What is direct (monolithic) coupling?

Assembling fluid and solid equations, including interface conditions, into one large system and solving them simultaneously. Interface temperature and heat flux always match automatically, so there's none of the back-and-forth iteration or convergence trouble of partitioned schemes that solve fluid and solid alternately. In exchange the single matrix is large, and combining separate fluid and solid codes is difficult.

Why can partitioned iteration diverge?

Passing interface temperature to the fluid to get heat flux, then that flux to the solid to get a new temperature, multiplies the error each pass by the solid-to-fluid resistance ratio R_s/R_f. Below 1 it converges, above 1 it diverges: ratio 0.1 took 7 iterations, 0.9 took 168, and 1.1 and 10 diverged.

Can partitioned schemes be stabilised?

Yes, with relaxation (applying a fraction of each update). The optimum factor is 1/(1 + R_s/R_f); in 1D it converged in one pass. Swapping which side receives temperature and which flux also works for large ratios (8 iterations at ratio 10). In real problems the optimum varies in space and time, so automatic schemes like Aitken relaxation are common.

Direct Coupling (Monolithic) Conjugate Heat Transfer

Category: Thermal – Conjugate Heat Transfer | Revised 2026-10-01

Monolithic conjugate heat transfer matrix system showing coupled solid-fluid temperature field

Direct Coupling (Monolithic) Conjugate Heat Transfer ― Simultaneously solving solid and fluid temperature fields in a single coupled equation system

Theory: solve together or alternate?

Overview

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Conjugate heat transfer can solve fluid and solid together or separately with exchanges. Do they give the same result?

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If they converge, yes — the question is whether they do. A partitioned scheme repeatedly passes interface temperature to the fluid for a heat flux, then that flux to the solid for a new temperature. Each pass multiplies the error by the solid-to-fluid resistance ratio; above 1, the error grows each pass and diverges. Direct coupling solves everything in one system, so the issue never arises. Let's count iterations and find when it diverges with a simple 1D model.

Error growth in partitioned iteration

$$ T_i^{(k+1)} - T_i^* = g\left(T_i^{(k)} - T_i^*\right),\qquad g = -\frac{R_s}{R_f},\qquad g_\omega = 1 - \omega\left(1 + \frac{R_s}{R_f}\right) $$

$T_i^*$ is the true interface temperature. $|g| < 1$ converges, $|g| > 1$ diverges. With relaxation ω the factor becomes $g_\omega$, which is zero at ω = 1/(1 + R_s/R_f).

The monolithic system

$$ \begin{pmatrix}\mathbf{A}_{ff} & \mathbf{A}_{fs}\\ \mathbf{A}_{sf} & \mathbf{A}_{ss}\end{pmatrix}\begin{pmatrix}\mathbf{T}_f\\ \mathbf{T}_s\end{pmatrix} = \begin{pmatrix}\mathbf{b}_f\\ \mathbf{b}_s\end{pmatrix} $$

Fluid and solid unknowns form one matrix including interface coupling terms A_fs and A_sf, solved at once. The matrix grows, but no interface back-and-forth is needed.

Coffee Break Trivia

Two schools of coupled analysis

Multiphysics coupling has two schools: monolithic, solving everything together, and partitioned, alternating between existing codes. Partitioned spread in industry because proven fluid and structural codes can be reused, but convergence issues follow. In the 1990s and 2000s, Aitken relaxation and quasi-Newton methods for setting relaxation automatically were developed, making partitioned schemes more robust. Meanwhile many commercial CFD codes solve conjugate heat transfer with direct coupling, treating solids as part of the same mesh.

Worked example: 1D conjugate heat transfer

Hot side 400 °C, cold side 300 °C. Fluid-side resistance set to 1, varying the solid-to-fluid ratio R_s/R_f. Initial interface guess 300 °C; converged at 1×10⁻⁶ K error.

Ratio R_s/R_f	No relaxation	Relaxation 0.5	Optimal relaxation	Roles swapped
0.1	7	21	1 (0.909)	Diverged
0.9	168	6	1 (0.526)	Diverged
1.1	Diverged	6	1 (0.476)	187
10	Diverged	Diverged	1 (0.091)	8

Direct coupling gives the correct interface temperature in one linear solve for every ratio.

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Just exceeding a ratio of 1 makes it diverge — and swapping roles makes the small-ratio cases diverge instead.

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In partitioned schemes, stability depends on the resistance ratio between the side receiving temperature and the side receiving flux; it's stable when the flux-receiving (Neumann) side has the smaller resistance. So a conductive metal with poorly transferring gas is stable with flux passed to the solid, as here, but insulation with liquid is the reverse. Direction and relaxation must suit the problem. In real 3D problems the ratio varies in space and time, so fixed relaxation can't cope; Aitken relaxation adjusts it each pass, or direct coupling is used. Direct coupling costs more per solve but removes the worry.

Choosing an approach

If one code handles both solid and fluid, make direct coupling the first choice.
In partitioned schemes, pass flux to the lower-resistance side.
Don't fix relaxation; adjust automatically (e.g. Aitken).
Monitor residuals of both interface temperature and flux to confirm convergence.
In transients, iterate enough within each time step rather than settling for weak coupling.

Coffee Break Trivia

“Changing the coolant broke the run”

In an electronics thermal analysis coupling a fluid code and a heat-transfer code in partitioned fashion, the run diverged as soon as the coolant changed from air to water. Water's heat transfer coefficient is tens of times air's, so fluid-side resistance fell sharply and the resistance ratio with the solid rose well above 1. Swapping which side received temperature and flux, and switching to automatic relaxation, restored convergence. Partitioned stability can change with small changes in conditions.

Common mistakes

Mistakes and fixes

Mistake	Effect	Fix
Ignoring exchange direction	Divergence	Choose by resistance ratio
Fixed relaxation	Slow or divergent	Automatic adjustment
Temperature-only check	Energy imbalance	Check flux too
Single exchange per step	Temporal error	Inner iterations
Non-matching interface meshes	Heat not conserved	Conservative interpolation

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