# A Novel Acoustic Ventilation Barrier Based on the Block-Shaped Space-Coiling Metamaterial
## 5 The Optimal Design of the Proposed Metamaterial  
This section provides the optimal design of the proposed metamaterial. The goal of this optimization is to maximize the noise reduction effect of the proposed acoustic barrier by maximizing the TL of the proposed metamaterial within a certain frequency range, \(F=[f_{i},f_{i}]\), where \(f_{i}\) and \(f_{i}\) are set to 500 and 1000 Hz, respectively, according to the measured urban rail transit wheel-rail noise spectrum. Three parameters are chosen for the optimization, namely \(L\), \(l\), and \(d_{3}\), while \(w\), \(d_{1}\), and \(d_{2}\) are considered constants. The detailed set-up of the parameters for this optimization is shown in Table 4.  
Figure 8: A comparison of the transmission losses obtained from the experimental test and numerical simulation.  
Figure 7: An experimental test on the transmission loss of the full-scale meta-unit sample. (**a**) Schematics of the experimental test; (**b**) the full-scale meta-unit sample; and (**c**) the BSWA F100 impedance tube.  
In order to obtain the optimal design for noise in a broad frequency range, the objective function, denoted by O, is defined as the averaged TL over \(F\), i.e.,  
\[\text{O}(L,l,d_{3})=\int_{f_{1}}^{f_{u}}\frac{TL(L,l,d_{3},f)}{f_{u}-f_{1}}\text {d}f \tag{4}\]  
where \(f\) is the frequency. Evidently, maximizing O yields the optimal design of the metamaterial.  
Considering that the objective function can be highly non-linear and the possibility of multiple local optimal solutions, a three-layer enumeration algorithm is adopted (shown in Figure 9), which consists of the following main steps: Step 1, set the initial values of O\({}_{\text{opt}}\); Step 2, enumerate each \(L\) in the range and execute Step 3; Step 3, enumerate each \(l\) in the range and execute Step 4; Step 4, enumerate each \(d_{3}\) in the range and execute Steps 5-7; Step 5, compute the TL for each \(f\) in \(F\); Step 6, compute O; and Step 7, update O\({}_{\text{opt}}\) if O \(<\) O\({}_{\text{opt}}\).  
The above optimization algorithm is realized through a Python script that calls COMSOL Multiphysics to compute the corresponding values of the TL. Figure 10 plots the simulation result of the objective function, O. The optimal parameters of the meta-unit were finally determined as \(L\) = 80 mm, \(l\) = 33 mm, and \(d_{3}\) = 9 mm, and the TL is shown in Figure 11. It can be seen from Figure 11 that, over the entire frequency range of interest, all the TLs are consistently greater than 11, and the average value of the TL in this range is 14 dB. Moreover, within the frequency range of 495-1063 Hz, the optimally designed metamaterial would provide a desirable noise reduction effect in which the corresponding TL is consistently over 10 dB. Compared to the metamaterial made of the benchmark meta-units, whose effective frequency range of noise reduction is 514-963 Hz (shown in Figure 4), the optimally designed metamaterial increased the width of the effective frequency range of  
\begin{table}
\begin{tabular}{c c c c c c c} \hline
**Design Parameter** & \(L\) & \(l\) & \(w\) & \(d_{1}\) & \(d_{2}\) & \(d_{3}\) \\ \hline Range & 70–100 mm & 0.42–0.50 \(L\) & 1 mm & 8 mm & 8 mm & 8–18 mm \\ \hline \end{tabular}
\end{table}
Table 4: The parameter set-up of the optimization.  
Figure 9: The flowchart of the optimization process.  
noise reduction by 26.5%. Such a result reveals that the above optimal design algorithm can substantially increase the noise reduction performance of the proposed metamaterial.