## 3 Block Iteration with Parallelization
### ONERA M6 Wing  
ONERA M6 wing is shown in Figure 7a. The initial surface mesh connecting the outside flow field with the wing is shown in Figure 7b. It consists of 39,496 internal points, 9951 moving boundary points inside the mesh. The motion of the wing is controlled by the sinusoidal function referred by [PERSON] et al. [21]:  
\[\Delta y=0.05y\sin\frac{2\pi y}{c}, \tag{34}\]  
where \(c\) is the length of root chord. The same as Section 4.1, for the motion, we compare the selection procedure of the first 0.0001 time unit.  
For the deformation of ONERA M6 wing, 560 control points are selected under the condition that \(\xi_{tol}=0.0005\). The distribution of boundary errors on wing surface are shown in Figure 8. Time comparison is shown in Table 5 and Figure 9. We can see that the time costs of parts \(a\), \(b\), \(d\) are efficiently reduced by more than 95% on average via the block iteration method. In addition, the parallel speedup of part \(d\), \(e\) is 10.33 on average.  
Figure 8: The distribution of boundary errors on wing surface.  
Figure 7: ONERA M6 wing and its initial surface mesh.  
For the deformation \(\xi_{tol}=0.00055\), there are 528 control points selected. The block iteration time cost of part \(a\) is about \(\frac{1}{148}\) compared with the conventional condition. It could be a validation that computational complexity is reduced from approximately \(\frac{1}{3}N_{c}^{3}\) to \(N_{c}^{2}\). For part \(d\), the block iteration time is \(\frac{1}{259}\) of the conventional time cost, reduced from \(\frac{\sigma}{2}N_{c}^{3}\) to \(\sigma N_{c}^{2}\). The same with parts \(a\) and \(d\), and the time cost of part \(b\) is reduced from \(O(N_{c}^{3})\) to \(O(N_{c}^{2})\). The time costs of part \(c\) and \(f\) are the same as Section 4.1. For the deformation \(\xi_{tol}=0.0005\), there are 560 control points selected. The computational complexities of block iteration for parts \(a\), \(b\), \(d\) are \(\frac{1}{168}\), \(\frac{1}{110}\), \(\frac{1}{278}\) of the conventional complexities, reduced from \(O(N_{c}^{3})\) to \(O(N_{c}^{2})\).  
The parallel speedup ratio is shown in Figure 10. We can see that parallelism could efficiently reduce the time cost of part \(d\) and \(e\). However, with the quantity increase of processors, the parallel efficiency rapidly declined.  
\begin{table}
\begin{tabular}{l c c c c c c} \hline \hline
**Time Comparison (s)** & \(a\) & \(b\) & \(c\) & \(d\) & \(e\) & \(f\) \\ \hline Origin (\(1.5\times 10^{-4}\)) & 15.58 & 249.76 & 5.98 & 407.40 & 172.95 & 17.80 \\ Block iteration (\(1.5\times 10^{-4}\)) & 0.10 & 2.69 & 6.02 & 1.57 & 184.66 & 21.86 \\ Block with parallel (\(1.5\times 10^{-4}\)) & 0.11 & 2.73 & 6.03 & 0.15 & 17.32 & 21.69 \\ \hline Origin (\(1.0\times 10^{-4}\)) & 18.50 & 315.40 & 7.01 & 460.04 & 195.31 & 19.11 \\ Block iteration (\(1.0\times 10^{-4}\)) & 0.11 & 2.84 & 7.11 & 1.65 & 205.34 & 23.42 \\ Block with parallel (\(1.0\times 10^{-4}\)) & 0.11 & 2.87 & 7.08 & 0.16 & 20.79 & 21.76 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Time comparison of ONERA M6 wing (value of \(\xi_{tol}\) is in the brackets, number of cores is 12).  
Figure 9: Time comparison of ONERA M6 wing.