Partitioned Conjugate Heat Transfer (CHT) Analysis Using CFD-FEM

Good question. Partitioned Conjugate Heat Transfer is a method that solves the fluid side with a CFD solver and the solid side with a FEM heat conduction solver independently, and exchanges temperature and heat flux iteratively at the fluid-solid interface.

The biggest advantage is that existing CFD codes and FEM codes can be used as-is. For example, in gas turbine blade cooling design, the external high-temperature gas flow path is solved with Ansys Fluent, and the internal solid heat conduction of the blade is solved with Abaqus—this kind of combination is standard in practice. With direct coupling, everything needs to be packed into a single solver, which skyrockets both development and computational costs.

If sufficiently iterated at the interface, it converges to the same accuracy as direct coupling. However, it can sometimes take time to converge—especially when the heat capacity ratio of the solid and fluid are close, convergence slows down. That's where techniques like relaxation factors and Aitken acceleration come in. Let's look at them in detail later.

Two conditions hold physically at the interface $\Gamma$. First, temperature continuity (Dirichlet condition):

And heat flux continuity (Neumann condition):

Exactly. In partitioned coupling, these two conditions are given alternately to each solver. In a typical Dirichlet-Neumann splitting method:

1. Give the interface temperature $T_\Gamma$ to the solid solver (Dirichlet condition) → Solve for the solid temperature field, calculate the interface heat flux $q_\Gamma$.
2. Give the interface heat flux $q_\Gamma$ to the fluid solver (Neumann condition) → Solve for the fluid temperature field, obtain the new interface temperature $T_\Gamma^{\text{new}}$.
3. Return to step 1 using $T_\Gamma^{\text{new}}$ as the new interface temperature.

This is repeated until the temperature converges.

If the newly obtained interface temperature $T_\Gamma^{\text{new}}$ is used directly in the next iteration, oscillations can occur, potentially leading to divergence. This is particularly pronounced when the solid's heat capacity is smaller than the fluid's. Therefore, under-relaxation is applied:

Yes. A small $\omega$ is stable but slow to converge, a large $\omega$ is fast but risks divergence. In practice, $\omega = 0.3 \sim 0.7$ is a starting point, adjusted according to the problem. This is the common "trade-off between stability and convergence speed" in numerical computation.

In cases like a blade where a thin solid is hit by high-temperature gas, the solid's heat capacity is relatively small, so we often start with $\omega = 0.3 \sim 0.5$. If we use Aitken acceleration, which we'll explain later, $\omega$ can be adjusted dynamically, so there's no need to struggle with a fixed value.

Convergence is judged when the change in interface temperature becomes sufficiently small. A common criterion is:

It depends on the nature of the problem and the relaxation method. With fixed relaxation at $\omega = 0.5$, it's typically around 10-30 iterations. Using Aitken acceleration often reduces it to 5-10 iterations. However, with strong nonlinearities (e.g., radiation or phase change), it can sometimes require 50+ iterations.

Yes. The stability of partitioned CHT has been rigorously analyzed by Causin, Gerbeau & Nobile (2005) and others. For Dirichlet-Neumann splitting with Neumann given to the fluid side, the stability condition boils down to the "added mass effect" at the interface. Roughly speaking, when the fluid's heat capacity $\rho_f c_{p,f}$ is relatively large compared to the solid's $\rho_s c_{p,s}$, it becomes unstable without relaxation.

Exactly. CHT analysis of liquid sodium-cooled reactor fuel rods is one of the cases where partitioned coupling struggles the most. In such cases, swapping the roles of Dirichlet and Neumann (using Robin-Neumann or Robin-Robin methods) can stabilize it. In practice, libraries like preCICE automatically handle switching between these acceleration techniques.

Here's the pseudo-code for Dirichlet-Neumann partitioned coupling for a steady problem:

1. Set initial estimate for interface temperature $T_\Gamma^{0}$
2. do $k = 0, 1, 2, \ldots$
   • Solid solver: Solve heat conduction equation with $T_\Gamma^k$ as Dirichlet BC → Calculate interface heat flux $q_s^k$
   • Fluid solver: Solve NS + energy equations with $q_s^k$ as Neumann BC → Obtain interface temperature $T_f^k$
   • Relaxation: $T_\Gamma^{k+1} = \omega \, T_f^k + (1 - \omega) \, T_\Gamma^k$
   • Convergence check: If $\| T_\Gamma^{k+1} - T_\Gamma^k \| / \| T_\Gamma^{k+1} \| < \varepsilon$, exit.
3. end do

Sharp observation. There are three challenges in data exchange:

• Mesh Mismatch: Node positions on the interface differ between CFD and FEM meshes → Interpolation is needed.
• Data Format Differences: Each solver has different output formats for temperature and heat flux.
• Inter-process Communication: For MPI parallel runs, data transfer between different process groups.

Coupling middleware like MpCCI or preCICE automates these tasks.

The most widely used is Aitken $\Delta^2$ acceleration. It automatically estimates the optimal relaxation factor each iteration:

Conceptually, yes. A more advanced method is the Quasi-Newton method (IQN-ILS). It approximates the interface "Jacobian" from the history of several past iterations to find the optimal update direction in multiple dimensions. In preCICE, IQN-ILS is the default acceleration method and shows the fastest convergence in many CHT benchmarks.

There are mainly three techniques:

• Nearest Neighbor Interpolation: Use the value of the closest node. Simple but low accuracy.
• RBF (Radial Basis Function) Interpolation: Construct a smooth interpolation surface from all node values. High accuracy but costly for large-scale problems.
• Projection Method (Mortar Method): Precisely calculate the overlap of interface elements to achieve conservative transfer. Highest conservation for heat flux.

It's important that Point-to-Point interpolation is fine for temperature, but conservative transfer (where the total heat amount matches across the entire interface) is needed for heat flux. Otherwise, energy can be created or destroyed.

That's the biggest computational cost factor in partitioned coupling. Interface iterations (sub-iterations) are needed within each time step $\Delta t$. However, there are countermeasures:

• Loose coupling: Exchange only once per step (no iteration). Lower accuracy but fast. Acceptable if the time step is sufficiently small.
• Tight coupling: Iterate until convergence each step. Accuracy guaranteed.
• Use of predictors: Extrapolate the next interface temperature from the previous step's solution to improve initial guess accuracy and reduce iteration count.