# A Kriging Model Based Finite Element Model Updating Method for Damage Detection
## 3 Numerical Example  
The proposed model updating method is first verified through a simulated numerical example of a cantilever beam of rectangular cross-section divided into 9 elements, as shown in Figure 3. The properties of the beam are 0.9 m in length, 200 GPa in Young's modulus, 7670 kg/m\({}^{3}\) in mass density, and 6 mm and 50 mm in height and width respectively. Figure 2: The flow chart of the proposed model updating method. A total of 9 Euler-Bernoulli beam elements are used and each node has two degrees of freedom of transverse displacement and rotation. It is assumed that the damage of the structure results in stiffness reduction in element 2 by 20%, in element 4 by 25% and in element 5 by 10%. The natural frequencies of the undamaged model and the damage model are shown in Table 1.  
The stiffness coefficients of all elements are used as the updating parameters and their change interval is set to (0.7, 1). The number of frequency selected for FRF should be more than the number of the updating parameters to reduce the effect of noise and excellent frequency selection away from resonance and anti-resonance can effectively improve the reliability and convergence speed of the algorithm. However, to check the robustness of the algorithm, the frequency of the FRF is simply set from 20 Hz to 800 Hz at every 50 Hz. A total of 90 samples are obtained by LHS as the initial sample population and the Kriging model based model updating is carried out with the stopping criterion of \(\varepsilon_{1}\ =\ 10^{-10}\) in Equation (15) and criterion of \(\varepsilon_{2}\ =\ 10^{-4}\) in Equation (16). At the same time, the model updating based on GA directly is also implemented as a benchmark to compare the results of the two methods. These two methods are herein after referred to as Kriging method and GA method respectively. The GA used in this paper is the ga function of MATLAB 2017 (R2017a, The Math Works Inc., Natick, MA, USA, 2017) whose Function Tolerance value is set to \(10^{-9}\) as the stopping criterion. For details, please consult Matlab Help menu. In this example the Kriging model is constructed after 358 iterations. As the dimension of the parameter is high, only the stiffness coefficients of element 1 and 2 are taken as variables while the stiffness coefficients of the other elements are set to 1 to demonstrate the similarity between the response surface predicted by the Kriging model and the real response surface as shown in Figure 4. It can be seen that when there are many parameters to be updated, the prediction error of the objective function of the Kriging model is large in the location away from the minimum. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Mode no.** & **Undamaged Model** & **Damaged Model** \\ \hline
1 & 6.11 & 5.81 \\
2 & 38.29 & 36.87 \\
3 & 107.26 & 104.95 \\
5 & 210.41 & 203.89 \\
5 & 348.63 & 332.91 \\
6 & 522.95 & 506.21 \\
7 & 734.98 & 712.38 \\
8 & 985.66 & 952.78 \\
9 & 1261.5 & 1216.71 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Natural frequencies of the theoretical and damaged model of the cantilever (Unit: Hz). Figure 3: The cantilever beam used for numerical simulation. Furthermore, the complexity of the real response surface when all the parameters are variables can be envisaged through Figure 4b. Even building that 2-dimensional surface requires a considerable number of sample points. That is why the EGO algorithm is adopted to save the computational cost. The final updating results are shown in Figure 5. For the sake of comparison, a conventional way of using FRF in the Kriging based model updating is also applied, in which the amplitude of FRF is used to construct the objective function, which is referred to as the AF approach in the paper. As illustrated in Figure 5, the proposed Kriging method can effectively identify the stiffness reduction caused by damage in this numerical simulation experiment. A great improvement has been made by introducing the FDAC in the objective function as the AF approach contains obvious errors.