A novel data-driven nonlinear reduced-order modeling framework is proposed for unsteady fluid–structure interactions (FSIs). In the proposed framework, a convolutional variational autoencoder model is developed to determine the coordinate transformation from a high-dimensional physical field into a reduced space. This enables the efficient extraction of nonlinear low-dimensional manifolds from the high-dimensional unsteady flow field of the FSIs. The sparse identification of a nonlinear dynamics (SINDy) algorithm is then used to identify the dynamical governing equations of the reduced space and the vibration responses. To investigate and validate the effectiveness of the proposed framework for modeling and predicting unsteady flow fields in FSI problems, the two-dimensional laminar vortex shedding of a fixed cylinder is considered. Furthermore, the proposed data-driven nonlinear reduced-order modeling framework is applied to the three-dimensional vortex-induced vibration of a flexible cylinder. Using the SINDy model to analyze the vibration responses, the dynamics of the flexible cylinder are found to be correlated with the flow wake patterns, revealing the underlying FSI mechanism. The present work is a significant step toward the establishment of machine learning-based nonlinear reduced-order models for complex flow phenomena, the discovery of underlying unsteady FSI physics, and real-time flow control.
References
1.
S. J.
Haward
and A. Q.
Shen
, “Fluid–structure interactions: From engineering to biomimetic systems
,” Phys. Fluids
32
, 120401
(2020
).2.
F.
Xie
, Y.
Yu
, Y.
Constantinides
, M. S.
Triantafyllou
, and G. E.
Karniadakis
, “U-shaped fairings suppress vortex-induced vibrations for cylinders in cross-flow
,” J. Fluid Mech.
782
, 300
–332
(2015
).3.
Q.
Yu
, X.
Li
, W.
Zhang
, and S.
Xu
, “Stability analysis for laminar separation flutter of an airfoil in the transitional flow regime
,” Phys. Fluids
34
, 034118
(2022
).4.
J.
Kou
and W.
Zhang
, “Data-driven modeling for unsteady aerodynamics and aeroelasticity
,” Prog. Aerosp. Sci.
125
, 100725
(2021
).5.
L.
Ma
, K.
Lin
, D.
Fan
, J.
Wang
, and M. S.
Triantafyllou
, “Flexible cylinder flow-induced vibration
,” Phys. Fluids
34
, 011302
(2022
).6.
F.
Liu
, G.
Liu
, and C.
Shu
, “Fluid–structure interaction simulation based on immersed boundary-lattice Boltzmann flux solver and absolute nodal coordinate formula
,” Phys. Fluids
32
, 047109
(2020
).7.
A.
Zhang
, W.
Wu
, Y.
Liu
, and Q.
Wang
, “Nonlinear interaction between underwater explosion bubble and structure based on fully coupled model
,” Phys. Fluids
29
, 082111
(2017
).8.
W.
Zhang
, K.
Chen
, and Z.
Ye
, “Unsteady aerodynamic reduced-order modeling of an aeroelastic wing using arbitrary mode shapes
,” J. Fluids Struct.
58
, 254
–270
(2015
).9.
G.
Liu
, H.
Li
, Z.
Qiu
, and Z.
Li
, “A comprehensive numerical analysis of cross-flow vortex-induced vibrations for top tension risers under different flows
,” Phys. Fluids
32
, 027102
(2020
).10.
K.
Lin
, J.
Wang
, H.
Zheng
, and Y.
Sun
, “Numerical investigation of flow-induced vibrations of two cylinders in tandem arrangement with full wake interference
,” Phys. Fluids
32
, 015112
(2020
).11.
P.
Han
, G.
Pan
, and W.
Tian
, “Numerical simulation of flow-induced motion of three rigidly coupled cylinders in equilateral-triangle arrangement
,” Phys. Fluids
30
, 125107
(2018
).12.
W.
Zhang
, J.
Kou
, and Y.
Liu
, “Prospect of artificial intelligence empowered fluid mechanics
,” Acta Aeronaut. Sin.
42
, 524689
(2021
) (in Chinese).13.
G.
Berkooz
, P.
Holmes
, and J. L.
Lumley
, “The proper orthogonal decomposition in the analysis of turbulent flows
,” Annu. Rev. Fluid Mech.
25
, 539
–575
(1993
).14.
T.
Bui-Thanh
, M.
Damodaran
, and K.
Willcox
, “Aerodynamic data reconstruction and inverse design using proper orthogonal decomposition
,” AIAA J.
42
, 1505
–1516
(2004
).15.
G.
Riches
, R.
Martinuzzi
, and C.
Morton
, “Proper orthogonal decomposition analysis of a circular cylinder undergoing vortex-induced vibrations
,” Phys. Fluids
30
, 105103
(2018
).16.
J. L.
Callaham
, K.
Maeda
, and S. L.
Brunton
, “Robust flow reconstruction from limited measurements via sparse representation
,” Phys. Rev. Fluids
4
, 103907
(2019
).17.
