} \\  & - & \(k_{\mathrm{b}}\) (N/m) & & & \\ \cline{1-1}  & \multicolumn{1}{c}{} & & & \\ \cline{1-1}  & \multicolumn{1}{c}{} & & & \\ \end{tabular}
\end{table} 
Table 4: Uncertainty parameters of model. Figure 13: First five mode shapes from the simulation: (**a**) Mode shape 1: f1 = 7.102 Hz, 1 st torsion; (**b**) Mode shape 2: f2 = 9.282 Hz, 1 st vertical bending; (**c**) Mode shape 3: f3 = 23.472 Hz, 2 nd torsion; (**d**) Mode shape 4: f4 = 38.227 Hz, 2 nd vertical bending; and (**e**) Mode shape 5: f5 = 46.105 Hz, 3 rd torsion. After obtaining the unnecessary parameters, update the model with the objective function. The update process is performed and processed on a computer with Intel(R) Core(TM) spec i5-10400F CPU @ 2.90 GHz 2.90 GHz, 8 GB RAM. In the PSO algorithm, the population is generated with 50 individuals. For the purpose of convergence, assign the value 0.3 to the inertial weight parameter (w), where social and cognitive learning coefficients are C\({}_{1}\) = 2 and C\({}_{2}\) = 2, respectively [37]. The fitness must not change by more than \(10^{-7}\) from one iteration to the next, or the number of iterations must not exceed 100, both of which were set as ending criteria for the PSO loops. The convergence curve when using PSO is shown in Figure 14.  
The updated uncertainty parameters and FE model after using the PSO are presented in Tables 5 and 6.