## 2 SPH-FEM Coupling Analysis Method  
### Introduction to SPH Method  
As a discrete element calculation method, the SPH method has gradually been applied to large analysis areas such as blasting and penetration due to its ability to handle large deformation properties. The system state in SPH is represented by a set of particles that can be assigned material properties and interact within a range of smoothing or weighting functions. These particle motions are controlled by the SPH formulation of the physical partial differential equation characterizing fluid flow; moreover, the SPH-controlled equation of motion can be transformed into the NS equation by replacing the SPH approximation with the function and its derivative with respect to the NS equation.  
\[\frac{D\rho_{i}}{Dt}=\sum_{j=1}^{N}m_{j}v_{ij}^{\beta}\frac{\partial W_{ij}}{ \alpha_{i}^{\beta}} \tag{1}\]  
\[\frac{Dv_{i}^{\alpha}}{Dt}=-\sum_{j=1}^{N}m_{j}(\frac{\sigma_{i}^{\alpha\beta }}{\rho_{i}^{2}}+\frac{\sigma_{j}^{\alpha\beta}}{\rho_{j}^{2}})\frac{\partial W _{ij}}{\partial x_{i}^{\beta}}+F_{i} \tag{2}\]  
\[\frac{De_{i}}{Dt}=\frac{1}{2}\sum_{j=1}^{N}m_{j}\left(\frac{P_{i}}{\rho_{i}^{2 }}+\frac{P_{j}}{\rho_{j}^{2}}\right)v_{ij}^{\beta}\frac{\partial W_{ij}}{ \partial x_{i}^{\beta}}+\frac{\mu_{i}}{2\rho_{i}}\varepsilon_{i}^{\alpha\beta }\varepsilon_{j}^{\alpha\beta} \tag{3}\]  
In the formula: \(\rho_{i}\), \(v_{i}^{\alpha}\), \(\varepsilon_{i}\) are the density, velocity component, and internal energy of particle \(i\), respectively; \(N\) denotes the number of particles; \(\alpha\) and \(\beta\) denote the coordinate directions; \(x_{i}^{\beta}\) is the coordinate component of \(x\) along \(\beta\); \(P_{j}\) denotes the isotropic pressure; \(W_{ij}\) denotes the smooth function between particles \(i\) and \(j\); \(m_{j}\) denotes the particle mass.  
From Equations (1)-(3), it can be found that SPH can calculate the velocity and energy changes of the particles inside the system and that the calculated data between the particles can be connected and act on each other; moreover, the whole action process varies with time. Further, the mesh-free nature of the SPH particles allows for the direct treatment of large deformations, avoiding the problem of large mesh distortions in numerical calculations.  
Specifically, when the SPH calculation method is applied to karst tunnel surge water under the coupled action of blasting load and water pressure, by dividing the groundwater in front of the tunnel palm face into SPH particles and applying gravity to the SPH particles, the groundwater in front of the palm face of the karst tunnel can be realized as a result of the blasting dynamic action and its own gravity action to produce a water inrush deformation along the tunnel interior, as shown schematically in Figure 1. Compared to the finite element, it is more intuitive to represent the evolution of the water inrushing while avoiding large deformation distortions in the mesh.  
Although SPH has the advantage of simulating large deformation, there is a serious boundary defect in the far field of the computational model due to particle inconsistency. In order to overcome this defect, a coupled analysis method using the SPH method in the near field and the Finite Element Method (FEM) in the far field is developed and applied, which combines the advantages of SPH (namely, being able to simulate large deformation) and FEM boundary calculation (being more stable and able to effectively overcome the problem of large deformation distortion and boundary instability).