On the numerical evaluation of wall shear stress using the finite element method

TL;DR

This work addresses variability in numerically evaluating wall shear stress (WSS) in patient-specific CFD by comparing two finite element discretizations—P1/P1 stabilized and Taylor–Hood P2/P1—and two WSS postprocessing approaches: $L_2$ projection and a boundary-flux reconstruction. It validates the methods on two benchmarks (2D Stokes on a unit square and 3D Poiseuille flow in a cylindrical pipe) and on two patient-specific aneurysm geometries, focusing on convergence, robustness, and sensitivity to mesh and pressure accuracy. The findings show that the P2/P1 element is more robust for average WSS and low shear area (LSA) in patient geometries, while the boundary-flux method can accelerate convergence in some cases but may be more sensitive to pressure fields; projection onto continuous spaces can introduce Gibbs artifacts, highlighting the need for careful FE-space selection for boundary stress computations. Together, these results advocate for standardizing WSS evaluation pipelines in patient-specific modeling to improve reliability and clinical translatability.

Abstract

Wall shear stress (WSS) is a crucial hemodynamic quantity extensively studied in cardiovascular research, yet its numerical computation is not straightforward. This work aims to compare WSS results obtained from two different finite element discretizations, quantify the differences between continuous and discontinuous stresses, and introduce a novel 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 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 patient-specific geometries compared to the projection method. In patient-specific cases, 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 results into continuous spaces can introduce artifacts, such as the Gibbs phenomenon. Consequently, it becomes crucial to carefully select the finite element space for boundary stress calculations - not only in applications involving WSS computations for aneurysms.

Figures (7)

• Figure 1: Computational geometries for both aneurysm cases considered in this study, with the inflow branch and two outflow branches indicated by arrows.
• Figure 2: 2D Stokes flow: Logarithmic plot of $L_2$ error in velocity, pressure, and as a function of edge length, with errors evaluated against the analytical solution. Red circles show boundary-flux evaluation; green, blue and brown triangles show computed by projection in , and space, respectively. Dashed lines represent computed convergence rate for corresponding datasets. (a) $L_2$ errors of $\boldsymbol{v}, p$ for P1/P1 stabilized element. (b) $L_2$ errors of $\boldsymbol{v}, p$ for P2/P1 element. (c) assessment for P1/P1 stabilized element. (d) assessment for P2/P1 element.
• Figure 3: 3D Poiseuille flow: Logarithmic plot of $L_2$ error in velocity, pressure, and as a function of edge length, with errors evaluated against the analytical solution. Red circles show boundary-flux evaluation; green, blue and brown triangles show computed by projection in , and space, respectively. Dashed lines represent computed convergence rate for corresponding datasets. (a) $L_2$ errors of $\boldsymbol{v}, p$ for P1/P1 stabilized element on meshes with boundary layers. (b) $L_2$ errors of $\boldsymbol{v}, p$ for P2/P1 element on uniform meshes. (c) assessment for P1/P1 stabilized element on meshes with boundary layers. (d) assessment for P2/P1 element on uniform meshes.
• Figure 4: Aneurysm case 1: Maximum, minimum and average values of over the aneurysm dome for each evaluation method. (a, c, e) P1/P1 stabilized element, evaluated on meshes with boundary layers. (b, d, f) P2/P1 element, evaluated on uniform meshes.
• Figure 5: Aneurysm case 2: Maximum, minimum and average values of over the aneurysm dome for each evaluation method. (a, c, e) P1/P1 stabilized element, evaluated on meshes with boundary layers. (b, d, f) P2/P1 element, evaluated on uniform meshes.