IQN-ILS partitioned multi-physics coupling

Degroote, Bathe, Vierendeels — "Performance of a new partitioned procedure versus a monolithic procedure in fluid-structure interaction," Computers & Structures 2009.

Chourdakis et al. — "preCICE v2: a sustainable and user-friendly coupling library," arXiv 2109.14470 (2021).

cad-sim::coupling ships the only MIT/Apache implementation of preCICE-style partitioned multi-physics coupling in the open-source ecosystem. preCICE itself is LGPLv3 — usable via dynamic linking but not directly portable into MIT/Apache codebases — so this module re-derives the algorithms from the public papers.

The problem

When two physics domains share an interface (a heated solid wall contacted by fluid; an oscillating airfoil in a transonic flow), solving them jointly as one monolithic system is expensive and inflexible. The partitioned approach keeps each solver independent and exchanges only the interface field — temperature, pressure, displacement — but introduces a numerical loop:

loop k:
  x_a  = solve_A(x_b)        # A's response to B's interface
  x_b  = solve_B(x_a)        # B's response to A's update
  if ‖x_a^new − x_a^old‖ < tol: converged

For stiff couplings (large added-mass FSI, conjugate heat transfer with mismatched conductivities) this loop diverges or crawls. Acceleration schemes choose how to combine the raw solver outputs into the next iterate.

The algorithms

cad-sim::coupling::RelaxationMode exposes three schemes:

Constant under-relaxation

x_{k+1} = x_k + ω · r_k with fixed ω ∈ (0, 1]. Always stable for small enough ω but slow when the optimal ω is unknown a priori.

Aitken's Δ²

Adapts ω from the previous two residuals (Mok 2001, used in preCICE):

ω_new = -ω_old · (r_prev · (r − r_prev)) / ‖r − r_prev‖²

Mathematically free (one extra inner product per iteration), and dramatically faster than constant ω on stiff couplings.

IQN-ILS (Interface Quasi-Newton with Inverse Least Squares)

The preCICE workhorse for tight FSI and conjugate heat transfer. Maintains column histories of past displacement and residual differences V = [Δr_1 … Δr_k], W = [Δx_1 … Δx_k]. On each iteration:

1. Solve the small least-squares problem min_c ‖V·c − (−r)‖² via normal equations (VᵀV + λI) c = −Vᵀr with Tikhonov regularisation (λ ≈ 10⁻¹²).
2. Compute the IQN-ILS update Δx = W·c.
3. Apply: x_{k+1} = x_k + Δx.
4. Truncate the oldest column when history exceeds history_max.

Substrate test (Wave 67.B)

On a stiff α = 0.9, β = 0.95 linear toy:

Calling it

{
  "method": "tools/call",
  "params": {
    "name": "simulate_coupled",
    "arguments": {
      "coupling_alpha": 0.9,
      "coupling_beta": 0.95,
      "relaxation": "iqn_ils",
      "iqn_history_max": 10,
      "max_iterations": 200,
      "tolerance": 1e-8
    }
  }
}

Returns iteration count, final residual, and the analytic fixed point for comparison — useful for choosing a relaxation scheme before plugging real cad-sim domains in.

Implementation notes

cad-sim::coupling::couple pre-primes field_b = solve_b(field_a) before the loop when RelaxationMode::IqnIls is selected. Without this, the first iteration's residual would lie off the fixed-point manifold (because field_b started as initial_b, not B(initial_a)), and the first (Δx, Δr) pair would poison the least-squares solve. The Aitken and constant-ω schemes don't need this since they're memoryless.

References

• Degroote, Bathe, Vierendeels (2009) — original IQN-ILS derivation.
• Mok (2001) — the Aitken Δ² ω-update form preCICE uses.
• Chourdakis et al. (2021) — preCICE v2 architecture survey (arXiv 2109.14470).