## 2 Methodologies for Parallel Numerical Computation
### Domain Decomposition Technique  
The domain decomposition method (DDM) is used in this study. Overlapping and non-overlapping methods are the two main classes of DDMs, and the non-overlapping method is more suitable for parallel computing architectures [29]. There exist two main kinds of non-overlapping DDMs for the finite element mesh, namely, the node-cut concept and the element-cut concept (see Figure 2). In the node-cut approach, the individual mesh elements are assigned to unique partitions (Figure 2a), and the shared nodes are duplicated to each partition who owns the nodes (filled circles in Figure 2a). On the contrary, the element-cut approach is based on a unique assignment of mesh nodes to unique partitions (Figure 2b), and the shared elements are duplicated to each partition to which their nodes belong (filled squares in Figure 2b). The communication among cores is through shared nodes or elements, respectively. Because the duplication effort is very little by sharing nodes among cores and the element-cut approach is generally less efficient, the node-cut approach has been used in the paper. The parallelization of LS-DYNA MPP (massive and parallel processors) version is based on the domain decomposition technique, and currently the RCB (Recursive Coordinate Bisection) is the default algorithm. RCB can decompose a model rapidly and effectively. It can easily be executed at run-time without significant costs and get a good balance of computational loads [30]. However, RCB also has some drawbacks. For instance, the disconnected domains may be created during the partition for models with complex geometric shapes. Furthermore, the RCB algorithm is only based on the geometric information of the model, and the load distributions and contact positions are not considered. Thus, the partition quality of the large-scale realistic simulation model is not so good. Therefore, for large-scale realistic models like the seismic analysis model of water conveyance tunnel, the original RCB is not the best DDM because of the need for extensive contact and FSI computation. The RCB method should be optimized for the specific analysis model to get better parallel efficiency. Modified Domain Decomposition Method (DDM) Considering Contact and Fluid-Structure Interaction (FSI) Load Balance  
Taking the structure characteristics of the large-scale water conveyance tunnel and the computation features of the seismic analysis into account, one modified DDM, which is derived from the basic principles of RCB and named as the MRCB (modified RCB), has been proposed. The MRCB is optimized for the supercomputer Dawning 5000A (Sugon, TianJin, China) and used for the seismic research of the specific tunnel. The MRCB is subject to the following rules to keep the balance of the total loads in each subdomain: firstly, the coupling loads including FSI and contact can be almost evenly distributed to each subdomain; secondly, the total numbers of nodes and elements can be balanced in every subdomain; thirdly, the amount of communication among adjacent subdomains can be minimized; lastly, the convergence rate in each subdomain is increased and nearly equivalent. The novel point of this parallelization is the balance of coupling loads in each subdomain, rather than just the equivalence of nodes and elements. Thus, the total computation loads in each subdomain for each processor core is almost equal and the parallel efficiency is improved. The basic idea of MRCB contains the following steps: firstly, a combined coordinate transformation is applied to the model according to its characteristics; then the domain decomposition approach is used to deal with the deformed model, and the model is divided into the required number of subdomains; finally, the original model is split by mapping the partitioned subdomains back. There exist a number of functions for coordinate transformation in LS-DYNA software (LSTC, Livermore, CA, USA), which provide aids to realize the MRCB.