# Investigation on Numerical Simulation of VIV of Deep-Sea Flexible Risers
## 2 Numerical Simulation Method of VIV of Riser
### Structural Dynamic Model of Riser  
The top tension flexible riser model was simplified as a Euler-Bernoulli beam, with the same material and section shape assumed along the riser's length. We took the in-line (IL) direction as the X axis, the crossflow (CF) direction as the Y axis and the riser length direction as the Z axis, as shown in Figure 1. The differential equations of structural motion in the flow direction are given by [26; 27]:  
\[\frac{\partial^{2}}{\partial z^{2}}\left[EI\frac{\partial^{2}u_{i}(z,t)}{ \partial z^{2}}\right]-\frac{\partial}{\partial z}\left[T(z)\frac{\partial u _{i}(z,t)}{\partial z}\right]+m\frac{\partial^{2}u_{i}(z,t)}{\partial t^{2}}+ c\frac{\partial u_{i}(z,t)}{\partial t}=F_{i}(z,t) \tag{5}\]  
where \(EI\) is the bending stiffness, EA is the tensile stiffness, \(m\) is the riser mass per unit length and \(c\) is the structural damping. \(u_{i}(z,t)\) denotes the CF and IL displacement, and \(F_{i}(z,t)\) denotes the CF direction and the IL hydrodynamic force (in which \(i=1\) is the IL direction, and \(i=2\) is the CF direction). \(T(z)\) is the riser top tension. [PERSON] and [PERSON] [28] studied the transverse vibration responses of a rigid cylinder. Their model allowed for three degrees of freedom, including axial and transverse displacements and a rotation about an axis perpendicular to the plane of the displacements. [PERSON] [29] also used the same three degrees of freedom per finite element node to investigate hydro-elastic interactions between oscillating flexible cylinders and fluid forces. However, this simplified finite element method can only reflect the riser's response in the transverse direction. To study the dynamic responses of deep-sea risers in both the transverse and downstream directions, we reduced the riser to a sufficient number of three-dimensional finite elements, each of which had six degrees of freedom. The finite element diagram is shown in Figure 2.  
The load vector of each element can be expressed as:  
\[F^{\varepsilon}=\left[F_{1x},F_{1y},F_{1z},F_{1\theta x},F_{1\theta y},F_{1 \theta z},F_{2x},F_{2y},F_{2z},F_{2\theta x},F_{2\theta y},F_{2\theta z}\right] ^{\mathrm{T}} \tag{6}\]  
Figure 1: Riser model diagram. The positions of some nodes were taken as the positions of the two-dimensional flow field slices, and the fluid acting force at the slice node was simplified as the uniformly distributed load plus the riser unit near the slice. The positional relationship between the slices and the finite element and the force exerted on the element are shown in Figure 3.  
By discretizing Equation (5) using the finite element method, we obtained the following equation [26; 27]:  
\[[M]\big{\{}\ddot{x}_{i}(t)\big{\}}+[C]\big{\{}\dot{x}_{i}(t)\big{\}}+([K_{E}]+[ K_{G}])\{x_{i}(t)\}=\{F_{i}(t)\} \tag{7}\]  
where \([M]\), \([C]\), \([K_{E}]\) and \([K_{G}]\) denote the global matrices of the mass, the damping, the elastic stiffness of the riser and geometric stiffness; \(\ddot{x}_{i}(t)\), \(\dot{x}_{i}(t)\) and \(x_{i}(t)\) represent the vectors of the accelerations, velocities and deflections of the nodes of the riser, respectively; and \(\{F_{i}(t)\}\) signifies the load vector in the x-y direction. The Newmark-\(\beta\) method was used to solve the motion equation. The structural damping usually depends on the natural frequency. In actual analysis, we usually adopt the Rayleigh damping matrix, which is a linear combination of \([M]\) and \([K]\)[30]:  
\[[C]=\alpha[M]+\gamma[K] \tag{8}\]  
where \(\alpha\) and \(\gamma\) are the Rayleigh damping coefficient, which is related to the natural frequency and damping ratio of the riser [30]:  
\[ \begin{bmatrix}\alpha\\ \gamma\end{bmatrix} =\frac{2\zeta}{\omega_{n}+\omega_{m}} \begin{bmatrix}\omega _{n}\omega_{m}\\ 1\end{bmatrix}  \tag{9}\]  
where \(\zeta=0.03\), \(\omega_{n}\) takes the first-order natural frequency and \(\omega_{m}\) takes the natural frequency with a relatively high contribution to the vibration response. Figure 3: Riser slice division and finite element layout.