The linear-time-invariance notion to the Koopman analysis is a recent advance in fluid mechanics [Li et al., “The linear-time-invariance notion to the Koopman analysis: The architecture, pedagogical rendering, and fluid–structure association,” Phys. Fluids 34(12), 125136 (2022c) and Li et al., “The linear-time-invariance notion of the Koopman analysis—Part 2. Dynamic Koopman modes, physics interpretations and phenomenological analysis of the prism wake,” J. Fluid Mech. 959, A15 (2023a)], targeting the long-standing issue of correlating nonlinear excitation and response phenomena in fluid–structure interactions (FSI), or, in the simplified case, flow over rigid obstacles. Continuing the serial research, this work presents a data-driven, Koopman-inspired methodology to decouple nonlinear FSI by establishing cause-and-effect correspondences between structure surface pressure and the flow field. Exploiting unique features of the Koopman operator, the new methodology renders dynamic visualizations of in-sync, fluid–structure-coupled Koopman modes possible, fostering phenomenological analysis and statistical quantifications of FSI energy transfers. Instantaneous contribution contours and densities offer new angles to evaluate pathways of energy amplification and diminution. The methodology enables better descriptions and interpretations of phenomena occurring in the flow and on the boundary (walls) of an FSI domain and readily applies to a broad spectrum of engineering problems given its data-driven nature.
REFERENCES
1.
Alford-Lago
, D. J.
, Curtis
, C. W.
, Ihler
, A. T.
, and Issan
, O.
, “ Deep learning enhanced dynamic mode decomposition
,” Chaos
32
, 033116
(2022
).2.
Amor
, C.
, Schlatter
, P.
, Vinuesa
, R.
, and Le Clainche
, S.
, “ Higher-order dynamic mode decomposition on-the-fly: A low-order algorithm for complex fluid flows
,” J. Comput. Phys.
475
, 111849
(2023
).3.
Arbabi
, H.
and Mezić
, I.
, “ Ergodic theory, dynamic mode decomposition, and computation of spectral properties of the Koopman operator
,” SIAM J. Appl. Dyn. Syst.
16
, 2096
–2126
(2017
).4.
Avila
, A. M.
and Mezić
, I.
, “ Data-driven analysis and forecasting of highway traffic dynamics
,” Nat. Commun.
11
, 2090
(2020
).5.
Axås
, J.
and Haller
, G.
, “ Model reduction for nonlinearizable dynamics via delay-embedded spectral submanifolds
,” Nonlinear Dyn.
111
, 22079
–22099
(2023
).6.
Bagheri
, S.
, “ Koopman-mode decomposition of the cylinder wake
,” J. Fluid Mech.
726
, 596
–623
(2013
).7.
Begiashvili
, B.
, Groun
, N.
, Garicano-Mena
, J.
, Le Clainche
, S.
, and Valero
, E.
, “ Data-driven modal decomposition methods as feature detection techniques for flow problems: A critical assessment
,” Phys. Fluids
35
, 041301
(2023
).8.
Bistrian
, D. A.
and Navon
, I. M.
, “ Efficiency of randomised dynamic mode decomposition for reduced order modelling
,” Int. J. Comput. Fluid Dyn.
32
, 88
–103
(2018
).9.
Bistrian
, D. A.
and Navon
, I. M.
, “ Randomized dynamic mode decomposition for nonintrusive reduced order modelling
,” Int. J. Numer. Methods Eng.
112
, 3
–25
(2017
).10.
Bloor
, M. S.
, “ The transition to turbulence in the wake of a circular cylinder
,” J. Fluid Mech.
19
, 290
–304
(1964
).11.
Bollt
, E. M.
and Santitissadeekorn
, N.
, Applied and Computational Measurable Dynamics
( Society for Industrial and Applied Mathematics
, 2013
).12.
Bramburger
, J. J.
and Fantuzzi
, G.
, “ Auxiliary functions as Koopman observables: Data-driven analysis of dynamical systems via polynomial optimization
,” J. Nonlinear Sci.
34
(1
), 8
(2024
).13.
Brunton
, S. L.
, Brunton
, B. W.
