## 3 Experiments and Simulations
### Simulation Methods  
For this paper, the BEM and FEM were used to simulate the test results, and to attempt to simulate the actual situation of the test. The simulation methods and ideas are hoped to provide references for others to more accurately simulate the STL of honeycomb sandwich materials tested with an impedance tube. A finite-element mesh was established for the honeycomb sandwich structure. As shown in Figure 6, a C\({}_{3}\)-type structure can be modeled using three components, all of which are shell elements. Two types of wall were found in the honeycomb core, each of which exhibit the same material properties but with different thicknesses. The model of the honeycomb sandwich panel was 30 mm in diameter, 20 mm in thickness, and the mesh side length was 0.4 mm. The side length of acoustical mesh was 2 mm, which satisfies the calculation requirements (less than 1/6 of the minimum wavelength) of the test frequency (6.3 kHz). As the honeycomb sandwich structures were complex, the simulations were carried out using an indirect BEM. In the method used, the sound fields inside and outside of the structure were calculated  
Figure 6: The mesh used to simulate a honeycomb sandwich structure. simultaneously, and the effect of the cavity in the honeycomb sandwich structure was also taken into account. Some specific details of the method are shown in Figure 7.  
The sound source generated plane waves on the left side of the diagram, and these were used to simulate the plane waves produced by the sound source in the impedance tube. An acoustical mesh was used to cover the structural mesh of the honeycomb sandwich structure so as to isolate the internal and external sound fields. The sound fields were also divided into two parts by a baffle, which stopped the back sound field of the honeycomb sandwich structure from influencing the front sound field. Four microphones were set in the mesh at the field points corresponding to the experimental locations so that the acoustic pressures at these field locations can be determined. The overlap between the acoustical mesh and baffle was set to be a transparent panel, that is, the transmitted acoustic waves could pass through the baffle at these points. Using modal-based vibro-acoustic response analysis, the acoustical meshes in front and inside of the honeycomb sandwich panel were set as Coupling Meshes. In practice, the honeycomb core needs to be processed into a circle. Due to the limit of processing accuracy, its actual diameter was slightly smaller than that of the impedance tube. So, the tube wall only constrained the aluminum plate, and the honeycomb core was constrained by the adhesive layer on the aluminum plate and the Teflon tape. Therefore, in the simulation, the edges of the aluminum plate and honeycomb core were constrained. The FEM model is shown in Figure 8. There is a plane wave sound source on the left, which simulated the plane wave generated by the sound source in the impedance tube. Two acoustic meshes of the air cavity in the impedance tube simulated the propagation of sound waves in the tube, and they were coupled with tube wall and structure mesh. The structural mesh and the cavity acoustic mesh were coupled. The end of the impedance tube was set as a sound-absorbing boundary, and the calculation of STL was the same as the BEM. The sound field behind the baffle was taken to be infinite, and therefore no reflected waves appeared at the acoustical boundary of the sound field. The transmission coefficients of the honeycomb sandwich structures could then be calculated using Equations (7) and (9), that is:  
\[\tau(\omega)=\frac{P_{t}}{P_{i}}=\frac{\sin k_{1}}{\sin k_{2}}\frac{p_{3} \mathrm{e}^{\mathrm{i}k_{2}}-p_{4}}{p_{1}-p_{2}\mathrm{e}^{-\mathrm{i}k_{1}}} \tag{15}\]  
Figure 8: Simulating the measurement of STL using the finite element method (FEM). Figure 7: Simulating the measurement of STL using the boundary element method (BEM).