## 5 Finite Element Simulation of Void Structure  
### Theory of Finite Element Models  
The internal interconnected voids of PAC and the damping of the material itself perform the functions of sound absorption and noise reduction. As a result, not only must the continuity equation for fluid momentum conservation (named the Navier-Stokes equation, N-S equation) be considered in this problem but also the structural dynamics equations. The following equations are available for a compressible Newtonian fluid [32]:  
\[\rho\bigg{(}\frac{\partial v}{\partial t}+v\cdot\
abla v\bigg{)}=-\
abla p+ \
abla\cdot\bigg{(}\mu\Big{(}\
abla v+(\
abla v)^{T}\Big{)}-\frac{2}{3}\mu( \
abla\cdot v)I\bigg{)}+F \tag{8}\]  
where \(v\) is the speed of sound; \(p\) is the sound pressure; \(\rho\) is the air density and \(\mu\) is the aerodynamic viscosity. From left to right, each part of the formula represents the inertial force, acoustic pressure, viscous force and external force. \[\frac{\partial p}{\partial t}+\
abla\cdot(\rho v)=0 \tag{9}\]  
Equations (17) and (18) should be solved simultaneously; they represent momentum conservation and mass conservation, respectively. By solving these two equations under specified boundary conditions, the fluid flow rate and pressure inside a given structure can be obtained. The analytical solution is often difficult to obtain accurately due to the complexity of the equations and the structure itself. As a result, solving the equations necessitates the simplification of the assumed conditions and the structure with the application of numerical methods [33]. When air is assumed to be a non-viscous compressible and homogeneous fluid, the N-S equation can be simplified to Equation (19), which means the acoustic fluctuation equation [34]. \[\frac{1}{v^{2}}\frac{\partial^{2}p}{\partial t^{2}}-\
abla^{2}p=0 \tag{10}\]  
In the finite element method, the discretized sound field fluctuation equations and structural dynamics equations are Equations (11) and (12). \[\Big{\{}M_{e}^{p}\Big{\}}\Big{\{}\ddot{p_{e}}\Big{\}}+\Big{\{}k_{e}^{p}\Big{\}} \big{\{}p_{e}\big{\}}+\rho_{0}[R_{e}]^{T}\{U\}=0 \tag{11}\]  
\[[M_{e}]\Big{\{}\ddot{U_{e}}\Big{\}}+[C_{e}]\Big{\{}\dot{U_{e}}\Big{\}}+[K_{e}] \{U_{e}\}=\{F_{e}\} \tag{12}\]  
The acoustic pressure term acting at the air-pavement interface is added to the right-hand side of the structural dynamic Equation (12) to couple the fluid to the solid and obtain Equation (13):  
\[[M_{e}]\Big{\{}\ddot{U_{e}}\Big{\}}+[C_{e}]\Big{\{}\dot{U_{e}}\Big{\}}+[K_{e}] \{U_{e}\}=\{F_{e}\}+\Big{\{}F_{e}^{pr}\Big{\}} \tag{13}\]  
We discretize the acoustic pressure vector \(\Big{\{}F_{e}^{pr}\Big{\}}\). \[\Big{\{}F_{e}^{pr}\Big{\}}=[R_{e}]\{p_{e}\} \tag{14}\]  
Then, Equation (13) can be simplified into equation (15). \[[M_{e}]\Big{\{}\dot{U_{e}}\Big{\}}+[C_{e}]\Big{\{}\dot{U_{e}}\Big{\}}+[K_{e}] \{U_{e}\}-[R_{e}]\{p_{e}\}=\{F_{e}\} \tag{15}\]  
Combining Equation (11) with Equation (15), the acoustic-solid coupling Equation (16) can be obtained as follows:\[ \begin{bmatrix}\begin{bmatrix}M_{e}\\ \end{bmatrix}&\begin{bmatrix}0\\ \end{bmatrix}\\ \begin{bmatrix}M^{g}\end{bmatrix}\end{bmatrix} \begin{\{} \begin{\}}\{\ddot{U}_{ \dot{e}}\\ \ddot{p}_{e}\end{\} }\end{bmatrix}\begin{\{}\ddot{U}_{\dot{e}}\\ \ddot{p}_{e}\end{\}}+ \begin{bmatrix}\begin{bmatrix}C_{e}\\ 0\\ \end{bmatrix}&\begin{bmatrix}0\\ \end{bmatrix}\\ \begin{bmatrix}\dot{U}_{e}\\ \ddot{p}_{e}\end{bmatrix}\end{bmatrix} \begin{\{}\dot{U}_{\dot{e}}\\ \ddot{p}_{e}\end{bmatrix}\end{bmatrix}\begin{\{}\dot{U}_{\dot{e}}\\ \ddot{p}_{e}\end{bmatrix}\end{bmatrix}\begin{\{}\dot{U}_{\dot{e}}\\ \dot{p}_{e}\end{\}}=\begin{\{}\dot{F}_{\dot{e}}\\ 0\end{\}} \tag{16}\]  
where \( \begin{bmatrix}M^{f_{s}}\end{bmatrix} =\rho_{0}[R_{e}]^{T}\) is the mass coupling matrix; \( \begin{bmatrix}K^{f_{s}}\end{bmatrix} =-[R_{e}]\) is the stiffness coupling matrix; \( \begin{bmatrix}M^{p}_{e}\end{bmatrix} \) is the fluid mass matrix; \( \begin{bmatrix}K^{p}_{e}\end{bmatrix} \) is the fluid stiffness matrix; \( \begin{bmatrix}M_{e}\end{bmatrix}