## 2 Basic Theory
### Parameter Identification  
Parameter identification methods are widely used in modeling problems [26; 27; 28; 29]. The key to solving parameter identification problems is optimization of the algorithm, the essence of which is an optimization problem to minimize the discrepancies between the measured and predicted parameters. The objective function and the constraint are defined as:  
\[\left\{\begin{array}{c}\text{Min }J(\mathbf{p})=\varepsilon^{T}\mathbf{W} \varepsilon=\parallel\mathbf{W}^{1/2}(\mathbf{z}^{m}-\mathbf{z}^{a}(\mathbf{ p}))\parallel_{2}^{2}\\ s.t.\ \ \mathbf{p}_{1}\leq\mathbf{p}\leq\mathbf{p}_{2}\end{array}\right. \tag{17}\]  
\(\mathbf{p}\) is a vector for identifying parameters. The expression of the objective function \(J(\mathbf{p})\) is the weighted and squared residual difference between the experimental and calculated modal parameters. The domain of the function \(J(\mathbf{p})\) is defined in a reasonable range of \(\mathbf{p}_{1}\leq\mathbf{p}\leq\mathbf{p}_{2}\). The minimum of the function is calculated in this domain. The \(\varepsilon\) is the residual difference of modal parameters, and \(\mathbf{z}^{m}\), and \(\mathbf{z}^{a}(\mathbf{p})\) are the modal parameters of the test and calculation, respectively. The weighted matrix \(\mathbf{W}\) is a diagonal matrix reflecting the relative weight of the residual difference of the modal parameters. Sensitivity analysis is used to solve the structural optimization and model updating problems with the iterative method. The \(j\)-th iteration can be described as follows:  
\[\mathbf{W}^{1/2}\Big{(}\mathbf{z}^{m}-\mathbf{z}_{j}^{a}\Big{)}=\mathbf{S}_{j }\Big{(}\mathbf{p}_{j+1}-\mathbf{p}_{j}\Big{)} \tag{18}\]  
\(\mathbf{S}_{j}\)= \(\mathbf{W}^{1/2}\partial\mathbf{z}_{j}/\partial\mathbf{p}_{j}\) represents the sensitivity matrix, which is the weighted Jacobian matrix of modal parameters. Using the numerical calculus, \(\mathbf{p}_{j+1}\), can be obtained by Equation (18). The modal parameters are taken as the objective function and the iterative optimization algorithm is used for the calculation so that the loss function is continuously reduced and the parameter \(\mathbf{p}\) converges. Finally, the precise material parameters of the contact surface can be obtained.  
Through optimization algorithm iteration, the parameters of thin layer elements can be converged and identified. The implementation procedure can be illustrated as follows:  
1. Initialize \(j=0\), construct an initial FE model of the bolt connection using the thin-layer element;
2. Select the elastic parameters \(\mathbf{p}\) through relative sensitivity analysis using the initial FE model;
3. Pair the experimental and numerical modal shapes using modal correlation analysis, calculating the modal assurance criterion (MAC) to achieve this;
4. Calculate the residual \(\mathbf{z}^{m}-\mathbf{z}_{j}^{a}\) between the experimental and numerical modal data, solve the iteration format Equation (18), and obtain the variation \(\Delta\mathbf{p}=\mathbf{p}_{j+1}-\mathbf{p}_{j}\);
5. If the variables \(\Delta\mathbf{p}\) are converged, go to step (6); otherwise set \(j=j+1\) and go to step (2), with the stop criterion \(\|\mathbf{p}_{j+1}-\mathbf{p}_{j}\|\leq 10^{-6}\);
6. The exact elastic parameters are obtained.