## 2 Analytical Modelling
### Mooring Load  
In the static analysis, the catenary equation [27] provides the essential idea of a relationship between the x-offset and the horizontal restoring force in the static stage:  
\[X=l-d\left(1+2\frac{T_{H}}{\omega d}\right)^{\frac{1}{2}}+\frac{T_{H}}{\omega} cosh^{-1}\left(1+\frac{\omega d}{T_{H}}\right) \tag{5}\]  
where \(X\) is the horizontal projection distance between fairlead and anchor, \(l\) is un-stretched mooring line length, \(T_{H}\) is the horizontal force from mooring line self-weight, \(d\) is the vertical distance from fairlead to seabed, and \(\omega\) is the per meter weight of the mooring line. Once the tension of each line is computed, the overall static force on the floating structure can be defined.  
In the dynamic analysis, a finite element model was used to simulate the mooring line in Orcalfex [26]. The line is divided into a series of line segments modelled by straight massless model segments with a node at each end, as illustrated in Figure 2. Line properties such as mass, weight, and buoyancy are lumped to the nodes. The index value of the line segment between nodes 0 and 1 is 1/2, and the node at the top of the fairlead node is N. In this study, there are 84 segments nodes in an 835.5 m mooring line. The mooring line model combines internal axial stiffness, damping force, buoyancy force, and hydrodynamic forces from Morison's equation.  
The equivalent mass of the mooring cable is discretised into point masses at each node by assigning each node half the combined mass of the two nearby segments. The 3 \(\times\) 3 mass matrix for node \(i\) can be represented by  
\[m_{i}=\frac{\pi}{4}d_{m}{}^{2}l\rho I \tag{6}\]  
where \(d_{m}\) is equivalent mooring line diameter, \(\rho\) is the density of the mooring cable, and \(I\) is the identity matrix.  
The added-mass matrix (which is included in the mass matrix) for node \(i\) is expressed as:  
\[a_{i}=a_{pi}+a_{qi}=\rho\frac{\pi}{4}d_{m}{}^{2}l\left[Can\left(1-q\dot{iq}i^{ T}\right)+Cat\left(q\dot{iq}i^{T}\right)\right] \tag{7}\]where \(a_{pi}\) is the corresponding transverse, \(a_{qi}\) is the tangential added-mass matrix, \(Can\) is the transverse added-mass coefficient, \(Cat\) is the tangential added-mass coefficient, and \(q\dot{i}\) is defined as the tangent direction at each node and can be approximated as the direction of a line passing between the two adjacent node points.  
The internal stiffness and damping force of the line segment \(i+1/2\) are expressed as \(T_{i+(\frac{1}{2})}\) and \(C_{i+(\frac{1}{2})}\), respectively. Meanwhile, the weight of the line segment lumped at each node \(i\) is denoted as \(W_{i}\), whereas \(B_{i}\) is the net buoyancy at node \(i\). \(D_{pi}\) and \(D_{qi}\) are drag forces in the transverse and tangential directions, respectively. Therefore, the motion equation of the mooring cable for each node \(i\) can be presented as below:  
\[\begin{split} F_{M}=\sum\limits_{i=1}^{N}(m_{i}+a_{i})\ddot{r_{ i}}\\ =S\sum\limits_{i=1}^{N}\Big{(}T_{i+(\frac{1}{2})}-T_{i-(\frac{1}{2 })}+C_{i+(\frac{1}{2})}-C_{i-(\frac{1}{2})}+W_{i}+B_{i}+D_{pi}+D_{qi}\Big{)} \end{split} \tag{8}\]  
where \(r_{i}\) is the position vector at node \(i\).