## 3 Numerical Model
### Governing Equations of Dynamic Simulation
In the equation, inertia and drag force terms are modified by introducing seismic velocity \(\mathbf{v}_{g}^{n}\) and acceleration \(\overset{n}{\mathbf{v}}_{g}^{n}\) as shown in Equation (4). The vertical component of seismic velocity and acceleration is considered only for the seaquake simulations. \[\mathbf{F_{d}}=-C_{A}\rho A_{E}\overset{-n}{\mathbf{r}}+C_{M}\rho A_{E}( \overset{n}{\mathbf{V}}^{n}+\overset{n}{\mathbf{v}}_{g}^{n})+\frac{1}{2}C_{D} \rho D\Big{|}\mathbf{V}^{n}+\mathbf{v}_{g}^{n}-\overset{n}{\mathbf{r}}^{n} \Big{|}(\mathbf{V}^{n}+\mathbf{v}_{g}^{n}-\overset{n}{\mathbf{r}}^{n}) \tag{4}\]  
Therefore, the final form of the equation of motion is given by Equations (5)-(9):  
\[\overset{-n}{\mathbf{r}}+C_{A}\rho A_{E}\overset{-n}{\mathbf{r}}+(EI\overset{ \pi}{\mathbf{r}})^{\prime\prime}-(\overset{\lambda}{\mathbf{r}}^{\prime})^{ \prime}=\widetilde{\mathbf{w}}+\widetilde{\mathbf{F}}_{d} \tag{5}\]  
\[\widetilde{\mathbf{F}}_{d}=C_{M}\rho A_{E}(\overset{n}{\mathbf{V}}^{n}+ \overset{n}{\mathbf{v}}_{g}^{n})+\frac{1}{2}C_{D}\rho D\Big{|}\mathbf{V}^{n}+ \mathbf{v}_{g}^{n}-\overset{n}{\mathbf{r}}^{n}\Big{|}(\mathbf{V}^{n}+\mathbf{ v}_{g}^{n}-\overset{n}{\mathbf{r}}^{n}) \tag{6}\]  
\[\widetilde{\lambda}=\widetilde{T}-EI\kappa^{2} \tag{7}\]  
\[\widetilde{\mathbf{w}}=\mathbf{w}+\mathbf{B} \tag{8}\]  
\[\widetilde{T}=T+P \tag{9}\]  
where \(\kappa\) is local curvature, \(\widetilde{\mathbf{w}}\) is wet weight of the rod per unit length, which is comprised of weight \(\mathbf{w}\) and buoyancy \(\mathbf{B}\), \(\widetilde{T}\) is effective tension in the rod, and \(P\) is the hydrostatic pressure, which is a scalar, at the position \(\mathbf{r}\) on the rod. Therefore, Equation (5) combined with the stretching condition given in Equation (2) are the governing equations for dynamic simulations. The governing equations are further formulated by [PERSON] finite element method [39; 43]. The position vector and Lagrange multiplier for a single element of the length \(L\) are expressed as follows:  
\[\mathbf{r}(s,t)=\sum_{m}A_{m}(s)\mathbf{U}_{m}(t) \tag{10}\]  
\[\lambda(s,t)=\sum_{n}P_{n}(s)\lambda_{n}(t) \tag{11}\]  
where \(A_{m}\) and \(P_{n}\) are shape functions defined on the interval \(0\leq s\leq L\). The weak form of the governing equation is generated by using the Galerkin method and integration by part:  
\[\int_{0}^{L}\Big{[}A_{m}(\overset{-n}{\mathbf{r}}+C_{A}\rho A_{E}\overset{-n} {\mathbf{r}}^{n})+EI\overset{\pi}{\mathbf{r}}^{\prime}+\widetilde{\lambda}A ^{\prime}_{m}r^{\prime}-A_{m}(\widetilde{\mathbf{w}}+\widetilde{\mathbf{F}}^{d })\Big{]}ds=EI\overset{\pi}{\mathbf{r}}^{\prime}A^{\prime}_{m}\Big{|}_{0}^{L} +\Big{[}\widetilde{\lambda}\overset{\prime}{\mathbf{r}}-(EI\overset{\pi}{ \mathbf{r}}^{\prime})^{\prime}\Big{]}A_{m}|_{0}^{L} \tag{12}\]  
\[\mathbf{F_{d}}=-C_{A}\rho A_{E}\overset{-n}{\mathbf{r}}+C_{M}\rho A_{E}( \overset{n}{\mathbf{V}}^{n}+\overset{n}{\mathbf{v}}_{g}^{n})+\frac{1}{2}C_{D} \rho D\Big{|}\mathbf{V}^{n}+\mathbf{v}_{g}^{n}-\overset{n}{\mathbf{r}}^{n} \Big{|}(\mathbf{V}^{n}+\mathbf{v}_{g}^{n}-\overset{n}{\mathbf{r}}^{n}) \tag{4}\]0  
where first and second terms of the right-hand side in Equation (12) are related to moment and force at the boundary. Cubic and quadratic shape functions, which are continuous on the element, are defined for the position vector and Lagrange multiplier, respectively:\[ \begin{array}{ll}A_{1}=1-3\xi^{2}+2\xi^{3},&A_{2}=L(\xi-2\xi^{2}+\xi^{3}),\\ A_{3}=3\xi^{2}-2\xi^{3},&A_{4}=L(-\xi^{2}+\xi^{3}),\\ P_{1}=1-3\xi+2\xi^{2},&P_{2}=4\xi(1-\xi),\\ P_{3}=\xi(2\xi-1)\end{array}  \tag{14}\]  
where \(\xi=s/L\). The position vector, tangent of the position vector, and Lagrange multiplier are chosen to be continuous at the node between the neighboring elements. Therefore, the parameters \(\mathbf{U}_{m}\) and \(\lambda_{n}\) can be written as:  
\[