# A Kriging Model Based Finite Element Model Updating Method for Damage Detection
## 2 Kriging Based Model Updating Method
### Efficient Global Optimization  
The EGO algorithm was proposed by [PERSON], [PERSON] and [PERSON] [38] as an effective method using response surfaces for global optimization. In that method, a Kriging model was constructed between the design variables and the objective function of the optimization problem, and only the accuracy of the location where the minimum value might exist was of interest. By only adding sample points in these effective areas, the purpose of solving the optimization problem using only a small number of sample points is achieved. After constructing the Kriging model by the initial sample points, it is easy to get the prediction and its MSE of any point by Equations (7) and (8). The EGO algorithm treats the uncertainty in the prediction as a Gaussian probability density function with the predicted value as mean value and the MSE as variance, as shown in Figure 1b. When finding a new point to be added in the sample points, the algorithm does not simply select the point with the minimum value of the current Kriging model, but rather, selects the point with the maximum value of the expected improvement (\(EI\)) relative to the minimum value of the current responses at all the sample points. Based on the Gaussian probability density function, \(EI\) is calculated by:  
\[\begin{array}{l}I(\mathbf{x})=y_{\text{min}}-y(\mathbf{x})\\ EI(\mathbf{x})=E[I(\mathbf{  
where \(y_{\min}\) is the minimum response value of the sample points, \(I(\mathbf{x})\) is the improvement of the minimum value by adding a new point, \(\text{erf}(\cdot)\) is the cumulative distribution function, and \(EI(\mathbf{x})\) is the expectation of that improvement. Usually, the smaller the predicted value of the unknown point is, the larger the MSE is, and the larger the value of \(EI\) is, as shown in Figure 1a, in which the right axis is only for the expected improvement. The calculation of expected improvement at \(x=0.7\) is illustrated in Figure 1b. Only the red region is valid for \(EI\) calculation, which is controlled by \(y_{\min}\).  
Thus, the infill criterion of the EGO algorithm not only finds the minimum value of the Kriging model (local search), but also explores the region with high MSE (global search). After adding enough sample points, the Kriging model will be accurate in the region where the minimum of the objective function exits and the global optimization problem is solved efficiently. In this paper, the solution for the location of the maximum expectation is also done by genetic algorithm. And the stopping criterion is set as:  
\[\max(EI(\mathbf{x}))<\varepsilon_{1} \tag{15}\]  
where \(\varepsilon_{1}\) is a small positive number that controls the precision and the convergence speed of the algorithm. However, a shortcoming of the EGO algorithm in this study is that its precision of the local minimum value is not particularly high since it puts a certain amount of effort into the global search. Thus it can be considered a step to find the area in which the global minimum exists and a supplementary process is implemented which keeps adding the point with the minimum value of the Kriging model as a new sample point until the following criterion is reached:  
\[\|\mathbf{x}_{new}-\mathbf{x}_{\min}\|\leq\varepsilon_{2} \tag{16}\]  
where \(\|\cdot\|\) is the Euclidean distance between the new sample point \(\mathbf{x}_{new}\) and \(\mathbf{x}_{\min}\) whose output response is minimum among current sample points, and small positive number \(\varepsilon_{2}\) denotes the tolerance for termination of iteration. Based on all methods mentioned above in Section 2, the flow chart of the complete model updating procedure is presented in Figure 2, and can be described as follows:  
1. Generate initial sample points of the updating parameters by DOE. 2. Run the FE analysis program to calculate the objective function output vector of the sample points and construct the initial Kriging Model.