## 2 FEM-SPH Conversion Coupling Algorithm
### Bank Slope Stability Analysis  
To evaluate the stability of the bank slope under the action of the reservoir water level cycle, the appropriate criterion should be selected to judge whether the bank slope is unstable or not. For the slope instability problem, the traditional criteria include a sudden change in displacement of characteristic points, non-convergence of numerical calculations, and plastic zone penetration. In the case of an unknown sliding zone, [PERSON] et al. [24] pointed out that it was more reasonable to use the sudden change in displacement increment at the top and bottom of the landslide as the criterion of landslide instability. Therefore, in this paper, the midpoint of the top and bottom of the bank slope was used as the characteristic point to record its displacement. From the sudden change in the displacement curve of the characteristic points at the top and bottom of the bank slope, the inflection point was the critical point of the bank slope instability. The displacement of the characteristic point of the bank slope is shown in Figure 7.  
As can be seen from Figure 7, the displacement of the characteristic point of the bank slope under the characteristic points of the reservoir level cycle and the displacement increment gradually increased in the first 15 cycles of reservoir level rise and fall; therefore, the action of the reservoir level cycle will accelerate the deformation of the bank slope. In the 16 th water level rise and fall cycle, the displacement of the two characteristic points had a cliff-like mutation and an evident inflection point appeared; thus, the bank slope was unstable and became damaged during the 16 th water level rise and fall cycle. After the 16 th water level rise and fall cycle, the sliding zone soil body separates from the original slope and slides to a new stable state. The displacement increments gradually decreased, and the displacement gradually tended to be stable. In the 20 th water level rise and fall cycle, the increment of displacement at the bottom of the slope did not change much, but the increment of displacement at the top of the slope abruptly changed again, so it was presumed that the second destabilization of the slope may have occurred. In order to confirm the above speculation, the finite element mesh of the sliding zone of the bank slope appeared to be excessively distorted and transformed into SPH particles. As a criterion, it was believed that when the finite element mesh of the sliding zone of the bank slope was completely transformed into SPH particles, the soil on the sliding zone would detach and slide down, and destabilization damage would occur. The deformation of the bank slope-bridge pier is shown in Figure 8.  
Figure 7: Displacement of characteristic points. As can be seen from Figure 8, the finite element mesh of the shore slope was not transformed into SPH particles in the first 15 water level cycles, and the shore slope was not destabilized. When N = 16, the finite element mesh of the sliding zone of the shore slope has been transformed into SPH particles, so in the 16 th water level lift cycle, the shore slope will be destabilized and damaged. After the shore slope is destabilized, the soil in the sliding zone slides downward and reaches a new equilibrium state. In the 17 th to 19 th water level lift cycles, the equilibrium state was basically unchanged, but the soil arch effect  
Figure 8: Deformation nephograms of bank slope-bridge pier: (**a**) 14 times, (**b**) 15 times, (**c**) 16 times, (**d**) 17 times, (**e**) 18 times, (**f**) 19 times, and (**g**) 20 times. (Unit: mm). on the back side of the pile became more and more obvious, indicating that the soil thrust on the back side of the pile gradually increased and the pile foundation played a blocking role in the slippage of the bank slope deformation instability. When N = 20, a new sliding zone was formed in the soil behind the pile, and a large number of finite element meshes were transformed into SPH particles.