physics.flu-dynSep 19, 2026

Learning Stiffness Dependent Fluid Structure Dynamics from Coarse Flow Representations

Authors: Chun-Jun Pu, Li-Wei Chen, Hai-Bo Huang

Organizations: University of Science and Technology of China, Hefei 230026, China · Beijing Institute of Aeronautical Systems Engineering, Beijing 100076, China

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Abstract

This paper develops a data-driven framework for long-term prediction of fluid--structure interaction (FSI) dynamics, focusing on the flow-induced vibration (FIV) of a flexible plate. A stiffness-conditioned neural evolution operator jointly represents the Eulerian flow field and Lagrangian structural state. The plate is represented by 101 ordered structural tokens carrying nodal coordinates and velocities, with nondimensional bending stiffness as a global conditioning variable. Bidirectional cross-attention couples fluid and structural representations within a hybrid CNN-Transformer architecture. Trained with staged multi-step autoregressive rollouts and symmetry-reflected trajectories, a single operator captures three stiffness-dependent response regimes: deflected--flapping, deflected, and flapping. The predicted trajectories preserve the principal flow structures, structural oscillations, and dominant frequencies, while blind 1000-step rollouts remain bounded. The operator also interpolates to stiffness values excluded from training. To reduce sensitivity to under-resolved near-wall gradients in force reconstruction, we develop a differentiable aerodynamic-force module based on the derivative-moment transformation (DMT). Conventional wall-stress surface integrals are replaced by an enclosed 2D curve integral around the core vortex region, enabling accurate reconstruction of lift and drag. A Signed Distance Function (SDF) and smoothed Dirac-delta formulation make the integration fully differentiable while preserving gradient flow. The proposed framework provides an accurate and differentiable surrogate for stiffness-dependent FSI dynamics, enabling efficient parameter studies and future stiffness optimization for flow-energy-harvesting applications.

