# 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 thus exclusively used to solve the CMSE system of linear equations hereafter. It is also noted from Figure 6 that the RCMSE method remains more accurate than the classical CMSE method both in damage localization and quantification, even in terms of an iterative Tikhonov solution. This improvement strongly benefits from the elimination of 25 redundant equations according to the fitness function given by Figure 7. It can also be seen from Figure 7 that the fitness value obtained by using the corresponding first 35 equations is much larger than that obtained using all 60 equations, which illustrates that the predicted damage state of the structure agrees better with the real state when some redundant equations are eliminated. For damage case B, Figure 8 shows a comparison between the damage identification results of the CMSE and RCMSE methods. The top panel of Figure 8 illustrates that the damage of element No. 19 can be located by using all equations, but its extent is overestimated. In comparison, the RCMSE method, as shown in the bottom panel of Figure 8, exhibits enhancement of damage quantification. Figure 6: Damage identification results of damage case A at a 3% level of noise, with an iterative Tikhonov solution: (**a**) CMSE method; (**b**) RCMSE method. Figure 7: Fitness function of damage case A at a 3% level of noise, for the ITR method. It should be noted that in this case only 40 equations are used to predict the preset damage location and extent of damage (see Figures 9 and 10), confirming that not all equations contribute to damage detection but complicate the CMSE system. Furthermore, by performing the analysis of CMSE equations' sensitivity to damage, numerous redundant equations are eliminated, thereby simplifying the CMSE system. For damage case C, the analysis of CMSE equations' sensitivity to damage indicates that 54 equations are most appropriate for solving the preset location and extent of damage (see Figures 11 and 12). Figure 8: Damage identification results of damage case B at a 3% level of noise, with an iterative Tikhonov solution: (**a**) CMSE method; (**b**) RCMSE method. Figure 10: Fitness function of damage case B at a 3% level of noise, for the ITR method. Figure 9: Sensitivity index of damage case B versus MCs \(N_{i}=20\) and \(N_{j}=3\), at a 3% level of noise. Following this, the CMSE method employing 60 equations and the RCMSE method employing 54 equations are used to identify the damage, respectively. Figure 13 shows that both methods can correctly locate the actual damage. However, two false-positive alarms of damage to elements Nos. 5 and 17 are produced when using the CMSE method. At the same time, both methods overestimate the preset extent of the damage. This reveals that the robustness of the CMSE method against noise is slightly improved by eliminating six redundant equations. Damage case D is a double-damage case, in which elements Nos. 14 and 17 both exhibit a 30% stiffness loss. For this case, 45 equations are suggested for identifying damage according to the sensitivity analysis (see Table 2). The sensitivity index and fitness function are not shown here because of space limitations. Figure 4:0 Sensitivity index of damage case C versus MCs \(N_{i}=20\) and \(N_{j}=3\), at a 3% level of noise. Figure 4:1 Fitness function of damage case C at a 3% level of noise, for the ITR method. Figure 4:2 Damage identification results of damage case C at a 3% level of noise, with an iterative Tikhonov solution: (**a**) CMSE method; (**b**) RCMSE method. Figure 14 provides a comparison between the damage identification results of the CMSE and RCMSE methods. It is observed that the CMSE method omits the actually damaged element No. 17 and the extent of damage of element No. 14 is clearly overestimated. The RCMSE method demonstrates good damage localization and quantification performances, while it slightly overestimates the extent of the preset damage. All results shown above demonstrate an improvement of damage identification by considering the sensitivity analysis scheme.