## 2 Materials and Methods
### Parameter Optimization by the Cuckoo Search Algorithm
This result indicates that sound absorption performance of the MC was insufficient, and it could not be improved by increasing the total thickness, which could also be judged from the result in Figure 3. Furthermore, sound absorption performance of the composite structure of MCPC was significantly improved by increasing the limited total thickness, as shown in Figure 3. It is interesting to note that the optimal length of the cavity for the composite structure of PC was 0, no matter what the limited total thickness was, as shown in Tables 3-5. It could also be observed that optimal average sound absorption coefficient of the porous metal rose along with increase of the thickness, which is consistent with the normal sound absorption principle of porous materials [8; 9]. \begin{table}
\begin{tabular}{l} \hline \hline
**Standard Cuckoo Search Algorithm** \\ \hline
1: Objective function \(f(x),x=(x_{1},x_{2},\ldots,x_{d})^{T}\); 2: Generate initial population of \(n\) host nests \(x_{i}(i=1,2,\ldots,n)\); 3: while (\(t\leq MaxGeneration\)) or (stop criterion) do 4: Get a cuckoo (say, \(i\)) randomly and generate a new solution by Levy flights; 5: Evaluate its quality/fitness \(F_{i}\); 6: Choose a nest among \(n\) (say, \(j\)) randomly; 7: if (\(F_{i}\geq F_{j}\)) then 8: replace \(j\) by the new solution; 9: end if 10: A fraction (\(P_{a}\)) of worse nests are abandoned and new ones are built at new locations; 11: Keep the best solutions (or nests with quality solutions); 12: Rand the solutions and find the current best; 13: end while 14: Postprocess results and visualization; \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Basic calculation process of the standard cuckoo search algorithm. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Parameters** & **PCMC** & **PC** & **MCPC** & **MC** & **PCMCPC** \\ \hline \(d\) (mm) & 0.15 & – & 0.17 & 0.16 & 0.16 \\ \(t\) (mm) & 0.10 & – & 0.10 & 0.10 & 0.10 \\ \(b\) (mm) & 1.00 & – & 1.00 & 1.00 & 1.00 \\ \(d_{b}\) (mm) & 21.02 & 30.00 & – & – & 21.13 \\ \(d_{f}\) (mm) & – & – & 12.10 & – & 8.69 \\ \(D_{a}\) (mm) & 8.88 & – & 0.00 & 12.38 & 0.00 \\ \(D_{b}\) (mm) & 0.00 & 0.00 & – & – & 0.00 \\ \(D_{f}\) (mm) & – & – & 0.00 & – & 0.07 \\ \hline Optimal average sound absorption coefficient & 98.15\% & 86.45\% & 89.99\% & 88.64\% & 98.37\% \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Summarized optimal parameter of composite structure with limited total thickness 30 mm. Detailed distributions of sound absorption coefficients of the five composite structures with the limited total thicknesses of 30 mm, 50 mm, and 100 mm are shown in the Figure 4a-c, respectively. It could be found that except the data for MC, the sound absorption performances of the other four composite structures were accordingly improved along with increase of the limited total thickness, which were consistent with the result shown in Figure 3. Meanwhile, it could be observed that undulation of the sound absorption coefficients was reduced along with increase of the limited total thickness. Especially when the limited total thickness reached 100 mm, minimum of the sound absorption coefficient in the (2000 Hz, 6000 Hz) for the investigated composite structures \begin{table}
\begin{tabular}{c c c c c c} \hline
**Parameters** & **PCMC** & **PC** & **MCPC** & **MC** & **PCMCPC** \\ \hline \(d\) (mm) & 0.15 & – & 1.80 & 0.16 & 0.16 \\ \(t\) (mm) & 0.10 & – & 0.10 & 0.10 & 0.10 \\ \(b\) (mm) & 1.00 & – & 1.90 & 1.00 & 1.00 \\ \(d_{b}\) (mm) & 26.17 & 50.00 & – & – & 26.21 \\ \(d_{f}\) (mm) & – & – & 49.90 & – & 10.49 \\ \(D_{a}\) (mm) & 10.77 & – & 0.00 & 12.38 & 0.00 \\ \(D_{b}\) (mm) & 0.00 & 0.00 & – & – & 0.00 \\ \(D_{f}\) (mm) & – & – & 0.00 & – & 0.00 \\ \hline Optimal average sound absorption coefficient & 99.09\% & 95.45\% & 95.14\% & 88.64\% & 99.21\% \\ \hline \end{tabular}
\end{table} 
Table 4: Summarized optimal parameter of composite structure with limited total thickness 50 mm.