## 3 Hybrid FE-SEA Modeling Parameter Identification Algorithm
### Coupling Matrix Solution of the Multi-System Coupling Model  
The power-balance equation of the mixed FE-SEA system can be expressed as:  
\[\omega \begin{bmatrix}(\eta_{1}+\eta_{d,1}+\sum\limits_{k=2}^{N}\eta_{1k}^{(Z) })n_{1}&-\eta_{12}^{(Z)}n_{1}&\cdots&-\eta_{1N}^{(Z)}n_{1}\\ -\eta_{21}^{(Z)}n_{N}&(\eta_{2}+\eta_{d,2}+\sum\limits_{k=1,k\
eq 2}^{N}\eta_{2k }^{(Z)})n_{2}&\cdots&-\eta_{2N}^{(Z)}n_{2}\\ \vdots&\vdots&\ddots&\vdots\\ -\eta_{N1}^{(Z)}n_{N}&-\eta_{N2}^{(Z)}n_{N}&\cdots&(\eta_{N}+\eta_{d,N}+\sum \limits_{k=1}^{N-1}\eta_{Nk}^{(Z)})n_{N}\end{bmatrix} = \begin{bmatrix}\frac{E _{1}}{n_{1}}\\ \frac{E_{2}}{n_{2}}\\ \vdots\\ \frac{E_{N}}{n_{N}}\end{bmatrix} = \begin{bmatrix}\frac{p_{in,1}^{ext}+P_{in,1}} \\ \frac{p_{in,2}^{ext}+P_{in,2}}{p_{in,2}}\\ \vdots\\ \frac{E_{N}}{n_{N}}\end{bmatrix}  \tag{71}\]  
Let the external excitation of the deterministic subsystem be 0, that is, \(p_{in,j}^{ext}=0\). Exciting each random subsystem in turn, the power-balance equation can be obtained as follows:\[\omega \begin{bmatrix}(\eta_{1}+\eta_{d,1}+\sum\limits_{k=2}^{N}\eta_{1k}^{(Z )})n_{1}&-\eta_{12}^{(Z)}n_{1}&\cdots&-\eta_{1N}^{(Z)}n_{1}\\ -\eta_{21}^{(Z)}n_{N}&(\eta_{2}+\eta_{d,2}+\sum\limits_{k=1,k\
eq 2}^{N}\eta_{2k}^{(Z )})n_{2}&\cdots&-\eta_{2N}^{(Z)}n_{2}\\ \vdots&\vdots&\ddots&\vdots\\ -\eta_{N1}^{(Z)}n_{N}&-\eta_{N2}^{(Z)}n_{N}&\ldots&(\eta_{N}+\eta_{d,N}+\sum \limits_{k=1}^{N-1}\eta_{Nk}^{(Z)})n_{N}\end{bmatrix}\begin{bmatrix}\frac{F_{1}^ {(1)}}{n_{1}}&\frac{F_{1}^{(2)}}{n_{1}}&\cdots&\frac{F_{1}^{(N)}}{n_{1}}\\ \frac{F_{1}^{(1)}}{n_{2}}&\frac{F_{1}^{(2)}}{n_{2}}&\cdots&\frac{F_{1}^{(N)}}{ n_{2}}\\ \vdots&\vdots&\ddots&\vdots\\ \frac{F_{N}^{(1)}}{n_{N}}&\frac{F_{N}^{(2)}}{n_{N}}&\cdots&\frac{F_{N}^{(N)}}{ n_{N}}\end{bmatrix} \] \[= \begin{bmatrix}P_{in,1}&0&\ldots&0\\ 0&P_{in,2}&\ldots&0\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\ldots&P_{in,N}\end{bmatrix}  \tag{72}\]  
where \(\omega\) is the center frequency and \(N\) represents the number of random subsystems, \(\eta_{j}\) represents the internal loss factor of the random subsystem, and \(\eta_{d,j}\) represents the loss factor caused by the coupling between the deterministic subsystem and the reverberation sound field of the random subsystem j. \(\eta_{j}\) represents the combined coupling loss factor of two random subsystems. \(E_{k}^{(j)}\) is the response energy of the \(K\)th subsystem when the Jth subsystem has power input. \(n_{k}\) represents the modal density of the random subsystem \(K\). \(P_{in,j}\) is the input power of the Jth subsystem; then, the loss-factor matrix can be obtained by inverting the energy matrix:  
\[\eta=\frac{ \begin{bmatrix}P_{in,1}&0&\ldots&0\\ 0&P_{in,2}&\cdots&0\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\cdots&P_{in,N}\end{bmatrix}\begin{bmatrix}\frac{F_{1}^{(1)}}{n_{1}}& \frac{F_{1}^{(2)}}{n_{1}}&\ldots&\frac{F_{1}^{(N)}}{n_{1}}\\ \frac{F_{2}^{(1)}}{n_{2}}&\frac{F_{2}^{(2)}}{n_{2}}&\ldots&\frac{F_{2}^{(N)}}{ n_{2}}\\ \vdots&\vdots&\ddots&\vdots\\ \frac{F_{N}^{(1)}}{n_{N}}&\frac{F_{N}^{(2)}}{n_{N}}&\ldots&\frac{F_{N}^{(N)}}{ n_{N}}\end{bmatrix} } \tag{73}\]  
For the loss factor of the complex-structure system with multiple subsystems, the input power is normalized by the subsystem energy, and then the normalized modal energy can be obtained:  
\[e_{i}^{(j)}=\frac{\frac{E_{i}^{(j)}}{n_{i}}}{P_{in,j}} \tag{74}\]  
Therefore, Equation (72) is transformed into:  
\[\omega\eta \begin{bmatrix}e_{1}^{(1)}&e_{1}^{(2)}&\cdots&e_{1}^{(N)}\\ e_{2}^{(1)}&e_{2}^{(2)}&\cdots&e_{2}^{(N)}\\ \vdots&\vdots&\ddots&\vdots\\ e_{N}^{(1)}&e_{N}^{(2)}&\cdots&e_{N}^{(N)}\end{bmatrix} =I \tag{75}\]  
The loss-factor matrix of each subsystem can be obtained by inverting the normalized modal energy matrix:  
\[\eta=\frac{1}{\omega} \begin{bmatrix}e_{i}^{(j)}\end{bmatrix} ^{-1} \tag{76}\]  
The above algorithm designed to identify the coupling loss factor of the mixed FE-SEA model can be realized by MATLAB.