## 4 Model Optimization and Validation  
In the finite element analysis, the bandwidth of the overall mass, stiffness, and damping matrices is determined by the node number. In order to improve the computational efficiency, the node number is optimized to reduce the difference between the related node numbers in the same element, thus, reducing the bandwidth of the overall matrix. The AD algorithm [36] is selected to optimize the node number and the algorithm is as follows. (1) Calculate the sum (\(M_{n}\)) of the number of the adjacent nodes of each node; (2) Calculate the node quotient \(Q_{n}=M_{n}/M_{e}\), where \(M_{e}\) is the number of related elements of the node; (3) Calculate the sum (\(M_{m}\)) of the maximum node number and the minimum node number in the adjacent nodes of each node; (4) Renumber all nodes according to the size of the node quotient (\(Q_{n}\)). If the \(Q_{n}\) of the two nodes is equal, the number of the two nodes is determined according to the size of the \(M_{m}\). With the AD algorithm, the bandwidth of the mass, stiffness, and damping matrices of the bridge was reduced from 2820 to 167, as shown in Figure 5.  
The two methods, which are of the general finite element software ANSYS (ANSYS Inc., Pittsburgh, PA, USA)adopting the Lancoz method [37] and the self-compiling program (SCP) adopting the subspace iteration method [38], are used in the numerical model to analyze the vibrational frequency and mode shapes of the bridge. The analysis results of the first six vibrational frequencies and mode shapes of the bridge are listed in Table 4. The maximum discrepancy of the frequency resulting from the two methods is only 0.3%. The first six modes of the bridge vibration are longitudinal displacement and lateral bending, and the minimum lateral bending frequency is 0.32 Hz, revealing that the lateral stiffness is smaller than the vertical stiffness. An in-situ train loading test was carried out to validate the finite element model [39] and the test data were used for model updating. In the process, the modeling parameters such as Young's modulus, mass density, and moments of inertia were modified using the sensitivity method [40] to make the correlation between the analytical and test results meet the practical requirements. The numerical and experimental results are compared in Figure 6. It can be seen that the numerical results are in reasonable agreement with the experimental results. The relative errors of the response values are less than 5% for the displacement and strain of the bridge. The good agreement between the measured and numerical results for the different bridge components and response types shows that the proposed \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Mode No.**} & \multicolumn{2}{c}{**Frequency (Hz)**} & \multirow{2}{*}{**Discrepancy**} & \multirow{2}{*}{**Vibration Mode**} & \multirow{2}{*}{**Mode Shape**} \\ \cline{2-2} \cline{4-5}  & **ANSYS** & & & & \\ \hline \multirow{2}{*}{1} & 0.317 & 0.318 & 0.3\% & \begin{tabular}{c} Transverse \\ deformation \\ \end{tabular} & & \\ \hline \multirow{2}{*}{2} & 0.388 & 0.389 & 0.3\% & \begin{tabular}{c} Longitudinal \\ deformation \\ \end{tabular} & & \\ \hline \multirow{2}{*}{3} & 0.419 & 0.420 & 0.1\% & \begin{tabular}{c} Transverse \\ deformation \\ \end{tabular} & & \\ \hline \multirow{2}{*}{4} & 0.515 & 0.516 & 0.1\% & \begin{tabular}{c} Longitudinal \\ deformation \\ \end{tabular} & & \\ \hline \multirow{2}{*}{5} & 0.537 & 0.537 & 0.0\% & \
\begin{tabular}{c} Longitudinal \\ deformation \\ \end{tabular} & & \\ \hline \multirow{2}{*}{6} & 0.609 & 0.610 & 0.2\% &
\begin{tabular}{c} Vertical \\ deformation \\ \end{tabular} & & \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The free frequency and mode shapes. Figure 5: The comparison of the bandwidth of the bridge characteristic matrix before and after optimization: (**a**) Before optimization; (**b**) After optimization. FE model of the vehicle-bridge coupled system works very well and has a high calculation precision.