F.
Gómez
and H. M.
Blackburn
, “Data-driven approach to design of passive flow control strategies
,” Phys. Rev. Fluids
2
, 021901
(2017
).18.
R.
Qiu
, R.
Huang
, Y.
Wang
, and C.
Huang
, “Dynamic mode decomposition and reconstruction of transient cavitating flows around a Clark-Y hydrofoil
,” Theor. Appl. Mech. Lett.
10
, 327
–332
(2020
).19.
K.
Taira
, S. L.
Brunton
, S. T.
Dawson
, C. W.
Rowley
, T.
Colonius
, B. J.
McKeon
, O. T.
Schmidt
, S.
Gordeyev
, V.
Theofilis
, and L. S.
Ukeiley
, “Modal analysis of fluid flows: An overview
,” AIAA J.
55
, 4013
–4041
(2017
).20.
X.
Zhang
, T.
Ji
, F.
Xie
, H.
Zheng
, and Y.
Zheng
, “Unsteady flow prediction from sparse measurements by compressed sensing reduced order modeling
,” Comput. Methods Appl. Mech. Eng.
393
, 114800
(2022
).21.
B. R.
Noack
, W.
Stankiewicz
, M.
Morzyński
, and P. J.
Schmid
, “Recursive dynamic mode decomposition of transient and post-transient wake flows
,” J. Fluid Mech.
809
, 843
–872
(2016
).22.
F.
Xie
, J.
Deng
, Q.
Xiao
, and Y.
Zheng
, “A numerical simulation of VIV on a flexible circular cylinder
,” Fluid Dyn. Res.
44
, 045508
(2012
).23.
F.
Xie
, D.
Pan
, Y.
Zheng
, and J.
Zou
, “Smoothed profile method and its applications in VIV
,” Int. J. Numer. Methods Heat Fluid Flow
27
, 1623
(2017
).24.
S.
Cai
, Z.
Wang
, F.
Fuest
, Y. J.
Jeon
, C.
Gray
, and G. E.
Karniadakis
, “Flow over an espresso cup: Inferring 3-D velocity and pressure fields from tomographic background oriented Schlieren via physics-informed neural networks
,” J. Fluid Mech.
915
, A102
(2021
).25.
S. L.
Brunton
, B. R.
Noack
, and P.
Koumoutsakos
, “Machine learning for fluid mechanics
,” Annu. Rev. Fluid Mech.
52
, 477
–508
(2020
).26.
X.
Zhang
, F.
Xie
, T.
Ji
, Z.
Zhu
, and Y.
Zheng
, “Multi-fidelity deep neural network surrogate model for aerodynamic shape optimization
,” Comput. Methods Appl. Mech. Eng.
373
, 113485
(2021
).27.
B.
Lusch
, J. N.
Kutz
, and S. L.
Brunton
, “Deep learning for universal linear embeddings of nonlinear dynamics
,” Nat. Commun.
9
, 4950
(2018
).28.
T.
Murata
, K.
Fukami
, and K.
Fukagata
, “Nonlinear mode decomposition with convolutional neural networks for fluid dynamics
,” J. Fluid Mech.
882
, A13
(2020
).29.
K.
Fukami
, T.
Nakamura
, and K.
Fukagata
, “Convolutional neural network based hierarchical autoencoder for nonlinear mode decomposition of fluid field data
,” Phys. Fluids
32
, 095110
(2020
).30.
D.
Fernex
, B. R.
Noack
, and R.
Semaan
, “Cluster-based network modeling-from snapshots to complex dynamical systems
,” Sci. Adv.
7
, eabf5006
(2021
).31.
M.
Schmidt
and H.
Lipson
, “Distilling free-form natural laws from experimental data
,” Science
324
, 81
–85
(2009
).32.
P.
Marzocca
, W. A.
Silva
, and L.
Librescu
, “Nonlinear open-/closed-loop aeroelastic analysis of airfoils via Volterra series
,” AIAA J.
42
, 673
–686
(2004
).33.
S.
Obeid
, G.
Ahmadi
, and R.
Jha
, “NARMAX identification based closed-loop control of flow separation over NACA 0015 airfoil
,” Fluids
5
, 100
(2020
).34.
W.
Yao
and R.
Jaiman
, “Feedback control of unstable flow and vortex-induced vibration using the eigensystem realization algorithm
,” J. Fluid Mech.
827
, 394
–414
(2017
).35.
H.
Eivazi
, H.
Veisi
, M. H.
Naderi
, and V.
Esfahanian
, “Deep neural networks for nonlinear model order reduction of unsteady flows
,” Phys. Fluids
32
, 105104
(2020
).36.
S. L.
Brunton
, J. L.
Proctor
, and J. N.
Kutz
, “Discovering governing equations from data by sparse identification of nonlinear dynamical systems
,” Proc. Natl. Acad. Sci.
