## 2 Simulation Methodology
### FEM-SPH Adaptive Algorithm  
#### 2.2.1 Conversion of Finite Elements into Smoothed Particles  
The FEM and SPH are crucial to numerical computation, but they have some limitations. The FEM delivers poor accuracy of solutions for the problem of large deformations such as damage to sea ice. The SPH is less efficient in computation, and difficult to apply the boundary conditions. In order to resolve these problems, an FEM-SPH adaptive algorithm is adopted in this paper to simulate sea ice so as to effectively integrate the strengths of both methods: high computational efficiency and stable boundary conditions of the FEM and the applicability to solve deformation problems of the SPH [38].  
The basic idea of the FEM-SPH adaptive algorithm is to adopt the FEM for the sea ice model at the initial stage and take strain as the basis for judging element conversion. In the computation, if the strain of any finite element exceeds its critical value, the finite element is converted into a smoothed particle. The smoothed particle, which inherits such physical information as pressure, density, and velocity from the finite element, will still play a part in the subsequent computation [39]. In this paper, the critical strain value is 0.35 [32].  
The conversion of finite elements into smoothed particles is illustrated in Figure 2. In the beginning, the FEM is employed to discretize the sea ice model. With the collision of the vehicle with sea ice, some finite elements experience mesh distortion. When the distortion of finite elements reaches a certain level (i.e., the critical strain of elements), the finite elements are converted into smoothed particles.  
After conversion, the smoothed particles will take the physical quantities of failed finite element nodes as their initial physical quantities.  
\[\left\{\begin{array}{l}m_{p}=m_{n}=\sum\limits_{i=1}^{N_{c}}\frac{\rho_{ci}V _{ei}}{N_{n}}\\ v_{p}=v_{n}\\ s_{p}=s_{n}\\ c_{p}=c_{n}\\ \sigma_{p}=\sigma_{n}=\frac{1}{N_{c}}\sum\limits_{i=1}^{N_{c}}\sum\limits_{j= 1}^{N_{c}}\omega_{xj}\sigma_{xj}\end{array}\right. \tag{17}\]  
where \(c\) and \(\sigma\) stand for the sound velocity and stress tensor, respectively; the subscript \(n\) represents the node of a finite element; the subscript \(p\) indicates the smoothed particle; the subscript \(\chi\) stands for the Gaussian integral point; the subscript \(e\) denotes the finite element; \(N_{n}\), \(N_{e}\), and \(N_{\chi}\) are the number of nodes in each finite element, the number of finite elements linked to the node, and the number of Gaussian integral points in the  
Figure 2: Schematic diagram for the conversion of finite elements into smoothed particles.  
finite element; \(w_{\chi j}\) and \(\sigma_{\chi j}\) are the coefficient and stress tensor at the Gaussian integral point, respectively.  
After obtaining the above physical quantities, we can calculate the strain rate tensor \(\varepsilon^{.\alpha\beta}\) of the SPH particles as follows:  
\[\varepsilon^{.\alpha\beta}=\frac{1}{2}\sum_{j=1}^{N}\frac{m_{j}}{\rho_{j}}[( \upsilon_{j}^{\alpha}-\upsilon_{i}^{\alpha})\frac{\partial W_{ij}}{\partial \varsigma_{i}^{\beta}}+(\upsilon_{j}^{\beta}-\upsilon_{i}^{\beta})\frac{ \partial W_{ij}}{\partial\varsigma_{i}^{\beta}}] \tag{18}\]  
Based on the conservation of mass, the smoothing length \(h_{p}\) of SPH particles can be calculated as follows:  
\[h_{p}=\frac{\zeta}{N_{e}}{\sum_{i}^{N_{e}}d_{0i}(\frac{\rho_{0i}}{\rho_{i}})}^ {1/3} \tag{19}\]  
where \(\rho_{0}\) is initial density; \(\zeta\) indicates the scale factor of the smoothing length and the finite element size; and \(d_{0}\) stands for the initial finite element size.