## 3 Numerical Examples
### Multi-Bubbles
#### 3.2.2 10 Bubbles  
Figure 9 shows a 3D multi-bubbles mesh grid. Table 6 shows the coordinates of each spherical and radius. The number of nodes and elements generated by two methods, as well as the calculation time cost on the same computer, are compared in Table 7. Select the section with dense bubbles to draw the potential map. The potential sectional at \(y=10\) mm and \(x=7.5\) mm of XFEM is shown in Figure 10.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Number** & **Center Position** & **Radius (mm)** \\ \hline Bubble 1 & (7.5, 7.5, 12.5) & 1.50 \\ Bubble 2 & (2, 3, 5) & 1.30 \\ Bubble 3 & (8, 8, 20) & 1.70 \\ Bubble 4 & (10, 10, 5) & 1.00 \\ Bubble 5 & (10, 10, 10) & 1.40 \\ Bubble 6 & (12.5, 2.5, 7.5) & 1.40 \\ Bubble 7 & (8, 4, 2) & 1.20 \\ Bubble 8 & (5, 11, 14) & 1.70 \\ Bubble 9 & (4, 5, 16) & 1.50 \\ Bubble 10 & (3, 9, 8) & 1.40 \\ \hline \hline \end{tabular}
\end{table}
Table 6: The center position and radius of 10 bubbles.  
Figure 9: (a) 3D 10 bubbles mesh grid of CFEM; (b) 3D 10 bubbles mesh grid of XFEM.  
Among the 1159 nodes, the maximum relative error is 7.93% and the average relative error is 2.52%. Ten nodes near the boundary of the sphere are selected to compare the calculation results. The comparison is shown in Table 8.  
Sections 3.1 and 3.2 show that the improved XFEM can save calculation costs and reduce calculation time compared with CFEM. At the same time, the accuracy of XFEM calculation results rises with the increase of model complexity. The calculation saved and the contrast of accuracy are shown in Table 9.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Method** & **Node Number** & **Element Number** & **Calculation Time** \\ \hline CFEM & 3582 & 19,851 & 15 min 47 s \\ XFEM & 1159 & 5626 & 34 s \\ \hline \hline \end{tabular}
\end{table}
Table 7: Contrast of number of nodes and elements.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Nodes** & **CFEM** & **XFEM** & **Relative Error** \\ \hline
1 & 20,288 & 19,907.7 & \(-\)1.87\% \\
2 & 22,955 & 24,775.6 & 7.93\% \\
3 & 46,363 & 46,548.6 & 0.40\% \\
4 & 35,053 & 35,484.3 & 1.23\% \\
5 & 28,163 & 28,853.9 & 2.45\% \\
6 & 38,021 & 38,472.6 & 1.19\% \\
7 & 7426.6 & 7017.14 & \(-\)5.51\% \\
8 & 12,749 & 12,978.8 & 1.80\% \\
9 & 3042.8 & 2818.45 & \(-\)7.37\% \\
10 & 20,027 & 20,461.1 & 2.17\% \\ \hline \hline \end{tabular}
\end{table}
Table 8: Contrast of the results of nodes near interface.  
Figure 10: (a) 10 bubbles potential diagram at \(y=10\) mm; (b) 10 bubbles potential diagram at \(x=7.5\) mm.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Method** & **Single Bubble** & \multicolumn{2}{c}{**5 Bubbles**} & \multicolumn{2}{c}{**10 Bubbles**} \\ \cline{2-5}  & **Elements** & **Calculation Time** & **Elements** & **Calculation Time** & **Elements** & **Calculation Time** \\ \hline CFEM & 7062 & 57 s & 12,997 & 4 min 51 s & 19,851 & 15 min 47 s \\ XFEM & 5626 & 37 s & 5626 & 34 s & 5626 & 34 s \\ Ratio & 79.66\% & 64.91\% & 53.28\% & 11.68\% & 28.34\% & 3.6\% \\ \hline \hline \end{tabular}
\end{table}
Table 9: Contrast of calculation saving and accuracy.  
As the complexity of the model increases, the ratio of element saving and time-saving increases. The calculation accuracy decreases. The average relative error of the single-bubble model is 0.39%, while the average relative error of the 10-bubble model increases to 2.52%.