Numerical simulations for blood flow problems using fluid-structure interaction with stabilized finite element methods

Abstract

The study and simulation of blood flow in the cardiovascular system (CS) have many applications such as pathologies studies, surgical planning, and design of medical devices. The blood flow in the CS can be considered as a fluid dynamics problem, and in the optimal case, a coupling between fluid and solid dynamics, named Fluid-Structure Interaction (FSI) problems. Firstly, this work focuses on solving different fluid dynamics problems using usual stabilizations for the Galerkin method in order to solve the Navier-Stokes equation. A comparison using CPU time and error for different levels of mesh refinement is made as a way to evaluate and explore the FEniCS library and the function of each stabilization term. This method may be used to perform blood flow simulations using a rigid wall domain assumption. However, some authors say that blood velocity and pressure in large arteries are greatly influenced by vessel wall dynamics. By this reason, in the second part of this work, it was used the Coupled Momentum Method of Fluid-Structure Interaction (CMM-FSI,) to solve blood flow problems in large arteries. The CMM-FSI simplifies the fluid-structure coupling process when using a finite element formulation due to be based on a strong coupling of degrees-of-freedom of the fluid and the solid domains. It could be made by coupling the vessel wall deformation at the variational level as a boundary condition for the fluid domain and by keeping the same description of motion (Eulerian) in both domains, and with a fixed mesh. One of the positive parts of CMM-FSI is that it demands a lot less computational effort, mainly for 3D models, than usual arbitrary Lagrangian-Eulerian methods. On the other hand, this method claims that the polynomial approximation of the pressure and velocity elements must be the same order. Thus, a stabilized finite element formulation is used. All numerical implementation is done using Python and FEniCS, a finite element library. Constitutive relationships of Newtonian fluid and thin-walled structure are considered. A comparison is made with Poiseuille’s blood flow theory and also with a rigid wall assumption. The impacts of choosing a rigid wall assumption or a deformable model are evaluated with hemodynamics parameters, as pressure peaks in diastole and systole and blood flow wave velocities. Finally, the CMM-FSI is used to perform a blood flow simulation in an idealized common carotid with stenosis. The results obtained in this work are consistent with the literature.