## 4 Acoustic Finite Element Analysis (AFEA) of Acoustic Wave Propagation in a Straight Pipe
### Simulation Results
#### 4.3.1 Verification of Results Using Time of Flight
This refinement is a bit challenging; as the model size will become bigger, the computation time will be too long and the grid cells will be too much to be numerically computed on a regular computer. In order to use a finer mesh size without increasing the model size, axisymmetric element types were also used in the simulation. AFEA of acoustic wave propagation in an intact air-filled pipe was conducted using solid axisymmetric element type ACAX8, as illustrated in Figure 6.3 With this axisymmetric element type, it was possible to use very refined element sizes of 1 mm, and even down to 0.01 mm, for the simulations on a desktop PC with standard specifications, whereas the use of \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 & 0.3 & 20.5 & 395.1 & 14.5 \\ \(s_{2}\) & 0.938 & 2.2 & 18.2 & 395.4 & 14.6 \\ \(s_{3}\) & 1.758 & 4.2 & 16.0 & 397.9 & 15.3 \\ \(s_{4}\) & 2.240 & 5.5 & 14.8 & 397.9 & 15.3 \\ \(s_{5}\) & 3.050 & 7.6 & 12.8 & 400.6 & 16.1 \\ \(s_{6}\) & 3.870 & 9.6 & 10.7 & 414.8 & 20.2 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Arrival Time of Compression and Rarefaction Wave-front using the acoustic continuum 3D 20-node brick elements (AC3D20). Figure 7: Enlarged view of pressure vs. time response predicted at sensor \(s_{1}\) using AC3D20 element type. the solid element types AC3D8 and AC3D20 was not possible for element mesh size less than 9 mm. The predicted wave propagation using element type ACAX8 for the smallest mesh size is shown in Figure 9. Similarly, the Time of Flight approach was used to verify the acoustic speed results from the arrival time predicted for sensor locations \(s_{1}\) extracted from Figure 10.  
Similarly, by using the properties of air used for the FEA analysis, the actual speed of sound in air is approximately 345 ms\({}^{-1}\). Comparison of the derived sound speed (derived from the predicted arrival times) with the actual sound speed used in the AFEA gives the errors (E) associated with each sensor location, as shown in the sixth column of Table 1. These errors are very small in comparison to the errors associated with the use of the AC3D20 element. The relatively very small errors associated with the use of the ACAX8 element type indicate that the axisymmetric elements are better suited for acoustic wave propagation simulations than the AC3D20 element type. Nevertheless, it is seen from Table 2 that the errors associated with sensor location \(s_{6}\) is about 3.5 to 7 times that of the errors associated with the other sensor locations. This is due to the closeness of sensor \(s_{6}\) to the end of the open-ended pipe outlet where the acoustic wave is reflected. Figure 8: A meshed air-filled pipe model using the acoustic continuum axisymmetric 8-node element (ACAX8). Figure 9: Stacked Plots of pressure vs. time predicted using FEA at sensor \(s_{1}\) to \(s_{6}\) on intact air–filled pipe using ACAX8 (Note: all plots start with zero amplitude; maximum peak amplitude of each plot is less than 1.0 Pa). \[c=\frac{2\times(4.12-0.128)}{(23.35-0.436)(\times 10^{-3})}\approx 348.43\;\text{ms}^{ -1}\]