Two-Way Fluid-Structure Interaction
Theory: added mass and coupling stability
Overview
My two-way FSI keeps diverging when the structure is underwater.
It's common in partitioned schemes that solve fluid and structure alternately in separate solvers. When the structure moves it must push fluid aside, so it behaves heavier than it is; the extra is added mass. In air it's negligible, but in water or blood it can exceed the structure's own mass. Once it does, the error grows with every exchange between fluid and structure, and the solve diverges however small the time step. The remedies are relaxing the exchanged displacement or solving fluid and structure together in a monolithic scheme.
Added-mass ratio and error factor
λ is the panel mode wavelength and m_s structural mass per area. In a simple model, each exchange multiplies the error by 1 − ω(1 + r). With ω = 1 (no relaxation), r > 1 diverges.
Convergence condition and Aitken relaxation
R is the residual of the exchanged displacement. Aitken relaxation recomputes ω each iteration from the last two residuals.
An instability learned from blood vessels
The added-mass instability gained attention in the 2000s as coupled blood-flow and vessel-wall simulations spread. Vessel walls are thin and about as dense as blood, so the added-mass ratio is large. Causin and colleagues showed in 2005 that explicit partitioned schemes do not become stable with smaller time steps, and in fact tend to become less stable. This drove research into relaxed iterations, monolithic schemes and quasi-Newton coupling such as IQN-ILS.
Worked examples
Example 1: added-mass ratio and iterations (residual 10⁻⁶)
| Case | Ratio r | ω limit | ω = 1 | ω = 0.25 | ω = 0.1 |
|---|---|---|---|---|---|
| 2 mm steel, air | 0.006 | 1.99 | 3 | 48 | 131 |
| 1 mm aluminium, air | 0.021 | 1.96 | 4 | 47 | 129 |
| 2 mm steel, water | 5.07 | 0.33 | diverges | 21 | 15 |
| 0.5 mm artery wall, blood | 5.78 | 0.30 | diverges | 39 | 13 |
Mode wavelengths are 0.5 m (steel) and 0.3 m (aluminium); the artery has radius 3 mm (circumferential mode). In air, no relaxation is fastest; relaxing just adds iterations.
Example 2: three vibration modes
| Case | r per mode | Best fixed ω | Aitken |
|---|---|---|---|
| 2 mm steel, water | 5.1 / 3.0 / 1.5 | 0.23 (16) | 13 |
| Artery wall, blood | 5.8 / 3.5 / 1.7 | 0.21 (17) | 13 |
| 2 mm steel, air | 0.01 / 0.00 / 0.00 | 1.00 (3) | 5 |
I assumed a smaller time step would stabilise it.
That's true for most flow calculations, but not for added-mass instability. The added-mass ratio depends on structure and fluid density, plate thickness and mode shape, not time step, so a smaller step leaves the error factor unchanged. It has even been reported that smaller steps make added-mass effects more prominent relative to structural inertia, worsening stability. Remedies are lower relaxation, smarter exchanges via Aitken or quasi-Newton methods, or a monolithic solver. When r is small, as in air, relaxation is unnecessary and only adds iterations, so estimate r before choosing settings.
Practical workflow
- Estimate added-mass ratio r from densities and thickness.
- If r is small, consider one-way coupling or unrelaxed partitioned coupling.
- If r exceeds 1, use strong coupling (iterating within each step), starting ω well below 2/(1 + r).
- Reduce iterations with Aitken or quasi-Newton methods.
- Check interface force balance and displacement continuity each step.
“A tenth of the time step didn't help”
A coupled analysis of a thin underwater partition diverged within a few steps, with displacement oscillating and growing. The engineer cut the time step by 10, then 100, but divergence only came sooner. The plate was 2 mm steel with an estimated added-mass ratio of about 5: the unrelaxed partitioned scheme multiplied the error fivefold each exchange. Switching to strong coupling with Aitken relaxation solved it stably with just over ten iterations per step, and the original time step could be restored.
Common mistakes
Mistakes and fixes
| Mistake | Effect | Fix |
|---|---|---|
| Shrinking time step to stabilise | Diverges sooner | Relaxation, strong coupling |
| Weak coupling in water | Added-mass divergence | Iterate within steps |
| Heavy relaxation in air | Wasted iterations | Start at ω = 1 |
| Mismatched interface meshes | Forces not conserved | Conservative mapping |
| Large fluid mesh deformation | Inverted cells | Choose mesh-motion method |
I'd like to learn related topics.
Related pages include one-way FSI, ALE FSI, immersed boundary method, vortex-induced vibration and aeroelasticity.
Related Topics
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