# Bridge Model Updating Based on Wavelet Neural Network and Wind-Driven Optimization
## 4 Verification of the Proposed Method
### Finite Element Model  
The numerical model is a continuous beam structure of three equal spans, and the length of each span is 4 m. Since the elastic modulus and density of materials could change along with time, and their changes could influence structural behavior significantly, these data were therefore selected for verification and demonstration purposes. The elastic modulus of the material of the beam is \(E\) = 3.25 \(\times\) 10\({}^{4}\) MPa, and the density is \(\rho\) = 25 kN/m\({}^{3}\). The beam's cross-section is a rectangular section of 0.2 m \(\times\) 0.15 m. The finite element model was established by finite element analysis software Midas Civil 2020, and the model was divided into 12 beam elements. The elastic moduli of each span of the three-span  
Figure 4: Calculation flow of WDO. continuous beam are denoted as \(E_{1}\), \(E_{2}\), and \(E_{3}\) and the midpoint of each span as \(C_{1}\), \(C_{2}\), and \(C_{3}\). The finite element model is shown in Figure 5.  
In practice, the material density of the structure may deviate, and the stiffness may change during service time. Therefore, for illustrative purposes, the material density \(\rho\) is assumed to be increased by 10%, and the elastic moduli \(E_{1}\), \(E_{2}\), and \(E_{3}\) are assumed to be decreased by 10%, 15%, and 20%, respectively, as the parameters of the actual structure. Values of these deviated parameters are called the actual values of the structure for simplicity, and the static and dynamic responses calculated by these deviated values are called the actual values of responses. The initial and actual values of parameters are shown in Table 2, where the difference is: (initial value \(-\) actual value)/actual value. The finite element model updating is carried out using the static and dynamic response of the structure, in which the dynamic response adopts the first six natural frequencies of the structure, and the static response is obtained by applying a vertical downward force of 30 kN at the two points C1 and C3 to obtain displacements at C1, C2, and C3. The initially calculated values obtained from the responses of the initial finite element model and the actual values of the finite element model with updated parameters are shown in Table 3. The difference in the table is: (initial value \(-\) actual value)/ actual value. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Parameter**} & \(E_{1}\) & \(E_{2}\) & \(E_{3}\) & \(\rho\) \\  & (\(\times 10^{4}\) MPa) & (\(\times 10^{4}\) MPa) & (\(\times 10^{4}\) MPa) & (\(\times 10^{4}\) kg/m\({}^{3}\)) \\ \hline Initial value & 3.250 & 3.250 & 3.250 & 2.500 \\ Actual value & 2.925 & 2.762 & 2.600 & 2.750 \\ Difference/\% & 11.11 & 17.67 & 25.00 & \(-9.09\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Initial value and actual value of physical parameters. Figure 5: The structural representation of the finite element model. It can be seen from Table 3 that the initial values of the initial finite element model show considerable differences from actual values, and the differences in natural frequencies and displacements are both larger than 11%; therefore, the initial finite element model needs to be updated. According to the basic process of model updating, the four parameters of the initial model, \(E_{1}\), \(E_{2}\), \(E_{3}\), and \(\rho\) were updated, and for illustrative purposes, the objective function was constructed without conducting sensitivity analysis to screen the updating parameters. Combined with static and dynamic responses of the finite element model, the objective function was constructed.