## 3 Establishment of the Finite Element Model and Simulation Results
### Finite Element Simulation Results
#### 3.2.2 Simulation Results of Multi-Angle Incident  
In most cases, the operating environment of the flange device is complex, and it is difficult for the ultrasonic probe to be perpendicular to the flange surface. Moreover, the placement of the probe is often limited by the size of the flange. Under these circumstances, it is necessary to change the incident angle of the ultrasonic probe to achieve optimal detection results. When the incident wave enters the surface of the flange cover at an angle of 40\({}^{\circ}\) through the background pressure field, reflection and projection phenomena occur at the interface between the flange cover and the silicone gasket. As shown in Figure 8, the figures depict the sound field of the background pressure field and the reflected sound field at the interface.  
As shown in Figure 8, the maximum sound pressure amplitude of the incident wave was 1 when it entered the upper flange cover, indicated by the arrow direction. When the incident sound field entered the silicone gasket through the upper flange cover, scattering and transmission phenomena occurred.  
A multi-angle measurement model was established to change the incident angle of the sound field and simulate the reflection spectra under different incident angles, as shown in Figure 9.  
As shown in Figure 9, as the driving frequency increased, an extremely low point approaching 0 appeared in the ultrasonic reflection coefficients for all incident angles. Under the same frequency condition, a larger incident angle resulted in a higher ultrasonic reflection coefficient. Therefore, selecting an appropriate incident angle can reduce the sound field attenuation and improve the detection efficiency. For example, when the air layer thickness was 50 \(\upmu\)m and the incident angle was 30\({}^{\circ}\), the ultrasonic reflection coefficient dropped to its lowest value of 0.1 as the frequency increased to 4.11 MHz. This indicates that most of the incident sound waves were transmitted and there was almost no reflected wave generated. At this driving frequency, the sound wave resonated with the air layer. Near the resonant frequency, the reflection coefficient tended toward 0, which means there was no reflected sound wave produced. Far from the resonant frequency, the reflection coefficient tended toward 1, which meant that the sound wave was completely reflected.  
In summary, for the same air layer thickness, as the incident angle increases, the resonant frequency also increases. However, in the detection process, large incident angles are not practical, and high-frequency sound waves attenuate rapidly during the propagation. Therefore, we set the incident angles to 10\({}^{\circ}\), 20\({}^{\circ}\), 30\({}^{\circ}\), 40\({}^{\circ}\), 50\({}^{\circ}\), 60\({}^{\circ}\), and 70\({}^{\circ}\), and added a simulation comparison group with increased air layer thickness to improve the reliability of the calculation and facilitate the derivation of their relationship curve.  
Figure 8: f = 0.01 MHz, \(\theta\) = 40\({}^{\circ}\). Acoustic field map: (**a**) incident sound field, (**b**) reflected sound field.  
Figure 9: Reflection coefficient spectra under different air layer thicknesses and incident angles: (**a**) reflection coefficient spectra at Q = 30 μm, (**b**) reflection coefficient spectra at Q = 50 μm, (**c**) reflection coefficient spectra at Q = 70 μm.