## 2 Methodology
### Numerical Simulation Methods  
The numerical simulation method, as a prevalent approach for analyzing spudcan-soil interactions, facilitates the high-fidelity simulation of soil flow dynamics and backflowing mechanisms during spudcan penetration. The size of the spudcan of new-generation WTIVs is large, and its penetration stage is often accompanied by a large range of extrusion and heave of the surrounding soil, representing a classical geotechnical large-deformation problem. The large deformation finite element (LDFE) method can simulate the process of unloading and backfilling of the restored soil more realistically by establishing an axisymmetric or three-dimensional finite element model. Its predictions exhibit strong concordance with empirical observations and centrifuge test data. At present, the CEL nonlinear material large deformation finite element method, which combines the advantages of the Lagrange method and the Euler method, effectively resolves challenges associated with continuous penetration-induced soil failure and mesh distortion [20]. This method can be implemented using the explicit calculation module of the general software Abaqus (Explicit), in which the soil is modeled using the Euler element, and the position is fixed without distortion. The mesh element can be occupied by one or more materials, and the flow trajectory of the soil material is described via element volume fraction (EVF) tracking. Compared with the implicit calculation, the explicit calculation does not need to be iterated, but its stability still needs to be guaranteed via the critical stability time increment. The estimation formula of (\(\Delta t_{crit}\)) is as follows [21]:  
\[\Delta t_{crit}=L_{min}\sqrt{\frac{\rho}{E}} \tag{3}\]  
Figure 8: Anti-overturning and sliding centrifuge model test: (**a**) general layout scheme; (**b**) overall loading arrangement and sensor layout diagram. where \(L_{min}\) is the minimum mesh size (m), \(\rho\) represents the material density (kg/m\({}^{3}\)), and \(E\) denotes the elastic modulus (Pa). Parametric sensitivity analyses of the mesh size and penetration rate are essential for computational optimization. Based on the data of the 150\(\times\)\(g\) centrifugal model test, a prototype-scale numerical model of the spudcan-soil interactions of WTIV was developed. A quarter-symmetry soil domain was implemented to enhance computational efficiency (Figure 9), with the soil discretized using Eulerian elements (EC3D8R). The model depth was set to 50 m, and a 10 m thick cavity layer was set on the surface to accommodate the soil heave induced by penetration and prevent the material boundary from overflowing. Based on the geometric parameters of the actual ship, the Lagrange element (C3D10M) was used to simulate the rigid characteristics (the elastic modulus of steel is significantly higher than that of soil, and the deformation can be ignored). The values of the parameters used in the numerical simulations are given in Table 4, and the detailed settings are described as follows:  
1. The grid division strategy of the computational domain: The core area of 1.5 \(L\)\(\times\) 1.5 \(L\) around the spudcan adopted a dense grid, and the unit size was determined to be 0.03 \(L\) (\(\approx\)0.25 m) using grid convergence analysis. 2. The boundary conditions and contact settings: The normal velocity constraint (\(v_{n}\) = 0) was applied to the lateral surfaces, and the bottom boundary was fixed in all degrees of freedom (a 5 m thick sand cushion was placed at the bottom of the clay model). The general contact algorithm was used to define the spudcan-soil interaction. The normal behavior adopted penalty function contact, and the tangential behavior was set as a frictionless model (based on the assumption of clay's low-permeability characteristics). 3.