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A partitioned fluid-structure interaction solver for two-phase sloshing and flexible spacecraft dynamics

Umberto Zucchelli, Miguel Alfonso Mendez, Annafederica Urbano, Sebastien Vincent-Bonnieu, Piotr Wenderski, Francesco Sanfedino

arXiv:2609.28355v1eess.SYcs.SYphysics.flu-dyn

TL;DR

Microgravity sloshing couples liquid motion to rigid spacecraft attitude and flexible appendages, making accurate prediction of exchanged loads important. The paper develops and assesses a two-way DNS–FSI framework, then replays a reduced-model controller in the nonlinear plant. For the considered manoeuvre, the reduced model captures principal attitude and actuator responses but not detailed nonlinear sloshing loads.

  • Problem

    Microgravity liquid redistribution and sloshing loads interact with spacecraft attitude and flexible appendages, while common reduced models have limited capability for large-amplitude motion and interface deformation.

    Source: §1 · p. 1

  • Method

    The paper couples a FLUIDICS-validated two-phase DNS solver, rigid-body dynamics, and a rotating flexible-panel model through a strongly coupled Dirichlet–Neumann fixed-point method.

    Source: §1 · p. 3, §2.1 · p. 5, §6 · p. 26

  • Results

    For the considered manoeuvre, the LTI model’s NRMSE was 0.267% for yaw rate and 3.77% for actuator torque, but 26.2% for sloshing torque.

    Source: §6 · p. 26

  • Takeaways & Limitations

    The framework provides an environment to replay controllers designed from reduced models in a nonlinear CFD–FSI plant and assess their coupled response and model-form errors.

    Source: §5.4 · p. 24, §6 · p. 26

    The closed-loop study examines only a single manoeuvre and does not demonstrate general controller optimality or robustness.

    Source: §5.4 · p. 24

Abstract

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This paper presents a high-fidelity direct numerical simulation (DNS)-fluid-structure interaction (FSI) framework for rigid-liquid-flexible spacecraft dynamics under microgravity conditions. The liquid-gas flow is simulated with the incompressible two-phase solver implemented in DIVA, validated against FLUIDICS experiments conducted aboard the International Space Station (ISS). The flexible appendages are described by a rotating assumed-mode plate model that accounts for geometric stiffening. The fluid and structural operators are coupled through a Dirichlet-Neumann fixed-point algorithm with Aitken relaxation, and a closed-system mechanical energy balance is used as an a posteriori diagnostic to assess the energy imbalance of the partitioned discretisation. The coupling strategy is validated against an experimental free-decay sloshing benchmark, and its numerical consistency is assessed through spatial sensitivity studies of the energy-balance defect. Prescribed-motion, rigid open-loop, and flexible open-loop simulations of a spin-up manoeuvre are compared to isolate the effect of structural feedback on the sloshing response. Reduced liquid models identified from the different simulation architectures exhibit different predictive capabilities when embedded in the same rigid-flexible plant. A controller synthesized from the reduced model identified from the flexible simulation is replayed in the nonlinear CFD-FSI environment. The reduced model reproduces the principal attitude and actuator responses for the considered manoeuvre but does not recover the detailed nonlinear sloshing-load history. The framework provides a high-fidelity environment for analysing coupled spacecraft dynamics, identifying control-oriented models, and assessing reduced-model-based control strategies beyond their linear design representation.

1. Introduction

Microgravity sloshing couples liquid redistribution, spacecraft attitude, and flexible-appendage dynamics, while common reduced liquid models have limited validity for large-amplitude motion and interface deformation. This paper develops a two-way DNS–FSI framework and applies it to spacecraft manoeuvres and reduced-model control assessment.

