## 3 Numerical Simulation
### Results of Numerical Simulation  
The purpose of this simulation was twofold: (1) By assessing multiple cases of structural damage, the study attempted to determine whether the system can successfully detect the location of damage. (2) Because the training database comprised cases of variable stiffness levels, the study attempted to detect the level of damage and then determine the sensitivity of the system. A regularized training function was selected to solve the overfitting problem. The applied parameters are listed in Table 3. The BPN achieved convergence when the momentum peaked or the sum squared error (SSE) remained unchanged. As presented in Figure 5, the SSE remained unchanged, and peak momentum was achieved in the training state. \begin{table}
\begin{tabular}{c c} \hline \hline
**Stiffness K** & **Health Index** \\ \hline
**K\({}^{\star}\)1 (Health)** & 1 \\ \hline
**K\({}^{\star}\)0.7** & 0.7 \\ \hline
**K\({}^{\star}\)0.35** & 0.35 \\ \hline
**K\({}^{\star}\)0** & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Relationship between stiffness loss and health index. Figure 4: Detailed structure of the artificial neural network (ANN). To realize the first purpose of the simulation, five different cases of damage were selected, which are outlined as follows: damage on the (1) first floor (case 1), (2) sixth floor (case 2), (3) third and fourth floors (case 3), (4) first floor to third floor (case 4), and (5) fourth floor to seventh floor (case 5). First, the cases of damage on a single floor, namely cases 1 and 2, are described. Figure 5: (**a**) The training state. (**b**) Regression distribution of the training, test. * Case 1: The HI evaluated on this floor was compared with the target value, as presented in Table 4. The evaluated HI on the damaged first floor was 0.026, which is almost equal to 0. The error between the evaluated HI and target index was 2.6%. As the global stiffness of the structure is decreased more seriously by the removal of the bracing in the lower floors than the other locations, a larger vibration is observed near these damaged lower floors. Therefore, the calculated entropy value and the corresponding HI may be influenced, and the impact between adjacent floors is amplified to cause higher error. Although the errors between the evaluated HI and target index on the second, third, and fourth floors were higher than that observed for the others, the overall diagnosis results indicate that the system can distinguish between healthy and damaged structures. As presented in Figure 6, the damage on the first floor can be easily identified. * Case 2: As presented in Table 5, the HI evaluated on the sixth floor was 0.014, which approached the target value. For other floors, slight errors were observed between the evaluated  
Second, the cases of damage on multiple floors, namely cases 3-5, are described. * Case 3: The system could clearly distinguish between healthy and damaged structures (Figure 8). The HI evaluated on the damaged floors was almost equal to zero, and the damage was identified successfully (Table 8). Among the healthy floors, all floors were identified successfully, except for the fifth floor. The reason for this is that during the calculation of RCMCSE, the damage of the fourth floor may have affected the HI evaluated for the fifth floor. While calculating the entropy of the fifth floor (high floor), the fourth floor (low floor) was needed to consider. * Case 4: A small error was observed between the HI evaluated for the damaged condition and the target value (Table 6). On the healthy floors, the evaluated HI approached the target HI. The overall diagnosis result is presented in Figure 9, demonstrating that the system could easily identify damage. * Case 5:  
Figure 6: Health index of damage on the first floor. Figure 7: Health index of damage on the sixth floor.