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arxiv: 2604.24439 · v1 · submitted 2026-04-27 · ⚛️ physics.flu-dyn · physics.comp-ph

Stable fluid-rigid body interaction algorithm using the direct-forcing immersed boundary method (DF-IBM)

E. Farah , A. Ouahsine , P. G. Verdin , B. Kaoui This is my paper

Pith reviewed 2026-05-08 01:30 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn physics.comp-ph
keywords fluid-rigid body interactiondirect-forcing immersed boundary methodimplicit couplingfixed relaxationNewton-Euler equationsPISO algorithmnumerical stabilityfree motion
0
0 comments X p. Extension

The pith

An extended direct-forcing immersed boundary method couples fluid flow to free rigid-body motion with an implicit algorithm and fixed relaxation.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper extends an earlier direct-forcing immersed boundary method so that rigid bodies can move freely under forces from the surrounding fluid. It adds an implicit coupling between the Navier-Stokes equations and the Newton-Euler rigid-body equations, iterating until the interface conditions are met. A fixed relaxation is applied to the rigid-body updates to stop instability when the body density approaches the fluid density or when rotation is strong. The scheme skips the momentum predictor step and repeated corrector loops of the PISO pressure algorithm to keep the cost low. A reader would care because many practical flows involve moving objects, and a reliable, efficient way to compute their motion opens the door to better predictions in engineering problems.

Core claim

The central claim is that an implicit partitioned coupling algorithm inside the DF-IBM framework, which links the Navier-Stokes equations to the Newton-Euler equations for rigid-body dynamics, produces stable and efficient simulations of flow-induced motion. Fixed relaxation on the rigid-body kinematics removes stability problems that arise at critical density ratios and from the approximation of internal mass in rotation, while reuse of the PISO splitting avoids extra predictor and corrector iterations.

What carries the argument

The implicit coupling algorithm with fixed relaxation applied to rigid-body kinematics inside the direct-forcing immersed boundary method.

If this is right

  • The method remains stable when solid-fluid density ratios approach unity.
  • Complex free motions such as falling, rising, or vortex-induced rotation can be captured without ad-hoc fixes.
  • Computational cost stays low because the momentum predictor and multiple PISO corrector loops are omitted from the coupling iterations.
  • Benchmark tests confirm robustness across a range of challenging interaction scenarios.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The reuse of an existing pressure solver suggests the same coupling could be added to other partitioned CFD codes without major rewrites.
  • If the relaxation works for rigid bodies, a similar technique might stabilize simulations of lightly deformable objects at low density contrast.
  • Long-time simulations of multiple bodies would become practical because each time step avoids repeated corrector loops.

Load-bearing premise

The fixed relaxation technique for rigid body kinematics sufficiently mitigates stability and convergence issues stemming from critical solid-fluid density ratios and the rigid body approximation of internal mass effects in rotational dynamics.

What would settle it

Running the algorithm on a body whose density is one-hundredth that of the fluid and observing divergence, growing oscillations in position, or non-physical rotation rates would show that the fixed relaxation has not removed the stability problems.

Figures

Figures reproduced from arXiv: 2604.24439 by A. Ouahsine, B. Kaoui, E. Farah, P. G. Verdin.