1. Introduction

  • Microgravity makes liquid motion consequential to attitude dynamics: capillary forces shape redistribution, and modest tank accelerations may produce large reorientations and persistent loads.Flexible appendages add low-frequency modes that may overlap sloshing frequencies, while rotation introduces gyroscopic, Coriolis, and geometric-stiffening effects.Source: §1 · p. 1
  • Frozen-mass models omit relative liquid motion, added-mass models do not resolve free-surface dynamics, and equivalent mechanical models are typically regime-specific.Their predictive capability is limited for large-amplitude motion, strong interface deformation, and capillary-driven redistribution in microgravity.Source: §1 · p. 2
  • Continuum and particle approaches address aspects of large-deformation sloshing, but cited formulations retain assumptions or challenges involving surface tension, viscosity, or long-time conservation.The paper motivates two-phase DNS as a high-fidelity reference for model validation and control-oriented assessment, rather than a real-time control model.Source: §1 · p. 2
  • The DIVA solver has been compared with FLUIDICS ISS measurements, reproducing dominant oscillation frequencies and measured tank-load trends in the investigated Bond-number range [1].This work extends DIVA from prescribed tank motion to two-way spacecraft coupling, in which spacecraft motion changes liquid forcing and sloshing loads feed back into the response.Source: §1 · p. 3
  • The coupled fluid solver exchanges tank kinematics and hydrodynamic loads with a rigid-body model and a rotating modal panel model that accounts for geometric stiffening [21].The framework is assessed against experimental data before application to microgravity spin-up manoeuvres.Source: §1 · p. 3
  • The paper formulates a microgravity DNS–FSI framework coupling nonlinear two-phase sloshing with rigid-body motion and flexible panels, then uses it for model development and control validation.Its three contributions are the coupled framework, experimental assessment and an energy-balance diagnostic, and spin-up simulations of modal interaction.Source: §1 · p. 3

2. Partitioned fluid–structure coupling

The framework partitions fluid and rigid–flexible structural operators, iterating within each physical time step to make tank kinematics and hydrodynamic loads consistent. DIVA resolves the two-phase flow and wall loads, while a rotating modal panel model returns structural response and feedback.

2.1. Dirichlet–Neumann partitioned coupling

  • The structural operator supplies tank kinematics to the fluid operator, which returns hydrodynamic loads; fixed-point iterations enforce consistency between these exchanges within each time step.The operators remain distinct, but the coupling is implicit because the interface equilibrium is enforced through sub-iterations rather than a single lagged exchange.Source: §2.1 · pp. 3, 5, §2 · p. 3
  • Each coupling iteration recomputes the structural state and repeats the DNS step from the beginning-of-step fluid state, while Aitken relaxation damps oscillatory load corrections and can accelerate convergence.The converged exchange matches the hydrodynamic wrench with structural acceleration and the non-inertial forcing used by the DNS solver.Source: §2.1 · p. 5

2.2. Fluid subsystem F

  • DIVA models incompressible, immiscible, isothermal two-phase flow on a tank-attached Cartesian grid, with a level-set interface and sharp capillary pressure jump.An immersed-boundary treatment represents the wall; the solver returns force and torque from pressure, wall-velocity enforcement, and viscous wall stress.Source: §2.2 · pp. 6–7

2.3. Structural subsystem S

  • The rotating assumed-mode plate model retains dominant panel bending modes and incorporates gyroscopic effects and centrifugal stiffening into the flexible dynamics.Four modes are used per panel; translational and angular accelerations excite the modes, whose accelerations feed reaction forces and torques back to the body.Source: §2.3 · pp. 8–9

3. System energy-balance verification

The paper uses a closed-system mechanical-energy balance as an a posteriori check on the coupled discretisation. It accounts for stored fluid and structural energy, external and dissipative work, and cancellation of internal wall exchange.

3. System energy-balance verification

  • The diagnostic compares changes in fluid and structural mechanical energies with accumulated external and dissipative work, using discrete numerical states to measure the unclosed energy ledger.The balance is applied to a closed tank coupled to a rigid structure with flexible appendages, with no wall mass transfer or phase change.Source: §3 · pp. 9–10

3.1. Stored energies and power ports

  • Total mechanical energy combines dry-structure energy with fluid kinetic, gravitational, and liquid–gas interfacial energy.The structural kinetic energy follows the assembled mass matrix, including rigid–flexible kinetic coupling; panel potential energy includes rotation-dependent geometric stiffness.Source: §3.1 · p. 10

3.2. Closed-system energy balance

  • Adding the subsystem balances cancels internal wall exchange, yielding the closed-system balance used to define the discrete energy-balance residual.The stated balances apply to the retained structural approximation; simultaneous translation and rotation require additional body-frame transport terms.Source: §3.2 · p. 11

4. Reference configuration

The reference configuration couples a spinning rigid–flexible spacecraft to a partially filled spherical tank, with the same parameter set used across spacecraft simulations. For control synthesis, the fluid and structural operators are replaced by yaw-port LTI models, with the liquid model calibrated on the post-spin-up response.

