## 2 Numerical Simulation Method of Riser Vortex Shock Vibration
### Establish the Pipe Structure Model  
Each riser unit has two nodes, each node having 6 degrees of freedom, and the node load vector is as follows:  
\[F_{e}=\left[\mathrm{F}_{1\times},\mathrm{F}_{2\times},\mathrm{F}_{3\times}, \mathrm{F}_{1\otimes}\mathrm{F}_{2\otimes}\right]^{T} \tag{2}\]  
The fluid force is applied to each riser unit in the form of a uniform load by a self-compiling program, and then Formula (2) is obtained by the equivalent load method. After discretizing the riser into a finite number of three-dimensional elements, Equation (1) can be discretized into the form of the following matrix:  
\[[M]\big{\{}\ddot{x}_{i}(t)\big{\}}+[C]\big{\{}\dot{x}_{i}(t)\big{\}}+([K_{E}]+[ K_{G}])\{x_{i}(t)\}=\{F_{i}(t)\} \tag{3}\]  
There, \([\mathbf{M}]\) is the mass matrix; \([\mathbf{C}]\) is a damping matrix, usually using Rayleigh damping matrix; \([\mathbf{K_{E}}]\) is the elastic stiffness matrix and \([K_{G}]\) is the geometric stiffness matrix, which constitute the stiffness matrix \([\mathbf{K}]\). Then, \(\{\mathbf{F_{i}(t)}\}\) is the load vector in the x and y directions. This paper uses the Newmark-beta method, with the advantages of good stability, high precision, fast calculation speed and high precision and reliability to solve the motion governing equation of riser structure, which is a numerical step-by-step integration method. By establishing the recursive relation between the displacement, velocity and acceleration of the previous time and that of the next time, the state variable of the finite element node of the riser at each time is solved step-by-step from time t = 0. When the control parameters \(\delta\geq\) 0.50 and \(\beta\geq\) 0.25 (0.50 + \(\delta\))\({}^{2}\), the Newmark-beta method is unconditionally stable. If the control parameters \(\delta\) and \(\beta\) take other values, the time step must meet the following Formula (4) to make the Newmark-beta method stable. \[\frac{\Delta t}{T}\leq\frac{1}{\pi\sqrt{2\gamma-4\beta}} \tag{4}\]  
The process of solving the riser vibration control equation based on the Newmark-beta method is as follows:  
1. The mass matrix of the element is assembled by Matlab programming, and the overall mass matrix \([M]\) and stiffness matrix \([K]\) of the riser are obtained. The overall damping matrix \([C]\) of the riser is obtained by the Rayleigh damping method. 2. Given acceleration \(\big{\{}\ddot{x}_{i}\big{\}}\), velocity \(\big{\{}\ddot{x}_{i}\big{\}}\), displacement \(\{x_{i}\}\) and load column vector \(F_{i}(t)\) at the initial time = 0,  
Figure 6: Background meshing diagram. Figure 7: Nesting of overlapping grids. 3. Select time step \(\Delta t\), control parameters \(\delta\) = 0.50 and \(\beta\) = 0.25 and calculate \(a_{0}\sim a_{7}\). \[ \begin{array}{c}a_{0}=\frac{1}{\beta\Delta t^{2}},\ a_{1}=\frac{\delta}{\beta \Delta t},\ a_{2}=\frac{1}{\beta\Delta t},\ a_{3}=\frac{1}{2\beta}-1,\ a_{4}= \frac{\delta}{\beta}-1,\\ a_{5}=\frac{\Delta t}{2}\Big{(}\frac{\delta}{\beta}-2\Big{)},\ a_{6}=(1- \delta)\Delta t,\ a_{7}=\delta\Delta t\end{array} \] (5)
4. Calculate the equivalent stiffness matrix: \[ \begin{array}{c}\big{[}K^{\prime}\big{]}=[K]+a_{0}[M]+a_{1}[C]\end{array} \] (6)
5. Calculate the equivalent load vector at time \(t+\Delta t\): \[ \begin{array}{c}\big{\{}F_{i}^{\prime}(t+\Delta t)\big{\}}=\big{\{}F_{i}(t+ \Delta t)\big{\}}+\big{[}M\big{]}\big{[}a_{0}x(t)+a_{2}\dot{x}(t)+a_{3}\ddot{x} (t)\big{]}\\ +\big{[}C\big{]}\big{[}a_{1}x(t)+a_{4}\dot{x}(t)+a_{5}\ddot{x}(t)\big{]}\end{array} \] (7)
6. Solve the equivalent static equilibrium equation and find the time displacement \(\{x_{i}(t+\Delta t)\}\) of \(t+\Delta t\): \[ \begin{array}{c}\big{[}K^{\prime}\big{]}\big{\{}x_{i}(t+\Delta t)\big{\}}= \big{\{}F_{i}^{\prime}(t+\Delta t)\big{\}}\end{array} \] (8)
7.