Nitsche-XFEM for the coupling of an incompressible fluid with immersed thin-walled structures

Abstract

International audienceIn this paper we introduce a Nitsche-XFEM method for fluid-structure interaction problems involving a thin-walled elastic structure (Lagrangian formalism) immersed in an incompressible viscous fluid (Eulerian formalism). The fluid domain is discretized with an unstructured mesh not fitted to the solid mid-surface mesh. Weak and strong discontinuities across the interface are allowed for the velocity and pressure, respectively. The fluid-solid coupling is enforced consistently using a variant of Nitsche's method with cut-elements. Robustness with respect to arbitrary interface intersections is guaranteed through suitable stabilization. Several coupling schemes with different degrees of fluid-solid time splitting (implicit, semi-implicit and explicit) are investigated. A series of numerical tests in 2D, involving static and moving interfaces, illustrates the performance of the different methods proposed

Monolithic cut finite element–based approaches for fluid‐structure interaction

“…Such an unfitted mesh technique has been considered in, for example, the works of Gerstenberger and Wall, Burman and Fernández, Alauzet et al, and Fernández and Landajuela and enables arbitrary solid motions within and interactions with the surrounding fluid. In contrast to Approach 1, the fluid mesh is assumed to be fixed over time and, thus, is not subjected to any distortion, and it does not necessarily need to account for mesh motion by an ALE technique.…”

“…A Nitsche‐based Cut FEM for Stokes flow interacting with linear elastic structures has been analyzed in the work of Burman and Fernández . A comparison of various coupling strategies of an incompressible fluid with immersed thin‐walled structures has been recently provided in the work of Alauzet et al Moreover, different formulations applicable to fixed‐grid methods based on utilizing an additional embedded fluid patch, which fits to the fluid‐structure interface but overlaps with a fixed background fluid grid in an unfitted fashion, have been developed in the works of Massing et al, Shahmiri et al, and Schott et al…”

“…40 A Nitsche-based CutFEM for Stokes flow interacting with linear elastic structures has been analyzed in the work of Burman and Fernández. 41 A comparison of various coupling strategies of an incompressible fluid with immersed thin-walled structures has been recently provided in the work of Alauzet et al 42 Moreover, different formulations applicable to fixed-grid methods based on utilizing an additional embedded fluid patch, which fits to the fluid-structure interface but overlaps with a fixed background fluid grid in an unfitted fashion, have been developed in the works of Massing et al, 29 Shahmiri et al, 43 and Schott et al 44 The objective of this paper is to provide a generalized unfitted framework for FSI solvers, which allows to relax strong limitations of most computational FSI approaches existing so far and might build the basis for future developments on more advanced FSI applications. While the structure is always approximated in a Lagrangian setting with an interface-fitted moving mesh, for the flow field, different fitted meshes and geometrically unfitted approaches, including Nitsche-based overlapping domain decomposition techniques, are provided and juxtaposed.…”

Unified one‐fluid formulation for incompressible flexible solids and multiphase flows: Application to hydrodynamics using the immersed structural potential method (ISPM)

“…The accurate transport of the level set vector field is necessary to correctly evaluate the indicator function vector field H in above Equations (25a) and (25b) when evaluating the smooth density (18) and viscosity (19) fields. Wherever necessary, re-initialisations strategies for the level set fields can be employed …”

“…An alternative recent trend to completely overcome these consistency issue is the combination of a local XFEM enrichment with a cut-finite element method (FEM) methodology and a Lagrange multiplier (or a Nitsche's method) treatment of the interface coupling. [17][18][19] The pioneering IBM was first introduced by Peskin 6 to simulate the deformation of heart valves. The distinguishing feature of this method is the fact that the simulation is carried out on a fixed Eulerian Cartesian grid, which does not conform to that of the current geometry of the deformed immersed structure.…”