113
, 3932
–3937
(2016
).37.
J.-C.
Loiseau
, B. R.
Noack
, and S. L.
Brunton
, “Sparse reduced-order modelling: Sensor-based dynamics to full-state estimation
,” J. Fluid Mech.
844
, 459
–490
(2018
).38.
S.
Li
, E.
Kaiser
, S.
Laima
, H.
Li
, S. L.
Brunton
, and J. N.
Kutz
, “Discovering time-varying aerodynamics of a prototype bridge by sparse identification of nonlinear dynamical systems
,” Phys. Rev. E
100
, 022220
(2019
).39.
J.-C.
Loiseau
, “Data-driven modeling of the chaotic thermal convection in an annular thermosyphon
,” Theor. Comput. Fluid Dyn.
34
, 339
–365
(2020
).40.
K.
Champion
, B.
Lusch
, J. N.
Kutz
, and S. L.
Brunton
, “Data-driven discovery of coordinates and governing equations
,” Proc. Natl. Acad. Sci.
116
, 22445
–22451
(2019
).41.
G. E.
Hinton
and R. R.
Salakhutdinov
, “Reducing the dimensionality of data with neural networks
,” Science
313
, 504
–507
(2006
).42.
D. M.
Blei
, A.
Kucukelbir
, and J. D.
McAuliffe
, “Variational inference: A review for statisticians
,” J. Am. Stat. Assoc.
112
, 859
–877
(2017
).43.
R.
Ranjan
, S.
Sankaranarayanan
, A.
Bansal
, N.
Bodla
, J.-C.
Chen
, V. M.
Patel
, C. D.
Castillo
, and R.
Chellappa
, “Deep learning for understanding faces: Machines may be just as good, or better, than humans
,” IEEE Signal Process. Mag.
35
, 66
–83
(2018
).44.
J.
Kim
and C.
Lee
, “Prediction of turbulent heat transfer using convolutional neural networks
,” J. Fluid Mech.
882
, A18
(2020
).45.
46.
47.
48.
I.
Higgins
, L.
Matthey
, A.
Pal
, C. P.
Burgess
, X.
Glorot
, M.
Botvinick
, S.
Mohamed
, and A.
Lerchner
, “beta-VAE: Learning basic visual concepts with a constrained variational framework
,” in ICLR
, 2017
.49.
F.
Santosa
and W. W.
Symes
, “Linear inversion of band-limited reflection seismograms
,” SIAM J. Sci. Stat. Comput.
7
, 1307
–1330
(1986
).50.
G. E.
Karniadakis
, M.
Israeli
, and S. A.
Orszag
, “High-order splitting methods for the incompressible Navier-Stokes equations
,” J. Comput. Phys.
97
, 414
–443
(1991
).51.
Y.
Bao
, R.
Palacios
, M.
Graham
, and S.
Sherwin
, “Generalized thick strip modelling for vortex-induced vibration of long flexible cylinders
,” J. Comput. Phys.
321
, 1079
–1097
(2016
).52.
D.
Newman
and G.
Karniadakis
, “Simulations of flow over a flexible cable: A comparison of forced and flow-induced vibration
,” J. Fluids Struct.
10
, 439
–453
(1996
).53.
S.
Komatsu
and H.
Kobayashi
, “Vortex-induced oscillation of bluff cylinders
,” J. Wind Eng. Ind. Aerodyn.
6
, 335
–362
(1980
).54.
S.
Laima
, H.
Li
, W.
Chen
, and F.
Li
, “Investigation and control of vortex-induced vibration of twin box girders
,” J. Fluids Struct.
39
, 205
–221
(2013
).55.
F.
Xie
, J.
Deng
, and Y.
Zheng
, “Multi-mode of vortex-induced vibration of a flexible circular cylinder
,” J. Hydrodyn.
23
, 483
–490
(2011
).56.
P.
Han
and E.
de Langre
, “There is no critical mass ratio for galloping of a square cylinder under flow
,” J. Fluid Mech.
931
, A27
(2022
).57.
C.
Zheng
, T.
Ji
, F.
Xie
, X.
Zhang
, H.
Zheng
, and Y.
Zheng
, “From active learning to deep reinforcement learning: Intelligent active flow control in suppressing vortex-induced vibration
,” Phys. Fluids
33
, 063607
(2021
).58.
J.
Kou
and W.
Zhang
, “An improved criterion to select dominant modes from dynamic mode decomposition
,” Eur. J. Mech.-B
62
, 109
–129
(2017
).59.
A.
Munir
, M.
Zhao
, H.
Wu
, L.
Lu
, and D.
Ning
, “Three-dimensional numerical investigation of vortex-induced vibration of a rotating circular cylinder in uniform flow
,” Phys. Fluids
30
, 053602
(2018
).© 2022 Author(s). Published under an exclusive license by AIP Publishing.
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