Figure 1
Figure 1. Figure 1: Schematic of the fluid-rigid body domain used in DF-IBM framework to simulate FSI problems. view at source ↗
Figure 2
Figure 2. Figure 2: Numerical challenges induced by the IME in the FSI partitioned coupling methods. view at source ↗
Figure 3
Figure 3. Figure 3: Flowchart of the proposed fluid-rigid body implicit coupling algorithm for one simulation time-step. view at source ↗
Figure 4
Figure 4. Figure 4: Computational domain and boundary conditions for the sedimenting 2D disk inside a confined channel. view at source ↗
Figure 5
Figure 5. Figure 5: Time history of the sedimenting 2D disk for di view at source ↗
Figure 6
Figure 6. Figure 6: Time history of the sedimenting 2D disk for di view at source ↗
Figure 7
Figure 7. Figure 7: Vorticity contours at different time t instances of the sedimenting 2D circular disk. 18 view at source ↗
Figure 8
Figure 8. Figure 8: Computational domain and boundary conditions for the shear flow-induced rotation of a 2D disk. view at source ↗
Figure 9
Figure 9. Figure 9: Non-dimensional time history of the angular velocity view at source ↗
Figure 10
Figure 10. Figure 10: Non-dimensional time history of the angular velocity view at source ↗
Figure 11
Figure 11. Figure 11: Computational domain and boundary conditions for the freely rotating 2D NACA0012 airfoil subjected to view at source ↗
Figure 12
Figure 12. Figure 12: Time history of the angle of attack θs (a) and angular velocity ωs (b) at two different time-steps for the freely rotating 2D NACA0012 airfoil for h = 1/9600 m. A phase shift is observed in the computed results compared to those obtained by Glowinski et al. [34]. In the present results, the airfoil’s rotation begins at approximately t ≈ 6.0 s, whereas for Glowinski et al., the rotation was initiated at t … view at source ↗
Figure 13
Figure 13. Figure 13: Number of implicit FSI iterations (a) and CPU time per time-step (b) at two di view at source ↗
Figure 14
Figure 14. Figure 14: Time history of the angle of attack θs (a) and angular velocity ωs (b) at two different mesh sizes for the freely rotating 2D NACA0012 airfoil for ∆t = 0.001 s. increase in the number of implicit iterations is observed, with the number of iterations almost tripling when the mesh resolution is doubled, as seen in Fig. (15a). Conversely, the CPU time required per time-step is higher for the finer grid compa… view at source ↗
Figure 15
Figure 15. Figure 15: Number of implicit FSI iterations (a) and CPU time per time-step (b) at two di view at source ↗
Figure 16
Figure 16. Figure 16: Computational domain and boundary conditions for the sedimenting 2D ellipse inside a confined channel. view at source ↗
Figure 17
Figure 17. Figure 17: Trajectory of the center of mass (Xs , Ys) (a) and orientation θs (b) of the 2D sedimenting ellipse in a confined channel. The current results are marked with solid lines ( ) and the numerical results extracted from the work of Xia et al. [35] are marked with symbols (◦). To compare the current Retrans solid with that obtained by Xia et al., the terminal velocity of the ellipse must be calculated. Fig. (1… view at source ↗
Figure 18
Figure 18. Figure 18: Vertical v-component of the velocity Vs of the 2D sedimenting ellipse in a confined channel. The current results are marked with solid lines ( ) and the numerical results extracted from the work of Xia et al. [35] are marked with symbols (◦). (a) t = 0.1 s (b) t = 0.25 s (c) t = 0.5 s (d) t = 0.75 s (e) t = 1.0 s (f) t = 1.5 s view at source ↗
Figure 19
Figure 19. Figure 19: Vorticity contours at different time t instances of the sedimenting 2D ellipse. this problem is considered challenging due to the critical density ratios chosen, which are known to destabilize the numerical coupling algorithm. The computed results are compared with refer￾26 view at source ↗
Figure 20
Figure 20. Figure 20: Computational domain and boundary conditions for the freely falling (a) and freely rising (b) 2D disk in view at source ↗
Figure 21
Figure 21. Figure 21: Non-dimensional vertical v-component (a) and horizontal u-component (b) of the disk velocity for the view at source ↗
Figure 22
Figure 22. Figure 22: Vorticity contours at different time instances for the freely falling (a) and rising (b) disk. coupling strategies, making it well suited for a parametric study of challenging FSI regimes. In addition, the present case is proposed as a benchmark problem for fluid-rigid body interaction at low density ratios, for which, to the best of the authors knowledge, no reference data are currently available for qua… view at source ↗
Figure 23
Figure 23. Figure 23: Time history of the vertical position Ys of the rising 2D disk for different solid to fluid density ratios ρs/ρf . convergence in the low density ratio regime. For the lowest density ratio considered, ρs/ρf = 0.3, even a relaxation of αr = 0.5 led to large velocity fluctuations and divergence of the solution. Con￾sequently, the velocity was over-relaxed using the smaller relaxation factor, αr = 0.25, to m… view at source ↗
Figure 24
Figure 24. Figure 24: Time history of the velocity v-component view at source ↗
read the original abstract

