## 2 Numerical Modeling of Ropes and Large Floating Structures Based on the ANCF
### The Coordinate and Parameter Description of Cable Elements  
The ANCF method has been employed in multibody dynamics for the past twenty years. This method imposes no restrictions on deformation or rotation, rendering it suitable for both linear and nonlinear problems. Generally, it is particularly suitable for large deformation and rotation problems and can precisely describe the position of rigid body motion with finite rotations [34]. As shown in Figure 2, the global position vector \(r\) of any point \(p\) on the neutral axis of a two-dimensional beam element is defined based on the nodal coordinates and element shape functions. \[\mathbf{r}= \begin{bmatrix}r_{1}\\ r_{2}\end{bmatrix} =\mathbf{N}\mathbf{q} \tag{1}\]  
where \(\mathbf{N}\) represents the global shape function with complete rigid body modes, forming a \(2\times 8\) matrix capable of describing arbitrary translations and rotations of rigid bodies; \(\mathbf{q}\) is the vector of nodal coordinates of the element  
\[\mathbf{q}= \begin{bmatrix}e_{1}&e_{2}&e_{3}&e_{4}&e_{5}&e_{6}&e_{7}&e_{8} \end{bmatrix} ^{T} \tag{2}\]  
where the superscript \({}^{T}\) indicates the transpose of the vector or matrix. Including global displacements and the global slopes of the element nodes, this is defined as  
\[e_{1}=r_{1}|_{x=0},\ e_{2}=r_{2}|_{x=0},\ e_{5}=r_{1}|_{x=l_{0}},\ e_{6}=r_{2 }|_{x=l_{0}} \tag{3}\]  
\[e_{3}=\frac{\partial r_{1}}{\partial x}|_{x=0},\ e_{4}=\frac{\partial r_{2}} {\partial x}|_{x=0},\ e_{7}=\frac{\partial r_{1}}{\partial x}|_{x=l_{0}},\ e_{8}=\frac{ \partial r_{2}}{\partial x}|_{x=l_{0}} \tag{4}\]  
where \(x\) denotes the coordinate of any point on the undeformed beam element; \(l_{0}\) represents the original length of the beam element (\(x=0\) at node one and \(x=l_{0}\) at node two, as illustrated in Figure 2); and \(x=[0,\ l_{0}]\). This method eliminates weak influence terms in the dynamic model of flexible beams, thereby reducing the absolute number of coordinates for the elements and enhancing computational efficiency. One-dimensional Hermite interpolation is employed for the interpolation of ANCF elements, effectively describing the two components of displacement. Therefore, the global shape function \(\mathbf{N}\) can be expressed as  
\[\mathbf{N}= \begin{bmatrix}n_{1}\mathbf{I}&n_{2}\mathbf{I}&n_{3}\mathbf{I}&n_{4} \mathbf{I}\end{bmatrix}  \tag{5}\]  
\[ \begin{cases}n_{1}=1-3\xi^{2}+2\xi^{3}\,,&n_{3}=l_{0}(\xi-2\xi^{2}+\xi^{3})\\ n_{2}=3\xi^{2}-2\xi^{3}\,,&n_{4}=l_{0}(-\xi^{2}+\xi^{3})\end{cases} \]  
where \(\mathbf{I}\) is the \(2\times 2\) identity matrix; and \(\xi=x/l_{0}\), \(\xi\in[0,1]\) is a mapping of local coordinates along the arc length of the element. The interpolation functions can be represented by \(x\), as depicted in Equations (6) and (7), utilizing prime notation in the upper right corner to perform interpolation during strain. \[\mathbf{N}^{{}^{\prime}}=\frac{\partial\mathbf{N}}{\partial x}= \begin{bmatrix} n_{1}^{\prime}\mathbf{I}&n_{2}^{\prime}\mathbf{I}&n_{3}^{\prime}\mathbf{I}&n_{4}^{ \prime}\mathbf{I}\end{bmatrix}  \tag{6}\]  
\[\mathbf{N}^{{}^{\prime}}=\frac{\partial^{2}\mathbf{N}}{\partial x^{2}}= \begin{bmatrix}n_{1}^{\prime\prime}\mathbf{I}&n_{2}^{\prime\prime}\mathbf{I}&n _{3}^{\prime\prime}\mathbf{I}&n_{4}^{\prime\prime}\mathbf{I}\end{bmatrix}