## 5 Finite Element Simulation of Void Structure
### Construction of the Finite Element Model
The model of the measuring instrument was established and the transfer function method was used to obtain the SAC, in which the sizes of the instrument, the position and the spacing of the microphone were designed according to the actual instrument to visualize the difference between the simulation model and the measured sound absorption effect. The model is shown in Figure 10.  
The working principle is that the two microphones measure the incident and reflected sound pressures, which are used to calculate the transfer function. The SAC is calculated from the incident sound reflection factor determined by the transfer function. The calculation process is shown as follows. If the side of the sample in contact with the acoustic wave is the reference surface (\(x=0\)), then the distance of microphone 1 from the reference surface is \(x_{1}\) and the distance of microphone 2 from the reference surface is \(x_{2}\), as the Figure. 10 shown. The incident sound pressure \(p_{i}\) and the reflected sound pressure \(p_{R}\) can be expressed as:  
\[p_{i}=\dot{p}_{i}e^{ik_{0}x} \tag{17}\]  
\[p_{R}=\dot{p}_{R}e^{-jk_{0}x} \tag{18}\]  
where \(\dot{p}_{i}\) is the amplitude of \(P_{i}\) on the reference plane; \(\dot{p}_{R}\) is the amplitude of \(P_{R}\) on the reference plane; \(k_{0}=k_{0}^{\prime}-jk_{0}^{\prime\prime}\) is the complex wavenumber. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Materials**} & **Density** & **Modulus** & **Poisson’s** & **Thermal** & **Specific Heat** & **Coefficient of** & **Sound** \\  & **(kg/m\({}^{3}\))** & **(MPa)** & **Ratio** & **Conductivity** & **Capacity** & **Thermal** & **Cyclity (m/s)** \\  & & & & **(W/(m-K))** & **(k/(kg-K))** & **Expansion (1/K)** & **343** \\ \hline Air & 1.205 & 0.142 & - & 0.0257 & 1.005 & 3.43 \(\times 10^{-3}\) & 343 \\ PAC-13 & 2039 & 2900 & 0.3 & 0.8705 & 1200 & 1.60 \(\times 10^{-5}\) & 3500 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Material parameters involved in the simulation. Figure 10: Model of the impedance tube. \(x_{1}\): the distance of microphone 1 from the reference surface, \(x_{2}\): the distance of microphone 2 from the reference surface. The sound pressures \(p_{1}\) and \(p_{2}\) at the two microphone positions are:  
\[p_{1}=\hat{p}_{i}e^{ik_{0}x_{1}}+\hat{p_{R}}e^{-jk_{0}x_{1}} \tag{19}\]  
\[p_{2}=\hat{p}_{i}e^{jk_{0}x_{2}}+\hat{p_{R}}e^{-jk_{0}x_{2}} \tag{20}\]  
If the distance between microphones 1 and 2 is \(x_{1}-x_{2}=s\), then the incident and reflected wave transfer functions \(H_{1}\) and \(H_{R}\) are:  
\[H_{1}=\frac{p_{21}}{p_{11}}=e^{-jk_{0}s} \tag{21}\]  
\[H_{R}=\frac{p_{2R}}{p_{1R}}=e^{jk_{0}s} \tag{22}\]  
If the incident acoustic reflection factor is \(r\), then \(\hat{p_{R}}=r\hat{p_{i}}\), and the total sound field transfer function \(H_{12}\) can be derived by combining Equations (19) and (20) as follows:  
\[H_{12}=\frac{p_{2}}{p_{1}}=\frac{e^{jk_{0}x_{2}}+re^{-jk_{0}x_{2}}}{e^{jk_{0}x _{1}}+re^{-jk_{0}x_{1}}} \tag{23}\]  
Combining Equation (21) with Equation (22), we can obtain \(r\) as Equation (24):  
\[r=\frac{H_{12}-H_{1}}{H_{R}-H_{12}}e^{2 jk_{0}x_{1}} \tag{24}\]  
Therefore, the SAC \(\alpha\) can be calculated by \(r\), as shown in Equation (25):  
\[\alpha=1-\left|r^{2}\right|=1-r_{r}^{2}-r_{i}^{2} \tag{25}\]  
where \(r_{r}\) is the real part of the incident acoustic reflection factor and \(r_{i}\) is the imaginary part of the incident acoustic reflection factor. The pressure acoustic module and the structural mechanics module were added to the model after it was established; the acoustic-solid coupling interface for both is generated by the COMSOL automatically, after which the mesh can be divided. Notably, if the size of the divided cells is too large, then the final model calculation results will be distorted and will have difficulty converging [24]. An acoustic medium cell was used in this study, whose size is smaller than 1/6 of the wavelength corresponding to the calculated maximum frequency (1500 Hz).