## 3 Establishment of the Finite Element Model and Simulation Results
### Establishment of the Finite Element Model
When the flange fastening device separates from the contact interface, a finite element simulation model can be established with different thicknesses of air layer media (0 \(\upmu\)m, 30 \(\upmu\)m, 50 \(\upmu\)m, 70 \(\upmu\)m, 90 \(\upmu\)m, 110 \(\upmu\)m) on interface \(I_{b}\). Different characteristic impedances are used to define the media on both sides of the interface, and the characteristic impedance of a material is generally characterized by its density and sound speed. To eliminate the multiple reflection and refraction effects of acoustic waves at the boundaries of the finite element model and reduce their impact on the calculation results, perfect matching layers (PML) are set at the top and bottom boundaries of the model. The thickness of the PML in the fluid domain is set to 1/3 wavelengths to completely absorb the reflected and transmitted acoustic fields from interface \(I_{a}\) and \(I_{d}\)[22]. To reduce the influence of the reflected and refracted acoustic waves at the left and right boundaries on the post-processing results, Floquet periodic boundary conditions are added to the left and right boundaries of the model to simulate the infinite extension of the calculation model in the horizontal direction [23]. This condition greatly reduces the simulation calculation workload without sacrificing simulation accuracy [24]. To obtain accurate numerical solutions of the model at all frequencies under study, the geometry of the periodic unit cell was set to depend on the wavelength. This means that the geometry was frequency-dependent, which is not allowed in the frequency-domain solver of COMSOL Multiphysics. To avoid this issue, a parametric scan was added, where the scan parameter was the frequency, in order to obtain the frequency and angle spectra of the acoustic reflection coefficient. During the process of finite element mesh generation, it is necessary to partition the mesh according to the sound speed of different media and the contact position between the media. Therefore, a reasonable refinement of the mesh is needed at each contact interface. Due to the introduction of periodic boundary conditions, the same mesh partitioning method is required at the symmetric boundary. In finite element simulation calculations, the cell partitioning is based on maximizing the computational efficiency while ensuring the calculation accuracy. The PML was set to have a thickness direction \"mapping\" distribution with 20 layers of elements. Considering the regular shape of the model, no stress changes, and the contact between solids, a free triangular mesh was chosen. According to the Courant-Friedrichs-Lewy (CFL) condition [25], the time step in finite element analysis must be smaller than a certain value, otherwise incorrect calculation results will be obtained. Therefore, the calculation step size of the incident sound wave must be smaller than the time it takes for the sound wave to propagate to adjacent meshes. As the mesh size decreases, the time step also decreases. The Courant-Friedrichs-Lewy condition in two-dimensional space is shown as follows. \[C=\Delta t\sum_{i=1}^{n}\frac{u_{xi}}{\Delta x_{i}}\leq C_{\text{max}} \tag{11}\]  
where \(C\) is the CFL number, which is a dimensionless quantity. \(C_{\text{max}}\) is the maximum CFL value, which indicates that the step displacement of the sound wave in unit time cannot exceed the unit length of the computational mesh. Generally, \(C_{\max}=1\). \(\Delta t\) is the time step, \(\Delta x\) is the mesh length, and \(u\) is the speed of sound wave. It should be considered that in finite element calculations, computational efficiency needs to be maximized while ensuring the convergence of the calculation. From Equation (11), the relationship between any mesh size and time step can be obtained. Since the water mainly acts as a coupling medium and an absorbing medium, the mesh size can be appropriately increased to reduce the calculation time.