## 3 Establishment of the Finite Element Model and Simulation Results  
### Establishment of the Finite Element Model  
During long-term service, the flange fastening device undergoes a transition from tight contact to small-scale separation between the flange cover and the silicone gasket. During the separation process, the contact interface is prone to air medium intrusion, forming a Q235-air-silicone multi-medium combination. With the aid of COMSOL multiphysics simulation software, a two-dimensional frequency domain analysis model of the multilayer media under a liquid immersion environment is established, as shown in Figure 4. In the figure, \(\mathbf{k}\) is the incident wave vector, \(\theta\) is the angle between the acoustic field and the axis, and \(\mathbf{I}\) is the contact interface between each medium, where interface \(I_{a}\) and \(I_{d}\) are liquid-solid interfaces and interfaces \(I_{b}\) and \(I_{c}\) are solid medium contact interfaces. The calculation interface used in the simulation process is the acoustic-solid module, where the acoustic analysis is performed using the pressure acoustics module. The fluid domain in the model is solved by calculating the Helmholtz equation in the pressure acoustics module, and the solid domain is solved by calculating [PERSON]'s law in the solid mechanics module. Once the material parameters in the model are assigned, the material parameters in the subsequent mathematical analysis should be consistent with those in the model. The upper and lower ends of the solid model are set as water domains with a density of 1000 kg/m\({}^{3}\) and a longitudinal wave speed of 1490 m/s [18]. The upper water domain is set as a plane wave with an incident pressure of 1 Pa as the background pressure field for the sound wave incident source. Assuming that the wave vector of the incident plane wave is \(\mathbf{k}\), and it propagates along the incident angle \(\theta\) to the surface of the flange cover, the expression for wave vector \(\mathbf{k}\) is as follows [19; 20]:  
\[\mathbf{k}=k_{0}(\sin(\theta)-\cos(\theta)) \tag{7}\]  
\[k_{0}=\omega/c \tag{8}\]  
where \(\omega\) represents the angular frequency, \(\omega=2\pi f\). The mathematical definition of the wave vector \(\mathbf{k}\) of the incident plane wave is shown below. \[P_{in}(\mathbf{x})=1 Pa\cdot\exp(-i(\mathbf{k}\cdot\mathbf{x})) \tag{9}\]  
where \(\mathbf{x}=(x,y)\), and the wave vector and incident wave are defined as variables in this multi-medium structure model. Due to the complexity of the incident sound wave after reflection and transmission, the reflection coefficient \(R\) is plotted for different combinations of incident angle and frequency (model parameters). The composite ratio of the reflected wave and incident wave at the  
Figure 4: Schematic of the finite element model. multi-layer medium interface is the reflection coefficient, which can be obtained using the following formula [21]:  
\[R=\frac{P_{sc}}{P_{in}}=\frac{P_{tot}-P_{in}}{P_{in}} \tag{10}\]  
where \(P_{sc}\) is the scattered pressure field on the interface \(I_{a}\) in the water, \(P_{tot}\) is the transmitted pressure field, and \(P_{in}\) is the incident pressure field in the water region on the interface \(I_{a}\). By using the average value of the reflection coefficient on the multi-layered interface, and matching with different combinations of excitation frequency \(f\) and incident angle \(\theta\), the variation of the interface reflection coefficient with incident angle and excitation frequency can be calculated. This model reflects the variation in reflection and transmission coefficients of the flange cover and silicone gasket on the flange bolted joint from a state of tight contact to separation, in order to analyze the contact condition of flange fasteners in practical engineering applications.