A time-consistent stabilized finite element method for fluids with applications to hemodynamics

Abstract

Several finite element methods for simulating incompressible flows rely on the streamline upwind Petrov–Galerkin stabilization (SUPG) term, which is weighted by $$\tau _{\text{SUPG}}$$ τ SUPG . The conventional formulation of $$\tau _{\text{SUPG}}$$ τ 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 resolve multiple scales of the blood flow. To overcome this issue, we propose a consistent method that is based on a new definition of $$\tau _{\text{SUPG}}$$ τ 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 $$\tau _{\text{SUPG}}$$ τ 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 involving an oscillatory flexible beam. These numerical experiments, which are performed using linear interpolation functions, 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.