# Numerical Simulation of a Submerged Floating Tunnel: Validation and Analysis
## 2 CEL Method for One- and Two-Way Fluid-Structure Coupling Validation  
### CEL Method  
Lagrangian methods are widely used for static analysis in geotechnical and structural engineering. [PERSON] and [PERSON] [21] simulated slope stability problems, and [PERSON] et al. [22] analyzed the ultimate bearing capacity of sandy soil foundations based on the limit analysis method. In the Lagrangian method, the cell grid is unified with the object to be analyzed. Thus, during calculation, the shape of the analyzed object remains the same, while that of the finite element mesh changes, and the material does not flow from cell to cell.  
In contrast, the Eulerian method is characterized by a fixed finite element mesh in terms of shape, size, and spatial location throughout the numerical computation. Material can flow between the mesh and the grid, so the method is ideal for dealing with material distortion and certain large deformation problems [23]. However, the Eulerian method struggles to accurately capture boundary information about the objects and materials, and so is typically used in the computational analysis of fluid dynamics.  
Pure Lagrangian and pure Eulerian methods each have strengths and weaknesses, and by combining these methods, it is possible to take advantage of their strengths while overcoming their weaknesses. The CEL method was devised by [PERSON] [24] for this purpose, and has since been used in fluid-solid coupling analysis, where the Lagrangian mesh is used to discretize the solid, whereas the Eulerian mesh is used to discretize the fluid domain. The Lagrangian mesh can pass through the fluid domain formed by the Eulerian mesh, thereby defining the contact between the fluid and solid materials. The CEL method has also been used to numerically simulate a variety of phenomena, including the scouring of vertical breakwaters [25] and a steel water-storage tank under blast loading [26], and has been employed for the development of a Reynolds-averaged Eulerian-Lagrangian model of vortex interaction that simulates the turbulent suspension of flow in a thin plate [27].  
Specifically, the basic idea of the CEL method is to divide the computational domain into small Eulerian regions, with the physical quantities within each Eulerian region being described by a Lagrangian function. During the time-stepping process, the physical quantities within the Eulerian regions are first calculated, and then these quantities are transferred to the adjacent Eulerian regions through interpolation, forming a complete Lagrangian description of the computational domain. Therefore, the CEL method can control the accuracy of different spatial scales through multi-level grids, and can adapt to different physical processes via the use of different time steps.  
The calculation process of the CEL method includes the following steps:  
1. Partitioning Eulerian regions: Divide the computational domain into several small Eulerian regions.
2. Constructing Lagrangian functions: For each physical quantity within each Eulerian region, construct a corresponding Lagrangian function.
3. Solving Euler equations: For each physical quantity within each Eulerian region, use the Euler method to solve its corresponding Euler equation.
4. Interpolation transfer: Use interpolation methods to transfer physical quantities within Eulerian regions to adjacent Eulerian regions, forming a complete Lagrangian description of the computational domain.
5. Time stepping: Repeat steps 3 and 4 to perform time-stepping calculations.