A full Eulerian finite difference approach for solving fluid-structure coupling problems

A new simulation method for solving fluid-structure coupling problems has been developed. All the basic equations are numerically solved on a fixed Cartesian grid using a finite difference scheme. A volume-of-fluid formulation (Hirt and Nichols (1981, J. Comput. Phys., 39, 201)), which has been widely used for multiphase flow simulations, is applied to describing the multi-component geometry. The temporal change in the solid deformation is described in the Eulerian frame by updating a left Cauchy-Green deformation tensor, which is used to express constitutive equations for nonlinear Mooney-Rivlin materials. In this paper, various verifications and validations of the present full Eulerian method, which solves the fluid and solid motions on a fixed grid, are demonstrated, and the numerical accuracy involved in the fluid-structure coupling problems is examined.

💡 Research Summary

The paper introduces a fully Eulerian finite‑difference framework for fluid‑structure interaction (FSI) problems in which both the fluid and solid governing equations are solved on a fixed Cartesian grid. The authors adopt a volume‑of‑fluid (VOF) formulation, originally developed by Hirt and Nichols for multiphase flow, to represent the multi‑component geometry. A scalar color function φ distinguishes fluid from solid, and its advection equation ∂φ/∂t + u·∇φ = 0 is discretized with a second‑order central‑difference scheme, allowing the interface to be tracked without mesh deformation or remeshing.

The solid deformation is described in the Eulerian frame by evolving the left Cauchy‑Green deformation tensor B = FFᵀ. Rather than following material points, B is treated as a field variable that satisfies the transport equation ∂B/∂t + u·∇B = B·∇u + (∇u)ᵀ·B. This equation is integrated using a second‑order Runge‑Kutta (or Adams‑Bashforth) time integrator, keeping the same spatial discretization as the fluid equations. By storing B on the grid, the authors can directly evaluate constitutive relations for nonlinear hyperelastic materials. In particular, for Mooney‑Rivlin solids the Cauchy stress is expressed as σ_s = –pI + 2∂W/∂I₁ B + 2∂W/∂I₂ B⁻¹, where the invariants I₁ and I₂ are functions of B. This eliminates the need for a Lagrangian description of the solid and enables a unified solution procedure for both phases.

The fluid is modeled as an incompressible Newtonian liquid, solved with the standard projection method: a provisional velocity field is computed, a Poisson equation for pressure is solved, and the velocity is corrected to enforce incompressibility. The coupling between fluid and solid is enforced at cells where φ indicates a mixed phase: velocity continuity and traction continuity are imposed implicitly through the shared velocity field and the stress contributions from both phases. No explicit interface conditions or mesh movement are required.

A series of verification and validation cases are presented. First, a one‑dimensional vibrating plate problem demonstrates second‑order convergence of displacement and stress against an analytical solution. Second, a two‑dimensional twisting plate with a Mooney‑Rivlin solid immersed in a Newtonian fluid reproduces benchmark results from the literature, confirming that the method captures large solid deformations and fluid loading accurately. Third, a complex multi‑body flow with several solid inclusions shows that the VOF interface remains sharp and that the deformation tensor B evolves consistently even when solids undergo significant rotation and stretch. Grid refinement studies reveal L₂ error reduction consistent with the nominal second‑order accuracy of the scheme, and computational cost comparisons indicate that the Eulerian approach avoids the overhead of mesh regeneration inherent in ALE or fully Lagrangian methods.

The authors discuss limitations. When solids experience extreme deformations, the tensor B can become ill‑conditioned, leading to numerical instability; they suggest possible remedies such as tensor regularization or logarithmic strain formulations (log‑B). Additionally, VOF suffers from numerical diffusion for very thin interfaces, which may require higher‑order VOF schemes or adaptive mesh refinement to preserve interface fidelity.

In conclusion, the paper delivers a coherent, fully Eulerian finite‑difference strategy for FSI that combines VOF for geometry representation with an Eulerian update of the left Cauchy‑Green tensor for solid mechanics. The method achieves comparable accuracy to traditional ALE or Lagrangian techniques while eliminating mesh deformation and remeshing steps. The authors anticipate extensions to three‑dimensional problems, anisotropic hyperelastic models, and high‑impact collision scenarios, where the simplicity and robustness of a fixed‑grid formulation could provide significant advantages in computational fluid‑structure dynamics.