# Structural Damage Identification Using a Modified Directional Bat Algorithm
## 4 Numerical Simulation
### Beam-Type Bridge Model  
The beam-type bridge model is shown in Figure 4, with a total of 10 elements and 11 nodes. The moment of inertia of the beam is \(I=7.617\times 10^{-7}\)m\({}^{4}\), the young's modulus of elasticity is \(E=2.1\times 10^{11}\) Pa, the mass density is \(\rho=7850\) kgm\({}^{-3}\) and the cross-sectional area is \(A=1.164\times 10^{-3}\) m\({}^{2}\). The first five natural frequencies and the corresponding modes of vertical degrees of freedom are used to identify the structural damage. The numerical simulation process is completed by matlab2020b software. The damage of the beam-type bridge model is shown in Figure 5. The stiffness of the second element and the seventh element are reduced by 20% and 25%, respectively. Figure 4: Beam-type bridge model. Figure 3: The flow chart of MDBA. The stiffness is identified by MDBA, PSO, DE and SFLA under the same population number and iteration times. The convergence graph and identification result without noise of different optimization algorithms are shown in Figure 6 and Table 1. It is clear that all the algorithms could identify the damage location. From the point of view of identification accuracy, MDBA had the highest identification accuracy, and its maximum relative error was only 0.05%. The maximum relative error of other algorithms was 1.36% (SFLA), 20% (PSO) and 10.95% (DE), respectively. The accuracy of SFLA was better than that of PSO and DE, but SFLA and PSO had small misjudgments in the undamaged area. As can be seen from Figure 6a, the MDBA algorithm had the fastest convergence speed of all the algorithms, and its number of iterations was only 10 when the objective function reached its optimal value. Though the SFLA algorithm shows better convergence than MDBA at the beginning, the iterations times with the optimal objective function was about twice that of MDBA. Therefore, the proposed MDBA had faster convergence speed and better identification accuracy than those of the other three algorithms in the case of small population and few iterations. In addition, when different swarm intelligence algorithms [42; 43; 45] were employed to identify the same structure of 10-element beam, a large number of populations and iterations were required. The total number of searches by the swarm intelligence algorithm depends on population and iterations, which reflects the efficiency of the algorithm [44]. In this example, MDBA showed high efficiency under the condition of a smaller population and fewer iterations. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Algorithm**} & \multicolumn{2}{c}{**Damage Severity (Relative Error)**} & \multicolumn{2}{c}{**Obvious Error Alarm**} \\ \cline{2-4}  & **Damage Element 2** & **Damage Element 7** & **Damage Severity @ Element** \\ \hline MDBA & 19.99\% (0.05\%) & 25.00\% (0\%) & none \\ SFLA & 20.16\% (0.80\%) & 24.66\% (1.36\%) & 5.15\%@5 \\ PSO & 20.08\% (0.40\%) & 30.00\% (20.00\%) & 3.00\%@1, 3.17\%@3 \\ DE & 17.81\% (10.95\%) & 23.53\% (5.88\%) & none \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The identification results of the damaged case by four different algorithms. Figure 5: Case of damage beam-type bridge model. Figure 6: (**a**) The convergence curves achieved by four different algorithms for objection function. (**b**) The identification results of the damaged case by four different algorithms. To verify that the MDBA algorithm has better superiority than other improved methods of BA, MDBA was compared with other bat variants algorithms (DBA, LBA, TMBA) in this simulation. The convergence graph and identification result without noise of different bat variants are shown in Figure 7 and Table 2. It is clear in Figure 7a that under the condition of the same small population and few iterations, MDBA had a faster convergence rate than other bat variants. All of the four algorithms could accurately find the damage location of structure, but the MDBA had the best identification accuracy in damage degree, and its maximum relative error was only 0.05%.