# 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  
In this section, damage detection analysis is performed by considering different damage locations. It starts from damage case A (\(\alpha_{6}=30\%\)) at a 3% level of noise. The least-squares method is firstly used to solve the CMSE system of linear equations. Figure 3 shows a comparison of the resulting location and extent of damage by using the classical CMSE and RCMSE methods. It is observed from Figure 3a that the localization indicator ambiguously locates the actually damaged element at a 3% level of noise, and both beam elements Nos. 2 and 25 seem to be candidates, as they are affected by the damage in the same way. Moreover, the estimated damage level of element No. 6 is far smaller than the expected value of 30%, indicating that the CMSE method cannot correctly detect stiffness reduction by using all 60 equations together. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Damage Case**} & \multirow{2}{*}{**Location**} & \multirow{2}{*}{**Extent**} & \multicolumn{3}{c}{**Natural Frequencies (Hz)**} \\ \cline{3-5}  & & & **1 st** & **2 nd** & **3 rd** \\ \hline
**Baseline** & N/A & N/A & 13.878 & 22.257 & 23.832 \\
**A** & 6 & 30\% & 13.845 & 22.074 & 23.634 \\
**B** & 19 & 30\% & 13.877 & 21.996 & 23.740 \\
**C** & 28 & 30\% & 13.701 & 22.166 & 23.601 \\
**D** & 14, 17 & 30\%, 30\% & 13.837 & 21.946 & 23.761 \\
**E** & 28 & 20\% & 13.774 & 22.202 & 23.686 \\
**F** & 28 & 10\% & 13.832 & 22.232 & 23.763 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The simulated damage cases of the truss structure. For the RCMSE method, a sensitivity analysis of the CMSE residual function with respect to damage is first performed to discriminate and eliminate redundant CMSE equations. Figure 4 shows the corresponding sensitivity indices of all 60 CMSE equations corresponding to 60 MCs. It is an expected result that the CMSE equations associated with the third mode of the damaged structure have larger sensitivity overall. The fitness function given in Figure 5 suggests that 43 equations are an optimum number for identifying the damage, so the CMSE system is reconstructed by eliminating 17 equations that correspond to the first 17 largest sensitivity indices, and damage identification analysis is performed again. It is shown in Figure 3b that the actual damage to element No. 6 is located accurately by the RCMSE method, whereas the estimated damage level is a little larger than the expected value, indicating that RCMSE outperforms the classical CMSE method both in damage localization and quantification. Note that in Figure 5, only the fitness functions corresponding to 32 tested \(N_{m}\left(N_{e}\leq N_{m}\leq N_{q}\right)\) are presented. This is because there are \(N_{e}=29\) elements simulated for the 2-D truss structure, which also means at least 29 equations are required to ensure that the reduced CMSE system has a unique solution. Figure 4: The sensitivity index of damage case A versus MCs \(N_{i}=20\) and \(N_{j}=3\), at a 3% level of noise. Figure 5: Fitness function of damage case A at a 3% level of noise for the least-squares method. Figure 3: The damage identification results of damage case A at a 3% level of noise, with the least-squares solution: (**a**) CMSE method; (**b**) RCMSE method. It is seen from Figure 3 that the damage identification results obtained by the least-squares method are not sparse, i.e., massive false-positive alarms of damage caused by noise contamination are produced, masking the actually damage-induced alarm. Here, an iterative Tikhonov solution can be tried to address this problem. Figure 6 shows a comparison between the identified results of the classical CMSE and RCMSE methods. By comparing Figures 3 and 6, the obtained Tikhonov solutions are clearly sparse and the estimated extents of the actual damage are both closer to the expected value. This comparison illustrates that the ITR method can eliminate false-positive alarms of damage and is more suitable for detecting isolated damage to the structure.