## 3 Numerical Validation  
### Finite Element Model  
In this study, a typical VBI system [29] was simulated to validate the proposed damage localization method. The VBI system consists of a simply supported beam and a moving sprung mass, which is a simplification of a vehicle. Physical properties of the bridge and the vehicle are listed in Tables 2 and 3, respectively.  
To simulate the VBI system, we employed the commercial software ABAQUS 2020 [36]. The beam was modeled using solid C3D8R elements. The dimensions of the solid elements used are 0.1 m \(\times\) 0.1 m \(\times\) 0.1 m, and the entire bridge structure is discretized into 160,000 such elements. The vehicle model includes a sprung mass, a wheel in contact with the top surface of the beam, and a linear spring element that connects the sprung mass to the wheel. Both the sprung mass and the wheel are modeled using solid C3D8R elements of the same size as those used for the beam structure. The interaction between the wheel and the beam is simulated using the penalty method, with \"Hard\" contact properties. For the contact simulation, no friction is specified between the wheel and the top surface of the bridge.  
The numerical calculation of the system's dynamic responses involves both a static analysis and a dynamic step. An implicit algorithm with the Hilber-Hughes-Taylor (HHT) method was employed in these steps. Initially, the vehicle was positioned at one end of the beam, and gravity was applied to the entire system. Subsequently, in the dynamic phase, the vehicle moved along the bridge at a constant speed. The time step for the simulation was set to 0.01 s. Given that the VBI system contains a manageable number of elements and the bridge structure is linear, even with the introduction of artificial local damage, the computation time on a standard personal computer is kept within 30 min. For more detailed information on the simulation parameters and methodology, one can refer to the existing study [37], where the results obtained using solid elements have been validated against theoretical analysis and field tests. It is acknowledged that computational efficiency  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Sprung Mass (kg)** & **600** \\ \hline Spring stiffness (N/m) & 500,000 \\ \hline Bouncing frequency (Hz) & 4.60 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Physical parameters of the simply supported beam [29].  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Mass per Meter (kg/m).** & **4800** \\ \hline EI (N/m2) & \(3.33\times 10^{9}\) \\ \hline First bending frequency (Hz) & 2.08 \\ \hline Second bending frequency (Hz) & 8.33 \\ \hline Length (m) & 25 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Parameters of the vehicle model [27].  
could potentially be enhanced by employing simpler elements, such as 2D beam elements, and reducing the total number of elements, once these finite element models have been validated. However, this topic is beyond the scope of the current study and is not discussed further herein.  
To simulate local damage, the material stiffness of a section of the beam is designed as half of the intact state. The damage location is from 8 m to 9 m (marked in Figure 4), while the material stiffness of the other sections is identical to the intact state. To validate the proposed method, we select five observation points (shown in Figure 4) whose displacements were measured.