## 5 Crack Identification Method Based on the Kriging Surrogate Model  
Crack identification is a typical inverse problem. The main idea of the proposed method is to transform the inverse problem into an optimization problem, based on the surrogate model established between crack parameters \(\mathbf{x}\) and super-harmonic components \(\mathbf{Y}\) from two measurement points. Therefore, crack identification can be stated as:  
\[\left\{\begin{array}{c}\text{find}\ \mathbf{x}^{*}\\ \text{min}\ \|\mathbf{Y}_{*}(\mathbf{x}^{*})-\mathbf{Y}_{\text{Target}}\|\\ \text{subject to}\ \mathbf{I}\mathbf{b}<\mathbf{x}^{*}<\mathbf{u}\mathbf{b} \end{array}\right. \tag{17}\]  
where, \(\mathbf{Y}_{*}\) is the vector of predicted super-harmonic components from two measurement points at the new set of crack parameters \(\mathbf{x}^{*}\); \(\mathbf{Y}_{\text{Target}}\) is the vector of amplitudes of the measured super-harmonic components; and \(\mathbf{I}\mathbf{b}\) and \(\mathbf{u}\mathbf{b}\) are vectors of the lower and upper bounds for \(\mathbf{x}^{*}\).  
As the surrogate model established by the initial samples will not guarantee the accuracy in the entire parameter range, the surrogate model should be updated for the more accurate parameters identification. Therefore, to estimate the accuracy of the surrogate model and identified parameters the following two equations are used:  
\[C_{1}=1-\frac{\sqrt{\sum_{i}\left(Y_{*}^{i}(\mathbf{x}^{*})-Y_{\text{FE}}^{i} (\mathbf{x}^{*})\right)^{2}}}{\sqrt{\sum_{i}\left(Y_{\text{FE}}^{i}(\mathbf{x }^{*})\right)^{2}}} \tag{18}\]  
\[C_{2}=1-\frac{\sqrt{\sum_{i}\left(Y_{*}^{i}(\mathbf{x}^{*})-Y_{\text{Target}}^ {i}\right)^{2}}}{\sqrt{\sum_{i}\left(Y_{\text{Target}}^{i}\right)^{2}}} \tag{19}\]  
where, \(C_{1}\) is the first criterion to evaluate the accuracy of the current surrogate model; and \(C_{2}\) is the second criterion to evaluate the matching degree of the identification results by the current surrogate model. The two criterions are together used to decide whether the current surrogate model needs to be updated or not. \(Y_{\text{FE}}^{i}\) is the \(i\)th element of \(\mathbf{Y}_{\text{FE}}\) with crack parameters \(\mathbf{x}^{*}\), where \(\mathbf{Y}_{\text{FE}}\) is the super-harmonic features of crack parameters \(\mathbf{x}^{*}\)calculated by the finite element model (i.e., the beam model). When the surrogate model and identified parameters do not satisfy the pre-set criterions, the surrogate model should be updated by adding the identified optimal solution in the current iteration, until the criterions are satisfied. Figure 6 illustrates the flowchart of the identification process. It should be noted that the surrogate model is composed of the sub-surrogate models from points A and B, and during the identification process, surrogate models from points A and B are operating individually to predict the features of untried parameters, and the two models will be merged when evaluating the fitness and the criterions.  
To search for the optimal crack parameters based on the surrogate model, global optimization methods, such as GA and Particle Swarm Optimization (PSO), can be used. Compared with GA, PSO utilizes the real coding in optimization and has many merits, such as good robustness and fast convergence rate [60]. In order to enhance the efficiency of PSO, an improved approach based on multi-point adding is proposed. Its main idea is to predict possible solutions in current iteration that the final optimal solution will be nearby, then to add the possible solutions and the optimal solution in the current iteration into the initial swarms for next iteration. The possibility is measured by the distance between the possible solution and the optimal solution in the current iteration. The nearer the distance is, the larger the possibility is. All the added points include three points, the optimal solution, the nearest and the second nearest solutions in the current iteration.