## Finite Element Modeling Approach
### Establishment of Local Model  
As shown in Figure 1d, the actual fracture position of the mid-phase jumper is the outlet position of the tension clamp. In order to accurately analyze the stress distribution of the actual fracture position, a local model was established. The local model established in this study consisted of four helical layers wound around a straight cylinder at the center. In addition, adjacent spiral layers were wound in the opposite directions to reduce the internal torque caused by the winding. The formula for calculating the radius of each helical layer is shown in Equation (1). The formula for calculating the helix angle is shown in Equation (2). In Equations (1) and (2), \(R_{i}\), \(d_{i}\), \(d_{\text{core}}\), \(m_{\text{i}}\), and \(ai\) indicate the layer radius (mm), wire diameter (mm), diameter of the central steel wire (mm), pitch diameter ratio, and helical angles of the helical layers, respectively.  
\[R_{i}=\frac{d_{\text{core}}}{2}+\frac{d_{i}}{2}+\sum_{k=1}^{k=i-1}d_{k} \tag{1}\]  
\[a_{\text{i}}=\arctan(\frac{m_{\text{i}}}{\pi}) \tag{2}\]  
Table 1 shows the mechanical properties of the materials and modeling parameters of the local finite element model. Figure 3a shows the sectional diagram of the local finite element model. As shown in Figure 3b, a section of all strands was fixed at one end of the local model. The fixed end was located at the outlet of the tension clamp. By setting the motion coupling constraints in Abaqus, the other end was coupled with the central point, all six degrees of freedom of the constrained area were selected to be coupled with the coupling points, and the constrained end face then became a rigid plane. There was no relative displacement between the nodes in the constrained area, only a rigid motion with the selected coupling point. In this case, the displacement boundary condition (Y\({}_{\text{a}}\)) could be applied to the coupling point.  
Figure 2: (**a**) Structural diagram of a mid-phase jumper, (**b**) side view of a mid-phase jumper on a tower, and (**c**) finite element model of a mid-phase jumper in Abaqus.  
The hexahedral incompatible element (C3D8I) was used in the local model. The number of elements and nodes were approximately \(1.3\times 10^{5}\) and \(1.8\times 10^{5}\), respectively. The default contact behavior in the Abaqus/Standard was the \"hard contact.\" The surface contact was set between each strand. Although numerous contact algorithms can be used to establish contact pairs, the contact is extremely complex owing to the large number of surface-to-surface contacts involved. Therefore, the contact algorithm was set as the augmented Lagrange algorithm, which improved the convergence efficiency of the model operation under the premise of ensuring the accuracy. The default value of the contact stiffness was adopted, and the expansion coefficient of the contact stiffness was 1.  
Although [PERSON] and [PERSON] (2014) explored the friction effect between steel wires, they were not specified regarding the friction coefficient. [PERSON] et al. (2015) and [PERSON] et al. (2019) applied a friction coefficient \(\mu\) of 0.115 to all contact points. The present simulation used the same coefficient value. When the mid-phase jumper sways significantly under wind load, the main deformation of the mid-phase jumper is bending. Therefore, the bending stiffness of the whole model was made the same as that of the local model by adjusting the Young's modulus parameters, and the adjusted Young's modulus was then input into the whole model for the calculations. Finally, the motion behavior of the whole model under wind load was obtained. The adjustment process of the Young's modulus of the whole model is described in detail in Section 3.