Figures & tables

Figure 1: Physical configuration and computational domains of the flexible-plate FSI problem. (a) Full non-uniform CFD domain and the location of the near-body prediction window. (b) Near-body Eulerian window used by the neural evolution operator.
Figure 2: Stiffness-dependent response regimes of the isolated inverted flexible plate. (a) Normalized peak-to-peak transverse displacement of the free tip. (b–d) Representative full-body envelopes of the deflected–flapping, deflected, and flapping modes, respectively. The dashed lines visually separate the response regimes represented by the discrete stiffness samples.
QuantityValue
Reynolds numberRe=200Re=200
Plate lengthL=1L=1
Mass ratiom∗=2.9m^{\ast}=2.9
Poisson ratioνs=0.25\nu_{s}=0.25
Fixed extensional stiffnessKS=1000K_{S}=1000
Training bending stiffnesses0.050.05 – 0.400.40 , Δκ=0.01\Delta\kappa=0.01 (36 cases)
Table 1: Summary of the physical system and multi-stiffness dataset.
Figure 3: Overview of the proposed framework. (a) The Eulerian flow field and the Lagrangian plate states, including nodal positions and velocities, are encoded as fluid and structural tokens, respectively. The normalized bending stiffness is introduced as a global conditioning token, and bidirectional cross-attention exchanges information between the two physical representations. (b) The coupled state predicted at each time step is recursively reused as the input for subsequent autoregressive evolution. (c) Aerodynamic lift and drag are reconstructed from the predicted flow field using a differentiable DMT operator defined on a fixed level-set control contour.
Figure 4: Flow-field reconstruction at rollout step 59 for three representative response regimes: (a) κ=0.05\kappa=0.05 ; (b) κ=0.10\kappa=0.10 ; and (c) κ=0.30\kappa=0.30 . Within each group, the columns show uu , vv , and pp , and the rows show the CFD ground truth, neural prediction, and pointwise absolute error.
Figure 5: Stepwise rollout errors for the three representative response regimes over 59 predicted states. The left column shows the relative L2L_{2} errors of uu , vv , and pp , while the right column shows the coordinate RMSE of the plate. The rows correspond to κ=0.05\kappa=0.05 , 0.100.10 , and 0.300.30 , respectively.
Figure 6: Prediction errors across the 36 training stiffnesses. (a) Mean fluid-field relative L2L_{2} error over the 59 predicted states. (b) Fluid-field relative L2L_{2} error at rollout step 59. (c) Mean flexible-plate position RMSE over the 59 predicted states. (d) Plate-position RMSE at rollout step 59. (e) Mean flexible-plate velocity RMSE over the 59 predicted states. (f) Plate-velocity RMSE at rollout step 59. The dashed separators indicate the deflected–flapping, deflected, and flapping regimes.
κ\kappaRegimeeue_{u} (%)eve_{v} (%)epe_{p} (%)exye_{xy}euve_{uv}
0.05Deflected–flapping1.895.395.412.91×10−32.91\times 10^{-3}4.35×10−34.35\times 10^{-3}
0.10Deflected1.414.083.369.17×10−49.17\times 10^{-4}1.68×10−31.68\times 10^{-3}
0.15Flapping1.043.264.211.19×10−31.19\times 10^{-3}1.54×10−31.54\times 10^{-3}
0.20Flapping0.862.123.547.75×10−47.75\times 10^{-4}1.09×10−31.09\times 10^{-3}
0.25Flapping0.872.203.636.15×10−46.15\times 10^{-4}8.10×10−48.10\times 10^{-4}
0.30Flapping1.132.633.941.53×10−31.53\times 10^{-3}2.23×10−32.23\times 10^{-3}
Table 2: Mean errors over the 59 predicted states for eight representative stiffnesses selected from the 36 training conditions. Fluid errors are relative L2L_{2} errors in percent; exye_{xy} and euve_{uv} are the nondimensional RMSEs of the plate position and velocity, respectively.
Figure 7: Free-tip displacement histories and frequency spectra for the three representative response regimes. The left column compares the predicted and CFD free-tip displacements, while the right column shows their corresponding amplitude spectra. The rows represent (a,b) κ=0.05\kappa=0.05 , (c,d) κ=0.10\kappa=0.10 , and (e,f) κ=0.30\kappa=0.30 .
Figure 8: Free-tip phase portraits for three representative response regimes: (a) κ=0.05\kappa=0.05 , deflected–flapping; (b) κ=0.10\kappa=0.10 , deflected; and (c) κ=0.30\kappa=0.30 , flapping. The transverse tip velocity is obtained from the time derivative of the free-tip displacement. The normalized RMSE values shown above each panel quantify the displacement and velocity differences between the prediction and CFD reference.
Figure 9: Flow-field reconstruction at rollout step 59 for the unseen stiffness κ=0.335\kappa=0.335 . The columns show uu , vv , and pp , while the rows show the CFD ground truth, neural prediction, and pointwise absolute error.
Figure 10: Evolution of the fluid-field relative L2L_{2} errors and plate-position RMSE over the 59-step autoregressive rollout at the unseen stiffness κ=0.335\kappa=0.335 .
Figure 11: Structural dynamics at the unseen stiffness κ=0.335\kappa=0.335 : free-tip displacement, Fourier spectrum, and phase portrait. CFD values are shown by open circles and neural predictions by solid lines.
Figure 12: Flow-field reconstruction at rollout step 59 for the unseen stiffness κ=0.105\kappa=0.105 . The columns show uu , vv , and pp , while the rows show the CFD ground truth, neural prediction, and pointwise absolute error.
Figure 13: Evolution of the fluid-field relative L2L_{2} errors and plate-position RMSE over the 59-step autoregressive rollout at the unseen stiffness κ=0.105\kappa=0.105 . The errors remain bounded, although their magnitudes are higher than those obtained within the flapping regime.
Figure 14: Structural dynamics at the unseen stiffness κ=0.105\kappa=0.105 : free-tip displacement, Fourier spectrum, and phase portrait. Although the dominant frequency and mean deflected branch are recovered, differences remain in the oscillation amplitude and phase-space trajectory.
Figure 15: Fixed DMT control contours for the three representative flexible-plate regimes. The background shows the predicted streamwise velocity at rollout step 59. The white curves show sampled plate configurations during the rollout, the black curve denotes the plate at step 59, and the orange rectangle is the fixed control contour used for force reconstruction.
Figure 16: Validation of the pressure-containing explicit Line DMT reconstruction using the original CFD flow fields at κ=0.30\kappa=0.30 . Open circles denote the direct CFD force and solid lines denote the DMT reconstruction: (a) drag coefficient and (b) lift coefficient.
Figure 17: Aerodynamic-force reconstruction from the autoregressively predicted flow fields for three representative structural regimes. The rows correspond to κ=0.05\kappa=0.05 (deflected-flapping), κ=0.10\kappa=0.10 (deflected), and κ=0.30\kappa=0.30 (flapping); the columns show CDC_{D} and CLC_{L} . Open circles denote the direct CFD force, dashed blue lines denote the explicit Line DMT reconstruction, and solid red lines denote the Level-set DMT reconstruction.
κ\kappaRegimeCoefficientLine DMTLevel-set DMT
   RMSENRMSE (%)RMSENRMSE (%)
0.05Deflected-flappingCDC_{D}0.05763.740.07654.96
0.05Deflected-flappingCLC_{L}0.06635.340.142411.49
0.10DeflectedCDC_{D}0.02663.270.04275.25
0.10DeflectedCLC_{L}0.02493.650.05588.19
0.30FlappingCDC_{D}0.05211.520.08842.58
Table 3: RMSE and range-normalized RMSE (NRMSE) of the aerodynamic coefficients reconstructed from the neural-predicted flow fields.
Figure 18: Drag and lift coefficients reconstructed from the predicted flow fields at the unseen stiffness κ=0.335\kappa=0.335 , compared with the forces reported directly by the CFD solver. Open circles denote the direct CFD force, dashed blue lines denote the explicit Line DMT reconstruction, and solid red lines denote the Level-set DMT reconstruction.
Figure 19: Drag and lift coefficients reconstructed from the predicted flow fields at the unseen stiffness κ=0.105\kappa=0.105 , compared with the forces reported directly by the CFD solver. Open circles denote the direct CFD force, dashed blue lines denote the explicit Line DMT reconstruction, and solid red lines denote the Level-set DMT reconstruction.
Appendix figures & tables2 assets

Supplementary material from the paper’s appendix.

Appendix

Figure A.1: Long-horizon stability diagnostics over 1000 fully autoregressive steps for κ=0.05\kappa=0.05 , 0.100.10 , and 0.300.30 . The left column shows the relative differences between consecutive predicted fluid states, and the middle column shows the coordinate RMSE between consecutive predicted plate states. The right column compares the normalized spectrum obtained from the final 60 predicted states with that of the corresponding 60-state CFD reference window. These quantities are used as stability diagnostics rather than errors against a synchronous 1000-step CFD trajectory.
Figure B.1: Analytical verification of the Level-set integration and discrete gradients: (a) circular-perimeter convergence; (b) geometric-gradient convergence; (c) Taylor-remainder test; and (d) translating-point-vortex force and gradient verification.
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