On the stability and conditioning of a fictitious domain formulation for fluid-structure interaction problems

TL;DR

This paper analyzes a fictitious domain method for fluid-structure interaction, demonstrating stability and well-conditioned behavior regardless of mesh refinement or interface position, without requiring conforming meshes.

Contribution

It provides a stability and conditioning analysis of a distributed Lagrange multiplier fictitious domain approach for fluid-structure interaction problems.

Findings

The formulation is stable regardless of small intersections.

Condition number remains unaffected by interface position.

Mesh refinement does not deteriorate conditioning.

Abstract

We consider a distributed Lagrange multiplier formulation for fluid-structure interaction problems in the spirit of the fictitious domain approach. This is an unfitted method, which does not require the construction of meshes conforming to the interface. We focus on the stationary problem arising from the time discretization and we analyze the behavior of the condition number with respect to mesh refinement. At the numerical level, the computation of the term coupling the fluid and the solid mesh requires the knowledge of the intersection between fluid and mapped solid elements and it might happen that a portion of the intersected elements is very small. We show that our formulation is stable independently of such intersections and that the conditioning is not affected by the interface position.