## 2 Methodology
### Response Surface-Based FE Model Updating Methodology
The unbiased estimate \(\mathbf{b}\) of the regression coefficient matrix \(\mathbf{B}\) can be obtained by the least square method:  
\[\mathbf{b}=\left(\mathbf{X}^{\prime}\mathbf{X}\right)^{-1}\mathbf{X}^{\prime} \mathbf{Y}, \tag{10}\]Finally, the response \(\hat{y}\) estimated by the response surface can be expressed as  
\[\hat{y}=b_{0}+\sum\limits_{i=1}^{n}b_{i}x_{i}+\sum\limits_{i=1}^{n}b_{ii}x_{i}^{2 }+\sum\limits_{i<j}b_{ij}x_{i}x_{j}, \tag{11}\]  
To ensure its reliability for future applications, the response surface model must be validated. This validation process involves two key aspects: assessing the model's fitting performance on design points and evaluating its predictive capability with unseen data. Several indices are used for this validation:  
\[R^{2}=\frac{\text{SS}_{M}}{SS_{T}}=1-\frac{\text{SS}_{E}}{SS_{T}}=1-\frac{ \sum\limits_{i=1}^{n}\left(\hat{y}_{i}-y_{i}\right)^{2}}{\sum\limits_{i=1}^{n }\left(y_{i}-\overline{y}\right)^{2}}, \tag{12}\]  
\[RMSE=\sqrt{\frac{1}{n}{\sum\limits_{i=1}^{n}\left(y_{i}-\hat{y}_{i}\right)^{2 }}}, \tag{13}\]  
\(R^{2}\) represents the interpretability of the model, with a range of [0, 1]. A value close to 1 indicates a high level of fit. A smaller RMSE indicates a smaller model error. The optimization problem in FE model updating can be expressed as minimizing the error between the field-measured results and the FE analysis results [39]:  
\[ \begin{split}\text{min }F(\mathbf{X})=F[FEM(\mathbf{P_{FEM}}, \mathbf{X})-EXP(\mathbf{P_{EXP}})]\\ s.t.\mathbf{X}\in[\mathbf{X}_{L},\mathbf{X}_{U}]\end{split} , \tag{14}\]  
where \(\mathbf{P_{FEM}}\) and \(\mathbf{P_{EXP}}\) represent the external excitations applied to the FE modal and the actual structure, respectively. \(FEM(\bullet)\) and \(EXP(\bullet)\) denote certain mappings of the FE model and the actual structure. \(F(\mathbf{X})\) represents the error function, and the feasible domain of the updating parameters \(\mathbf{X}\) is [\(\mathbf{X_{L}}\), \(\mathbf{X_{U}}\)]. FE model updating requires minimizing the error function. In this study, the first four modal frequencies are chosen as the response features: \(f=\left[f_{1}\,f_{2}\,f_{3}\,f_{4}\right]^{T}\). Each response feature is fitted with an independent response surface function \(\mathbf{RSM}(\mathbf{X})=\left[RSM_{1}(\mathbf{X})\,\,RSM_{2}(\mathbf{X})\,\,RSM _{3}(\mathbf{X})\,\,RSM_{4}(\mathbf{X})\right]^{\mathsf{T}}\). So, Equation (14) can be rewritten as  
\[ \begin{split}\text{min }F_{1}(\mathbf{X})=\text{min}& \left|RSM_{i}(\mathbf{X})-f_{i,E}\right|,\,i=1,2,3,4,\\ s.t.\mathbf{X}\in[\mathbf{X}_{L},\mathbf{X}_{U}]\end{split}  \tag{15}\]  
Equation (15) represents a multi-objective optimization problem, which can be transformed into a single-objective optimization problem:  
\[ \begin{split}\text{min }\sum\limits_{i=1}^{4}\frac{\left|RSM_{i}( \mathbf{X})-f_{i,E}\right|}{f_{i,E}},\,i=1,2,3,4,\\ s.t.\mathbf{X}\in[\mathbf{X}_{L},\mathbf{X}_{U}]\end{split} , \tag{16}\]  
The single-objective optimization problem under the constraint of Equation (16) is subsequently solved using PSO [46]. In PSO, \(N\) particles are randomly initialized in a D-dimensional parameter space. Each particle \(i\) has two attributes: a velocity \(V_{id}=\left(v_{i1},v_{i2},\dots,v_{iD}\right)\), representing the direction and magnitude of iteration, and a position \(X_{id}=\left(x_{i1},x_{i2},\dots,x_{iD}\right)\), representing a combination of updating parameters.