# Numerical Analysis on Spanwise Correlation of Vortex-Induced Force of Split Double-Box Beam
## 2 Basic Theory and Methods
### Numerical Solution Method Based on FLUENT
#### 2.3.2 Newmark-\(\beta\) Algorithm  
The Newmark-\(\beta\) algorithm is essentially a time-domain stepwise integration method, whose basic principle is to replace the equilibrium conditions at any time by satisfying the structural dynamics equations at a finite number of momentary points, thus achieving the purpose of replacing the true solution accuracy with the displacements, velocities, and accelerations at particular time points [25].  
The procedure for writing the Newmark-\(\beta\) algorithm in this paper is as follows.  
1. Preparation of basic data and calculation of initial conditions  
(1) Select the time step \(\Delta t\) and parameters \(\beta\gamma\), and calculate the integration constant  
\[\begin{array}{l}a_{0}=\frac{1}{\beta\Delta t^{2}},\ a_{1}=\frac{\gamma}{ \beta\Delta t},\ a_{2}=\frac{1}{\beta\Delta t},\ a_{3}=\frac{1}{2\beta}-1,\ a_{0}=\frac{ \gamma}{\beta}-1,\\ a_{5}=\frac{\Delta t}{2}\left(\frac{\gamma}{\beta}-2\right),\ a_{6}=\Delta t(1- \gamma),\ a_{7}=\gamma\Delta t\end{array} \tag{12}\]  
(2) Given the initial values \(\left\{x\right\}_{0}\), \(\left\{\dot{x}\right\}_{0}\) and \(\left\{\ddot{x}\right\}_{0}\).  
2. Formation of mass matrix \([M]\), damping matrix \([C]\), and stiffness matrix \([K]\).  
3. Formation of the equivalent stiffness matrix \(\left\lfloor\dot{K}\right\rfloor\):  
\[\left\lfloor\dot{K}\right\rfloor=[K]+a_{o}[M]+a_{1}[C] \tag{13}\]  
Figure 1: Flowchart of the calculation of the Newmark-\(\beta\) method embedded in FLUENT.  
4. Calculation of the equivalent load at each time step \(t_{i+\Delta t}\) moment:  
\[\begin{array}{l}\left\{\hat{P}\right\}_{i+\Delta t}=\left\{\hat{P}\right\}_{i+ \Delta t}+\left[M\right]\left[a_{0}\left\{x\right\}_{i}+a_{2}\left\{\dot{x} \right\}_{i}+a_{3}\left\{\ddot{x}\right\}_{i}\right]\\ +\left[C\right]\left[a_{1}\left\{x\right\}_{i}+a_{4}\left\{x\right\}_{i}+a_{5} \left\{x\right\}_{i}\right]\end{array} \tag{14}\]  
Calculation of the displacement at each time step \(t_{i+\Delta t}\) moment:  
\[\left\lvert\left\lvert\left\lvert\left\lvert\left\lvert\left\lvert\left\lvert \left\lvert\left\lvert\left\lvert\left\lvert\left\lvert\left\lvert\left\lvert \left\lvert\left\lvert\left\lvert\left\lvert\left\lvert\left\lvert\left\lvert \left\lvert\left\lvert\left\lvert\left\lvert\left\lvert\left\lvert\left\lvert\left\lvert \left\lvert\left\lvert\left\lvert\left\lvert\left\lvert\left\lvert\left\lvert\left\lvert \left\lvert\left\lvert\left\lvert\left\lvert\left\lvert\left\lvert\left\right\right\right\right\right\right\right\right\rvert \right\right\rvert\right\right\rvert\right\right\rvert\right\right\rvert\right..\right.\right.\right.\right.\right.\right.\right.\right.\right. \tag{15}\]  
Solving for the velocity and acceleration at each time step moment:  
\[\left\{\ddot{x}\right\}_{i+\Delta t}=a_{0}\left(\left\{x\right\}_{i+\Delta t} -\left\{x\right\}_{i}\right)-a_{2}\left\{\dot{x}\right\}_{i}-a_{3}\left\{ \ddot{x}\right\}_{i} \tag{16}\]  
\[\left\{\dot{x}\right\}_{i+\Delta t}=\left\{\dot{x}\right\}_{i}+a_{6}\left\{ \ddot{x}\right\}_{i}+a_{7}\left\{\ddot{x}\right\}_{i+\Delta t} \tag{17}\]  
The dynamic response of the structural system at any point in time is obtained by cycling the calculation steps 4 to 6 until the calculation converges.