## 3 Geometric Model, Computational Mesh, and Physical Parameter
### Model for the Jack-Up Platform-Seabed Foundation Dynamics  
The finite element mesh for the jack-up platform-seabed foundation system is shown in Figure 3. There are a total of 394,697 elements. Among them, the seabed foundation is discretized by the three-dimensional eight-nodes element C3D8P, which includes the degree of freedom for pore pressure. There are a total of 369,308 elements and 388,740 nodes in the domain of the seabed foundation. Since the seabed foundation surrounding the pile legs of the jack-up platform is a key area of concern, the mesh in these zones is refined, and the minimum mesh size is only 0.1 m. In the zone far away from the pile legs of the jack-up platform, the mesh size is generally 0.5 to 3.5 m. The jack-up platform is discretized by the three-dimensional eight-nodes element C3D8 and the ten-nodes tetrahedral element C3D10, in which there is no degree of freedom for pore pressure. There are a total of 20,500 C3D8 elements and 4889 C3D10 elements in the domain of the jack-up offshore platform. In this study, the Biot \"u-p\" equation is used to describe the stress balance and pore water seepage for the seabed foundation. Meanwhile, the Mohr-Coulomb (MC) model is used to describe the mechanical behavior of the seabed soil. It should be noted that the MC model is just an ideal elastoplastic model that can only approximately describe the behavior of soil. The physical property parameters of the seabed soil used are listed in Table 3. It is known that there are many constitutive soil models available in the numerical simulation of geotechnical engineering problems. Among them, the Mohr-Coulomb model is one of the most widely used in practical engineering design. The yield surface of the MC model is:  
\[f=\tau-\sigma\tan\varphi-C=0 \tag{1}\]  
where \(\sigma\) is the normal stress, \(\tau\) is the shear stress, \(C\) is the cohesion of soil, and \(\varphi\) is the internal friction angle of soil. The expression of the MC model in the principal stress space is:  
\[f=(\sigma_{1}-\sigma_{3})+(\sigma_{1}+\sigma_{3})\sin\varphi-2C\cos\varphi=0 \tag{2}\]  
where \(\sigma_{1}\) and \(\sigma_{3}\) are the maximum and minimum principal stresses, respectively. Figure 4 shows the shape of the yield surface of the MC model in the principal stress space and on the \(\pi\) plane. It can be seen in Figure 4 that the yield surface of the MC model in the principal stress space is an open hexagonal pyramid, and its projection on the \(\pi\) plane is a hexagon. In numerical modeling, since the plastic flow direction at the six vertexes is not unique, it will bring great difficulty to the numerical computation. However,  
Figure 3: Generated mesh for the offshore jack-up platform and its seabed foundation (noted: the four pile legs are thin-walled and hollow). the ABAQUS software has smoothed the six vertexes of the hexagon, guaranteeing the uniqueness of the plastic flow direction on the yield surface of the MC model. As a result, the numerical stabilization in modeling can be greatly improved. The designed structural weight of the whole jack-up platform is 4000 t. The load imposed on the upper surface of the box-type platform is set as 4000 t, to consider the weight of other functional structures and the active loads on the platform. Since the end of the pile legs are designed as tapered closed form, the four partially submerged pile legs can provide strong buoyancy, which is beneficial to reduce the requirement for the bearing capacity of the seabed foundation. According to the water depth and the geometrical size of the model, the buoyancy of the whole jack-up platform is estimated as 25,120 kN. Considering the dead weight of the whole platform, and the vertical load on the upper surface of the box-type platform, as well as the buoyancy effect, the equivalent density of the entire jack-up platform is determined to be 41 kg/m\({}^{3}\). Since the whole Jack-up platform is made of steel, the elastic model is adopted to describe its behavior, with [PERSON]'s modulus = 200 GPa and [PERSON]'s ratio = 0.25.