, Proctor
, J. L.
, and Kutz
, J. N.
, “ Koopman invariant subspaces and finite linear representations of nonlinear dynamical systems for control
,” PLoS One
11
, e0150171
(2016
).14.
Brunton
, S. L.
, Noack
, B. R.
, and Koumoutsakos
, P.
, “ Machine learning for fluid mechanics
,” Annu. Rev. Fluid Mech.
52
, 477
–508
(2020
).15.
Budišić
, M.
, Mohr
, R.
, and Mezić
, I.
, “ Applied Koopmanism
,” Chaos
22
, 047510
(2012
).16.
Brunton
, S. L.
, Proctor
, J. L.
, Tu
, J. H.
, and Nathan Kutz
, J.
, “ Compressed sensing and dynamic mode decomposition
,” J. Comput. Dyn.
2
, 165
–191
(2015
).17.
Chen
, K. K.
, Tu
, J. H.
, and Rowley
, C. W.
, “ Variants of dynamic mode decomposition: Boundary condition, Koopman, and Fourier analyses
,” J. Nonlinear Sci.
22
, 887
–915
(2012
).18.
Chen
, Z.
, Bai
, J.
, Wang
, S.
, Xue
, X.
, Li
, K.
, Tse
, K. T.
, Li
, C. Y.
, and Lin
, C.
, “ The role of transverse inclination on the flow phenomenology around cantilevered prisms and the tripole wake mode
,” J. Fluids Struct.
118
, 103837
(2023a
).19.
Chen
, Z.
, Chen
, X.
, Liu
, J.
, Cen
, L.
, and Gui
, W.
, “ Learning model predictive control of nonlinear systems with time-varying parameters using Koopman operator
,” Appl. Math. Comput.
470
, 128577
(2024
).20.
Chen
, Z.
, Fu
, X.
, Xu
, Y.
, Li
, C. Y.
, Kim
, B.
, and Tse
, K. T.
, “ A perspective on the aerodynamics and aeroelasticity of tapering: Partial reattachment
,” J. Wind Eng. Ind. Aerodyn.
212
, 104590
(2021a
).21.
Chen
, Z.
, Huang
, H.
, Tse
, T. K. T.
, Xu
, Y.
, and Li
, C. Y.
, “ Characteristics of unsteady aerodynamic forces on an aeroelastic prism: A comparative study
,” J. Wind Eng. Ind. Aerodyn.
205
, 104325
(2020
).22.
Chen
, Z.
, Wang
, Y.
, Wang
, S.
, Huang
, H.
, Tse
, K. T.
, Li
, C. Y.
, and Lin
, C.
, “ Decoupling bi-directional fluid–structure interactions by the Koopman theory: Actualizing one-way subcases and the role of crosswind structure motion
,” Phys. Fluids
34
, 095103
(2022
).23.
Chen
, Z.
, Xu
, Y.
, Hua
, J.
, Xu
, F.
, Tse
, K. T.
, Huang
, L.
, and Xue
, X.
, “ Unsteady aerodynamic forces on a tapered prism during the combined vibration of VIV and galloping
,” Nonlinear Dyn.
107
(1
), 599
–615
(2021b
).24.
Chen
, Z.
, Zhang
, L.
, Li
, K.
, Xue
, X.
, Zhang
, X.
, Kim
, B.
, and Li
, C. Y.
, “ Machine-learning prediction of aerodynamic damping for buildings and structures undergoing flow-induced vibrations
,” J. Build. Eng.
63
, 105374
(2023b
).25.
Colbrook
, M. J.
, Ayton
, L. J.
, and Szőke
, M.
, “ Residual dynamic mode decomposition: Robust and verified Koopmanism
,” J. Fluid Mech.
955
, A21
(2023
).26.
Colbrook
, M. J.
, Li
, Q.
, Raut
, R. V.
, and Townsend
, A.
, “ Beyond expectations: Residual dynamic mode decomposition and variance for stochastic dynamical systems
,” Nonlinear Dyn.
112
, 2037
–2061
(2024
).27.
Deem
, E. A.
, Cattafesta
III, L. N.
