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

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 analysis of our stabilized fluid formulation, including studies of Nitsche’s parameter and the ghostpenalty 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.     «

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 ad...     »