The direct-forcing immersed boundary method (DF-IBM) algorithm previously developed by the authors is extended by coupling the Navier-Stokes equations with the Newton-Euler equations for rigid body dynamics within the DF-IBM framework. This coupling broadens the applicability of the previous development, from stationary or prescribed motion to flow-induced (free) motion cases. To address fluid-rigid body interactions under a partitioned approach, an implicit coupling algorithm is developed to handle strongly coupled interface conditions. Stability and convergence issues, particularly stemming from critical solid-fluid density ratios and from the rigid body approximation of internal mass effects in rotational dynamics, are mitigated using a fixed relaxation technique for the rigid body kinematics to ensure numerical robustness. Additionally, the proposed algorithm leverages the previously developed DF-IBM formulation and the predictor-corrector strategy of the pressure implicit with splitting of operators (PISO) algorithm by omitting the momentum predictor step and the costly corrector loops from the implicit iterations. The method is validated against several benchmark cases, demonstrating robustness, stability, and efficiency in capturing complex fluid-rigid body interactions across a range of challenging scenarios.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The paper extends the authors' prior direct-forcing immersed boundary method (DF-IBM) by coupling the incompressible Navier-Stokes equations to the Newton-Euler rigid-body equations under a partitioned implicit scheme. A fixed relaxation step is applied to the rigid-body kinematics to stabilize the coupling at critical solid-fluid density ratios and to compensate for the rigid-body approximation of internal mass in rotation; the PISO pressure-velocity coupling is simplified by dropping the momentum predictor and inner corrector loops. The resulting algorithm is tested on several benchmark FSI problems and is claimed to be robust, stable, and computationally efficient across challenging regimes.

Significance. If the stability claims are substantiated, the work would provide a practical, low-cost extension of DF-IBM to free rigid-body motion that avoids the expense of fully monolithic or strongly coupled solvers. The omission of PISO inner loops and the use of a single fixed relaxation parameter could yield measurable efficiency gains for density ratios near unity, a regime that remains numerically delicate in many partitioned FSI codes.

major comments (3)
  1. [Method section] Method section (description of the implicit coupling and relaxation step): the fixed relaxation factor applied to the rigid-body velocity and angular velocity updates is presented without derivation, stability analysis, or bounds. The text states that it “mitigates” added-mass and rotational-mass approximation problems, yet no von Neumann or energy estimate is supplied to show that a constant factor suffices for all density ratios.
  2. [Validation section] Validation section (benchmark results and density-ratio tests): the reported cases do not include a systematic sweep of solid-fluid density ratios, especially in the interval 0.5 < ρ_s/ρ_f < 2 where added-mass effects are strongest. Without such data or comparison against an adaptive relaxation or monolithic reference, the headline claim of robustness “across a range of challenging scenarios” rests on an unverified assumption that the chosen constant works universally.
  3. [Method section] PISO simplification (omission of momentum predictor and corrector loops): the decision to drop these steps reduces per-iteration cost but removes the inner-loop damping that normally stabilizes partitioned schemes when the added-mass operator is ill-conditioned. No a-posteriori error or convergence-rate comparison is given between the reduced PISO scheme and the full predictor-corrector version on the same test problems.
minor comments (2)
  1. [Method section] Notation: the symbol for the relaxation factor is introduced without an explicit equation number or definition of its numerical value; readers must infer it from the algorithm box.
  2. [Results] Figure captions: several benchmark figures lack quantitative error norms or grid-convergence data in the caption, making it difficult to judge the accuracy of the reported trajectories without consulting the main text.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive and detailed review of our manuscript. We address each major comment point by point below, indicating the revisions we intend to incorporate in the revised version.

read point-by-point responses

  1. Referee: [Method section] Method section (description of the implicit coupling and relaxation step): the fixed relaxation factor applied to the rigid-body velocity and angular velocity updates is presented without derivation, stability analysis, or bounds. The text states that it “mitigates” added-mass and rotational-mass approximation problems, yet no von Neumann or energy estimate is supplied to show that a constant factor suffices for all density ratios.

