Simultaneous solution of incompressible Navier-Stokes flows on multiple surfaces

TL;DR

This work develops a unified framework for solving incompressible Stokes and Navier–Stokes flows simultaneously on all level-set–defined surfaces embedded in a 3D bulk domain, using a Bulk Trace FEM that is conforming to the boundary but non-conforming to internal level sets. By formulating strong and weak forms for single surfaces and then aggregating them over all level sets via the co-area formula, the authors implement a stable, higher-order finite element method with either Taylor–Hood or equal-order elements plus PSPG/Brezzi–Pitkáranta stabilization. Numerical results demonstrate higher-order convergence for stationary problems and good agreement with Surface FEM solutions across multiple test cases, including flow around obstacles and driven cavity flows on manifolds, as well as instationary flows on a torus. The approach enables efficient analysis of geometry variations and has potential applications in design optimization and biomechanics, with future work focusing on stabilization tuning and coupling with structural membranes.

Abstract

A mechanical model and finite element method for the simultaneous solution of Stokes and incompressible Navier-Stokes flows on multiple curved surfaces over a bulk domain are proposed. The two-dimensional surfaces are defined implicitly by all level sets of a scalar function, bounded by the three-dimensional bulk domain. This bulk domain is discretized with hexahedral finite elements which do not necessarily conform with the level sets but with the boundary. The resulting numerical method is a hybrid between conforming and non-conforming finite element methods. Taylor-Hood elements or equal-order element pairs for velocity and pressure, together with stabilization techniques, are applied to fulfil the inf-sup conditions resulting from the mixed-type formulation of the governing equations. Numerical studies confirm good agreement with independently obtained solutions on selected, individual surfaces. Furthermore, higher-order convergence rates are obtained for sufficiently smooth solutions.

Figures (23)

• Figure 1: A generic example: (a) The volumetric bulk domain in blue, some level sets shown in different colours. In (b) and (c), some example mesh and nodes with no-slip conditions (blue) and those on the inflow (red) are seen.
• Figure 2: Vector fields in the domain $\Omega$ and on the boundary $\partial\Omega$ shown on some level set $\Gamma_{\!c}$ with $c \in \left[\phi_{\min}, \phi_{\max}\right]$. The right figure shows a zoom of the left one. Normal vectors $\boldsymbol n$ with respect to the level sets $\Gamma_{\!c}$ in $\Omega$ are shown in blue. Normal vectors $\boldsymbol m$ with respect to $\partial\Omega$ are red, tangential vectors $\boldsymbol t$ are gray and co-normal vectors $\boldsymbol q$ are green.
• Figure 3: Setup for the first numerical example: (a) The level sets which define the bulk domain $\Omega$, (b) some arbitrary mesh and Dirichlet boundary conditions at the inflow, and (c) the bulk domain in gray and some selected level sets in yellow and the level set with $r_0=1.0$ in blue.
• Figure 4: Velocity magnitudes and pressure of the first numerical example are shown. (a) to (d) show results obtained with the Bulk Trace FEM, while in (a) and (b) some arbitrarily selected level sets are shown, (c) and (d) show the surface with $r_0 = 1.0$. (e) and (f) show results obtained with the Surface FEM where the one considered surface has a radius of $1.0$ at $z=0$ and, therefore, (c) to (f) can be used to compare a Bulk Trace FEM solution with a Surface FEM solution.
• Figure 5: Convergence results for the simultaneous solution of the axisymmetric test case, (a) $L_2$-error of the velocities, (b) (pseudo-)energy error, (c) residual error in the momentum equations, and (d) residual error in the continuity equation. The numbers in the legends are the polynomial orders $\{q_{\mathrm{geom}},\, q_{\boldsymbol u},\,q_p\}$ of the FE function spaces.