4. Reference configuration

  • The reference spacecraft combines a rigid hub, two opposing solar panels, and an offset spherical tank, with coupled yaw, panel, and liquid dynamics.The tank centre is 0.30 m from the spacecraft reference point; after torque switch-off, the system evolves freely under coupled reactions.Source: §4 · p. 12

4.1. Control-oriented LTI representation and controller synthesis

  • The fluid and structural operators are replaced by yaw-port LTI models, retaining four yaw-coupled panel modes and two equivalent-pendulum liquid modes.The LTI plant uses the zero-spin yaw projection of the panel model; the liquid model maps yaw acceleration to sloshing torque.Source: §4.1 · pp. 12–14
  • The liquid fit uses two dominant torque-spectrum peaks as frequency seeds, then estimates modal weights before fitting physically interpretable mass and lever parameters.The frequency seeds are separated by at least 0.15 Hz; fitted frequencies remain within 5% of their seeds.Source: §4.1 · pp. 13–14
  • The liquid-surrogate parameters are calibrated over the post-spin-up window using a bounded trust-region least-squares fit initialized from spectral seeds.The imposed spin-up transient is excluded, and the fit enforces physical mass and inertia constraints.Source: §4.1 · pp. 13–14
  • The combined 13-state plant uses commanded yaw torque as input, measured yaw rate for feedback, and predicted sloshing torque as a diagnostic output.Its state includes yaw rate, liquid-surrogate coordinates and velocities, and retained panel coordinates and velocities.Source: §4.1 · pp. 12, 14
  • An LQG servo combines integral yaw-rate error and state feedback, with a steady Kalman–Bucy observer estimating the plant state from measured yaw rate.The yaw command is limited to 0.35 N m, but the reported replay peak of 2.20 × 10^-2 N m does not activate saturation.Source: §4.1 · p. 14

5. Results and discussion

The results validate the coupled solver against a free-decay experiment, assess its energy-balance consistency under spatial refinement, and compare liquid responses across prescribed-motion and FSI architectures. A closed-loop replay reproduces attitude and actuator responses for the tested manoeuvre, but not its detailed nonlinear sloshing-load history.

5.1. Coupling validation

  • The DIVA coupled simulation reproduces the experimental oscillation frequency and remains in good phase agreement, while the frozen-liquid response isolates sloshing-induced frequency reduction.Hydrodynamic forces generally oppose the container's instantaneous motion, consistent with liquid lag.Source: §5.1 · p. 16, Figure 5 · p. 16

5.2. Grid convergence of the energy balance

  • 1.35%, 0.914%, and 0.312% energy-residual magnitudes on the 643, 1283, and 2563 grids show monotonic reduction with refinement.The residual is predominantly fluid-associated; accepted steps met the Dirichlet–Neumann convergence criterion.Source: §5.2 · p. 19, Figure 9 · p. 19
  • The 1283 grid is retained for matched PM, rigid-FSI, and flexible-FSI runs as a compromise between late-time signal retention and computational feasibility.Aitken-relaxed coupling still requires about four to five additional CFD resolves per physical time step.Source: Figure 10 · p. 19, §5.2 · p. 20

5.3. Prescribed-motion, rigid open-loop, and flexible open-loop comparisons

  • During final coast (60 ≤ t ≤ 120 s), mean-removed flexible-FSI sloshing-torque RMS was 1.47 × 10−4 N m versus 8.11 × 10−5 N m for rigid FSI, an 81.5% increase.During early coast (10 ≤ t ≤ 60 s), RMS values were 1.21 × 10−3, 9.79 × 10−4, and 9.56 × 10−4 N m for PM, rigid FSI, and flexible FSI, respectively.Source: §5.3 · p. 20
  • The dominant liquid-response peaks are 0.450, 0.463, and 0.458 Hz for PM, rigid FSI, and flexible FSI; flexible FSI also has a 0.518 Hz component in both sloshing and panel-reaction spectra.The 0.518 Hz component has normalized amplitude 0.984 and is negligible in PM and rigid FSI.Source: §5.3 · p. 20, Figure 12 · p. 22, Figure 13 · p. 22
  • The flexible-FSI-identified liquid model gives the lowest common-plant prediction errors for sloshing torque, flexible torque, and yaw rate.The advantage is most evident in flexible-load and yaw-rate responses; the rigid-FSI model generally provides intermediate accuracy.Source: §5.3 · pp. 20–21, Table 2 · p. 21, Figure 15 · p. 24

