## 3 Case Study of a Damaged Simply Supported Beam  
A damaged simply supported beam is selected as an example to demonstrate the efficiency and accuracy of the proposed hybrid method. The structure is a 10 m-span beam, shown in Figure 3, with a rectangle section of 1 \(\times\) 1 m. The elastic module and density of material are 3.45 \(\times\) 1010 Pa and 2600 kg/m\({}^{3}\), respectively. The gravity acceleration is 9.806 m/s\({}^{2}\) and the damping ratio of beam is 0.05. BEAM 44 element is used to establish the FE model based on ANSYS (12.0, ANSYS Inc., Canonsburg, PA, USA) and the model consists of 11 nodes and 10 elements. To get the experimental data of mode frequencies, two damaged locations are assumed in this beam, of which the moment inertia of section in elements 3 and 8 reduce to 70% of original value, the aim of using the proposed method is yielding precise FE model which can reflect this damage. Ten parameters of ratio coefficients (real value/original value) of moment inertia of those 10 element sections in Figure 3 (for example, initial FE model with parameter vector of [1,1,1,1,1,1,1,1,1,1] and objective model with parameter vector of [1,1,0.7,1,1,1,1,0.7,1,1]) are selected as modified parameters. In this case study, both direct GMPSO and proposed hybrid method are implemented. Figure 2: Flowchart of proposed hybrid method for model updating. For direct GMPSO with a particle described as \(\mathbf{X}_{i}=[x_{i1},x_{i2},\cdot\cdot\cdot,x_{i10}]\), the lower and upper boundary of each parameter is assumed to be 0.6 and 1.1, respectively. The amount of particles \(N\) is 60, the inertia weight \(w\) is 0.55, the learning coefficients \(c_{1}\) and \(c_{2}\) are both 2, the probability of \(P_{\text{m}}\) is set as 0.1 and the maximum iteration number is 60. Objective function is chosen as the sum of residuals of frequencies of first two vertical bending mode and first two transverse bending mode between FE value and experimental value. The modified parameters shown in Table 1 are obtained and the frequency comparison of first 10 modes between the updated and experimental model are shown in Table 2. Finally, this optimization process takes nearly five hours. For the proposed hybrid method, the lower and upper boundary of each parameter is the same. Firstly, using LHS technique, 200 input samples of selected parameters are yielded. Secondly, after a series of FE analyses, output datasets, and mode frequencies of first two vertical bending modes and first two transverse bending modes, are obtained. This process needs a total time of approximately 400 s. Thirdly, the Kriging meta-model of each mode frequency represents the explicit expression between parameters and those frequencies which needs a total calculation time of 3140 s. Finally, GMPSO technique is introduced in the updating process with an objective function of the sum of the four frequencies residuals between explicit expression value and experimental value. 100 particles, 100 maximum iteration number, and other coefficients and objective functions chosen as the same as aforementioned are selected for the GMPSO algorithm. The final updated parameters shown in Table 1 are obtained after time of 3 s and the frequency comparison of first 10 modes between the updated model and experimental model is shown Table 2. The total model updating time of the proposed method is less than one hour, which is only 0.2 times of direct GMPSO. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Parameter** & **1** & **2** & **3** & **4** & **5** & **6** & **7** & **8** & **9** & **10** \\ \hline Experimental model & 1.000 & 1.000 & 0.700 & 1.000 & 1.000 & 1.000 & 1.000 & 0.700 & 1.000 & 1.000 \\ Direct GMPSO & 0.991 & 0.945 & 0.717 & 0.978 & 1.005 & 1.037 & 0.982 & 0.729 & 0.980 & 1.034 \\ Hybrid method & 1.025 & 1.016 & 0.735 & 0.932 & 0.932 & 1.078 & 1.008 & 0.668 & 0.968 & 0.993 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameter values of updated and experimental model with different methods.