## 3 Three-Dimensional Finite Element Simulation
### Results and Analysis of Simulation  
The simulation compares the microcrack positioning results at different locations and verifies that the localization algorithm has strong adaptability. As shown in Figure 9, the darkest area in the positioning maps represents the area where the amplitude of the _Img_ is largest, which is considered to be the area where the microcrack is located. The red area represents the area where a microcrack may exist. The light blue and blue area represent the absence of a microcrack in the area. We count the ratio (_Area-A_) of the darkest area to the total scan area, the ratio (_Area-B_) of the red area to the total scan area and the farthest distance \(L_{d\text{-}max}\) from the center of the microcrack to the edge of the darkest area. As shown in Table 1:  
Figure 8: Simulation model: (**a**) Schematic diagram of ABAQUS simulation model; (**b**) ABAQUS simulation model. Figure 7: Schematic diagram of result evaluation. _Appl. Sci._**2019**, \(9\), 1874 \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Figure**} & \(\boldsymbol{L_{d-max}}\) & _Area-A_ & _Area-B_ \\ \cline{2-4}  & **(m)** & **\%** & **\%** \\ \hline (a) & 0.019 & 0.060 & 0.160 \\ (b) & 0.015 & 0.039 & 0.142 \\ (c) & 0.004 & 0.015 & 0.059 \\ (d) & 0.050 & 0.053 & 0.516 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters used to evaluate the results of Figure 9.  
Figure 9: _Cont._  
As shown in Figure 9a when the microcrack is in the lower right corner of the scanning area and in Figure 9b when the microcrack is in the right side of the scanning area, the microcracks are effectively positioned. But as shown in Figure 10, the values of \(L_{d\_max}\), _Area-A_, and _Area-B_ in Figure 9a,b are all larger than the values in Figure 9c. The resulting positioning accuracy is low. There is only one scanning center line through the microcrack in Figure 9d, and the microcrack positioning failure can be found in Figure 9d. In the four positioning maps, the minimum value of \(L_{d\_max}\) is 0.004. The minimum value of _Area-B_ is 0.059, and the minimum value of _Area-A_ is 0.015. The minimum values all appear in Figure 9c. After simulation, it can be known that the scanning center line passing through the microcrack should be two or more, and the positioning result is better when the microcrack is in the center of the scanning area. The positioning algorithm proposed in this paper has strong adaptability. Figure 9: Positioning results when microcracks are in different positions in the scanning area: (**a**) microcrack in the lower right corner; (**b**) microcrack in the right side; (**c**) microcrack in the middle; (**d**) microcrack at the bottom. The arrangement changes of the excitation and reception probe are shown in Figure 11. When the probe arrangement was changed, it was found that the microcracks could still be effectively positioned. The evaluation parameters of the positioning result are shown in Table 2: \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Figure**} & \(\mathbf{L_{d-max}}\) & \(\mathbf{Area-A}\) & \(\mathbf{Area-B}\) \\ \cline{2-4}  & **(m)** & **\%** & **\%** \\ \hline (a) & 0.006 & 0.015 & 0.048 \\ (b) & 0.008 & 0.018 & 0.089 \\ (c) & 0.011 & 0.019 & 0.061 \\ (d) & 0.007 & 0.018 & 0.081 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameters used to evaluate the results of Figure 11.  
Figure 10: Evaluation chart. Figure 11: Positioning results when the excitation and reception probes are arranged differently: (**a**) Excitation1–4&Reception1–5; (**b**) Excitation1–3&Reception1–5; (**c**) Excitation1–5&Reception1–3; (**d**) Excitation1–5&Reception2–4. It can be seen from Figure 12 that the positioning error in Figure 11a is small compared with Figure 11b-d, that is, increasing the number of probes can effectively reduce the positioning error. Comparing Figure 11a with Figure 9c, the positioning error of Figure 11a is slightly larger than that of Figure 9c.