A new face-oriented stabilized XFEM approach for 2D and 3D incompressible Navier–Stokes equations

Abstract

We develop a new stabilized XFEM based fixed-grid approach for the transient incompressible Navier–Stokes equations using cut elements. Our framework is based on a new flexible strategy to handle several degrees of freedom (DOFs) at nodes required for complex shaped embedded meshes as they occur for fixed-grid fluid–structure interaction problems. Boundary conditions on embedded boundaries are imposed weakly using a Nitsche type approach. Ghost penalty terms for velocity and pressure are added for stability reasons and to improve the conditioning of the system matrix. The idea of ghost penalties, previously developed for Stokes problems, is extended to the incompressible Navier–Stokes equations by using face-oriented fluid stabilizations for both viscous and convective dominated flows. Furthermore, the need for additional terms to enforce boundary conditions for non-viscous flows is addressed. A detailed numerical study of our stabilized fluid formulation, including studies of Nitsche’s parameter and the ghost penalty parameter, shows optimal error convergence and a good system conditioning in the viscous and the convective dominated cases. Compared to other fixed-grid fluid formulations, our method is much more accurate and less sensitive to the location of the interface. Regarding the accuracy and the sensitivity with respect to the interface position a clear improvement could be observed for the viscous and pressure fluxes as well as for the imposition of the boundary condition. Results of a transient two-dimensional high Reynolds-number flow over a cylinder and the flow over a thick plate in 3D confirm the applicability of the proposed method.