## 4 Simulation and Analysis of the DLLFP  
### Comparison of Numerical and Analytical Solutions  
Given that the dispersion curves vary as the physical fields differ, and to verify the correctness of the calculation procedure, numerical simulations are performed on the DLLFP shown in Figure 4 under different physical fields and the numerical solution is compared with the analytical solution. The analytical solution is used to embed the wave number formula derived in the paper into COMSOL Multiphysics are used for calculation to obtain the analytical results of the dispersion curve. Through parametric scanning, the excitation frequency is scanned from \(f_{0}\) = 10 Hz to \(f_{\max}\) = 25 KHz, and the scan step is set to 249.9 Hz. The mode analysis frequency is given as \(f_{0}\) = 10,000 Hz for the interior of the pipe, the mode search method is set to the manual search method, the APPACK solver is selected as the mode solver, and the mode search base \(k_{0}\) = 2\(\pi f_{0}\)/c\({}_{0}\). The numerical simulation is solved using the MUMPS direct solver, which requires higher memory usage compared to iterative solvers, but is more robust. The free triangular meshing technique is chosen to divide the mesh in the pressure acoustic domain. The solid mechanics domain is selected to divide the mesh by mapping distribution, and the number of cells of the whole geometric mesh is 6638. The physical fields are the pressure acoustic domain, solid mechanics domain, and acoustic-structural coupling domain. The fluid inside the DLLFP is water, the \(x\)-axis is the scanning frequency in Hz, and the \(y\)-axis is the phase velocity value in m/s. The black \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Materials** & **Wall Thickness** & **Density \(\rho\)** & **Sound Velocity c** & **Young’s Modulus E** & **Poisson’s Ratio \(\mu\)** \\  & **[mm]** & **[kg/m\({}^{3}\)]** & **[m/s]** & **[pa]** & **Poisson’s Ratio \(\mu\)** \\ \hline Structural steel & 2.5 & 7850 & Disregard & 205 \(\times\) 10\({}^{9}\) & 0.30 \\ Titanium beta-21 Sl & 3 & 4940 & Disregard & 105\(\times\) 10\({}^{9}\) & 0.33 \\ Water & Disregard & 1000 & 1500 & Disregard & Disregard \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Material parameters. Figure 4: Model diagram of the DLLFP. asterisk line indicates the dispersion curve obtained with analytical calculation, and the red dot line indicates the dispersion curve obtained with numerical simulation. The dispersion curves of different physical fields are shown in Figure 5.  
From Figure 5, it can be seen that, except for slight deviations in the pressure acoustic domain, the analytical and numerical solutions of the main guided wave modes are in good agreement in the solid mechanics domain and the acoustic-solid coupling domain. The difference between the numerical and analytical solutions in Figure 5b is also related to the number of modes solved, increasing the number of modes solved for the numerical solution results in more accurate results. Finite element calculations have good accuracy, which validates the correctness of the simulation method and the multi-layer pipe model. Figure 5a shows that the number of guided wave modes increases with the scanning frequency. The phase velocities of the four guided wave modes decrease with increasing frequency (except for the phase velocity of the straight-through mode, which does not vary with scanning frequency), and tend to be similar to the phase velocity of the straight-through mode, indicating significant dispersion characteristics. When the phase velocity approaches the cut-off frequency of the guided wave [50], the phase velocity tends to infinity. From Figure 5b,c, it can be observed that there are more propagating guided wave modes in the acoustic-solid coupling domain than in the solid mechanics domain, indicating that the additional modes are generated during coupling, and the propagation of guided wave modes at the same frequency is more complex under the action of acoustic-solid coupling.