# Damage Identification Method Using Additional Virtual Mass Based on Damage Sparsity
## 2 Damage Identification Method Based on Additional Virtual Mass and Damage Sparsity
### Damage Identification Method Based on Sparsity
#### 2.2.1 Basic Theory and Regression Model  
1. Traditional regression model  
Based on the initial premise that structural damage is sparse, a new objective function (9) is derived to obtain a sparse solution of the objective function consistent with the actual damage condition. Equation (9) is derived by introducing the \(l_{p}\) norm of damage-factor variation as a regular constraint penalty term, which gradually approximates the solution of the actual damage structure. \[\min f(\mathbf{\mu})=\left\|\Delta\mathbf{\omega}-R\Delta\mathbf{\mu}\right\|_{2}^{2}+ \lambda\left\|\Delta\mathbf{\mu}\right\|_{p} \tag{9}\]  
In the damage identification process, \(\lambda\) is a regularization coefficient that limits sparse degree of the damage-factor variation \(\Delta\mathbf{\mu}\). When \(\lambda\) is high, the penalty degree of the objective function for frequency residual is significant, and the sparsity of the optimization results will be significant, resulting in deviation from the least-square solution of frequency residual. When \(\lambda\) is low, the fitting degree of the damage-factor variation \(l_{p}\) norm penalty term is minimal, and the result is close to the least-square solution of frequency residuals, but the sparsity of the solution will not be significant. Based on the different constraint norm, different optimization iteration methods can be adopted for the objective function. When the constraint term is the \(l_{1}\) norm, the objective function is the Lasso regression model, which means that the absolute value of the damage-factor variation is used as a constraint. It is easy to update and iterate \(\Delta\mu\) to zero, so the Lasso regression model can easily generate sparse solutions that conform to the sparse characteristics of structural damage. The Lasso regression model can be solved using the coordinate axis descent method or minimum angle regression method. Besides, when the constraint term is the \(l_{2}\) norm, the objective function is the ridge regression model. Each update of \(\Delta\mu\) is an overall change based on a particular proportion, which only reduces it, and hard changes to zero. Therefore, the ridge regression model shows a slight constraint on damage sparsity. The ridge regression model can be solved using the Tikhonov regularization method. However, regardless of using Lasso regression model or ridge regression model, the \(\lambda\) value will have a decisive influence on the final results. Exist engineering structures have large scale with much degree of freedom, which also means that the FEM model is complex. So, the sensitivity matrix \(R\) is difficult to calculate according to Equation (6). A non-parameter Gaussian kernel regression model is adopted. The predicted function between structural frequency \(\omega\) and the damage factor \(\mu\) is expressed as follows:  
\[\omega=\widetilde{\omega}(\mu)+\varepsilon \tag{10}\]  
This function is performed Taylor expansion at \(\mu_{0}\) to develop the local linear form of non-parameter regression, which is also consist with the above approximate linear relationship in Equation (8). \[\omega=\widetilde{\omega}(\mu_{0})+\sum_{p=1}^{n}\frac{\partial}{\partial\mu_ {p}}\widetilde{\omega}(\mu_{0})\big{(}\mu_{p}-\mu_{0,p}\big{)}=\widetilde{ \omega}(\mu_{0})+R(\mu-\mu_{0}) \tag{11}\]  
The \(\widetilde{\omega}(\mu_{0})\) and \(R\) are fitted from N groups known data (\(\mu_{i}\), \(\omega_{i}\)) by optimizing the local linear form non-parameter Gaussian kernel regression function as shown in Equation (12) [30]. \[