论文标题
用于血液动力学应用的流体稳定有限元方法
A time-consistent stabilized finite element method for fluids with applications to hemodynamics
论文作者
论文摘要
模拟不可压缩流的几种有限元方法依赖于由tau_supg加权的流线上的彼得罗夫 - 盖尔金稳定（SUPG）项。 tau_supg的常规公式包括一个常数，该常数取决于时间步长，而随着时间步长接近零，产生的总体方法变得非常准确。在实践中，这种方法不一致会在解决方案中引入重大错误，尤其是在心血管模拟中，其中可能需要较小的阶梯尺寸。为了克服这个问题，我们提出了一种基于tau_supg的新定义的一致方法。该方法可以轻松地在现有的流线上端点彼得罗夫 - 加盖尔（Petrov-Galerkin）和压力稳定的彼得罗夫 - 盖尔金（Petrov-Galerkin）方法的顶部实现，涉及用物理时间尺度更换tau_supg中的时间步长。该时间尺度是在一个简单的操作中每次步骤计算的，每次时间步骤从加速度和速度的L2-norm的比率中为整个计算域而言。使用四种情况将所提出的方法与常规方法进行比较：稳定的管道流动，血管解剖结构，正方形障碍物上的外部流动以及流体结构的相互作用情况。这些数值实验表明，在所有情况下，提出的配方消除了与常规配方相关的不一致问题。虽然所提出的方法比常规方法稍高，但它大大降低了误差，尤其是在较小的时间步长尺寸。对于可以使用精确溶液的管道流量，我们显示常规方法可以过度预测压力下降三倍。所提出的公式几乎完全消除了这一大误差，所有时间步长和雷诺数的数字下降到约1％。
Several finite element methods for simulating incompressible flows rely on the streamline upwind Petrov-Galerkin stabilization (SUPG) term, which is weighted by tau_SUPG. The conventional formulation of tau_SUPG includes a constant that depends on the time step size, producing an overall method that becomes exceedingly less accurate as the time step size approaches zero. In practice, such method inconsistency introduces significant error in the solution, especially in cardiovascular simulations, where small time step sizes may be required. To overcome this issue, we propose a consistent method that is based on a new definition of tau_SUPG. This method, which can be easily implemented on top of an existing streamline upwind Petrov-Galerkin and pressure stabilizing Petrov-Galerkin method, involves the replacement of the time step size in tau_SUPG with a physical time scale. This time scale is calculated in a simple operation once every time step for the entire computational domain from the ratio of the L2-norm of the acceleration and the velocity. The proposed method is compared against the conventional method using four cases: a steady pipe flow, a blood flow through vascular anatomy, an external flow over a square obstacle, and a fluid-structure interaction case. These numerical experiments show that the proposed formulation eliminates the inconsistency issue associated with the conventional formulation in all cases. While the proposed method is slightly more costly than the conventional method, it significantly reduces the error, particularly at small time step sizes. For the pipe flow where an exact solution is available, we show the conventional method can over-predict the pressure drop by a factor of three. This large error is almost completely eliminated by the proposed formulation, dropping to approximately 1% for all time step sizes and Reynolds numbers considered.
