# Robust Structural Damage Detection Using Analysis of the CMSE Residual's Sensitivity to Damage
## 4 Numerical Simulation
### Robustness Performance Investigation
#### 4.3.1 Effects of Damage Location
It is worth noting that when the equation subsets are eliminated from the original CMSE system, the corresponding coefficient matrix of the reduced CMSE system, determining the accuracy of solution to some extent, is simplified. Hence, one would expect to explain this improvement from an ill-conditioned problem viewpoint. The condition number of the coefficient matrix **C**, defined as  
Figure 4:3  
measures how ill-conditioned a CMSE system is, where **C\({}^{++}\)** is the Moore-Penrose inverse of **C\({}^{\ast}\)** because it is usually not a square matrix. Table 2 shows the number of equations and the corresponding condition number of the original and reduced CMSE systems. It can be seen that in all simulated cases, the condition number is reduced as some redundant equations are eliminated. Taking damage case A as an example, the condition number is reduced from 126.36 to 95.92 as 25 equations are eliminated. This might confirm that a reasonable reduction of the dimension of the CMSE system yields a better-conditioned system and a better damage identification. To further investigate the noise robustness of the proposed scheme, 1000 Monte Carlo simulations are performed by varying the level of noise from 0.5% to 5%. The damage identification probability [24] is used to measure the prediction accuracy, which is defined as  
Figure 4:4 \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Damage Case**} & \multicolumn{2}{c}{**Original System**} & \multicolumn{2}{c}{**Reduced System**} \\ \cline{2-5}  & \(\textbf{N}_{m}\) & \(\kappa\) & \(\textbf{N}_{m}\) & \(\kappa\) \\ \hline A & 60 & 126.36 & 35 & 95.92 \\ B & 60 & 126.79 & 40 & 28.23 \\ C & 60 & 128.45 & 54 & 22.31 \\ D & 60 & 123.86 & 45 & 55.15 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The number of equations (\(N_{m}\)) and the corresponding condition number (\(\kappa\)) of the original and reduced CMSE systems for damage cases A to D.  
Figure 14: Damage identification results of damage case D at a 3% level of noise, with an iterative Tikhonov solution: (**a**) CMSE method; (**b**) RCMSE method. where \(n_{s}=1000\) is the total number of Monte Carlo simulations for a given level of noise and \(n_{d}\) is the number of realizations in which the actual damage is detected. Figure 15 shows the obtained damage identification probability of damage case A to damage case C, in which at least two findings are observable. First, it is expected that the damage localization performance of the RCMSE method is better than that of the CMSE method at every level of noise, thereby confirming the effectiveness of eliminating redundant equations from the classical CMSE method. Second, it is observed that for the CMSE method, the damage identification probability of DB element No. 28 and HB element No. 6 is always greater than that of HB element No. 19 in the examined range of noise level. When using a vibration-based method, the detectability of damage depends highly upon the modal parameter changes that the damage caused. It is observed by recalling Table 1 that the overall change in the first three natural frequencies due to the damage of element No. 19 is the smallest among these three damage cases, which may account for the difference in the damage localization performance of the CMSE method for different damage locations. However, for the RCMSE method, the damage identification probability of DB element No. 28 comes last. It is realized by comparing Figures 4, 9 and 11 that the MCs associated with the third mode of the damaged structure, on the whole, have larger sensitivity of the residual function to damage. In other words, the CMSE equations constructed by the third mode of the damaged structure are likely classified as redundant and eliminated from damage detection analysis. It can also be observed from Table 1 that the damage of element No. 28 mainly changes the third mode rather than the others. The discrepancy of the employed MCs may explain the different performances of these two methods.