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

TL;DR

This paper analyzes a fictitious-domain fluid–structure interaction method with a distributed Lagrange multiplier, highlighting its unconditional stability in time and well-posed discretization regardless of interface placement. It provides rigorous bounds on the condition number of the resulting saddle-point systems, showing that conditioning depends on mesh sizes $h_\Omega$ and $h_\mathcal B$ and on the choice of coupling form, but not on interface position or small cut cells. The authors compare exact and inexact coupling, deriving convergence guarantees that include quadrature-error terms, and validate the theory with extensive numerical tests on shifted squares and multiple immersed geometries. The results guide choosing the coupling form to balance conditioning and computational efficiency, and demonstrate that the method remains stable and accurate across challenging unfitted discretizations with nonmatching grids.

Abstract

We consider a distributed Lagrange multiplier formulation for fluid-structure interaction problems in the spirit of the fictitious domain approach. Our previous studies showed that the formulation is unconditionally stable in time and that its mixed finite element discretization is well-posed. In this paper, we analyze the behavior of the condition number with respect to mesh refinement. Moreover, we observe that our formulation does not need any stabilization term in presence of small cut cells and conditioning is not affected by the interface position.

Figures (13)

• Figure 1: Our geometrical setting. $\Omega$ and $\mathcal{B}$ are fixed domains, independent of time. A Lagrangian point $\mathbf{s}\in\mathcal{B}$ is mapped into $\mathbf{x}\in\Omega^s_t$ through the map $\mathbf{X}$.
• Figure 2: Mapping of a solid element $\mathrm{T}_s\in\mathcal{T}_h^\mathcal{B}$ into the fluid mesh $T_{h/2}^\Omega$ showing the support mismatch of fluid basis function (yellow) with the immersed solid element and quadrature nodes for inexact integration.
• Figure 3: The geometrical configuration of the shifted square. The value of $\sigma$ gives the shift between the fluid mesh and the mapped solid one.
• Figure 4: Condition number of the shifted square problem as a function of the shift $\sigma$. The value of $\sigma$ does not affect the condition number. For $\mathbf{c}=\mathbf{c}_0$ or $\mathbf{c}_{0,h}$, $\mathsf{cond}(\mathcal{A}_\square)$ is five order of magnitude larger than for $\mathbf{c}=\mathbf{c}_1$ or $\mathbf{c}_{1,h}$.
• Figure 5: Error for the shifted square test plotted with respect to $\sigma$. In all cases, results are not affected by the value of the shift. The curves related to the coupling $\mathbf{c}_{1,h}$ are plotted separately since they have a different scale due to the quadrature error.

Theorems & Definitions (22)

• Proposition 1
• Proposition 2
• Theorem 1
• Proposition 3
• Theorem 2
• Theorem 3
• Theorem 4
• Lemma 1
• proof
• Proposition 4