5.4. Closed-loop CFD replay

  • The closed-loop replay's total energy-balance residual is −4.70 × 10−4 J, or −0.457% of maximum absolute cumulative control work.Subsystem residuals remain bounded and the fixed-point coupling residual remains controlled throughout the replay.Source: §5.4 · pp. 22–23, Figure 16 · p. 24
  • 0.267% yaw-rate NRMSE and 3.77% actuator-torque NRMSE accompany a 26.2% full-record sloshing-torque NRMSE in the flexible-FSI closed-loop replay.The LTI model captures the dominant low-frequency attitude effect but not the observed nonlinear liquid and flexible-load histories.Source: §5.4 · pp. 23–24, Figure 17 · p. 25

6. Conclusions

The framework’s coupling implementation was assessed against experiment and through an energy balance, while simulations showed that structural feedback affects liquid response and reduced-model predictions. In one closed-loop replay, the reduced model reproduced principal attitude and actuator responses but not the detailed nonlinear sloshing-load history.

6. Conclusions

  • The study developed a partitioned DNS–FSI framework combining a FLUIDICS-validated two-phase solver, rigid-body dynamics, and a rotating flexible-panel model with strongly coupled load–motion exchange.The framework provides a nonlinear environment for investigating coupled response, quantifying model-form discrepancies, and assessing control strategies beyond their synthesis model.Source: §6 · pp. 26–27
  • The coupled calculation reproduced the free-decay benchmark’s dominant oscillation frequency and tank-displacement phase, including the frequency reduction relative to the frozen-liquid response.The reduction is consistent with effective dynamic compliance introduced by relative liquid motion.Source: §6 · p. 26
  • Spatial refinement consistently reduced the final energy-balance deficit, with the dominant contribution associated with liquid-domain, interface, immersed-boundary, and fluid-load representation.The closed-system mechanical energy balance served as an a posteriori verification of the coupled simulations.Source: §6 · p. 26
  • Two-way coupling changed the liquid response despite relatively close attitude histories, and structural flexibility altered sloshing when structural and liquid frequencies were close.Liquid surrogates identified from different simulation architectures produced different long-time predictions in the same rigid–flexible plant; the flexible-FSI model was most representative of its corresponding coupled system.Source: §6 · p. 26
  • Yaw-rate, actuator-torque, and sloshing-torque NRMSEs were 0.267%, 3.77%, and 26.2%, respectively, in the closed-loop replay.The two-mode reduced model tracked the considered manoeuvre’s low-frequency attitude and control response but not its detailed nonlinear load history.Source: §6 · p. 26

CRediT authorship contribution statement

The contribution statement assigns conceptualization and methodology across the author team, with software, validation, analysis, supervision, resources, and writing responsibilities distributed among individuals.

CRediT authorship contribution statement

  • Umberto Zucchelli contributed conceptualization, methodology, software, validation, formal analysis, investigation, data curation, original-draft writing, and visualization.He also contributed to the original draft.Source: CRediT authorship contribution statement · p. 27
  • Miguel Alfonso Mendez contributed conceptualization, methodology, formal analysis, supervision, and review and editing; Annafederica Urbano contributed methodology, software, formal analysis, and review and editing.Source: CRediT authorship contribution statement · p. 27
  • Sebastien Vincent-Bonnieu and Piotr Wenderski contributed conceptualization and resources, while Francesco Sanfedino contributed conceptualization, methodology, formal analysis, supervision, project administration, and review and editing.Source: CRediT authorship contribution statement · p. 27

Appendix A. Control-oriented state-space matrices

The appendix specifies matrices that convert the second-order reduced model into a first-order LTI plant. It also identifies the control-study outputs and relates the equations to the chosen state ordering.

Appendix A. Control-oriented state-space matrices

  • The appendix gives explicit matrices for converting the second-order reduced model into the first-order LTI plant using the state ordering of Eq. (52).The listed matrix entries include the flexible-state blocks and identity matrices used in that conversion.Source: Appendix A · p. 31
  • The appendix defines the vector of ones and presents the first-order equations for the flexible and sloshing states.These equations include the flexible stiffness and damping terms and the sloshing-state relations.Source: Appendix A · p. 31
  • The control study uses measured angular rate and reconstructed sloshing torque as its two outputs.The yaw–flexible blocks of the inverse generalized mass matrix include h00, h0f, hf0, and Hff.Source: Appendix A · p. 31
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