On the Numerical Evaluation of Wall Shear Stress Using the Finite Element Method

Wall shear stress (WSS) is a crucial hemodynamic quantity extensively studied in cardiovascular research, yet its numerical computation is not straightforward. This work compares WSS results obtained from two different finite element discretizations, quantifies the differences between continuous and discontinuous stresses, and introduces a modified variationally consistent method for WSS evaluation through the formulation of a boundary-flux problem. Two benchmark problems are considered: a 2D Stokes flow on a unit square and a 3D Poiseuille flow through a cylindrical pipe. These are followed by investigations of steady-state Navier-Stokes flow in two image-based, patient-specific aneurysms. The study focuses on P1/P1 stabilized and Taylor-Hood P2/P1 mixed finite elements for velocity and pressure. WSS is computed using either the proposed boundary-flux method or as a projection of tangential traction onto first order Lagrange (P1), discontinuous Galerkin first order (DG-1), or discontinuous Galerkin zero order (DG-0) space. For the P1/P1 stabilized element, the boundary-flux and P1 projection methods yielded equivalent results. With the P2/P1 element, the boundary-flux evaluation demonstrated faster convergence in the Poiseuille flow example but showed increased sensitivity to pressure field inaccuracies in image-based geometries compared to the projection method. Furthermore, a paradoxical degradation in WSS accuracy was observed when combining the P2/P1 element with fine boundary-layer meshes on a cylindrical geometry, an effect attributed to inherent geometric approximation errors. In aneurysm geometries, the P2/P1 element exhibited superior robustness to mesh size when evaluating average WSS and low shear area (LSA), outperforming the P1/P1 stabilized element. Projecting discontinuous finite element functions into continuous spaces can introduce artifacts, such as the Gibbs phenomenon. Consequently, it is crucial to carefully select the finite element space for boundary stress calculations, not only in applications involving WSS computations for aneurysms.