Staggered Coupling Method
Theory: the lag from alternating solves
Overview
We couple thermal and structural analyses by running them alternately. Does that give the right answer?
With weak coupling, nearly so. In alternating solves, within each time step the side solved first uses the other's previous-step values. This lag causes error, small for weak coupling but large for strong. More important is stability: even if each solver is unconditionally stable implicit, the exchange behaves like an explicit step, so the coupled system can diverge at large time steps. Let's check with a simple two-variable coupled equation.
A coupled two-variable model
u and v represent the two physics; b and c set coupling strength. Weak coupling (b = c = 0.5) is compared with strong oscillatory coupling (b = 5, c = −5). Initial u = 1, v = 0; solutions to time 2 are compared with the exact solution.
Alternating scheme (Gauss–Seidel)
Each equation uses backward Euler (implicit). The Jacobi variant uses u^n (old value) in the v equation too. G is the one-step amplification matrix; spectral radius ρ below 1 means stability.
The era of coupling solvers
Fluid, structural, thermal and electromagnetic solvers developed separately over decades in their own fields. From the 1990s demand grew to combine them for coupled problems, and staggered (partitioned) coupling, reusing existing solvers, became widespread. Meanwhile Farhat and colleagues, studying aircraft aeroelasticity, analysed the accuracy and stability of alternating schemes in detail and proposed predictors to compensate lag and strongly coupled methods iterating within each step.
Worked examples
Example 1: error with weak coupling (b = c = 0.5, t = 2)
| Scheme | Δt = 0.2 | Δt = 0.1 | Δt = 0.05 | Δt = 0.025 | Observed order |
|---|---|---|---|---|---|
| Monolithic | 2.34×10⁻² | 1.19×10⁻² | 6.00×10⁻³ | 3.01×10⁻³ | 0.99 |
| Gauss–Seidel | 2.34×10⁻² | 1.21×10⁻² | 6.16×10⁻³ | 3.11×10⁻³ | 0.99 |
| Jacobi | 3.47×10⁻² | 1.79×10⁻² | 9.12×10⁻³ | 4.60×10⁻³ | 0.99 |
Example 2: stability with strong oscillatory coupling (b = 5, c = −5)
| Time step | Monolithic | Gauss–Seidel | Jacobi |
|---|---|---|---|
| 0.5 | 0.329 | 0.577 | 1.555 (diverges) |
| 1.0 | 0.180 | 3.283 (diverges) | 2.082 (diverges) |
| 2.0 | 0.093 | 6.122 (diverges) | 2.595 (diverges) |
Values are spectral radii of the amplification matrix. Stability limits are Δt = 0.567 for Gauss–Seidel and 0.130 for Jacobi. The system's eigenvalues are −1.5 ± 4.98i, a physically decaying oscillation.
Each is solved implicitly, yet alternating makes it diverge.
Each solver is implicit internally, but the partner's values come from the previous step, so the coupling terms are treated explicitly, and explicit schemes have time-step limits that shrink as coupling strengthens. Weakly coupled problems are fine to alternate, but strongly coupled ones need smaller steps or iterating the alternation within each step to convergence (strong coupling). Iterated to convergence, the result matches monolithic and stability returns. Coupling strength depends on each physics' time scale and mutual influence, so first check a stable step on a small problem solved both ways.
Coupled-solve workflow
- Estimate each physics' time scale and mutual influence.
- Use one-way or staggered coupling for weak coupling; strong coupling or monolithic for strong.
- Confirm results don't change with time step and don't diverge.
- For strong coupling, set convergence criteria and relaxation for sub-iterations.
- Check exchanged quantities (temperature, heat flux, displacement, force) are conserved.
“It started oscillating with a larger time step”
In a coupled electromagnetic–thermal analysis, doubling the time step to save run time made temperature swing up and down each step until it diverged. Each solver was implicit and stable alone even with large steps. The cause was a material whose resistance varied strongly with temperature, coupling temperature and heating tightly. Iterating electromagnetics and heat about three times within each step to convergence made the doubled step stable, and total run time still fell.
Common mistakes
Mistakes and fixes
| Mistake | Effect | Fix |
|---|---|---|
| Assuming implicit means stable | Coupled divergence | Check coupled stability |
| Jacobi-type exchange | Narrow stability range | Gauss–Seidel |
| Single exchange with strong coupling | Error and divergence | Iterate within steps |
| No time-step study | Lag error missed | Compare step sizes |
| Non-conservative exchange | Energy drift | Conservative mapping |
I'd like to learn related topics.
Related pages include convergence of coupled solves, co-simulation, mapping between dissimilar meshes, model order reduction and multiphysics digital twins.
Related Topics
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