, Hemati
, M. S.
, Zhang
, H.
, Rowley
, C.
, and Mittal
, R.
, “ Adaptive separation control of a laminar boundary layer using online dynamic mode decomposition
,” J. Fluid Mech.
903
, A21
(2020
).28.
Ding
, J.
, Zhao
, P.
, Liu
, C.
, Wang
, X.
, Xie
, R.
, and Liu
, H.
, “ From irregular to continuous: The deep Koopman model for time series forecasting of energy equipment
,” Appl Energy
364
, 123138
(2024
).29.
Dotto
, A.
, Lengani
, D.
, Simoni
, D.
, and Tacchella
, A.
, “ Dynamic mode decomposition and Koopman spectral analysis of boundary layer separation-induced transition
,” Phys. Fluids
33
, 104104
(2021
).30.
Duke
, D.
, Soria
, J.
, and Honnery
, D.
, “ An error analysis of the dynamic mode decomposition
,” Exp. Fluids
52
, 529
–542
(2012
).31.
Erichson
, N. B.
and Donovan
, C.
, “ Randomized low-rank Dynamic Mode Decomposition for motion detection
,” Comput. Vision Image Understanding
146
, 40
–50
(2016
).32.
Erichson
, N. B.
, Mathelin
, L.
, Kutz
, J. N.
, and Brunton
, S. L.
, “ Randomized dynamic mode decomposition
,” SIAM J. Appl. Dyn. Syst.
18
, 1867
–1891
(2019
).33.
Fan
, X.
, Zhang
, X.
, Weerasuriya
, A. U.
, Hang
, J.
, Zeng
, L.
, Luo
, Q.
, Li
, C. Y.
, and Chen
, Z.
, “ Numerical investigation of the effects of environmental conditions, droplet size, and social distancing on droplet transmission in a street canyon
,” Build. Environ.
221
, 109261
(2022
).34.
Froyland
, G.
, Gottwald
, G. A.
, and Hammerlindl
, A.
, “A computational method to extract macroscopic variables and their dynamics in multiscale systems,” SIAM J. Appl. Dyn. Syst.
13
(4
), 1816
–1846
(2014
).35.
Fu
, Y.
, Lin
, X.
, Li
, L.
, Chu
, Q.
, Liu
, H.
, Zheng
, X.
, Liu
, C.-H.
, Chen
, Z.
, Lin
, C.
, Tse
, T. K. T.
, and Li
, C. Y.
, “ A POD-DMD augmented procedure to isolating dominant flow field features in a street canyon
,” Phys. Fluids
35
, 025112
(2023
).36.
Garicano-Mena
, J.
, Li
, B.
, Ferrer
, E.
, and Valero
, E.
, “ A composite dynamic mode decomposition analysis of turbulent channel flows
,” Phys. Fluids
31
, 115102
(2019
).37.
Han
, W.
and Stankovic
, A. M.
, “ Model-predictive control design for power system oscillation damping via excitation—A data-driven approach
,” IEEE Trans. Power Syst.
38
, 1176
–1188
(2023
).38.
He
, G.
, Wang
, J.
, and Pan
, C.
, “ Initial growth of a disturbance in a boundary layer influenced by a circular cylinder wake
,” J. Fluid Mech.
718
, 116
–130
(2013
).39.
He
, Y.
, Zhang
, L.
, Chen
, Z.
, and Li
, C. Y.
, “ A framework of structural damage detection for civil structures using a combined multi-scale convolutional neural network and echo state network
,” Eng. Comput.
39
, 1771
–1719
(2022
).40.
Helmholtz
, H.
, “ Über integrale der hydrodynamischen gleichungen, welche den wirbelbewegungen entsprechen
,” J. Reine Angew. Math.
1858
, 25
–55
.41.
Herrmann
, B.
, Baddoo
, P. J.
, Semaan
, R.
, Brunton
, S. L.
, and McKeon
, B. J.
, “Data-driven resolvent analysis,” J. Fluid Mech.
918
, A10
(2021
).42.
Hong
, S.
, Huang
, G.
, Yang
, Y.
, and Liu
, Z.
