# Bridge Model Updating Based on Wavelet Neural Network and Wind-Driven Optimization
## 5 Finite Element Model Updating of Ningbo Bund Bridge
### Results and Discussion  
Combined with the well-trained WNN and WDO algorithm, seven parameters were updated. The seven updating parameters correspond to the seven dimensions of air particles. One hundred air particles were used for optimization calculation, and the velocity limit is \([-1,1]\), the maximum number of iterations is 100, and the coefficients \(a\), \(g\), \(RT\), and \(c\) are, respectively, 0.85, 0.65, 0.1, and 2. The optimization process of the WDO algorithm is shown in Figure 14.  
The objective function value reaches convergence in 57 iterations, and the convergence error is 5.6 \(\times\) 10\({}^{-3}\); thus, the optimization effect is obvious. The comparison between updated values and initial values is shown in Table 9. In the table, the difference is: (updated value \(-\) initial value)/initial value.  
It can be seen from the table that the updating ranges of seven parameters are small, which are all within the range of \(-\)10-10%. Thus, it can be inferred that the decrease in values of \(E_{1}\) and \(E_{2}\) may be caused by the adverse effects of the structural assembly; the increase of \(E_{3}\) and \(E_{4}\) might be caused by the favorable effects of auxiliary structures and detailed components; the increase in the value of \(\rho_{1}\) is likely to be caused by the addition of bridge deck pavement load and auxiliary structures or the deviation caused by the mass conversion method during modeling; the decrease in the value of \(\rho_{2}\) may be caused by considering the extra mass of the detailed components in the initial  
Figure 14: The iterative curve of WDO.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Updated Value** & **Initial Value** & **Difference (\%)** \\ \hline \(E_{1}/\times 10^{4}\) MPa & 1.95 & 2.06 & \(-\)5.3\% \\ \(E_{2}/\times 10^{4}\) MPa & 1.92 & 2.06 & \(-\)6.8\% \\ \(E_{3}/\times 10^{4}\) MPa & 2.09 & 2.06 & 1.5\% \\ \(E_{3}/\times 10^{4}\) MPa & 2.15 & 2.06 & 4.4\% \\ \(\rho_{1}/\times 10^{3}\) kg/m\({}^{3}\) & 12.68 & 12.59 & 0.7\% \\ \(\rho_{2}/\times 10^{3}\) kg/m\({}^{3}\) & 7.88 & 8.05 & \(-\)1.9\% \\ \(\rho_{3}/\times 10^{3}\) kg/m\({}^{3}\) & 8.43 & 8.05 & 4.7\% \\ \hline \hline \end{tabular}
\end{table}
Table 9: Comparison between updated values and initial values.  
the deviation of the mass conversion method during modeling; the increase in the value of \(\rho_{3}\) may be caused by ignoring part of the mass of detailed components or the deviation of the quality conversion method during modeling. The selected seven updating parameters have reasonable changes in value, and the updated values of \(\rho_{1}\), \(\rho_{2}\), and \(\rho_{3}\) are all greater than 7.85 \(\times\) 10\({}^{3}\) kg/m\({}^{3}\), which still lie in reasonable ranges.  
The seven updated parameters were brought into the initial finite element model to obtain the updated value of the dynamic response of the structure. The comparison results with initial values are shown in Table 10. In the table, the difference is: (updated value \(-\) measured value)/measured value.  
It can be seen from the table that after updating the selected seven parameters, the differences between the updated values and measured values all fell within the range of 5%, and the maximum difference was reduced from \(-\)10.9% to \(-\)3.6%; thus, the updating is proven to be effective. The finite element model updating method based on the WDO algorithm and WNN has proven to be applicable and practical for multi-parameter bridge model updating.