# Improving Low-Frequency Panel Absorbers with Two-Dimensional Acoustic Black Hole
## 2 Numerical Modeling
### Meshing  
The numerical modeling of the DMA with the ABH optimized front panel was done with a Finite Element Analysis (FEA) using ANSYS Mechanical. The model was created using a multi-physics approach by combining acoustic and mechanical elements in a single model. For simulating the front panel behavior, quadrilateral 2D shell elements were used. The choice of thickness and panel material influences the flexural wave velocity, which determines the required resolution of the mesh. In the area of the ABH, a finer mesh is required due to the decreasing wavelengths. It is recommended to use at least ten shell elements per flexural wavelength for accurate results [32]. For a residual thickness of the ABH \(h_{\text{ABH}}=h_{0}\cdot\frac{1}{10}=0.3\) mm, the smallest flexural wavelength \(\lambda_{\text{flex}}\) of a plate is computed by Equation (4) from [33], with [PERSON]'s modulus \(E\), density \(\rho\), [PERSON]'s ratio \(\
u\) from Table 1 and the upper frequency limit \(\omega_{u}=2\pi f_{u}=2\pi\cdot 500\) Hz of the analysis.  
\[\lambda_{\text{flex}}=2\pi\sqrt[4]{\frac{Eh_{\text{res}}^{2}}{12\rho\omega_{u }^{2}(1-\
u^{2})}} \tag{4}\]  
The required length for the shell elements is calculated with Equation (5). The resulting shell element length of \(l_{\text{shell}}\) was rounded so that it is an integer divisor of the panel height \(L_{y}\).  
\[l_{\text{shell}}=8\,\text{mm} \leq\frac{\lambda_{\text{flex}}}{10}\approx 9.22\,\text{mm} \tag{5}\]  
For simulating the air back cavity, hexagonal 3D elements were used. Acoustic elements are chosen to simulate the stiffness and losses introduced by the enclosed air in the back cavity. It is required to have at least ten elements per wavelength for the chosen acoustic simulation with selected ANSYS element type FLUID220, cf. [32]. The maximum edge length \(l_{\text{solid}}\) of a solid acoustic element is calculated by Equation (6), with upper frequency limit \(f_{u}\), its wavelength \(\lambda_{f_{u}}=0.483\) m and the speed of sound c = 343 ms\({}^{-1}\).  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Material Property** & **HPL 3 mm** \\ \hline Young’s Modulus \(E/\text{GPa}\) & 10 \\ Poisson’s Ratio \(\
u\) & 0.4 \\ Density \(\rho/\text{kg}\,\text{m}^{-3}\) & 1470 \\ Loss Factor \(\zeta/\%\) & 2.54 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Material properties and geometry parameters used in ANSYS for modeling the HPL front plate of the DMA.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Material Property** & **Air** \\ \hline Temperature \(T/\text{K}\) & 295.15 \\ Density \(\rho/\text{kg}\,\text{m}^{-3}\) & 1.225 \\ Speed of Sound \(c/\text{m}\,\text{s}^{-1}\) & 346.250 \\ Viscosity \(\eta/\text{kg}\,\text{m}^{-1}\,\text{s}^{-1}\) & \(1.789\times 10^{-5}\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Properties of air used for modeling in the FE simulations.  
\[l_{\text{solid}}=5\cdot l_{\text{shell}}=40\,\text{mm}\leq\frac{\lambda_{f_{u}}}{10 }=\frac{c}{10f_{u}}=48.3\text{ mm} \tag{6}\]  
A multiple of five times the shell element size was chosen for the fluid element size, to ensure good interaction between the mechanical and acoustical domain. The depth was chosen to fit four elements inside the back cavity. This leads to hexagonal elements with an edge length of \(l_{\text{solid,$x$}}=40\) mm, \(l_{\text{solid,$y$}}=40\) mm and \(l_{\text{solid,$z$}}=30\) mm.