, “ Introduction of DMD method to study the dynamic structures of a three-dimensional centrifugal compressor with and without flow control
,” Energies
11
, 3098
(2018
).43.
Huhn
, Q. A.
, Tano
, M. E.
, Ragusa
, J. C.
, and Choi
, Y.
, “ Parametric dynamic mode decomposition for reduced order modeling
,” J. Comput. Phys.
475
, 111852
(2023
).44.
Hunt
, J. C. R.
, Wray
, A. A.
, and Moin
, P.
, “ Eddies, streams, and convergence zones in turbulent flows
,” in Studying Turbulence Using Numerical Simulation Databases, Proceedings of the Summer Program
( NASA
, 1988
), pp. 193
–208
.45.
Hussain
, A. K. M. F.
, “ Coherent structures and turbulence
,” J. Fluid Mech.
173
, 303
–356
(1986
).46.
Jang
, H. K.
, Ozdemir
, C. E.
, Liang
, J. H.
, and Tyagi
, M.
, “ Oscillatory flow around a vertical wall-mounted cylinder: Dynamic mode decomposition
,” Phys. Fluids
33
, 025113
(2021
).47.
Jeon
, D.
and Gharib
, M.
, “ On the relationship between the vortex formation process and cylinder wake vortex patterns
,” J. Fluid Mech.
519
, 161
–181
(2004
).48.
Jovanović
, M. R.
, Schmid
, P. J.
, and Nichols
, J. W.
, “ Sparsity-promoting dynamic mode decomposition
,” Phys. Fluids
26
, 24103
(2014
).49.
Kareem
, A.
and Cermak
, J. E.
, “ Pressure fluctuations on a square building model in boundary-layer flows
,” J. Wind Eng. Ind. Aerodyn.
16
, 17
–41
(1984
).50.
Koopman
, B. O.
, “ Hamiltonian systems and transformation in Hilbert space
,” Proc. Natl. Acad. Sci. U. S. A.
17
, 315
–318
(1931
).51.
Koopman
, B. O.
and Neumann
, J. V.
, “ Dynamical systems of continuous spectra
,” Proc. Natl. Acad. Sci. U. S. A.
18
, 255
–263
(1932
).52.
Korda
, M.
and Mezić
, I.
, “ On convergence of extended dynamic mode decomposition to the Koopman operator
,” J. Nonlinear Sci.
28
, 687
–710
(2018
).53.
Kurdila
, A. J.
, Paruchuri
, S. T.
, Powell
, N.
, Guo
, J.
, Bobade
, P.
, Estes
, B.
, and Wang
, H.
, “ Approximation of discrete and orbital Koopman operators over subsets and manifolds
,” Nonlinear Dyn.
112
, 6291
–6327
(2024
).54.
Kuttichira
, D. P.
, Gopalakrishnan
, E. A.
, Menon
, V. K.
, and Soman
, K. P.
, “ Stock price prediction using dynamic mode decomposition
,” in International Conference on Advances in Computing, Communications and Informatic (ICACCI)
( IEEE
, 2017
), pp. 55
–60
.55.
Kutz
, J. N.
, Brunton
, S. L.
, Brunton
, B. W.
, and Proctor
, J. L.
, “ Dynamic mode decomposition: Data-driven modeling of complex systems
” (Society for Industrial and Applied Mathematics, Philadelphia, PA, 2016a
).56.
Kutz
, J. N.
, Fu
, X.
, and Brunton
, S. L.
, “ Multiresolution dynamic mode decomposition
,” SIAM J. Appl. Dyn. Syst.
15
, 713
–735
(2016b
).57.
Lander
, D. C.
, Moore
, D. M.
, Letchford
, C. W.
, and Amitay
, M.
, “ Scaling of square-prism shear layers
,” J. Fluid Mech.
849
, 1096
–1119
(2018
).58.
Le Clainche
, S.
and Vega
, J. M.
, “ Analyzing nonlinear dynamics via data-driven dynamic mode decomposition-like methods
,” Complexity
2018
, 6920783
.59.
Li
, C. Y.
, Chen
, Z.
, Tse
, T. K. T.
