# Improving Low-Frequency Panel Absorbers with Two-Dimensional Acoustic Black Hole
## 3 Simulation Results  
A parametric study on ABH design was performed to show and understand the effect of ABH variations on the panel behaviour. Therefore, different positions, sizes and depths were modeled and analyzed. The results are evaluated and the most promising design option was selected for manufacturing and validating the numerical analysis. The aim of this study is to improve the diffuse sound absorption \(\alpha_{S}\) of the DMA. The diffuse sound absorption \(\alpha_{S}\) is measured according to the standard DIN EN 20354 [34] inside a reverberant chamber. Due to the statistical nature of this method, various influences e.g., edge diffraction, non-diffuseness and the Sabine formulation can lead to values of \(\alpha_{S}>1\)[2]. To set a benchmark for the sound absorption results, a target curve is developed in the following. From various studies e.g., [4] it is known that porous and Helmholtz absorbers are sufficient for high-frequency sound absorption. For targeting specific low-frequency absorption, the classical panel absorbers and Helmholtz resonators are an established option. These systems are widely commercially available. The corresponding absorption coefficients are shown in Figure 3 for one example per category. However, broad-band low-frequency absorption is rarely found among commercially available absorbers. The goal of this study is to improve the sound absorption in the gap from 50 Hz \(<f<\) 250 Hz between the widely available absorber types. The desired frequency-dependent absorption coefficient is displayed in Figure 3 as the DMA target (passive). From the target curve for the sound absorption, some parameters for the mechanical FE simulation of the DMA can be derived. The highest frequency to be analyzed was set to be \(f_{u}=500\) Hz, which determines the FE size of the model. The identification of sound absorption takes place in the logarithmic frequency domain [34]. To save calculation time in the low frequencies, the numerical FE analysis was performed in the linear frequency domain with frequency steps of \(\Delta f=5\) Hz. This is possible because the behavior of the DMA panel at low frequencies is determined by the first mode, where no significant change is to be expected due to the ABH. [PERSON] [35] showed that the absorption behaviour of a vibrating panel in front of an enclosed air volume is dependent on incident, reflected and scattered sound pressure as well as the spatial distribution of the panel velocity. Therefore, it was decided to evaluate the simulation results via the frequency response of the arithmetic mean velocity \(\overline{v}_{x,y}(f)\) for the \(z\)-component of the whole vibrating surface. The calculation is done via Equation (7),  
Figure 3: Comparison of the diffuse sound absorption \(\alpha_{S}\) of different absorber types using examples available on the market including DMA target absorption curve. where \(n\) is the total number of elements on the surface and \(|\vec{v_{z,i}}|(f)\) is the amplitude of the element velocity in the \(z\)-direction. \[\overline{v}_{x,y}(f)=\frac{1}{n}\sum_{i=1}^{n}|\vec{v_{z,i}}|(f) \tag{7}\]  
The resulting graphs are evaluated by comparing the \(\overline{v}_{x,y}(f)\) of the panels with the embedded ABH to the reference panel. If the mean velocity of the ABH panel exceeds the mean velocity of the reference panel at some frequencies, the sound absorption is expected to be increased at the same frequencies. This way, the target curve shown in Figure 3 defines the frequencies of interest in evaluating the different simulations. Additionally, an equal spacing on the logarithmic frequency axis of the modal peaks is considered best. The ABH geometry is dependent on four main parameters. Formulas (1)-(3) presented in the previous chapter show the relationships between the parameters. While the panel thickness \(h_{0}\) is predestined by the material choice for the front plate, the residual height \(h_{\text{res}}\), radius \(a\) and power-law factor \(m\) are free to be chosen by the designer.