## 3 Hybrid FE-SEA Modeling Parameter Identification Algorithm  
### Deterministic Subsystem Response Derivation of Three-Coupled Systems  
Figure 2 is a schematic diagram of the three-coupled system. It consists of a deterministic subsystem and a stochastic subsystem. In the figure, components 1 and 2 represent the random subsystem. The middle grid represents the deterministic subsystem, the dashed line represents the random boundary, the thick solid line represents the deterministic boundary, and the line segments a and b represent the hybrid connection of the two. \(f_{ext}\) represents the force applied to the deterministic subsystem, and \(f_{in,i}\) represents the force applied to the random subsystem. In the model, the number of random subsystems is \(N=2\), and \(f_{rev}^{(k)}\) is the reverberation field force received by the random subsystem (\(k=2\)). \(f_{ext}\) represents the excitation force of the external load on the deterministic subsystem and the frequency of the force is \(\omega\), \(q\) represents the degree of freedom of response of the deterministic subsystem after excitation, and \(D_{tot}\) represents the dynamic stiffness matrix of the deterministic subsystem. Then,the motion equation of the deterministic subsystem of the three-coupled system is as follows [6; 7; 8]:  
\[D_{tot}q=f_{ext}+\sum\limits_{k=1}^{N}f_{rec}^{(k)}=f_{ext}+\left(f_{rec}^{(1)}+f _{rev}^{(2)}\right) \tag{26}\]  
In the three-coupled system, \(D_{tot}\) is the dynamic stiffness matrix of the deterministic subsystem. The superposition of the matrix \(D_{d}\) and the dynamic stiffness matrix (\(D_{dir}^{(k)}\)) of the other random subsystems on the coupling boundary can be expressed as:  
\[D_{tot}=D_{d}+\sum\limits_{k=1}^{N}D_{dir}^{(k)}=D_{d}+\left(D_{dir}^{(1)}+D_{ dir}^{(2)}\right) \tag{27}\]  
In general, \(D_{d}\) is obtained by the finite-element method and \(D_{dir}^{(k)}\) is obtained by the boundary-element method or the analytic method. The reverberation field force needs to be confirmed before solving the displacement response of the hybrid model. Therefore, the reverberation field force (\(\sum\limits_{k=1}^{N}f_{free}^{(k)}\)) can be obtained through the reciprocal relationship between the diffusion field and the reverberation field:  
\[\left\{ \begin{array}{l}E\left[f_{rev}^{(1)}f_{rev}^{(1)H}\right]=\frac{4E_{ 1}}{\omega\pi n_{1}}\text{Im}\Big{\{}D_{dir}^{(1)}\Big{\}}\\ E\left[f_{rev}^{(2)}f_{rev}^{(2)H}\right]=\frac{4E_{2}}{\omega\pi n_{2}}\text{ Im}\Big{\{}D_{dir}^{(2)}\Big{\}}\end{array} \right. \tag{28}\]  
where \(E[*]\) represents the population average of the system energy, \(n_{1}=1,2\) represents the modal density of the random subsystem (\(k=2\)), and \(n_{1}=1,2\) represents the energy response of the random subsystem (\(k=2\)). Therefore, a relationship is established between the deterministic subsystem and the random subsystem, which can be obtained from Equation (26):  
\[q=D_{tot}^{-1}\Bigg{(}f_{ext}+\sum\limits_{k=1}^{N}f_{rev}^{(k)}\Bigg{)}=D_{ tot}^{-1}\Big{(}f_{ext}+\left(f_{rev}^{(1)}+f_{rev}^{(2)}\right)\Big{)} \tag{29}\]  
where the output displacement \(q\) of the deterministic subsystem can be expressed in the form of a cross spectrum:  
\[ \begin{array}{l}E\left[qq^{H}\right]=D_{tot}^{-1}E\bigg{[}ff^{H}+\sum \limits_{k=1}^{N}f_{rev}^{(k)}f^{H}+f\sum\limits_{k=1}^{N}f_{rev}^{(k),H}+\sum \limits_{k=1}^{N}f_{rev}^{(k)}\sum\limits_{k=1}^{N}f_{rev}^{(k)-H}\bigg{]}=\\ D_{tot}^{-1}E\Big{[}ff^{H}+\left(f_{rev}^{(1)}f^{H}+f_{rev}^{(2)}f^{H}\right) +\left(f_{rev}^{(1),H}f+f_{rev}^{(2),H}f\right)+\left(f_{rev}^{(1)}+f_{rev}^{ (2)}\right)(f_{rev}^{(1)-H}+f_{rev}^{(2)-H})\Big{]}\end{array}  \tag{30}\]  
where \(*^{-H}\) represents the solution of the conjugate matrix after the transposition of the matrix and then the inverse of the conjugate matrix. According to the hypothesis of  
Figure 2: Diagram of coupling between subsystems. maximum entropy value, the interaction between the random subsystems is small and can be ignored. Then, Equation (30) can be rewritten as:  
\[