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

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.

💡 Research Summary

The paper presents a fictitious‑domain formulation for fluid‑structure interaction (FSI) problems that relies on a distributed Lagrange multiplier to enforce the kinematic coupling between an incompressible Newtonian fluid and an incompressible visco‑elastic solid. Unlike fitted ALE approaches, the fluid and solid are discretized on independent, unfitted meshes that do not need to conform to the moving interface. The solid deformation is described in a Lagrangian reference domain B and mapped to the current configuration by a deformation map X(s,t). The fluid equations are solved on a fixed Eulerian domain Ω that contains both the fluid and the solid region; the solid velocity is imposed in Ω via the constraint ∂X/∂t = u∘X, which is weakly enforced by a Lagrange multiplier λ belonging to a Hilbert space Λ. Two possible choices for Λ and the associated bilinear form c(·,·) are discussed: (i) Λ₀ = H¹(B)′ with c₀ as the duality pairing, and (ii) Λ₁ = H¹(B) with c₁ as the H¹ inner product. Both satisfy the non‑degeneracy condition c(μ,Z)=0 ⇒ Z=0.

After a backward Euler time discretization (the convective term is omitted for analysis), the semi‑discrete problem reduces at each time step to a stationary saddle‑point system involving four unknowns: fluid velocity u, fluid pressure p, solid displacement X, and multiplier λ. The variational formulation contains the standard fluid bilinear form a_f, a solid bilinear form a_s (including elastic stiffness κ and solid viscosity ν_s), the divergence constraints for incompressibility, and the coupling terms c(λ, v∘X) and c(μ, X−u∘X).

The authors prove that the continuous and discrete bilinear forms satisfy a uniform inf‑sup condition independent of the mesh sizes h_Ω and h_B. Consequently, the condition number of the global matrix can be bounded above by a constant that depends only on physical parameters (density difference δρ, viscosities ν_f, ν_s, elastic modulus κ) and the time step Δt, but not on how the interface cuts the background mesh. This result implies that the presence of “small cut cells” – cells that are only partially intersected by the interface and may have arbitrarily small measure – does not degrade the spectral properties of the system. No additional stabilization (e.g., ghost‑penalty) is required.

Numerical experiments confirm the theory. Two integration strategies for the coupling term are examined: (a) exact integration on each intersected element (requiring precise geometric intersection computation) and (b) a simplified rule using a single quadrature point per mapped solid element, which may miss very small cut cells. In both cases the condition number remains essentially unchanged as the mesh is refined and as the interface moves, even when sliver cells appear. Convergence studies show optimal rates for velocity, pressure, and solid displacement. A time‑dependent test demonstrates unconditional stability with respect to Δt, corroborating the earlier analytical stability result.

Overall, the work establishes that the fictitious‑domain, unfitted FSI formulation with a distributed Lagrange multiplier is robust: it is stable without artificial penalty terms, its conditioning is mesh‑independent, and it handles arbitrarily small intersected cells gracefully. These properties make the method attractive for complex biomedical simulations, large‑deformation problems, and multiphysics applications where mesh generation and interface tracking are major bottlenecks.