## 2 Materials and Methods
### CEL Method  
The slamming simulation of the steel plate included the significant deformation of a fluid, the stress analysis of the structure, and the contact between the fluid and structure. Modeling a fluid region using the classical traditional Lagrangian formulation, which is generally applicable for solid mechanics applications, is almost impossible owing to excessive distortion in the fluid element. Therefore, the fluid region is typically modeled using the Eulerian formulation, which can express the flow of matter within a fixed mesh system. A Lagrangian formulation, in which the nodes are fixed to the material and the element deforms against material deformation, is typically applied to calculate the stress and strain levels for a steel plate. In Eulerian analysis, a material is expressed using the volume-of-fluid technique, and the elements do not deform the material. In the Lagrangian method, the governing equation is based on momentum conservation. The principle of virtual work in finite element methods for this study can be expressed as shown in Equations (1)-(3). \[\int_{V}\rho\ddot{x}\delta xdV=F^{\text{ext}}-F^{\text{int}} \tag{1}\] \begin{table}
\begin{tabular}{c c} \hline \hline
**Model** & **Steel** \\ \hline Dead-rise angle (\({}^{\circ}\)) & 0 \\ \hline Length (m) & 2 \\ \hline Width (m) & 1.2 \\ \hline Height (m) & 0.3 \\ \hline Mass (kg) & 340 \\ \hline Thickness (mm) & \(3,5,8\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Principal dimension of steel plate used in current study [3]. Figure 1: Wet drop test setup [3]. where \(\Phi\) is the flux term, and \(S\) is the source term. In the CEL method, the operator split method is adopted to solve partial differential equations. This method separates Equation (13) into two equations, i.e., Equation (14) for the Lagrangian step involving the source term, and Equation (15) for the Eulerian step, which involves the convective term [21]. \[\frac{\partial\varphi}{\partial t}=S \tag{14}\]  
\[\frac{\partial\varphi}{\partial t}+\
abla\cdot\Phi=0 \tag{15}\]  
Figure 2 shows the operator split method for each step of the CEL method. The deformed mesh based on Equation (14) is remapped to the original position in the Eulerian fixed mesh based on Equation (15). The Lagrangian solution variables, such as mass, momentum, and energy, are adjusted for material flow between adjacent elements [30]. To describe the hydrodynamic response of water, a constitutive equation based on the governing equations shown in Equations (10) to (12) is required. The constitutive equation of water assumed in this study is shown in Equation (16). \[\sigma=-P(x,\ t)I+2\mu R \tag{16}\]  
where \(P(x,\ t)I\) is the pressure at location x and time t, \(\mu\) the dynamic viscosity of water, and R the strain-rate tensor. The hydrodynamic properties of water can be described using an equation of state (EOS). ABAQUS/Explicit uses the Mie-Gruneisen EOS, as shown in Equation (17), with the linear \(U_{s}-U_{p}\) Hugoniot form, as shown in Equation (18). \[P-P_{H}=\Gamma\rho(E_{m}-E_{H}), \tag{17}\]  
where \(P\) denotes the pressure stress, which is defined as positive in compression; \(\Gamma=\Gamma_{0}\frac{\rho_{0}}{\rho}\), where \(\Gamma_{0}\) is a material constant, and \(\rho_{0}\) is the reference density; \(E_{m}\) is the internal energy per unit mass, where \(P_{H}\) and \(E_{H}\) are the Hugoniot pressure and specific energy, respectively; \(E_{H}=\frac{P_{H}\mu}{2\rho_{0}}\), with \(\eta=1-\frac{\rho_{0}}{\rho}\). A typical fit to the Hugoniot data is shown in Equation (18). \[P_{H}=\frac{\rho_{0}c_{0}^{2}\eta}{(1-s\eta)^{2}} \tag{18}\]  
where \(c_{0}\) and \(s\) relate the linear shock velocity \(U_{s}\) and particle velocity \(U_{p}\), as shown in Equation (19). \[U_{s}=c_{0}+sU_{p} \tag{19}\]  
By combining Equations (17) and (18), the linear \(U_{s}-U_{p}\) Hugoniot form can be written as shown in Equation (20).