# Structural Damage Identification Using a Modified Directional Bat Algorithm
## 4 Numerical Simulation
### Truss-Type Bridge Model  
To further verify the performance of MDBA in structural damage identification, a truss-type bridge model was used. The young's modulus of elasticity was \(E=2.1\times 10^{11}\) Pa, the mass density was \(\rho=7800\) kgm\({}^{-3}\) and the cross-sectional area was \(A=7.854\times 10^{-5}\) m\({}^{2}\), as shown in Figure 9. The first five natural frequencies and the corresponding modes of vertical degrees of freedom were used to identify the structural damage.  
Two kinds of damaged cases were set; case 1 was the complex damaged case with multiple damage of members; case 2 was the small damage cases with multiple damage of members, as shown in Figure 10a,b. The MDBA was verified by applying different degrees of noise to the modal information.  
This structure was also identified by several bat variants with the same population number and iterations times in the complex multiple damage cases. The convergence  
Figure 8: The identification results of the damaged case by MDBA under different noise conditions.  
Figure 10: **(a)** case 1: complex damage case with multiple damage of members. **(b)** case 2: small damage case with multiple damage of members.  
Figure 9: Truss-type bridge model.  
graph and identification result diagram of the objective function of different bat variants algorithms in damage case 1 are shown in Figure 11. The detailed results of identification are shown in Table 4.  
In the complex multi-damage cases without noise, the MDBA converged to the optimal value of the function when the number of iterations reached 25. The structural damage identification results of MDBA at damaged elements were 19.99% (element 6), 20.00% (element 7), 24.99% (element 14) and 14.99% (element 22), respectively, and the relative error was 0.07%-0%. Under the condition of a small population and few iterations, all four algorithms could accurately find the location of structural damage, but the MDBA had the best identification accuracy in damage degree and the best and fastest convergence. Furthermore, MDBA had no error alarm where the structure was not damaged. Therefore, MDBA was superior to the other three bat variants in convergence rate and recognition accuracy.  
Figure 11: (**a**) The convergence curves achieved by other variations of BA for the objection function. (**b**) The identification results of the damaged case by other variations of BA.  
To verify the anti-noise performance of the MDBA, the identification of two damage cases by MDBA at different noise levels were studied. The detailed results of identification are shown in Figure 12 and Table 5. It is clear that the relative error of the identification results also gradually increased with the increase of noise level, and there are two tiny misjudgments in element 11 and element 13 in case 2. In the two damage cases, the maximum relative error was only 6.53%. Especially in cases of minor damage in case 2, the maximum relative error was only 5.5%. Therefore, under the condition of a small population and few iterations, the MDBA not only showed accurate identification results for complex and multiple damage cases at different noise levels, but had a high sensitivity for a minor damage case and could accurately locate and quantify the structural damage.