# Bridge Model Updating Based on Wavelet Neural Network and Wind-Driven Optimization
## 6 Conclusions  
Finite element model updating makes a significant contribution to the healthy operation and damage detection of structures. This paper mainly focuses on model updating and proposes a WNN combined with the WDO algorithm as the surrogate model of the finite element model. The following conclusions are drawn:  
(1) A structural finite element model updating method using a WNN as the surrogate model and WDO algorithm is proposed. The finite element model of the structure was updated based on static and dynamic responses of structures. The updated results show that the WNN can reflect the nonlinear relationship between the structural response and the parameters and has good simulation performance; the wind-driven optimization algorithm has the advantages of high optimization efficiency and fast convergence speed, and the method can effectively update the finite element model;  
(2) The selection of updating parameters is important for improving the quality and efficiency of model updating. To avoid selecting unnecessary parameters, in this paper, the elastic modulus and mass density of the main beam and the elastic modulus and mass density of the triangular tower were preliminarily selected as updating parameters according to the engineering and modeling experience; then, sensitivity analysis was conducted, and the seven updating parameters were finally determined;  
(3) The finite element model updating of the Ningbo Bund Bridge was studied. Combined with measured dynamic responses of the structure, the initial model was updated based on the WDO algorithm and WNN. The results show that the differences between measured dynamic responses and dynamic responses calculated by the updated finite element model were greatly reduced: The maximum difference was reduced from \(-\)10.9% to \(-\)3.6%. It can be concluded that the finite element model updating method based on the WDO algorithm and WNN is applicable and practical for multi-parameter bridge model updating and has the advantages of high updating efficiency, reliability, and practical significance.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Structural Response** & **Updated Value** & **Measured Value** & **Difference (\%)** \\ \hline  & 1 st order & 0.53 & 0.55 & \(-\)3.6\% \\  & 2 nd order & 0.75 & 0.73 & 2.7\% \\  & 3 rd order & 0.87 & 0.86 & 1.1\% \\ Natural & 4 th order & 1.22 & 1.26 & 3.2\% \\ frequency/Hz & 5 th order & 1.35 & 1.34 & 0.7\% \\  & 6 th order & 1.44 & 1.40 & 2.9\% \\  & 7 th order & 1.69 & 1.64 & 3.0\% \\  & 8 th order & 2.01 & 1.98 & 1.5\% \\ \hline \hline \end{tabular}
\end{table}
Table 10: Comparison of structural responses.  
For future research, as evidenced in Section 2 of the paper, since the fitting accuracy and stability of WNNs are affected by the initial value of coefficient matrix, the number of hidden layer nodes, the selected excitation function, the network learning method, and the number of hidden layers, currently, the information and scale of the network are often determined based on experience and trial calculation. How to correctly select the wavelet function, the number of hidden layers, and the number of hidden layer nodes still needs research.  
Methodology, H.H.; software, Y.Z.; validation, B.Z.; investigation, T.Z.; data curation, Y.S.; funding acquisition, H.S. and J.W. All authors have read and agreed to the published version of the manuscript.  
This work was financially sponsored by the National Key Research and Development Program of China (Grant No. 2022 YFC3005305), and the Open Project of National Engineering Laboratory of Bridge Safety and Technology (Grant No. 003-04150201122, 76140-42990014).  
Not applicable.  
Data Availability Statement: Data is available upon request.  
The authors declare no conflict of interest.