Partitioned and Monolithic Solution Methods
Theory: two ways to solve
Overview
For fluid–structure coupling, I can't decide between a partitioned and a monolithic solver.
A partitioned solver solves fluid and structure separately, exchanging forces and displacements at the interface and iterating. Its big advantage is combining familiar solvers as they are. A monolithic solver assembles both into one system and solves them together — stable even with strong coupling, but it needs a dedicated solver. The choice usually comes down to coupling strength, and a key measure of that is the ratio of the fluid's added mass to the structural mass.
The added-mass model
When a mass $m_s$ on a spring $k$ moves in incompressible fluid, the fluid pushes back with force $-m_a \ddot x$. If a partitioned scheme computes the fluid force from the previous acceleration, the error is multiplied by $-m_a/m_s$ at each exchange. Above a ratio of 1, the error keeps growing.
Scaling the update by relaxation $\omega$ changes the error factor to $1 - \omega(1 + m_a/m_s)$, so even large ratios converge with a small enough $\omega$. Aitken relaxation sets $\omega$ automatically from the last two residuals.
An instability made famous by blood vessels
The added-mass instability became widely known in the 2000s as coupled blood-flow and vessel-wall analysis grew. Where the structure is far heavier than the fluid, like an aircraft wing in air, partitioned schemes ran without trouble. But a vessel wall has nearly the same density as blood and is thin. Added mass exceeded structural mass, and cases were even reported where smaller time steps made things less stable. This spurred major development of stabilising methods for partitioned schemes: relaxation, quasi-Newton methods and Robin-type coupling with modified interface conditions.
Worked example: varying the added-mass ratio
With a 1 kg mass and 100 N/m spring, a loosely coupled scheme exchanging force once per step was compared with the monolithic solution (initial displacement 0.01 m, 400 steps):
| Added-mass ratio | Loose, Δt = 0.01 s | Loose, Δt = 0.001 s | Monolithic |
|---|---|---|---|
| 0.5 | Stable | Stable | Stable |
| 0.9 | Stable | Stable | Stable |
| 1.1 | Diverges | Diverges | Stable |
| 1.5 | Diverges | Diverges | Stable |
It diverges even with a time step ten times smaller?
Yes — that's what sets it apart from ordinary numerical instability. The error factor depends only on $m_a/m_s$, with no time step in it, so shrinking the step doesn't help. An incompressible fluid responds instantly as a whole when the structure accelerates, so delivering the force late is itself the problem. If the added-mass ratio may exceed 1, treat loose coupling as unusable and choose sub-iteration or a monolithic solver.
Example 2: sub-iteration counts
Iterations to reach 10⁻⁸ error when exchanges are repeated within one time step:
| Added-mass ratio | No relaxation | Relaxation 0.5 | Aitken |
|---|---|---|---|
| 0.1 | 8 | 24 | 2 |
| 0.5 | 27 | 14 | 2 |
| 0.9 | 175 | 7 | 2 |
| 1.5 | Diverges | 14 | 2 |
| 10 | Diverges | Diverges | 2 |
With a single unknown, Aitken finds the exact relaxation factor in 2 iterations. Real coupled analyses have many interface unknowns and added mass is not a single number, so more iterations are needed — though usually still fewer than with fixed relaxation. At small ratios such as 0.1, fixed relaxation 0.5 actually slows convergence.
Choosing a strategy
- Estimate the added-mass ratio from density ratio and thickness (small for heavy structures in gas, large for thin structures in liquid).
- If the ratio is small, loose coupling suffices; near or above 1, use sub-iteration with relaxation (Aitken, quasi-Newton).
- If sub-iteration counts make run time a problem, consider a monolithic solver.
- Check that interface data transfer (non-matching meshes) conserves total force.
- Record convergence tolerances and per-step iteration history.
“It ran in air but crashes in water”
Vibration of a thin metal plate was solved without trouble in air using loosely coupled FSI; running the same setup in water diverged within a few steps, and a ten-times smaller time step gave the same result. Air's added mass may be a few percent of the plate mass, but water is about 800 times denser, so for a thin plate added mass far exceeds the plate's own. Adding sub-iterations with Aitken relaxation made it solve stably. When changing the medium, re-estimate the added-mass ratio.
Common mistakes
Mistakes and fixes
| Mistake | Effect | Fix |
|---|---|---|
| Shrinking the time step to fix it | Added-mass instability persists | Sub-iteration and relaxation |
| Loose coupling in liquid | Diverges in a few steps | Estimate added-mass ratio |
| Fixed relaxation at large ratios | Diverges | Aitken or quasi-Newton |
| Loose convergence tolerance | Energy creeps up | Check residual history |
| Non-conservative force transfer | Total force off | Conservative mapping |
I'd like to learn related topics.
Check added mass for various shapes with the added mass coefficient calculator. For concepts see strong versus weak coupling, staggered coupling and convergence in coupled analysis; for implementation, co-simulation and mapping between dissimilar meshes.
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