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

Area of Science:

• Computational fluid dynamics
• Numerical analysis
• Scientific computing

Background:

• Incompressible flow simulations often use streamline upwind Petrov-Galerkin (SUPG) stabilization.
• The conventional SUPG formulation has an inconsistency issue related to time step size, causing significant errors in simulations requiring small time steps, such as cardiovascular flows.

Purpose of the Study:

• To propose a consistent finite element method for incompressible flows by redefining the SUPG stabilization term.
• To eliminate the method inconsistency associated with the conventional SUPG formulation, particularly at small time step sizes.

Main Methods:

• A new definition of the SUPG stabilization term is introduced, replacing the time step size with a physical time scale.
• The physical time scale is computed as the ratio of the L2-norm of acceleration to velocity.
• The proposed method was tested against the conventional method in steady pipe flow, vascular blood flow, external flow over an obstacle, and fluid-structure interaction simulations.

Main Results:

• The proposed formulation successfully eliminated the inconsistency issue present in the conventional SUPG method across all tested cases.
• While slightly more computationally expensive, the new method significantly reduced simulation errors, especially for small time step sizes.
• For steady pipe flow, the conventional method over-predicted pressure drop by a factor of three, an error reduced to approximately 1% with the proposed formulation.

Conclusions:

• The proposed consistent finite element method effectively addresses the SUPG formulation's time step dependency issue.
• This new approach offers a significant reduction in numerical error for incompressible flow simulations, particularly in complex scenarios like cardiovascular modeling.
• The method is easily implementable and provides a more accurate and reliable solution across various flow conditions and time step sizes.