A stabilised immersed boundary method on hierarchical b-spline grids

In this work, an immersed boundary finite element method is proposed which is based on a hierarchically refined cartesian b-spline grid and employs the non-symmetric and penalty-free version of Nitsche's method to enforce the boundary conditions. The strategy allows for h- and p-refinement and employs a so-called ghost penalty term to stabilise the cut cells. An effective procedure based on hierarchical subdivision and sub-cell merging, which avoids excessive numbers of quadrature points, is used for the integration of the cut cells. A basic Laplace problem is used to demonstrate the effectiveness of the cut cell stabilisation and of the penalty-free Nitsche method as well as their impact on accuracy. The methodology is also applied to the incompressible Navier-Stokes equations, where the SUPG/PSPG stabilisation is employed. Simulations of the lid-driven cavity flow and the flow around a cylinder at low Reynolds number show the good performance of the methodology. Excessive ill-conditioning of the system matrix is robustly avoided without jeopardising the accuracy at the immersed boundaries or in the field.