深海立管涡激振动模拟中块状线性方程组的预处理及算法研究

Deepwater risers are indispensible engineering equipment in exploring and drilling for oil and gas. Under the circumstance of complex ocean environment, the vortex-induced vibration (VIV) caused by ocean flow tends to result in the fatigue damage of the risers, which may further lead to a huge economic loss and environmental disaster. For this reason, it remains a difficult subject on how to predict the VIV accurately by numerical methods. Among all candidates, the computational fluid dynamic approaches, which are based on the Navier-Stokes equations (N-S Eqs.), have received much attention in related fields. However, it is hard to solve the N-S Eqs. due to the existence of the nonlinear term. In this project, we first use the finite element methods to discretize the governing equations and obtain the block linear systems. Then we propose preconditioners and improve the corresponding algorithms for the block linear systems derived from VIV simulation via numerical linear algebra techniques. In light of the structures and properties of the coefficient matrices, we present three types of preconditioners, and optimizing the algorithms by taking into account of the optimal parameters and the different convergence rate between unknown vectors. The achievement of this project will have a positive impact on improving the simulation accuracy, reducing the simulation time and the economic cost.

深海立管系统是油气开采不可或缺的工程设备。在复杂的海洋环境下，由海流引发的涡激振动往往会导致立管的疲劳损伤，从而造成巨大的经济损失和环境危害。因此, 如何从数值角度准确预测涡激振动目前仍是一个较难的课题。在众多数值模拟方法中，以Navier-Stokes方程（N-S方程）为基础的计算流体力学法因其模拟精度高，已受到相关领域的重视。然而，由于N-S方程中的非线性项往往导致此类方法较难实施。本项目首先利用有限元将涡激振动数值模拟过程中的控制方程（N-S方程）转化为具有分块结构的线性代数方程组的数值求解问题。然后从数值线性代数的角度研究这类线性方程组的预处理技术和算法优化问题。针对系数矩阵的结构特性，提出三类预处理子，并从最优参数的选择和子未知量收敛速率方面优化算法。该项目的实施对于立管涡激振动模拟精度的提高、数值模拟时间和经济成本的节约等方面有积极的影响。

作为海面平台与海底的重要连接通道，立管是深海开发必不可少的关键设备。本项目以深海立管涡激振动模拟中衍生的块状线性方程组为研究对象，重点考虑其预处理技术及算法，具体工作如下：(1) 针对非对称鞍点问题，提出一类尽可能保持原始系数矩阵结构的松弛型PSS预处理子，充分研究了对应定常迭代法的无条件收敛性及预处理矩阵的特征信息；(2) 针对具有3x3块状结构的鞍点问题，提出了一种广义位移分裂预处理子，研究了相应定常迭代法的无条件收敛性及预处理矩阵的谱分布；(3) 针对多位移线性方程组，提出了一种高效的Krylov子空间方法及其预处理变体，明显改善现有子空间方法的收敛偏慢甚至停滞的现状；(4) 针对深海立管涡激振动，从立管固有频率、主频率等角度研究了顶张力对立管涡激振动的影响, 并利用拖曳水池模拟试验加以验证。