# Damage Identification Method Using Additional Virtual Mass Based on Damage Sparsity
## 4 Numerical Simulation of Simply Supported Beam and Space Truss
### Simply Supported Beam Model
#### 4.1.3 Damage Identification  
The assumed damage factors of substructures 3, 5, and 8 were 70%, 80%, and 60%, respectively. The OMP method, Lasso regression model with the \(l_{1}\) norm, and ridge regression model with the \(l_{2}\) norm, which are three traditional damage identification methods based on sparsity, and the proposed IOMP method, were combined with the additional virtual quality method to identify each damage substructure and determine the degree of damage. As shown in Figure 4, the damage recognition results for the objective function without sparse constraints showed damage to substructures 3, 5, 7, 8, and 9, indicating inconsistency with the actual local damage. When the regularization coefficient was 0.1 [28], the Lasso regression model with the \(l_{1}\) norm and the ridge regression model with the \(l_{2}\) norm could locate the damage positions more accurately. Under the \(l_{1}\) norm constraint, the damage factors to substructures 3, 5, and 8 were 69.3%, 79.4%, and 62.2%, respectively, while substructure 7 is misjudged with 96.9% damage. Under the \(l_{2}\) norm constraint, 70.4%, 79.7%, and 60.9% damage factors were identified in substructures 3, 5, and 8, respectively, while substructures 6 and 7 are misjudged as 99.6% and 98.1% damage, respectively. Both methods showed damage misjudgment, which directly resulted in larger deviations in identifying the degree of actual adjacent damage to the substructures. Nevertheless, the results significantly improved compared to the results for cases without constraint. Figure 3: Frequency-domain responses of original and virtual structures. Figure 2: Relative sensitivity of first fourth-order frequencies with virtual mass in substructure 5. The damage identification results for the OMP method were relatively ideal from Figure 5. The damage extent identification of damaged substructures 3, 5, and 8 were 77.5%, 79.4%, and 55.6%, respectively, reflecting the actual damage conditions, but the damage to substructure 2 was misjudged. The corresponding identification values, 78.4%, 79.2%, and 54.9%, obtained using the IOMP method based on the residual variance criterion were relatively accurate, and no misjudgment of the undamaged substructures was observed. As shown in Figure 6, the OMP method misjudged the damage for substructures 6, and there was a significant difference in the identification between damage factors of actually damaged substructures and that of the IOMP method based on the sensitivity correlation criterion. The IOMP method based on the sensitivity correlation criterion showed 72.3%, 80.1%, and 59.0% damage factors recognition for substructures 3, 5, and 8, respectively. The identification accuracy satisfied the requirements, and no misjudgment of the undamaged substructures was observed. Figure 4: Identification results of Lasso regression model with \(l_{1}\) norm and ridge regression model with \(l_{2}\) norm. Figure 5: Identification results obtained using OMP and IOMP method (residual variance criterion). As shown in Figure 7, when the regression model is non-parameter Gaussian kernel regression, the IOMP method is more accurate than OMP method. Because the non-parameter regression model is approximate, its accuracy is worse than the FEM model. However, the damaged substructures-selected process of the IOMP method has stronger integrality. Both the OMP and IOMP methods determined the location and number of damaged substructures, and it was assumed that the remaining substructures were undamaged. The damage identification results indicated significant sparseness, consistent with the local damage conditions of actual structures. The residual vector \(\epsilon_{k}(\epsilon_{0,k})\) norms were compared to establish the superiority of the two methods, where \(k\) is the number of damaged substructures. Under multiple damage conditions, the residual vector norm was large when the number of identified damaged substructures was one. The curve in Figure 8 starts when the number of damaged substructures is two to improve the image contrast.