## 3 Model Verification  
Dubee University has conducted numerous classic FSI experiments on liquid-filled pipelines, encompassing both straight and elbow pipelines [32]. The resulting data have provided effective support for researchers to validate the accuracy of their models. In this section, we focus on a straight pipeline and an L-shaped elbow pipeline as examples to assess the performance of the proposed method. Both types of pipes are free-hanging, water-filled structures with closed ends. One end of the straight pipeline is subjected to a transverse exciting force, while one end of the elbow pipeline experiences an axial exciting force, as illustrated in Figure 4. The parameters for both the pipeline and fluid are detailed  
Figure 3: A flow chart of the proposed method. in Table 1. The selection of these two cases was motivated by their representativeness and their ability to directly demonstrate the instability phenomena present in the TMM. To comprehensively evaluate the theoretical method's performance, we employed an FSI module integrated into commercial finite element software to predict the frequency domain response of liquid-filled pipelines for cross-verification purposes. The parameter settings of the simulation model are the same as those described in the literature [8]. As depicted in Figure 5, the TMM effectively predicts the frequency domain response of pipelines within low-frequency ranges, aligning closely with the prediction results of the FEM. However, when assessing the transverse velocity response of the straight pipeline at frequencies exceeding 800 Hz, significant instability is observed. Similarly, instability arises in the frequency domain response of the L-elbow pipeline above 500 Hz. This indicates that the TMM possesses inherent limitations when addressing long-span pipelines; the explanations regarding these deficiencies can be found in Section 2.2. Figure 4: The liquid-filled pipelines used in the Dubee University experiment. (**a**) Straight pip –line, (**b**) L –elbow pipeline. Figure 5: Frequency domain response of Dubee pipelines. (**a**) Transverse velocity response of straight pipeline, (**b**) Pressure response of L –elbow pipeline, (**c**) Axial velocity response of L –elbow pipeline. The instability observed is associated with both the length of the pipeline and the frequency band utilized for solving. To address this instability issue, the MVTMM is employed to predict the frequency domain response of Dubee pipelines. The results are presented in Figure 6. It is evident that the processing outcomes from the MVTMM and the FME are largely consistent, effectively mitigating the instability problem. Tables 2 and 3 provide a comparison between the natural frequencies calculated using the MVTMM, the TMM, the FEM, and those obtained from experiments. It can be seen that the results calculated by the FEM are basically consistent with the experiment, which proves the correctness of the simulation model. Then, it can be observed that the maximum error between the MVTMM and the experimental model is 3.03%, which substantiates that not only does our method resolve the instability issue, but it also demonstrates high predictive accuracy. In addition, it is worth noting that the calculatio \begin{table}
\begin{tabular}{l l l} \hline \hline
**Pipeline** & & \\ \hline Density: 7985 kg/m\({}^{3}\) & L0: 4.502 m & Inner radius: 0.02601 m \\ Young’s modulus: 168 GPa & L1: 4.435 m & Wall thickness: 0.003945 m \\ Poisson’s ratio: 0.29 & Rw: 0.1013 m & \\ Shear coefficient: 0.53 & L2: 1.263 m & \\ \hline Fluid & & \\ \hline Density: 999 kg/m\({}^{3}\) & Bulk modulus: 2.14 GPa & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The parameters of the pipeline and fluid. Figure 6: Frequency domain response of Dubee pipelines. (**a**) Transverse velocity response of straight pipeline, (**b**) Pressure response of L-elbow pipeline, (**c**) Axial velocity response of L-elbow pipeline. fluid noise source to generate pressure pulsation throughout the pipeline.