An unconditionally stable semi-implicit CutFEM for an interaction problem between an elastic membrane and an incompressible fluid

TL;DR

This paper presents a novel, unconditionally stable semi-implicit CutFEM for simulating fluid-membrane interactions, allowing high accuracy and flexible meshing without stability restrictions, demonstrated through theoretical proofs and numerical tests.

Contribution

The paper introduces a high-accuracy, unconditionally energy stable CutFEM for fluid-structure interaction with membranes, overcoming previous parameter restrictions.

Findings

The method is unconditionally energy stable.

The scheme works with non-aligned structured meshes.

Numerical simulations confirm theoretical stability and accuracy.

Abstract

In this paper we introduce a finite element method for the Stokes equations with a massless immersed membrane. This membrane applies normal and tangential forces affecting the velocity and pressure of the fluid. Additionally, the points representing this membrane move with the local fluid velocity. We design and implement a high-accuracy cut finite element method (CutFEM) which enables the use of a structured mesh that is not aligned with the immersed membrane and then we formulate a time discretization that yields an unconditionally energy stable scheme. We prove that the stability is not restricted by the parameter choices that constrained previous finite element immersed boundary methods and illustrate the theoretical results with numerical simulations.