## 2 Theoretical Background
### Coupled Eulerian-Lagrangian Method  
In simulations involving fluid-structure interactions, such as sloshing in a liquid tank, it is essential to account for significant fluid deformation and the interface between the fluid and the solid structure. Directly applying the traditional Lagrangian formulation to model the fluid domain can be impractical due to excessive distortion of fluid elements, particularly under high-deformation conditions. Instead, the Eulerian formulation is typically used for the fluid, as it maintains a fixed mesh while allowing fluid flow to be described within this mesh system. For the solid structure, the Lagrangian formulation is typically employed, where the mesh moves with the material, enabling an accurate representation of its deformation and the calculation of stress and strain within the solid. The fluid region is modeled using techniques such as the volume-of-fluid method, which prevents mesh deformation and allows for the tracking of fluid flow through fixed grid points [16]. The Eulerian formulation for fluid dynamics is described by the conservation equations for mass, momentum, and energy, as presented in Equations (7)-(9). To convert the Lagrangian descriptions of these variables into a Eulerian form, the relationship between the time derivatives in the two frameworks is given by Equation (10). \[\frac{D\rho}{Dt}+\rho\
abla\cdot v=0 \tag{7}\]  
\[\rho\left(\frac{Dv}{Dt}\right)=\
abla\cdot\sigma+\rho b \tag{8}\]  
\[\frac{DE}{Dt}=\
abla\cdot(\sigma\cdot v)+\rho b\cdot v \tag{9}\]  
\[\frac{D\varphi}{Dt}=\frac{\partial\varphi}{\partial t}+v\cdot(\
abla\varphi) \tag{10}\]  
where \(\rho\) is the density, \(v\) is the material velocity, and \(\sigma\) is the Cauchy stress. \(E\) is the total energy per unit volume. \(\varphi\) is an arbitrary solution variable. \(D\varphi/Dt\) and \(\partial\varphi/\partial t\) are the material and spatial-time derivatives of \(\varphi\), respectively. To simplify the fluid dynamics, the conservation equations are further modified, leading to a system of equations as shown in Equations (11)-(13), where \(R\) is the strain-rate tensor. The system is expressed in the general conservation form as presented in Equation (14), which governs both mass and energy conservation in the Eulerian formulation. \[\frac{\partial\rho}{\partial t}+\
abla\cdot(\rho v)=0 \tag{11}\]  
\[\frac{\partial\varphi v}{\partial t}+\
abla\cdot(\rho v\otimes v)=\
abla \cdot\sigma+\rho b \tag{12}\]  
\[\frac{\partial e}{\partial t}+\
abla\cdot(ev)=\sigma:R \tag{13}\]  
\[\frac{\partial\varphi}{\partial t}+\
abla\cdot\Phi=S \tag{14}\]  
where \(\Phi\) is the flux term, and \(S\) is the source term. The CEL method solves these equations by using the operator split method, which divides the solution process into two distinct steps: one for the Lagrangian update of material properties and the other for the Eulerian update of the convective terms. In the Lagrangian step, the material properties are updated by solving the source terms, while in the Eulerian step, the convective terms are evaluated based on the fixed grid. This coupling enables the accurate simulation of complex fluid-structure interactions, where both fluid flow and solid structure deformation are considered simultaneously. In a typical simulation, the deformed mesh is remapped back to the original Eulerian grid, with material flow between adjacent elements adjusted accordingly. The hydrodynamic response of the water is described by the constitutive equation, which accounts for the pressure and strain-rate of the fluid, as shown in Equation (15). The behavior of the water is modeled using the Mie-Gruneisen equation of state (EOS), as shown in Equation (16), incorporating the linear \(U_{s}-U_{p}\) Hugoniot form, as shown in Equation (17). This formulation is integrated into the solver to capture the dynamic pressure and energy relationships during the fluid-structure interaction process.