## 4 Acoustic Finite Element Analysis (AFEA) of Acoustic Wave Propagation in a Straight Pipe
### Simulation Results
#### 4.3.2 Sensitivity of Acoustic Finite Elements  
In this section, a sensitivity analysis is developed for the meshing of the acoustic medium. The aim is to investigate the influence of mesh element type and mesh size in acoustic FEA. The AC3D20 element in the Abaqus FEA software produced a lot of computational errors in the derived acoustic speed, as seen in the sixth column of Table 1. Most of the errors in the acoustic speed are more than 15%. To address this much more closely, the two axisymmetric element types ACAX4 and ACAX8 and the two solid element types AC3D8 and AC3D20 were used. Also, various mesh densities were employed involving element seed sizes of 120 mm down to 0.01 mm. While it was possible to run these analyses using the axisymmetric element types ACAX4 and ACAX8 on a regular desktop PC (which has a single processor), the analyses using the solid element types AC3D8 and AC3D20 could only be carried out on supercomputers using multiple cores (between 4 and 12 parallel processors). The results of these sensitivity analyses are shown in Figures 11-13. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(s_{n}\) & \(l_{s}\) (m) & \(t_{s}\) (ms) & \(t_{p}\) (ms) & \(c\) (ms\({}^{-1}\)) & Error, E (\%) \\ \hline \(s_{1}\) & 0.128 & 0.43 & 23.3 & 348.4 & 0.9 \\ \(s_{2}\) & 0.938 & 2.76 & 21.0 & 348.5 & 1.0 \\ \(s_{3}\) & 1.758 & 5.08 & 18.7 & 346.9 & 0.5 \\ \(s_{4}\) & 2.240 & 6.49 & 17.2 & 348.2 & 0.9 \\ \(s_{5}\) & 3.050 & 8.81 & 14.9 & 347.7 & 0.7 \\ \(s_{6}\) & 3.870 & 11.1 & 12.6 & 357.1 & 3.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Arrival Time of Compression and Rarefaction Wave-front using the acoustic continuum axisymmetric 8-node element (ACAX8). Figure 10: Enlarged view of pressure vs. time response predicted at sensor \(s_{1}\) using ACAX8 element type. Figure 11: **(a,c,e,g)** Stacked Plots of pressure vs. time predicted at \(s_{1}\) using ACAX4, ACAX8, AC3D8 and AC3D20 elements respectively. **(b,d,f,h)** Magnified view of pressure vs. time response predicted at \(s_{1}\) using ACAX4, ACAX8, AC3D8 and AC3D20 elements respectively (All plots start with zero amplitude). Figure 12: (**a**,**c**,**e**,**e**,**g**) Stacked Plots of pressure vs. time predicted at \(s_{3}\) using ACAX4, ACAX8, AC3D8 and AC3D20 element respectively. (**b**,**d**,**f**,**h**) Magnified view of pressure vs. time response predicted at \(s_{3}\) using ACAX4, ACAX8, AC3D8 and AC3D20 elements respectively (All plots start with zero amplitude). Figure 13: (**a**,\(c\),**e**,**g**) Stacked Plots of pressure vs. time predicted at \(s_{6}\) using ACAX4, ACAX8, AC3D8 and AC3D20 elements respectively. (**b**,\(d\),\(f\),**h**) Magnified view of pressure vs. time response predicted at \(s_{6}\) using ACAX4, ACAX8, AC3D8 and AC3D20 elements respectively (All plots start with zero amplitude). Figures 11-13 show the pressure vs. time histories predicted using the two axisymmetric element types, ACAX4 and ACAX8, and the two solid element types, AC3D8 and AC3D20, predicted at sensor locations \(s_{1},~{}s_{3}\) and \(s_{6}\) respectively. These pressure-time histories manifest the relationships between the element types chosen for the analyses, the mesh densities used, and the computational errors. The first waveforms for each set of pressure-time histories were also magnified in order to clearly show the differences in the predictions using different element types and mesh densities. Firstly, the axisymmetric element type ACAX4 was used with a very coarse mesh density of element length L\({}_{\rm e}\) = 120 mm to discretise the model. It can be seen from Figures 11a, 12a and 13a, and the corresponding enlarged plots of Figures 11b, 12b and 13b, that the predicted waveforms are not well-separated pulses, but are rather wavy and very noisy curves. The model was refined using L\({}_{\rm e}\) = 30 mm; this resulted in a less noisy waveforms, but not well-defined pulses. By further refinement using L\({}_{\rm e}\) = 9 mm, the waveforms become a little clearer, with well-defined pulses and only a little noise.