## 2 Numerical Modeling of Ropes and Large Floating Structures Based on the ANCF
### Elastic Internal Forces and Stiffness Matrix  
In the static and dynamic analysis of flexible bodies, the stiffness matrix plays a crucial role. Furthermore, deriving elemental elastic forces is a prerequisite for obtaining the elemental stiffness matrix. For flexible body elements, strain energy is regarded as the sum of axial strain energy \(U_{E}\) and bending strain energy \(U_{Q}\). Assuming material isotropy, the  
Figure 2: ANCF framework and coordinate system. total strain energy can be obtained by integrating these corresponding strain energies as follows:  
\[U=U_{E}+U_{Q}=\frac{1}{2}\int_{0}^{l_{0}}\left(EA\varepsilon^{2}+E\imath k^{2} \right)dx \tag{11}\]  
where \(E\) represents the Young's modulus, \(A\) and \(I\), respectively, denote the cross-sectional area and moment of inertia of the structure, \(\varepsilon\) represents only the axial Green strain, and \(\kappa\) represents the curvature to be integrated to the local coordinate \(x\) in the element. According to continuum mechanics, the Green strain tensor stipulates that the Green strain at any point can be expressed by Equation (12) [35]:  
\[\varepsilon=\frac{1}{2}\left(\mathbf{J}^{\mathsf{T}}\mathbf{J}-\mathbf{I}\right) \tag{12}\]  
where \(\mathbf{J}\) is the Jacobian matrix of the displacement gradient vector. Here, expressed using the Frechet derivative with respect to both global absolute coordinates and local coordinates, the expression can be represented in the form of a Jacobian matrix as follows. \[\mathbf{J}=\frac{\partial\mathbf{r}}{\partial\mathbf{x}}= \begin{bmatrix} \frac{\partial\mathbf{r}_{1}}{\partial x_{1}}&\frac{\partial\mathbf{r}_{1}}{ \partial x_{2}}\\ \frac{\partial\mathbf{r}_{2}}{\partial x_{1}}&\frac{\partial\mathbf{r}_{2}}{ \partial x_{2}}\end{bmatrix}  \tag{13}\]  
However, when considering deformation solely along the length direction of the beam, the Green strain tensor has only one component, which is \(\varepsilon_{11}\). The axial strain at any point \(p\) can be represented by the following formula. \[\varepsilon_{11}^{p}=\frac{1}{2}\Bigg{[}\Bigg{(}\frac{\partial\mathbf{r}^{p} }{x_{1}}\Bigg{)}^{T}\frac{\partial\mathbf{r}^{p}}{x_{1}}-1\Bigg{]} \tag{14}\]  
In Figure 3, the first-order derivative of interpolation function \(\mathbf{N}\) with respect to absolute coordinate \(\mathbf{r}^{p}\) represents the tangent vector, which is used for calculating axial strain; the second-order derivative of interpolation function \(\mathbf{N}\) with respect to absolute coordinate \(\mathbf{r}^{p}\) represents the normal vector, which is used for calculating curvature. The axial strain and curvature of the highly deflected beam are represented as follows. \[\varepsilon=\frac{1}{2}\Bigg{[}\Bigg{(}\frac{\partial\mathbf{r}^{p}}{x_{1}} \Bigg{)}^{T}\frac{\partial\mathbf{r}^{p}}{x_{1}}-1\Bigg{]}=\frac{1}{2}\Big{(} \mathbf{q}^{\mathsf{T}}\mathbf{N}^{\mathsf{T}}\mathbf{N}^{\mathsf{`}}\mathbf{ q}-1\Big{)} \tag{15}\]  
\[\kappa=\frac{\left|\mathbf{r}^{\mathsf{`}}\times\mathbf{r}^{\mathsf{`}} \right|}{\left|\mathbf{r}\right|^{3}} \tag{16}\]  
Figure 3: Tangent and normal directions denoted by \(\mathbf{r}^{\mathsf{`}}\) and \(\mathbf{r}^{\mathsf{`}}\). However, the denominator in Equation (16) is typically set to 1 because the curvature is approximated as the second spatial derivative of the beam lateral deflection [36]. For two-dimensional problems, Equation (16) can be expressed as follows:  
\[\kappa=\frac{\mathbf{r}^{\ast T}\widetilde{\mathbf{I}}\mathbf{r}^{\prime}}{ \left|\mathbf{r}^{\ast T}\mathbf{r}^{\prime}\right|^{3/2}}=\frac{\mathbf{r}^{ \ast T}\widetilde{\mathbf{I}}\mathbf{r}^{\prime}}{f^{3}} \tag{17}\]  
where \(\widetilde{\mathbf{I}}= \begin{bmatrix}0&-1\\ 1&0\end{bmatrix} \). In the special case of minor longitudinal deformation, the expression for curvature simplifies. This occurs only when \(f^{3}\simeq 1\), representing the second derivative of curvature, is derived from the approximate value of curvature.