, Weerasuriya
, A. U.
, Zhang
, X.
, Fu
, Y.
, and Lin
, X.
, “ The linear-time-invariance notion of the Koopman analysis—Part 2. Dynamic Koopman modes, physics interpretations and phenomenological analysis of the prism wake
,” J. Fluid Mech.
959
, A15
(2023a
).60.
Li
, C. Y.
, Chen
, Z.
, Tse
, T. K. T.
, Weerasuriya
, A. U.
, Zhang
, X.
, Fu
, Y.
, and Lin
, X.
, “ A parametric and feasibility study for data sampling of the dynamic mode decomposition: Range, resolution, and universal convergence states
,” Nonlinear Dyn.
107
, 3683
–3707
(2022a
).61.
Li
, C. Y.
, Chen
, Z.
, Tse
, T. K. T.
, Weerasuriya
, A. U.
, Zhang
, X.
, Fu
, Y.
, and Lin
, X.
, “ A parametric and feasibility study for data sampling of the dynamic mode decomposition: Spectral insights and further explorations
,” Phys. Fluids
34
, 035102
(2022b
).62.
Li
, C. Y.
, Chen
, Z.
, Tse
, T. K. T.
, Weerasuriya
, A. U.
, Zhang
, X.
, Fu
, Y.
, and Lin
, X.
, “ Establishing direct phenomenological connections between fluid and structure by the Koopman-Linearly Time-Invariant analysis
,” Phys. Fluids
33
, 121707
(2021
).63.
Li
, C. Y.
, Chen
, Z.
, Weerasuriya
, A. U.
, Zhang
, X.
, Lin
, X.
, Zhou
, L.
, Fu
, Y.
, and Tse
, T. K. T.
, “ Best practice guidelines for the dynamic mode decomposition from a wind engineering perspective
,” J. Wind Eng. Ind. Aerodyn.
241
, 105506
(2023b
).64.
Li
, C. Y.
, Chen
, Z.
, Zhang
, X.
, Tse
, T. K. T.
, and Lin
, C.
, “ Koopman analysis by the dynamic mode decomposition in wind engineering
,” J. Wind Eng. Ind. Aerodyn.
232
, 105295
(2023c
).65.
Li
, C. Y.
, Chen
, Z.
, Lin
, X.
, Weerasuriya
, A. U.
, Zhang
, X.
, Fu
, Y.
, and Tse
, T. K. T.
, “ The linear-time-invariance notion to the Koopman analysis: The architecture, pedagogical rendering, and fluid–structure association
,” Phys. Fluids
34
, 125136
(2022c
).66.
Li
, C. Y.
, Tse
, T. K. T.
, and Hu
, G.
, “ Dynamic Mode Decomposition on pressure flow field analysis: Flow field reconstruction, accuracy, and practical significance
,” J. Wind Eng. Ind. Aerodyn.
205
, 104278
(2020a
).67.
Li
, C. Y.
, Tse
, T. K. T.
, and Hu
, G.
, “ Reconstruction of flow field around a square prism using dynamic mode decomposition
,” in The 4th Hong Kong Wind Engineering Society Workshop (HKWES4)
, 7–8 Feb, Hong Kong, China, 2020b
.68.
Li
, Q.
, Dietrich
, F.
, Bollt
, E. M.
, and Kevrekidis
, I. G.
, “ Extended dynamic mode decomposition with dictionary learning: A data-driven adaptive spectral decomposition of the Koopman operator
,” Chaos
27
, 103111
(2017
).69.
Liu
, C.
, Gao
, Y.
, Tian
, S.
, and Dong
, X.
, “ Rortex—A new vortex vector definition and vorticity tensor and vector decompositions
,” Phys. Fluids
30
, 035103
(2018
).70.
Liu
, C.
, Wang
, Y.
, Yang
, Y.
, and Duan
, Z.
, “ New omega vortex identification method
,” Sci. China Phys., Mech. Astron.
59
, 684711
(2016
).71.
Lu
, H.
and Tartakovsky
, D. M.
, “ Data-driven models of nonautonomous systems
,” J. Comput. Phys.
507
, 112976
(2024
).72.
