## 4 Acoustic Finite Element Analysis (AFEA) of Acoustic Wave Propagation in a Straight Pipe
### Simulation Results
#### 4.3.3 Comparisons of Prediction Errors Due to Acoustic Element Type  
Table 4 shows the acoustic wave speed derived from the waveforms predicted by the four acoustic element types using different mesh densities based on the Time of Flight approach. The percentage errors associated with these predicted velocities are also shown in the table. A summary of the requirements, performance, and associated errors for the four acoustic element types in the predictions of the acoustic wave propagation is shown in Table 5. \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Element Type**} & \multicolumn{6}{c}{**Acoustic Wave Speed (ms\({}^{-1}\))**} & \multicolumn{6}{c}{**Error (\%)**} \\ \cline{2-11}  & \multicolumn{6}{c}{\(\mathbf{L_{e}}\) (mm)} & \multicolumn{6}{c}{**L\({}_{e}\) (mm)**} \\ \cline{2-11}  & **120** & **30** & **9** & **1** & **0.1** & **0.01** & **120** & **30** & **9** & **1** & **0.1** & **0.01** \\ \hline ACAX4 & 354.1 & 353.7 & 353.2 & 352.4 & 349.3 & 348.4 & 2.6 & 2.5 & 2.4 & 2.1 & 1.2 & 0.9 \\ ACAX8 & 352.6 & 352.4 & 352.4 & 351.6 & 349.0 & 348.4 & 2.2 & 2.1 & 2.1 & 1.9 & 1.1 & 0.9 \\ AC3D8 & 396.3 & 395.4 & 395.3 & 395.4 & 381.0 & 350.5 & 14.9 & 14.6 & 14.5 & 14.6 & 10.4 & 1.5 \\ AC3D20 & 395.9 & 395.1 & 395.0 & 395.0 & 380.4 & 349.9 & 14.8 & 14.5 & 14.4 & 14.4 & 10.2 & 1.4 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Acoustic mesh sensitivity study showing the acoustic speed computation error. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Element Type** & **Mesh Seed L\({}_{\text{w}}\) (mm)** & **Total No. of Element** & **Total No. of Nodes** & **Total No. of Variables** & **Analysis Time (h)** \\ \hline \multirow{6}{*}{ACAX4 (\({}^{1}\))} & 120 & 35 & 72 & 72 & 3 \\  & 30 & 136 & 274 & 274 & 4.5 \\  & 9 & 1374 & 1836 & 1836 & 6 \\  & 1 & 103,000 & 107,146 & 107,146 & 7 \\  & 0.1 & 2,475,400 & 2,612,980 & 2,612,980 & 8.5 \\  & 0.01 & 13,305,460 & 13,937,450 & 13,937,450 & 10 \\ \hline \multirow{6}{*}{ACAX8} & 120 & 35 & 178 & 178 & 3.5 \\  & 30 & 136 & 683 & 683 & 5 \\  & 9 & 1374 & 3836 & 3836 & 8 \\  & 1 & 103,000 & 317,291 & 317,291 & 9.5 \\  & 0.1 & 2,475,400 & 3,085,430 & 3,085,430 & 11 \\  & 0.01 & 13,305,460 & 15,135,560 & 15,135,560 & 12 \\ \hline \multirow{6}{*}{AC3D8 (\({}^{2}\))} & 120 & 432 & 629 & 629 & 24 \\  & 30 & 1584 & 2261 & 2261 & 36 \\  & 9 & 5328 & 7565 & 7565 & 46 \\  & 1 & 128,000 & 164,041 & 164,041 & 60 \\  & 0.1 & 22,776,000 & 23,235,670 & 23,235,670 & 132 \\  & 0.01 & 63,085,430 & 64,587,604 & 64,587,604 & 220 \\ \hline \multirow{6}{*}{AC3D20} & 120 & 432 & 2277 & 2277 & 30 \\  & 30 & 1584 & 8229 & 8229 & 46 \\ \cline{1-1}  & 9 & 5328 & 27,573 & 27,573 & 57 \\ \cline{1-1}  & 1 & 128,000 & 616,113 & 616,113 & 71 \\ \cline{1-1}  & 0.1 & 22,776,000 & 28,488,752 & 28,468,752 & 179 \\ \cline{1-1}  & 0.01 & 63,085,430 & 113,654,300 & 113,654,300 & 240 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Basic acoustic mesh sensitivity study. Figure 14 shows the variations of the percentage acoustic speed errors with the element sizes for the four acoustic element types. It is observed that the axisymmetric element type ACAX4 and ACAX8 produce less computational errors in the acoustic FEA. Also, it is clearly seen that the linear interpolation elements (ACAX4 and AC3D8) have less errors than the quadratic interpolation elements (ACAX8 and AC3D20). Figure 14 shows that very large computational errors occur in the AFE analysis result when solid element types AC3D8 and AC3D20 are used. But when the axisymmetric element types ACAX4 and ACAX8 are used, the errors are minimised. It is thus concluded that the solid element types AC3D8 and AC3D20 are not optimal in terms of speed, accuracy, and computation-resource requirements for AFE analysis of acoustic wave propagation in an air-filled pipe. Furthermore, considering the computational time and resources for each model as illustrated in Table 5, the ACAX4 is the best element type for acoustic FEA.