## 5 Conclusions  
The acoustic finite element approach has been used to predict the wave propagation characteristics in a fluid-filled pipe. This paper has detailed the use of Finite Element Analysis (FEA) to simulate the  
Figure 14: Relationship between the Acoustic speed error and the element type/element size. \begin{table}
\begin{tabular}{c c c c c c c c} \hline
**Element Type** & **Refinement** & **Quality** & **NE** & **NN** & **TC (h)** & **C (ms\({}^{-1}\))** & **E (\%)** \\ \hline AC3D20 & Coarse & Noisy & 1584 & 8229 & 46 & 395.1 & 14.5 \\ AC3D8 & Coarse & Noisy & 1584 & 2261 & 36 & 395.4 & 14.6 \\ AC3D20 & Fine & Smooth & \(6.31\times 10^{7}\,{}^{(1)}\) & \(1.13\times 10^{8}\,{}^{(3)}\) & 240 & 349.9 & 1.4 \\ AC3D8 & Fine & Smooth & \(6.31\times 10^{7}\,{}^{(1)}\) & \(6.45\times 10^{7}\,{}^{(4)}\) & 220 & 350.5 & 1.5 \\ ACAX8 & Coarse & Noisy & 136 & 683 & 5 & 352.4 & 2.1 \\ ACAX4 & Coarse & Noisy & 136 & 274 & 4.5 & 353.7 & 2.5 \\ ACAX8 & Fine & Smooth & \(1.33\times 10^{7}\,{}^{(2)}\) & \(1.51\times 10^{7}\,{}^{(5)}\) & 12 & 348.4 & 0.9 \\ ACAX4 & Fine & Smooth & \(1.33\times 10^{7}\,{}^{(2)}\) & \(1.39\times 10^{7}\,{}^{(6)}\) & 10 & 348.4 & 0.9 \\ \hline \end{tabular} NE = number of elements generated; NN = number of nodes generated; E = Error; C = Speed; TC = computational time, exact values: \({}^{(1)}\) 63,085,430, \({}^{(2)}\) 13,205,460, \({}^{(3)}\) 113,654,300, \({}^{(4)}\) 64,587,604, \({}^{(5)}\) 15,135,560, \({}^{(6)}\) 13,937,450.  
\end{table} 
Table 5: Sensitivity analysis- key variable performance evaluation. acoustic wave propagation and acoustic wave reflectometry of a pipe. The wave velocity predicted by the FEA has been found to be consistent with the actual velocity when the right element type is chosen for the numerical computation. The primary conclusions of this paper are as follows. * The acoustic FEA of a fluid-filled pipe can be achieved with the simulation of only the fluid content of the pipe and without the inclusion of the pipe in the model. Thus, this eliminates contact interaction of the solid pipe wall and the fluid, and as a consequence, reduces computational time and resources. Since the specific acoustic impedance of the air medium is much smaller than the specific impedance of the wall of a metallic or plastic pipe, this assumption is reasonable. The results confirm that the acoustic FEA of an air-filled pipe can be achieved with the simulation of only the fluid content of the pipe and without the inclusion of the pipe in the model. However, when a denser fluid medium is used, and when there is high flow velocity, fluid-structure interaction will become very significant. Then, the fluid medium and the pipe wall will need to be modelled together. * The main source of error in an acoustic FEA is the element type chosen and the mesh density. The results indicate that the ACAX4 element type with a fine mesh should be used in any acoustic FEA simulation of wave propagation in a symmetrical pipe in order to obtain the smallest computational errors, fastest computations, and least usage of computer static and dynamic storage resources. For a nonsymmetrical pipe, the 3D solid hexahedral elements can be used at the part of the pipe that is unsymmetrical and the 2D-like axisymmetric element should be used at the symmetric part of the pipe to obtain the smallest computational errors, fastest computations, and least usage of computer static and dynamic storage resources. * It has been shown that the use of the Time of Flight approach enables the verification of the accuracy of the acoustic FEA simulations using any of the four element types investigated, namely: ACAX4, ACAX8, AC3D8, and AC3D20. * Of these four element types, the best element for AFE simulations of wave propagation in a symmetric air-filled pipe was found to be the axisymmetric element type ACAX4, whereas the worst element type was found to be the solid element type AC3D20.