## 2 Numerical Modeling of Ropes and Large Floating Structures Based on the ANCF
### Elastic Internal Forces and Stiffness Matrix
#### 2.2.2 Bending Force  
The first derivative of curvature can be derived by substituting Equations (9) and (10) into Equation (16).  
\[\begin{array}{ll}\frac{\partial\epsilon}{\partial\mathbf{q}}&=\frac{\left| \mathbf{r}\right|^{3}\cdot\frac{\left(\mathbf{r}\cdot\mathbf{r}^{\prime} \right)^{T}}{\left|\mathbf{r}^{\prime}\right|^{6}}\cdot\left(\mathbf{\vec{r }}\cdot\mathbf{N}^{\prime^{*}}-\mathbf{\vec{r}}^{*}\cdot\mathbf{N}^{\prime} \right)-3\left|\mathbf{r}\right|^{2}\left(\mathbf{q}^{T}\mathbf{N}^{\prime T} \mathbf{N}^{\prime}\right)}{\left|\mathbf{r}\right|^{6}}\\ &=\frac{\left(\mathbf{\vec{r}}\cdot\mathbf{r}^{\prime}\right)^{T}}{\left| \mathbf{r}\right|^{6}}\cdot\left(\mathbf{\vec{r}}^{\prime}\cdot\mathbf{N}^{ \prime^{*}}-\mathbf{\vec{r}}^{*}\cdot\mathbf{N}^{\prime}\right)-\frac{3\kappa ^{2}}{\left|\mathbf{r}\right|^{2}}\left(\mathbf{q}^{T}\mathbf{N}^{\prime T} \mathbf{N}^{\prime}\right)\end{array} \tag{28}\]  
The bending force of the ANCF flexible beam element can be derived from Equation (21) as follows:  
\[\begin{array}{ll}\mathbf{F}_{Q}^{e}&=EI\int_{0}^{l_{0}}\kappa\frac{\partial \kappa}{\partial\mathbf{q}^{T}}dx\\ &=EI\int_{0}^{1}\biggl{[}\frac{\left(\mathbf{\vec{r}}\times\mathbf{r}^{\prime }\right)^{T}}{\left|\mathbf{r}\right|^{6}}\cdot\left(\mathbf{\vec{r}}^{\prime }\cdot\mathbf{N}^{\prime^{*}}-\mathbf{\vec{r}}^{*}\cdot\mathbf{N}^{\prime} \right)-\frac{3\kappa^{2}}{\left|\mathbf{r}\right|^{2}}\left(\mathbf{q}^{T} \mathbf{N}^{\prime T}\mathbf{N}^{\prime}\right)\biggr{]}d\xi\end{array} \tag{29}\]  
where an antisymmetric matrix is used for cross product operations. For the bending force induced by the tangent stiffness matrix, the second derivative of curvature is specified in Equation (18). This approximation significantly reduces the computational costs associated with the second derivative of curvature and the tangent stiffness matrix [38].  
\[\frac{\partial^{2}\kappa}{\partial\mathbf{q}^{T}\partial\mathbf{q}}\approx \frac{\left.\partial\left|\mathbf{r}^{\prime}\right|\right.}{\partial\mathbf{q }^{T}\partial\mathbf{q}}=\left(\mathbf{N}^{\prime}\right)^{T}\biggl{(} \frac{\mathbf{I}}{\left|\mathbf{r}^{\prime}\right|}-\frac{\mathbf{\vec{r}}^{* }\left(\mathbf{r}^{\prime}\right)^{T}}{\left|\mathbf{r}^{\prime}\right|^{3}} \biggr{)}\mathbf{N}^{\prime^{*}} \tag{30}\]Similarly, the tangent stiffness matrix \(\mathbf{K}_{Q}^{e}\) for the bending force can be obtained based on Equations (22), (29), and (31).  
\[\begin{array}{ll}\mathbf{K}_{Q}^{e}&=EI\int_{0}^{l_{0}}\Biggl{[} \left(\frac{\partial\kappa}{\partial\mathbf{q}}\right)^{T}\frac{\partial \kappa}{\partial\mathbf{q}}+\kappa\frac{\partial^{2}\kappa}{\partial\mathbf{q }^{T}\partial\mathbf{q}}\Biggr{]}dx\\ &\approx Ell_{0}\int_{0}^{1}\Biggl{[}\left(\frac{\partial\kappa}{ \partial\mathbf{q}}\right)^{T}\frac{\partial\kappa}{\partial\mathbf{q}}+\kappa \frac{\partial^{2}\kappa}{\partial\mathbf{q}^{T}\partial\mathbf{q}}\Biggr{]}d \xi\end{array} \tag{32}\]  
Due to the nonlinearity of the force and tangent stiffness matrix of the ANCF element, the Gauss-Legendre integration method is required to solve the integrations in the stretch and curvature terms of the stiffness matrix. The accuracy requirements are typically met by using five Gauss points:  
\[\int_{-1}^{1}f(\zeta)d\zeta\approx\sum_{i=1}^{n}w_{i}f(\zeta_{i}) \tag{33}\]  
where \(\zeta\) is the orthogonal node; \(f(\zeta)\) is the function containing the integration point \(\zeta\); and \(w_{i}\) is the weighting coefficient. There is a linear mapping relationship between \(\zeta\) and \(\zeta\), allowing the coefficient corresponding to \(\zeta\) to be directly utilized for Gauss-Legendre integration, and enabling integral transformation.