## 2 Numerical Modelling
### Validation of the Basic Model  
FLAC3D was employed to build the numerical model. In this model, soil is treated as an elastoplastic material using the Mohr-Coulomb failure criterion. Soil-pile interaction is simulated using interface elements, whose friction and cohesion are 60% of the soil parameters [13]. This study is focused on the effect of soil-pile interaction and soil nonlinear on the natural frequency, so pile and structure are assumed to be linear and elastic. The damping of structure is only considered 1.5% [24], but aerodynamic damping or other damping are not considered in this model. [PERSON] [25] derived how soil relative density impacts its elastic modulus using soil unit volume tests across different stress states. [PERSON] et al. [26] proposed an equation to reflect soil elastic modulus in the different stress state by reviewing these experimental results. Equation (1) was used to obtain the elastic modulus at different depths in the numerical model. The boundary condition is a fixed boundary. \[E_{0}=k\cdot\sigma_{at}\cdot\left(\frac{\sigma_{3}}{\sigma_{at}}\right)^{\lambda} \tag{1}\]  
\[\sigma_{3}=K_{0}\gamma z \tag{2}\]  
where \(E_{0}\) is the initial elastic modulus of the soil (kPa). Coefficients \(k\), \(\lambda\), and \(n\) are defined by sandy soil relative density functions, while \(\sigma_{3}\) denotes soil minimum principal stress (kPa), and \(\sigma_{\text{at}}\) represents atmospheric pressure with a value of \(10^{5}\) kPa. \(z\) is the soil depth (m). \(K_{0}\) denotes the coefficient of lateral earth pressure at rest. Monopile is the common foundation in the offshore wind farm, so the primary objective here was to analyze the natural frequency of monopile-supported offshore wind turbines. Specifically, this study calculated the natural frequency of a Vestas 3 MW turbine at the Belwind wind farm. Hub height is 94 m, and the transition section is 22 m. Density of the tower structure is 8500 kg\(\cdot\)m\({}^{-3}\). The dimension of the monopile foundation is 5 m in diameter, 60 mm in wall thickness, and 20.9 m in embedment depth. The density of the monopile is 7800 kg\(\cdot\)m\({}^{-3}\), and the elastic modulus is 210 GPa. Figure 2 shows all the specific parameter details. The specific parameters of this wind turbine were presented in Refs. [27; 28]. Site soil condition is mainly sand. Soil depth 0-15 m is sand, and the angle of internal friction is 40\({}^{\circ}\). Soil depth 15-17 m is clay, and the ultimate resistance is 175 kPa. Soil depth 17-26 m is sand, and the angle of internal friction is also 40\({}^{\circ}\). Based on the above parameters of structure and soil, which is as shown in Figure 3. The dimension of soil site is 100 m \(\times\) 100 m \(\times\) 50 m. The overall natural frequency of offshore wind turbine structures was computed using the numerical model established above. A sensitivity analysis of mesh size is conducted to obtain the well mesh. The numerical model is shown in Figure 4a. Wedge grid is used to create soil, and the number  
Figure 2: Geometry of wind turbine structures (**(a)**tower; **(b)**transition piece; **(c)** monopile). of mesh is 44,172. A brick grid serves as the basis for creating the wind turbine structure, and the sum of mesh is 1056. The detail model is shown in Figure 4.  
The free vibration method was employed to derive the natural frequency. First, lateral displacements of 10 cm and 5 cm were imposed at the tower's top, respectively. Subsequently, the tower top was released so as to capture the dynamic response displacement of the structure at the tower's top, with the natural frequency calculated as the tower base being fixed, as shown in Figure 5. Following FFT processing of Figure 5a, the derived tower-top response power spectrum (Figure 6a) reveals a structural natural frequency of 0.353 Hz. This outcome arises because the soil surrounding the monopile remains elastic, exhibiting no plastic deformation. The foundation stiffness is the same, so the calculated natural vibration period is the same.