Numerical methods for immersed FSI with thin-walled structures

Abstract

The numerical simulation of a thin-walled structure immersed in an incompressible fluid can be addressed by various methods. In this paper, three of them are considered: the Arbitrary Lagrangian-Eulerian (ALE) method, the Fictitious Domain/Lagrange multipliers (FD) method and the Nitsche-XFEM method. Taking ALE as a reference, the advantages and limitations of FD and Nitsche-XFEM are carefully discussed on three benchmark test cases which have been chosen to be representative of typical difficulties encountered in valves or living cells simulations. Résumé :La simulation numérique d'une structure mince immergée dans un fluide incompressible peutêtre abordée par différentes méthodes. Dans cet article, trois d'entre elles sont considérées: la méthode Arbitrairement Lagrangienne Eulérienne (ALE), la méthode de domaine fictif avec multiplicateurs de Lagrange (FD) et la méthode Nitsche-XFEM. En prenant la méthode ALE comme référence, les avantages et les limites des méthodes FD et Nitsche-XFEM sont soigneusement discutés sur trois cas tests qui ontété choisis pourêtre représentatifs des difficultés typiques rencontrées dans les simulations de valves ou de cellules. Mots-clés :interaction fluide-structure, fluide incompressible, structure mince immergée, maillages non compatibles, maillages compatibles, méthode de domaine fictif, méthode XFEM, méthode de Nitsche, méthode ALE.

A Loosely Coupled Scheme for Fictitious Domain Approximations of Fluid-Structure Interaction Problems with Immersed Thin-Walled Structures

“…In order to overcome the artificial interfacial mass losses induced by the continuous nature of the pressure approximations considered in (3.4), we will consider (notably when dealing with enclosed fluid domains) the approach proposed in [34] for an immersogeometric method, which consists in boosting the grad-div stabilization while reducing the SUPG/PSPG stabilization near the interface by taking (see also [16,12]):…”

A Nitsche-based formulation for fluid-structure interactions with contact

“…For the latter, a theoretical stability and convergence analysis has been derived [42] . The reader is referred to Boilevin-Kayl et al [47] for a comparative study on the accuracy of some of these approaches.…”

Non-body-fitted fluid–structure interaction: Divergence-conforming B-splines, fully-implicit dynamics, and variational formulation

“…Moreover, the exact solution of the Eulerian pressure is usually discontinuous at the fluid-solid interface, which leads to poor approximation properties of the discrete pressure spaces used in immersed FSI methods. As a result, immersed FSI methods, whether they follow Peskin's idea or not (see [35,36,37,38,39]), often have profound difficulties to accurately impose the incompressibility constraint at both Eulerian and Lagrangian levels [40,41,6,34,38,25,42]. This issue is extremely important since large errors in the incompressibility constraint are able to even alter the qualitative behavior of numerical solutions in challenging FSI applications, such as heart valves [40,38] and cell-scale blood flow [25].…”

“…However, as the authors mentioned in [34,38], the modified stabilization will lead to locking and other instabilities if the scaling parameters are large enough and there is no rule to know a priori when this will happen. In addition, the modified stabilization increases the condition number of the final system of equations and it becomes a challenge to find a scalable solver in order to work with highly-refined three-dimensional meshes [38,42,25]. In [43], an ad hoc volume correction strategy at the Lagrangian level, to be carried out every ten time steps, was proposed to reduce the spurious volume change of co-dimension zero solids in immersed FSI methods.…”

“…Although the authors acknowledge that the introduction of this correction strategy, which is not derived from physical laws, may negatively impact the accuracy of numerical solutions. Interesting options to capture the pressure discontinuity at the interface are to combine the fictitious domain method with adaptive meshing following the interface as done in [37] and to use an extended finite element discretization as done in [44,42]. However, the automatic application of these two methods to complex three-dimensional problems remains an open problem.…”