## 2 Simulation Methodology
### S-ALE Fluid/Structure Coupling Algorithm  
The fluid/structure coupling effect occurs among the vehicle, ice, and fluid. In the conventional Lagrangian method, a finite element mesh can be adopted to describe the geometry of an object, but it deforms with the motion of the object. Therefore, mesh distor  
Figure 3: Flowchart of the FEM-SPH adaptive algorithm.  
tion is easily caused in the analysis of large deformation, which lowers the computational accuracy and even terminates the procedure. For this reason, it is not applicable to the simulation of water and air flow characteristics. Differently, the arbitrary Lagrangian-Eulerian (ALE) algorithm allows the mesh to move arbitrarily and vary flexibly in space so that it has been widely employed in the calculation of fluid/structure interaction [41].  
The structured arbitrary Lagrangian-Eulerian (S-ALE) algorithm is an improved algorithm based on the ALE algorithm, which keeps the same interface reconstruction method as the ALE algorithm. However, it is superior in the following aspects: (1) the S-ALE algorithm can generate orthogonal meshes internally, and it is and easy to maintain and occupies less memory; (2) it is more efficient because of parallel computation; (3) the solution is stable and can effectively solve the fluid leakage problem that often occurs in fluid/structure interaction algorithms. During fluid/structure interaction calculations, the fluid mesh deforms as the fluid moves and the S-ALE algorithm maintains the boundary conditions of the deformed fluid unchanged, performs the remeshing of the internal elements, keeps the topological relationships of the mesh unchanged, and transfers physical quantities such as density and energy from the pre-deformed mesh to the new mesh. By remeshing at the fluid/structure interface, the impact of mesh deformation on computational accuracy can be reduced.  
The S-ALE algorithm adopts a reference coordinate system, and the time derivative of the material in connection with the reference coordinate system can be written as  
\[\frac{\partial f(X,t)}{\partial t}=\frac{\partial f(x^{a},t)}{\partial t}+w^{ a}\frac{\partial f(x^{a},t)}{\partial x^{a}} \tag{25}\]  
where \(X\) is the displacement in the Lagrangian coordinate system; \(x\) is the displacement in the Eulerian coordinate system; and \(w=v-u\), where \(v\) is the velocity of the material and \(u\) is the moving speed of the reference coordinate system. The control equations of the S-ALE algorithm are as follows:  
\[\left\{\begin{array}{l}\rho\frac{\partial v^{a}}{\partial t}=\sigma^{a\beta} +\rho b^{a}-\rho w^{a}\frac{\partial v^{\beta}}{\partial x^{\beta}}\\ \frac{\partial v}{\partial t}=-\rho\frac{\partial v^{\beta}}{\partial x^{a}}- w^{a}\frac{\partial v}{\partial x^{a}}\\ \rho\frac{\partial U}{\partial t}=\sigma^{a\beta}\tau^{a\beta}+\rho b^{a}v^{a}- \rho w^{b}\frac{\partial U}{\partial x^{b}}\end{array}\right. \tag{26}\]  
where \(b\) is the unit volume force, and \(\tau^{a\beta}\) is the viscous shear force. The S-ALE algorithm uses overlapping grid technology to handle interpolation issues between fluid grids and solid grids.  
In order to solve the energy dissipation and pressure oscillation problems caused by artificial numerical damping in the fluid/structure interaction process, this paper adds an artificial viscosity formula to the S-ALE algorithm.  
\[P_{q}=\begin{cases}\rho(C_{q}cd_{a}|\dot{e}_{V}|+C_{l}\dot{r}\dot{e}_{V}^{2})& \dot{e}_{V}<0\\ 0&\dot{e}_{V}\geq 0\end{cases} \tag{27}\]  
where \(P_{q}\) is the artificial viscosity pressure added to the actual pressure, \(d_{a}\) is the feature length, \(C_{q}\) is linear artificial viscosity coefficient, \(C_{l}\) is secondary artificial viscosity coefficient, \(\dot{\epsilon}_{V}^{2}\) is volume strain rate. In this paper, \(C_{q}\) is 1.5, and \(C_{l}\) is 0.06.