## 3 Numerical Simulation Analysis
### Establishment and Validation of a Micro-Perforated Finite Element Model  
Directly simulating micro-perforations would drastically increase simulation time and computer memory requirements due to their small size. Therefore, an alternative approach is used to integrate the sound absorption properties of micro-perforations into the structure. This alternative approach uses the \"Internal Perforated Plate\" module in COMSOL to characterize the acoustic performance of a micro-perforated plate. The experimental parameters for the acoustic impedance of the composite micro-perforated plate are taken from the literature [25]. The following parameters are employed: the circular micro-perforated plate has a diameter of 100 mm, a thickness of 1 mm, a micro-perforation hole diameter of 0.8 mm, and a perforation ratio of 1.67%. Moreover, the Johnson-Champoux-Allard (JCA) model is utilized to characterize the acoustic properties of the porous material. The fibrous acoustic foam used as a porous material exhibits a porosity of 0.97, an air flow resistivity of 73,145.7 Pa\(\cdot\)s/m\({}^{2}\), and thermal and viscous characteristic lengths of 0.594 mm. Additionally, it has a tortuosity factor of 1.67. A micro-perforated plate is closely incorporated into the acoustic foam to create a composite structure, with a 10 mm air gap positioned behind it. Figure 8a displays the finite element computational model formulated using these parameters. From Figure 8b, it can be observed that when the solving frequency exceeds 1000 Hz, the accuracy of the finite element simulation method decreases due to the higher modal density of the composite structure in the high-frequency range. This leads to deviations between the simulation results and experimental data, which are also outside the research frequency range. However, overall, the simulation results are in good agreement with the experimental data. Therefore, the finite element method demonstrates the capability to accurately predict the acoustic performance of micro-perforated composite structures. Figure 7: Comparison of sound pressure level calculation results. The Influence of Micro-Perforated Shells on the Vibration and Acoustic Radiation of Micro-Perforated Sandwich Cylindrical Shells  
The Influence of a Sandwich Structure on the Vibration and Acoustic Radiation Performance of Cylindrical Shells  
A preliminary design for the micro-perforated ring-stiffened sandwich cylindrical shell, as shown in Figure 9a, has been proposed. Additionally, three comparative models have been designed to investigate the vibration and acoustic radiation performance of different sandwich structures in the free air field within the medium-to-low-frequency range (1-1000 Hz). As shown in Figure 9a, the micro-perforated ring-stiffened sandwich cylindrical shell consists of upper and lower aluminum thin cover plates, a micro-perforated inner shell, an outer shell, and four ring stiffeners. The geometric parameters of the structure are as follows: the length of the cylindrical shell is \(l=720\) mm, the inner shell radius is \(R_{1}=250\) mm, the outer shell radius is \(R_{2}=274.7\) mm, and the thickness of both the inner and outer shells is \(h_{1}=2.7\) mm; the intermediate ring stiffener has a thickness of \(h_{r}=1.6\) mm and the spacing is \(h_{dis}=144\) mm; the micro-perforation hole diameter is 0.1 mm and the perforation ratio is 0.1%. Additionally, both ends of the sandwich cylindrical shell are sealed with cover plates, where the cover plate radius is \(R_{a}=350\) mm and the thickness is \(h_{p}=2\) mm. The structural material parameters are as follows: the inner and outer shells, upper and lower cover plates, as well as the annular ribs are all made of aluminum. Simultaneously, the material for the inner and outer acoustic fields is air. The material parameters for aluminum are listed in Table 2, while the material parameters for air are found in Table 1.  
Figure 8: Verification graph of a micro-perforated finite element model.