## 3 Numerical Model
### Governing Equations of Dynamic Simulation
\begin{array}{ll}\mathbf{U}_{1}=\mathbf{r}(0,t),&\mathbf{U}_{2}=\mathbf{r} ^{{}^{\prime}}(0,t),\\ \mathbf{U}_{3}=\mathbf{r}(L,t),&\mathbf{U}_{4}=\mathbf{r}^{{}^{\prime}}(L,t), \\ \lambda_{1}=\lambda(0,t),&\lambda_{2}=\lambda(L/2,t),&\lambda_{3}=\lambda(L,t) \end{array}  \tag{15}\]  
The position and its tangent vectors are obtained at both ends of the element while the Lagrange multiplier are computed at both ends and the middle point of the element. The final finite element formulation of the governing equation for the 3 dimensional problem are presented in Equation (16). \[(M_{ijlk}+M^{a}_{ijk})\ddot{\mathbf{U}}_{jk}+(K^{1}_{ijlk}+\lambda_{n}K^{2}_{ iijlk})\mathbf{U}_{jk}=\mathbf{F}_{il} \tag{16}\]  
For \(\mathbf{U}_{jk}\), subscript \(j\) is dimension, which is 1-3 for the 3 dimensional problem, and subscript \(k\) is for 1-4 given in Equation (15). In Equations (17)-(21), the general mass, the added mass, the general stiffness from the bending stiffness and rod tension, and external force matrices are defined with Kronecker Delta function \(\delta_{ij}\):  
\[M_{ijlk}=\int_{0}^{L}mA_{1}A_{k}\delta_{ij}ds \tag{17}\]  
\[M^{a}_{ijlk}=C_{A}\rho A_{E}\bigg{[}\int_{0}^{L}A_{I}A_{k}\delta_{ij}ds-(\int_ {0}^{L}A_{I}A_{k}A^{\prime}{}_{s}A^{\prime}{}_{t}ds)\mathbf{U}_{it}\mathbf{U}_ {js}\bigg{]} \tag{18}\]  
\[K^{1}_{ijlk}=\int_{0}^{L}EIA^{\prime\prime}{}_{t}A^{\prime\prime}{}_{k}\delta _{ij}ds \tag{19}\]  
\[K^{2}_{iijlk}=\int_{0}^{L}P_{n}A^{\prime}{}_{l}A^{\prime}{}_{k}\delta_{ij}ds \tag{20}\]  
\[\mathbf{F}_{il}=\int_{0}^{L}\big{(}\widetilde{\mathbf{w}}_{i}+\widetilde{ \mathbf{F}}^{d}_{i}\big{)}A_{l}ds \tag{21}\]  
In addition, the stretching condition can be formulated as given in Equation (22):  
\[G_{m}=A_{mil}\mathbf{U}_{kl}\mathbf{U}_{ki}-B_{m}-C_{mt}\lambda_{t} \tag{22}\]  
where  
\[A_{mil}=\frac{1}{2}\int_{0}^{L}P_{m}A_{i}{}^{\prime}A_{l}{}^{\prime}ds \tag{23}\]  
\[B_{m}=\frac{1}{2}\int_{0}^{L}P_{m}ds \tag{24}\]  
\[C_{mt}=\frac{1}{A_{I}E}\int_{0}^{L}P_{m}P_{t}ds \tag{25}\]  
A dummy 6 DOF rigid body, which is equipped with negligible properties, is introduced to conveniently connect the tunnel and mooring lines. The dummy mass means negligible mass (1 kg in proto type) of dummy rigid body used only for connection purpose. Therefore, force and moment are transferred by using both linear and rotational springs of very large stiffness from the tunnel and mooring lines through the rigid body. Force and moment transmitted from the mooring line to the rigid body are computed as follows [43]:  
\[M_{ijlk}=\int_{0}^{L}mA_{1}A_{k}\delta_{ij}ds \tag{17}\]0  
where \(\widetilde{K}\) and \(\widetilde{C}\) represent coupling stiffness and damping matrices, \(\widetilde{T}_{P}\) denotes a transformation matrix between the rigid body origin and the connection location, \(\widetilde{\mathbf{u}}_{P}\) and \(\widetilde{\mathbf{u}}_{I}\) are the displacements of the rigid body and the connecting location. Infinite stiffness values are used in the coupling stiffness matrix to tightly connect lines, and damping matrix is not utilized in the simulations. Therefore, the entire stiffness matrix that couples tunnel elements with mooring lines is created as shown in \[M_{ijlk}=\int_{0}^{L}mA_{1}A_{k}\delta_{ij}ds \tag{17}\]1  
[PERSON]'s iteration method is used in static analysis of the SFT. The Adams-Moulton implicit integration method, which has 2 nd-degree of accuracy, is used for the time-domain-integration method. Since instantaneous velocity and acceleration are required to calculate hydrodynamic force from Morison equation, the Adam-Bashforth explicit scheme is combined with the Adams-Moulton implicit scheme to avoid iteration.