# A Perforated Plate with Stepwise Apertures for Low Frequency Sound Absorption
## 2 Models and Methods
### FEM Simulation  
In the thermo-viscous and pressure acoustics frequency domain interface of COMSOL Multi-physics, 3D finite element models are performed to simulate the acoustic behavior of the PPSA absorber under normal incidence. Considering viscosity and thermal loss, the acoustic field of single PPSA is built up in the thermo-viscous-acoustics, frequency domain interface, as shown in Figure 2a. Due to the stepped hole is a symmetric structure, to reduce the simulation time, 1/4 unit of the stepped hole is constructed. This numerical element model is composed of a stepped hole domain, two air domains and two perfectly matched layers (PML) at the ends. Moreover, the background pressure field is defined in an air domain, and PML acts as an infinite air field. The domains marked by the yellow and red line are the inlet and outlet surfaces of acoustic wave, respectively. In the simulation, the maximum frequency is chosen as 1000 Hz, the calculated frequency range is 100-500 Hz and the frequency interval is 5 Hz. For the stepped hole domain, the maximum element size of free tetrahedral meshes is set as the viscous boundary layer thickness size, and for the background sound field and air domain, the maximum element size of free tetrahedral meshes are equal to 1/3 of the small hole radius. In addition, the distribution of mesh elements for PML is at least six layer using the swept node. The mechanical and thermal boundary conditions for the sidewalls of the steppe hole are no slip and isothermal,respectively. For a stepped hole with \(d_{1}\) = 1.5 mm, \(d_{2}\) = 4 mm, \(t\) = 2 mm and hole spacing of 5 mm, the 3D finite model is meshed with a total of 72,269 tetrahedral meshes and the maximum element for the stepped hole domain is 0.07 mm.  
The transfer impedance \(Z_{trans}\) of the PPSA is defined as [6]:  
\[Z_{trans}=\frac{\Delta p}{\overline{u}} \tag{11}\]  
where \(\Delta p\) is the pressure drop between the inlet and outlet surfaces, and \(\overline{u}\) is the mean velocity in the stepped hole. Then, define the numerical value of \(Z_{trans}\) as a global variable using derived values.  
After that, a finite element model of the PPSA absorber is established in the pressure acoustics frequency domain interface. The upper PML acts as an infinite air domain, the sound incident is placed in background pressure field and the air cavity behind the PPSA is set as air domain, as shown in Figure 2b. The values of the transfer impedance \(Z_{trans}\) are then imported using the Interpolation function command in the Definitions toolbar, thus the acoustic impedance of the PPSA can be defined by the built-in impedance. Thus, the sound pressure reflection coefficient \(R\) is expressed as:  
\[R=\frac{p_{sc}}{p_{in}} \tag{12}\]  
where \(p_{sc}\) and \(p_{in}\) are the scattered and incident sound pressure, respectively. Finally, the sound absorption coefficient of the PPSA absorber can be obtained from Equation (10).  
Figure 3 plots the theoretical and simulated sound absorption of PPSA absorbers with different perforation ratio and cavity depth. The parameters for three PPSA absorbers are listed in Table 1, for PPSA1 and PPSA2 absorbers, the theoretical sound absorption coefficient curves are the same as that of the numerical simulation. In addition, for PPSA3 there is an offset of about 2 Hz between the theoretical and the numerical resonances. In addition, in Figure 3b, there exists a difference between the theoretical and simulated acoustic resistance at frequencies far from resonance, which may be further modified by model mesh refinement and considering the thermal effects of air in the calculations, but its acoustic reactance is almost identical. Overall, the finite element simulation is in agreement with the theoretical calculation.  
Figure 2: Finite element simulation. (**a**) Thermos-viscous model of PPSA; (**b**) Pressure acoustic model of PPSA absorber.