## 2 Hybrid Model
### Acoustic FEM Characterization Sub-Model  
As depicted in Figure 2, a certain fluid domain is described with multiple connecting pipes. In each pipe, pressure wave could be decomposed linearly in a natural way as a forward acoustic pressure wave \(p_{FW_{i}}^{*}\) and a backward acoustic pressure wave \(p_{BW_{i}}^{*}\) by means of acoustic theory. For the sake of convenience, positive direction denotes the orientation to configuration. As shown in this figure, we may interpret that \(p_{FW_{i}}^{*}\) takes information with regard to the flow toward configuration. Conversely, \(p_{BW_{i}}^{*}\) carries the resulting effect of the configuration on pressure wave propagation. It is summed up that \(p_{BW_{i}}^{*}\) can be expressed in a linear way as a function of \(p_{FW_{i}}^{*}\) with correlative transmission and reflection coefficients:  
\[\left\{ \begin{array}{l}p_{BW_{1}}^{*}=rc_{1}p_{FW_{1}}^{*}+tc_{21}p_{FW_{2}} ^{*}+\ldots+tc_{i1}p_{FW_{i}}^{*}+\ldots+tc_{n1}p_{FW_{n}}^{*}\\ p_{BW_{2}}^{*}=tc_{12}p_{FW_{1}}^{*}+rc_{2}p_{FW_{2}}^{*}+\ldots+tc_{i2}p_{FW_{ i}}^{*}+\ldots+tc_{n2}p_{FW_{n}}^{*}\\ \vdots\\ p_{BW_{i}}^{*}=tc_{1i}p_{FW_{1}}^{*}+tc_{2i}p_{FW_{2}}^{*}+\ldots+rc_{i1}p_{FW _{i}}^{*}+\ldots+tc_{ni}p_{FW_{n}}^{*}\\ \vdots\\ p_{BW_{n}}^{*}=tc_{1n}p_{FW_{1}}^{*}+tc_{2n}p_{FW_{2}}^{*}+\ldots+tc_{in}p_{FW_{ i}}^{*}+\ldots+rc_{n}p_{FW_{n}}^{*}\\ \end{array} \right. \tag{13}\]  
Here, \(rc_{i}=p_{BW_{i}}^{*}/p_{FW_{i}}^{*}\) and \(tc_{ij}=p_{BW_{i}}^{*}/p_{FW_{i}}^{*}\) are respectively the reflection and transmission coefficients; \(p_{FW_{i}}^{*}\) and \(p_{BW_{i}}^{*}\) are the resultant of acoustic pressure \(p_{i}^{*}\) and particle velocity \(u_{i}^{*}\):  
\[p_{FW_{i}}^{*}=\frac{p_{i}^{*}+Y_{i}u_{i}^{*}}{2},\quad p_{BW_{i}}^{*}=\frac{p _{i}^{*}-Y_{i}u_{i}^{*}}{2} \tag{14}\]  
where \(Y=\rho a\) is acoustic impedance. To obtain all the transmission and reflection coefficients, \(n\) FEM simulations are needed by imposing the incident pressure \(p_{FW_{i}}^{*}\) at one boundary and anechoic termination to other boundaries. At\(i\)-th simulation, the reflection coefficient \(rc_{i}\) and transmission coefficient \(tc_{ij}\) are produced. Equation (13) could be written more clearly in a matrix form:  
\[\left\{ \begin{array}{l}p_{BW_{1}}^{*}\\ p_{BW_{2}}^{*}\\ \vdots\\ p_{BW_{i}}^{*}\\ \vdots\\ p_{BW_{n}}^{*}\end{array} \right\}=\left[ \begin{array}{ccccc}rc_{1}&tc_{21}& \ldots&tc_{i1}&\ldots&tc_{n1}\\ tc_{12}&rc_{2}&\ldots&tc_{i2}&\ldots&tc_{n2}\\ \vdots&\vdots&&\vdots&&\vdots\\ tc_{1i}&tc_{2i}&\ldots&rc_{i}&\ldots&tc_{ni}\\ \vdots&\vdots&&\vdots&&\vdots\\ tc_{1n}&tc_{2n}&\ldots&tc_{in}&\ldots&rc_{n}\end{array} \right]\left\{ \begin{array} []{l}p_{FW_{1}}^{*}\\ p_{FW_{2}}^{*}\\ \vdots\\ p_{FW_{i}}^{*}\\ \vdots\\ p_{FW_{n}}^{*}\end{array} \right\} \tag{15}\]  
Acoustic FEM has been extensively employed in compressor field [12; 13]. The thereon based 3D Helmholtz equation is given by:  
\[\
abla^{2}p^{*}+\sigma p^{*}=-j\rho\omega q_{f} \tag{16}\]  
where \(\
abla^{2}=\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial y ^{2}}+\frac{\partial^{2}}{\partial z^{2}}\) is Laplace Operator; \(\sigma=\omega/a\) is wave number; \(j=\sqrt{-1}\). When calculating Equation (16), boundary conditions (i.e., acoustic pressure, particle velocity or acoustic impedance) in the fluid domain must be defined firstly. Subsequently, the fluid domain is discretized into finite elements to obtain the matrices of each element respect to acoustic stiffness, acoustic mass and acoustic damping. Finally, the direct-response frequency-analysis procedure can be employed to evaluate each nodal acoustic pressure by solving the total algebraic matrices determined by summarizing each element matrix.