## 4 Results and Discussion
### Domain Decomposition Comparison and Parallel Performance Evaluation  
All simulations were carried out using the proposed MRCB (modified Recursive Coordinate Bisection) based parallel computing method, which was embedded in the LS-DYNA optimized to run on the Dawning 5000 A at the Shanghai Supercomputer Center. The evaluation of the developed parallel method applied in the seismic analysis of the specific water conveyance tunnel will be presented in this section. To evaluate the proposed MRCB domain decomposition method (DDM), calculations were carried out with 1, 2, 4, 8, 16 and 32 cores using RCB and MRCB, separately. The topologies of domain decomposition with 16 subdomains for the dynamic model using these two DDMs are shown in Figure 21. From this figure, one can easily find that the topologies from these two DDMs are different, more specifically, by using RCB the model is split in both \(x\) and \(y\) direction along which the model is longest, while by using MRCB the model is divided only in the, \(x\) direction along which the coupling (contact and FSI) load is distributed. Thus the MRCB may lead to a more balanced distribution of coupling loads from a qualitative perspective. For further quantitative analyses of the differences between these two domain decomposition methods, Table 4 lists the load details (number of nodes participating in calculation) of each subdomain using these two methods, respectively. As shown in this table, for each subdomain of both these two DDMs, the total number of nodes (N\({}_{\text{tol}}\)) is almost equal. However, the subdomains obtained from RCB have a very unbalanced distribution of nodes involved in contact (N\({}_{\text{con}}\)) and FSI (N\({}_{\text{fsi}}\)), in contrast with those obtained from MRCB. As is known to all, the calculation overheads of nodes participating in contact or FSI far outweigh those of common nodes. Therefore, for the traditional RCB method, the loads of each partition cannot achieve balance, and cores processing partitions with heavier loads will take extra time when cores processing partitions with less loads are idle and waiting. As a result, the expensive computing resources are wasted and the total calculation time increases unnecessarily. On the contrary, the nodes in contact and the FSI of the partitions from the MRCB method are well distributed, and the number of these nodes in every subdomain is approximately equal. Therefore, the total loads in every subdomain are balanced. Besides, the running time of cores processing each subdomain is almost the same, so the computing resources are used fully and reasonably, and the total calculation time decreases greatly. Figure 21: Description of domain decomposition topology for 16 subdomains using: (a) recursive coordinate bisection (RCB) method; (b) modified recursive coordinate bisection (MRCB) method. Table 5 shows the time consumption and parallel performance of different approaches with different numbers of cores to perform a 20 s real-time seismic analysis, where \(\mathit{Ti}\), \(\mathit{Su}\) and \(\mathit{Ef}\) represent total computation time, speedup and parallel efficiency, respectively. As shown in the table, MRCB presents a more balanced load distribution and a higher speed up and parallel efficiency compared to those of RCB. Therefore, the total computation time decreases. The comparison of speedup for the parallel dynamic analysis is also shown in Figure 22. The increment of speedup for both RCB and MRCB is not linearly proportional to the increment of cores involved in computation, which means that the parallel efficiency drops when the number of processor cores grow. Compared with RCB, the speedup of MRCB is larger and closer to the theoretical one and this superiority increases as the number of cores grows.