An XFEM‐based embedding mesh technique for incompressible viscous flows

Abstract

SUMMARYThis paper presents a finite element (FE) embedding mesh technique to efficiently embed arbitrary fluid mesh patches into Cartesian or unstructured background fluid grids. Our motivating application for such a technique is to efficiently resolve flow features like boundary layers around structures, which is achieved by attaching fluid boundary layer meshes around these structure surfaces. The proposed technique can be classified as a non-overlapping domain decomposition method. The particular feature is that the embedded patch cuts a void region into the background grid independently of background element edges. Since the embedded fluid domain ends in the middle of background elements, extended FE techniques are used to model a sharp separation between active and inactive regions on the background grid. The active background region is coupled to the boundary layer mesh using a mixed/hybrid Lagrange multiplier technique as proposed in Gerstenberger and Wall (Int. J. Numer. Meth. Engng. 2010; 82:537-563). The coupling formulation works without stabilization for the Lagrange multiplier unknowns and the Lagrange multiplier can be completely condensed on the element level. Within this paper, the approach is derived for incompressible, viscous flows. Three-dimensional examples using linear and quadratic shape functions demonstrate the correctness and the versatility of the proposed approach.

The tetrahedral finite cell method for fluids: Immersogeometric analysis of turbulent flow around complex geometries

“…Baaijens [18] and Parussini et al [19,20] combined the fictitious domain approach with Lagrange multiplier fields at the interface for immersed thin and volumetric structures. Gerstenberger, Wall and coworkers [21][22][23] combined Lagrange multiplier fields with interface enrichments of the velocity and pressure fields in the sense of the extended finite element method to ensure the separation of physical and fictitious domains. Rüberg and Cirak [24,25] combined weak Nitsche-type coupling methods at the interface with Cartesian B-spline finite elements for moving boundary and FSI problems.…”

The non-symmetric Nitsche method for the parameter-free imposition of weak boundary and coupling conditions in immersed finite elements

“…In contrast to Lagrange multiplier methods [8,29,54,70], the symmetric Nitsche formulation is free of auxiliary fields, which simplifies the theory and reduces computational cost. The variational consistency of the symmetric Nitsche method allows the reinterpretation of the penalty parameter as a mesh dependent stabilization parameter that needs be chosen sufficiently large as to maintain stability of the bilinear form.…”

A stabilized Nitsche‐type extended embedding mesh approach for 3D low‐ and high‐Reynolds‐number flows

“…For the 3D‐1Z case, the flow in a channel with dimensions [0,2.5] × [0,0.41] × [0,0.41] is computed, where a cylinder with circular cross‐section of diameter d = 0.1 and a length of 0.41 is centered at (0.5,0.2,0.205). Similar to , a boundary layer mesh around the cylinder is embedded into a regular background mesh , as visualized in Figure for the different configurations.…”

“…The additional use of a strain‐rate‐balance coupling equation renders this approach in a stabilized formulation. In , we applied this method to embed fluid patches into background fluid meshes for mainly viscous flows. A symmetric variant of this approach has been proposed in , and its close relation to Nitsche's method, as originally proposed in , has been discussed for the Poisson and Stokes problem therein.…”