## 2 Numerical Methodology  
### Implementation of the CEL Approach  
The CEL approach is provided by the commercially available software Abaqus, in which an Eulerian domain can be coupled with Lagrangian domains in the same model. [PERSON] et al. [29] and [PERSON] [19] verified its applications in geotechnical problems and spudcan penetration, respectively. In the present model, the spudcan is modeled as a Lagrangian part, while the soil is modeled as an Eulerian part. Eulerian mesh is fixed in the space and allows the soil material to flow through the elements, which can avoid mesh distortion induced in the Lagrangian approach. An Eulerian element can comprise several materials, of which the element volume fraction (EVF) is employed to describe the composition of each element. In contrast, a Lagrangian element can only have one material and deforms with the material. However, during analysis, the spudcan is constrained as a rigid body, as its deformation is not significant and of concern. Moreover, the EVF of each soil material in each Eulerian element is tracked and recorded. The CEL approach with Abaqus can only support 8-node linear hexahedron elements, which implies that only a 3D model is supported by the CEL approach. Due to the symmetry of the problem, only one-fourth of the spudcan and soil are modeled. The Eulerian part is initially divided into the void region and soil region, as shown in Figure 2a. To allow soil sperm development at the soil surface, the void region is initially configured with no soil. The soil region is filled with different soils in regard to the problem. Figure 1: Comparison of simulated resistance profiles in dense sand overlying clay of the existing numerical models: (**a**) loose sand overlying clay [7; 18; 19]; (**b**) medium-loose sand overlying clay [11; 12]; (**c**) dense sand overlying clay [9; 20; 24]. Figure 2: Diagram of the quarter symmetric CEL model: (**a**) Schematic diagram; (**b**) Mesh. In the present model, the distance between the outer boundary to the spudcan center is 6D (\(D\), the spudcan diameter), and the vertical distance between the soil base and the spudcan tip in all the cases is larger than 1.5\(D\), as shown in Figure 2b. According to the studies by [PERSON] et al. [30, 31], the boundary effect can be neglected in the present study. Therefore, unlike some previous models [32, 33, 34], the shearing on the boundaries can be neglected, and the boundary conditions of the model are set symmetric, which can avoid soil leakage during analysis. The Eulerian domain is discretized with 8-node linear hexahedron elements with reduced integration and hourglass control (EC3D8R). As shown in Figure 2b, the mesh of the region within 2\(D\) to the spudcan center is refined, and the influence of mesh density will be discussed later. During each incremental step of the CEL approach, an updated Lagrangian calculation is firstly conducted with an explicit integration scheme, followed by an advective Eulerian phase to map the solution variables (such as material properties, stresses, strains, velocities, and accelerations). The Eulerian material boundaries and interfaces are tracked and updated by updating the element volume fraction of each material. Similarly, the contact interface between the soil and spudcan is updated continuously with the \"general contact\" algorithm, which is based on a frictional contact with the Coulomb friction model using the penalty method. The frictional contact is described by defining a roughness factor in Equation (1):  
\[\alpha=\tan\delta/\tan\phi^{\prime} \tag{1}\]  
in which \(\delta\) is the friction angle between sand and spudcan, \(\phi^{\prime}\) is the effective friction angle of sand. The spudcan will always be in contact with the sand after penetrating the sand layer. [PERSON] et al. (2014) have demonstrated that the shape and volume of the sand plug depend on the roughness of the spudcan [11]. The rougher the spudcan is, the less the strength mobilization is induced in the sand layer, and the more sand is trapped under the spudcan.