A discontinuous Nitsche‐based finite element formulation for the imposition of the Navier‐slip condition over embedded volumeless geometries

Abstract

This work describes a novel formulation for the simulation of incompressible Navier–Stokes problems involving nonconforming discretizations of membrane‐like bodies. The new proposal relies on the use of a modified finite element space within the elements intersected by the embedded geometry, which is represented by a discontinuous (or element‐by‐element) level set function. This is combined with a Nitsche‐based imposition of the general Navier‐slip boundary condition, to be intended as a wall law model. Thanks to the use of an alternative finite element space, the formulation is capable of reproducing exactly discontinuities across the embedded interface, while preserving the structure of the graph of the discrete matrix. The performance, accuracy and convergence of the new proposal is compared with analytical solutions as well as with a body fitted reference technique. Moreover, the proposal is tested against another similar embedded approach. Finally, a realistic application showcasing the possibilities of the method is also presented.

An embedded Finite Element framework for the resolution of strongly coupled Fluid–Structure Interaction problems. Application to volumetric and membrane-like structures

“…More details on the implementation, convergence and accuracy analysis of the discontinuous Nitsche Navier-slip imposition can be found in [46]. This approach is also validated with data from biomedical in vitro experiments in [8].…”

“…1 This makes possible to represent the velocity and pressure discontinuities arising from the immersion of any body, regardless of its type (volumetric or volumeless). This formulation is enhanced in [46] by using a Nitsche method to impose a Navier-slip condition, allowing thus the representation of any wall behaviour from the no-slip to the slip limits.…”

“…This subsection presents the two embedded methods that we employ to weakly impose the fluid BCs over the level set intersections. On the one hand, we use the modified Nitsche method presented in [40] for those bodies that have a well defined internal volume, On the other hand, we use the discontinuous formulation described in [46] for the membrane-like bodies that feature no internal volume. In the following, we briefly describe the particularities of each one of these approaches.…”