    Authors: We agree that a formal derivation or von Neumann analysis of the fixed relaxation factor is absent and would strengthen the theoretical foundation. The factor was selected empirically through numerical testing to stabilize the partitioned coupling for the density ratios in our benchmarks, consistent with relaxation strategies reported in other partitioned FSI studies. In the revised manuscript we will add a dedicated paragraph in the method section explaining the parameter choice, citing relevant literature, and reporting empirical stability bounds obtained from additional tests across a wider density-ratio range. revision: partial

  2. Referee: [Validation section] Validation section (benchmark results and density-ratio tests): the reported cases do not include a systematic sweep of solid-fluid density ratios, especially in the interval 0.5 < ρ_s/ρ_f < 2 where added-mass effects are strongest. Without such data or comparison against an adaptive relaxation or monolithic reference, the headline claim of robustness “across a range of challenging scenarios” rests on an unverified assumption that the chosen constant works universally.

    Authors: The existing benchmarks were selected from standard FSI test cases that include density ratios near unity, but we acknowledge that a systematic parametric sweep would provide clearer evidence. We will expand the validation section with a new subsection presenting results for a density-ratio sweep from 0.1 to 10 (with focus on 0.5–2), using the same relaxation parameter. Where literature data for monolithic or adaptive-relaxation solvers exist, we will include comparative error metrics to support the robustness claim. revision: yes

  3. Referee: [Method section] PISO simplification (omission of momentum predictor and corrector loops): the decision to drop these steps reduces per-iteration cost but removes the inner-loop damping that normally stabilizes partitioned schemes when the added-mass operator is ill-conditioned. No a-posteriori error or convergence-rate comparison is given between the reduced PISO scheme and the full predictor-corrector version on the same test problems.

    Authors: The simplification is justified by the stabilizing effect of the outer implicit coupling loop and the fixed relaxation, as previously validated in our DF-IBM framework. To quantify any trade-off, we will add a short comparison in the validation section for one representative benchmark, reporting iteration counts, convergence rates, and L2 error norms between the reduced PISO scheme and the full predictor-corrector implementation. This will demonstrate that the omitted steps do not degrade accuracy under the proposed coupling. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation builds on standard equations with independent algorithmic extensions

full rationale

The paper extends the authors' prior DF-IBM formulation by coupling Navier-Stokes with Newton-Euler equations under a partitioned implicit scheme, adding a fixed relaxation on rigid-body kinematics and omitting PISO momentum loops. No load-bearing step reduces by construction to a fitted parameter, self-citation, or ansatz; the relaxation is presented as a pragmatic choice to address density-ratio issues without deriving its value from the target result. Validation on benchmarks provides external checks independent of the cited prior work. The self-citation supplies the base immersed-boundary discretization but does not force the new coupling or stability claims.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard governing equations plus one numerical stabilization parameter whose value is not derived from first principles.

free parameters (1)
  • fixed relaxation factor
    Applied to rigid body kinematics to ensure numerical robustness; its specific value is chosen to mitigate density-ratio instabilities but is not derived within the paper.
axioms (2)
  • standard math Navier-Stokes equations govern incompressible fluid flow
    Invoked as the base fluid model throughout the DF-IBM framework.
  • standard math Newton-Euler equations govern rigid body translation and rotation
    Used to couple fluid forces to object motion under the partitioned approach.

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Reference graph

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