4. Interface kinematics and dynamics

In a partitioned FSI simulation, the fluid and structural solvers are coupled only through the interface. Everything the two domains "know" about each other is exchanged as kinematic quantities (displacement, velocity) and dynamic quantities (traction, pressure, viscous stress) on that shared surface. If the interface conditions are formulated incorrectly, or if the mesh motion violates geometric consistency, the coupled solution will drift, stall, or diverge — even when each solver is individually stable. This chapter defines the four interface conditions that a correct FSI coupling must satisfy, then examines the mesh motion strategies that keep the fluid domain valid as the boundary moves.

4.1 The Four Interface Conditions

Let Γ_FSI denote the wetted interface, with unit normal n pointing from the fluid into the solid. The following conditions must hold at every point of Γ_FSI at every coupling instant.

Kinematic continuity (no-slip and no-penetration). For a viscous fluid, the fluid velocity at the wall equals the wall velocity:

u_f(x, t) = d_s_dot(x, t)   on Γ_FSI
u_f · n = d_s_dot · n        (no-penetration)
u_f × n = d_s_dot × n        (no-slip, tangential)

Here u_f is the fluid velocity and d_s_dot is the structural velocity. The no-penetration part prevents mass flux through the wall; the no-slip part is what generates wall shear stress and drives boundary-layer separation. In a partitioned scheme, the structural displacement is transferred to the fluid mesh, and the fluid solver imposes the wall velocity as a Dirichlet condition on the momentum equations.

Dynamic continuity (traction equilibrium). Newton's third law requires that the traction exerted by the fluid on the solid equals and opposes the traction exerted by the solid on the fluid:

σ_f · n = −σ_s · n   on Γ_FSI
σ_f · n = −p n + τ · n

where σ_f and σ_s are the Cauchy stress tensors of fluid and solid, p is fluid pressure, and τ is the viscous stress tensor. In practice the fluid solver exports the total traction (pressure plus viscous) and the structural solver applies it as a Neumann boundary condition. A common error is to transfer only pressure and neglect the viscous traction; this is acceptable only when the viscous contribution is negligible relative to pressure, which is rarely true near separation or in low-Reynolds-number flows.

Geometric consistency. The fluid and solid meshes must share the same interface geometry at every coupling step. If the structural solver deforms the interface by Δd but the fluid mesh only moves by 0.9Δd, a spurious gap or overlap appears. This violates mass conservation and introduces an artificial leakage path. Geometric consistency is enforced by transferring the same displacement field to both sides and by using a consistent interpolation/projection operator.

Energy consistency. The work done by the fluid traction on the interface must equal the work absorbed by the structure, with no spurious energy injection or dissipation at the coupling. This is a consequence of the first three conditions when the transfer operators are adjoint-consistent, but it is worth checking explicitly in strongly coupled problems.

4.2 Transfer Operators: Interpolation and Projection

The fluid and solid meshes are generally non-matching: different element types, different resolutions, and different node distributions along Γ_FSI. Two transfer directions must be defined.

Fluid-to-solid (load transfer). The fluid traction is known at fluid face centers or nodes. To apply it to the structural mesh, a conservative projection is used. For a node i on the structural side, the consistent nodal load is

F_i = ∫_Γ N_i(x) · t_f(x) dΓ

where N_i is the structural shape function. This integral is evaluated by projecting each fluid face onto the structural surface and integrating. A simple and robust alternative for matching meshes is node-to-node transfer; for non-matching meshes, a mortar or nearest-neighbor-with-conservation approach is preferred.

Solid-to-fluid (displacement transfer). The structural displacement is known at structural nodes. The fluid interface nodes receive an interpolated displacement:

d_f(x) = Σ_j N_j(x) · d_s_j

where the sum runs over structural nodes whose shape functions have support at x. For non-matching meshes, a radial basis function (RBF) interpolation or a nearest-neighbor inverse-distance weighting is commonly used. The key requirement is that the transfer must be conservative: the total force and total moment transferred must be preserved.

4.3 Mesh Motion Strategies for Moving Boundaries

Once the interface displacement is known, the fluid mesh must deform to accommodate it. Three families of strategies are used in practice.

Arbitrary Lagrangian–Eulerian (ALE) with smoothing. The mesh nodes move with a velocity w that is independent of the fluid velocity. The ALE form of the momentum equation is

ρ (∂u/∂t + (u − w) · ∇u) = −∇p + ∇ · τ

The mesh velocity w is obtained by solving a Laplace or linear-elasticity equation for the mesh displacement field, with the interface displacement as a Dirichlet condition and the far-field boundaries fixed. This is the most common approach for moderate boundary motion. The mesh quality degrades as the boundary moves; remeshing is triggered when the minimum cell volume or maximum skewness exceeds a threshold.

Overset (Chimera) grids. The fluid domain is decomposed into a background grid and one or more component grids that move rigidly with the structure. Interpolation between grids is performed at each time step. This avoids mesh deformation entirely and is well suited to large rigid-body motion (e.g., a pitching airfoil or a flapping flag). The cost is the interpolation overhead and the need to maintain adequate overlap.

Remeshing. When the boundary motion is too large for ALE smoothing, the fluid mesh is regenerated periodically. This is the most robust but also the most expensive option, and it introduces interpolation errors each time the solution is mapped from the old mesh to the new one. Remeshing is typically combined with ALE: smoothing is used until a quality threshold is reached, then remeshing is triggered.

4.4 A Worked Example: Flow Past a Flexible Cantilever

Consider a 2D channel of height H = 0.1 m and length L = 0.5 m. A flexible cantilever of length l = 0.08 m and thickness h = 0.002 m is attached to the bottom wall at x = 0.1 m. The fluid is water (ρ = 1000 kg/m³, μ = 1.0×10⁻³ Pa·s), inflow velocity is 0.5 m/s, giving Re ≈ 50,000 based on channel height. The solid has Young's modulus E = 2.0 MPa, Poisson ratio ν = 0.3, and density ρ_s = 1200 kg/m³.

Setup. The fluid mesh uses 200×40 quadrilateral cells with ALE smoothing. The solid mesh uses 40×4 quadrilateral elements. The coupling is partitioned with a fixed-point iteration and Aitken relaxation. Time step Δt = 1×10⁻⁴ s. The interface normal points from fluid into solid, i.e., upward along the cantilever's wetted surface.

What to observe. At each coupling step, the structural solver computes the tip displacement d_tip. The fluid solver receives the interface displacement and updates the mesh. The fluid traction is transferred back to the solid. The iteration continues until the interface residual (change in tip displacement between coupling iterations) falls below 1×10⁻⁶ m.

Typical failure symptoms and diagnosis.

4.5 Practical Guidelines

• Always verify conservation. After building transfer operators, apply a unit constant traction and check that the integrated force and moment are preserved to within 1–2%.
• Match the displacement field exactly. Use the same interpolation order on both sides of the interface, or use a conservative projection that preserves the integral of displacement.
• Choose the mesh motion strategy based on expected boundary amplitude. ALE smoothing handles displacements up to roughly 20–30% of the local cell size. Beyond that, overset or remeshing is more robust.
• Monitor mesh quality continuously. Track minimum cell volume, maximum skewness, and maximum aspect ratio. Trigger remeshing before quality degrades below solver tolerance.
• Use under-relaxation or Aitken acceleration for the coupling iteration when the added-mass effect is significant (ρ_s/ρ_f < 10).

The interface conditions and mesh motion strategy together determine whether a partitioned FSI simulation remains stable and accurate. The next chapter builds on this foundation by examining the coupling algorithms — explicit, implicit, and semi-implicit — that enforce these conditions at the discrete level.