## 4 Acoustic Finite Element Analysis (AFEA) of Acoustic Wave Propagation in a Straight Pipe
### Simulation Results
#### 4.3.2 Sensitivity of Acoustic Finite Elements
Very refined mesh density of L\({}_{\rm e}\) = 0.1 mm and L\({}_{\rm e}\) = 0.01 mm was further used, and the waveforms become very smooth, clear, identical, and well-defined. Thus, it can be deduced that the AFEA solution becomes more accurate when element type ACAX4 with length L\({}_{\rm e}\) = 0.1 mm is used for the analyses. These analyses were repeated using the axisymmetric element ACAX8 with the same mesh densities as those previously used with the element type ACAX4. The predicted waveforms are shown in Figure 11c, Figure 12c, and Figure 13c, and the corresponding enlarged plots of the first pulses are shown in Figure 11d, Figure 12d, and Figure 13d, respectively. These figures show that the characteristics predicted by the ACAX8 element type are similar to those predicted by the ACAX4 element type. That is, the predicted waveforms are not well-separated pulses, but are rather wavy and very noisy curves when coarse mesh densities are used. However, when fine mesh densities are used, the waveforms become very smooth, clear, and well-defined. Furthermore, the converged waveforms predicted by the ACAX8 element type are identical to those predicted by the ACAX4 element type. The main difference between the two element types is that the ACAX8 element type required greater computational times. In the introduction, it was stated that [PERSON] and [PERSON] (2018) used 3D hexahedral acoustic elements which they expected to give accurate predictions, just as the 3D hexahedral structural elements give accurate predictions for structural problems. But it was shown in the previous sub-section and in Table 1 that the hexahedral acoustic element type AC3D20 gives large prediction errors even when the element size used is deemed to be reasonably small. In order to ascertain the mesh sensitivity versus accuracy of the hexahedral acoustic element types AC3D8 and AC3D20, they were also used in the acoustic simulations for the same mesh densities as those used with the axisymmetric element types ACAX4 and ACAX8. Figure 11e-h, Figure 12e-h, and Figure 13e-h show that when coarse mesh densities (i.e., L\({}_{\rm e}\) = 120 mm, L\({}_{\rm e}\) = 30 mm and L\({}_{\rm e}\) = 9 mm) are used with element types AC3D8 and AC3D20, the predicted waveforms are worse than those predicted by the axisymmetric elements. The predicted waveforms are not well-separated pulses, but are rather very wavy and very noisy curves. When a fine mesh is used (L\({}_{\rm e}\) = 1 mm), the waveforms become less noisy and show well-defined pulses. Figure 11 also shows that when the mesh is coarse, the waveforms predicted by the solid element type AC3D20 are less noisy than the waveforms predicted by the solid element types AC3D8. When the mesh is very fine (i.e., L\({}_{\rm e}\) = 0.1 mm and L\({}_{\rm e}\) = 0.01 mm), the predictions of the two element types are identical. However, the analysis using element type AC3D20 takes much more time to complete than with element type AC3D8. Furthermore, the use of AC3D20 elements generates a much bigger model and requires more static and dynamic storage than the use of AC3D8 elements. In addition to the predicted waveforms not being sufficiently smooth with well-defined pulses when a fine mesh is used (L\({}_{\rm e}\) = 1 mm), the figures also show that the predicted time intervals between the pulses is also larger, resulting in large errors in the predicted acoustic wave velocity. Thus, while it is important for the waveforms to be as smooth as possible, it is not sufficient to judge the accuracy of an acoustic wave propagation analysis by the waveform alone; it is also necessary to consider the accuracy of the Time of Flight. The performance of the FEA solutions for different element types and mesh sizes is evaluated using three key variables, namely: (1) the waviness of the waveforms; (2) the computational time of the analysis; and (3) the predicted acoustic speed error. The variations of these variables are examined using different acoustic element types and sizes in the Abaqus FEA software.