## 5 Design of the Low-Frequency Broadband High Sound-Absorbing Structure
### Optimization Objectives Are Constrained  
When the optimization objective is not constrained, the optimization direction is diverse, as demonstrated in the previous section. The final Pareto optimal solution will present a variety of types. However, fixed indicators are frequently employed in the actual design of acoustic structures. Consequently, based on the optimization model that is presented in the preceding subsection, the following objective constraints are added in this subsection:  
\[\left\{ \begin{array}{l}\frac{1}{3}min(f_{3}+f_{4}+f_{5})\geq 0.6\\ \frac{1}{5}min(f_{6}+\ldots+f_{10})\geq 0.9\end{array} \right. \tag{22}\]  
The meaning of Formula (22) is that the average absorption coefficient for the frequency range of 700-1500 Hz is greater than 0.6, and the average absorption coefficient for the frequency range of 1600-3000 Hz is greater than 0.9. The NSGA-III algorithm's primary parameters are set in the same manner as they are in the preceding section, and the optimization outcomes are depicted in Figure 13. With a constrained optimization objective, the optimization outcomes are extremely stable, and all of the results satisfy the objective constraints. In addition, the algorithm performs diversity attempts after reaching the predetermined constraint target, and the average absorption coefficient from 700 Hz to 1500 Hz varies between 0.6 and 0.65. The average absorption coefficient between 1600 Hz and 3000 Hz varies between 0.94 and 0.9. Figure 12: Optimized results (optimization goals are unconstrained). Figure 13: Optimized results (optimization goals are constrained). In addition, a subset of the optimization results is analyzed using the parameters shown in Table 5, and the model depicted in Figure 14.  
According to [PERSON] et al. [36], this cavity absorption mechanism is primarily composed of resonance energy loss, scattering, and waveform conversion. The low-frequency absorption is primarily determined by resonance energy loss from the three concepts. The absorption peak of Unit-cell model one is 1 kHz and the absorption coefficient is 0.993, whereas the absorption peak of Model two is 1.9 kHz, and the absorption coefficient peaks at 0.969. Figure 15 illustrates the vibrational velocity of the model [25], and the vibrational velocity \(\mathbf{v}_{tt}\) (23) is derived from the structural displacement \(\mathbf{u}_{tt}\) in Equation (4). \[\mathbf{v}_{tt}=\frac{d\mathbf{u}_{tt}}{dt} \tag{23}\]  
Figure 14: Optimized structure (one of them). \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Parameter Name** & **Structure 1** & **Structure 2** & **Structure 3** & **Structure 4** \\ \hline \(r_{1}\) (mm) & 9.39 & 7.69 & 16.83 & 16.12 \\ \(r_{2}\) (mm) & 15.33 & 12.67 & 8.46 & 5.10 \\ \(r_{3}\) (mm) & 16.46 & 12.82 & 8.95 & 10.08 \\ \(h_{1}\) (mm) & 11.49 & 7.06 & 1.00 & 1.38 \\ \(h_{1}\) (mm) & 12.89 & 11.93 & 4.79 & 3.50 \\ \(h_{1}\) (mm) & 10.74 & 12.05 & 4.67 & 3.95 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Optimized parameters (one of them). Figure 15: Vibrational velocity diagram at the maximum absorption coefficient, (**a**) Unit-cell model 1, 1000 Hz; (**b**) Unit-cell model 2, 1900 Hz. In comparing the vibrational velocities of Unit-cell models one and two at the maximum absorption coefficient, Model two has resonance energy loss, waveform conversion, and scattering, in addition to resonance energy loss compared to Model one. Shear waves are more likely to dissipate in solid media than in longitudinal waves. The cavity in Model two of the unit-cell transforms the incident longitudinal waves into shear waves and increases the propagation distance, resulting in a wider absorption peak width in Model two.