Luo
, X.
and Kareem
, A.
, “ Dynamic mode decomposition of random pressure fields over bluff bodies
,” J. Eng. Mech.
147
, 04021007
(2021
).73.
Luo
, X.
and Kareem
, A.
, “ Bayesian deep learning with hierarchical prior: Predictions from limited and noisy data
,” Struct. Saf.
84
, 101918
(2020
).74.
Luo
, X.
and Kareem
, A.
, “ Deep convolutional neural networks for uncertainty propagation in random fields
,” Comput.-Aided Civil Infrastruct. Eng.
34
, 1043
–1054
(2019
).75.
Lusch
, B.
, Kutz
, J. N.
, and Brunton
, S. L.
, “Deep learning for universal linear embeddings of nonlinear dynamics,” Nat. Commun.
9
(1
), 4950
(2018
).76.
Magionesi
, F.
, Dubbioso
, G.
, Muscari
, R.
, and Di Mascio
, A.
, “ Modal analysis of the wake past a marine propeller
,” J. Fluid Mech.
855
, 469
–502
(2018
).77.
Mann
, J.
and Kutz
, J. N.
, “ Dynamic mode decomposition for financial trading strategies
,” Quant. Finance
16
, 1643
–1655
(2016
).78.
Mauroy
, A.
and Mezić
, I.
, “ Global stability analysis using the eigenfunctions of the Koopman operator
,” IEEE Trans. Autom. Control
61
, 3356
–3369
(2016
).79.
Mauroy
, A.
and Mezić
, I.
, “ On the use of Fourier averages to compute the global isochrons of (quasi)periodic dynamics
,” Chaos
22
, 033112
(2012
).80.
Menon
, K.
and Mittal
, R.
, “ Dynamic mode decomposition based analysis of flow over a sinusoidally pitching airfoil
,” J. Fluids Struct.
94
, 102886
(2020
).81.
Mezić
, I.
, “ Analysis of fluid flows via spectral properties of the Koopman operator
,” Annu. Rev. Fluid Mech.
45
, 357
–378
(2013
).82.
Mezić
, I.
, “ Spectral properties of dynamical systems, model reduction and decompositions
,” Nonlinear Dyn.
41
, 309
–325
(2005
).83.
Mezić
, I.
, Drmač
, Z.
, Črnjarić
, N.
, Maćešić
, S.
, Fonoberova
, M.
, Mohr
, R.
, Avila
, A. M.
, Manojlović
, I.
, and Andrejčuk
, A.
, “ A Koopman operator-based prediction algorithm and its application to COVID-19 pandemic and influenza cases
,” Sci. Rep.
14
(1
), 5788
(2024
).84.
Morgenthal
, G.
, “ Fluid-structure interaction in bluff-body aerodynamics and long-span bridge design: Phenomena and methods
,” Tribol. Int.
5
, 83
–84
(1972
).85.
Muld
, T. W.
, Efraimsson
, G.
, and Henningson
, D. S.
, “ Mode decomposition on surface-mounted cube
,” Flow, Turbul. Combust.
88
, 279
–310
(2012
).86.
Noack
, B. R.
, Stankiewicz
, W.
, Morzyński
, M.
, and Schmid
, P. J.
, “ Recursive dynamic mode decomposition of transient and post-transient wake flows
,” J. Fluid Mech.
809
, 843
–872
(2016
).87.
Nüske
, F.
, Peitz
, S.
, Philipp
, F.
, Schaller
, M.
, and Worthmann
, K.
, “ Finite-data error bounds for Koopman-based prediction and control
,” J. Nonlinear Sci.
33
(1
), 14
(2023
).88.
Otto
, S. E.
and Rowley
, C. W.
, “ Linearly recurrent autoencoder networks for learning dynamics
,” SIAM J. Appl. Dyn. Syst.
18
, 558
–593
(2019
).89.
Page
, J.
and Kerswell
, R. R.
, “ Koopman mode expansions between simple invariant solutions
,” J. Fluid Mech.
879
, 1
–27
(2019
).90.
Paidoussis
, M. P.
, Price
, S. J.
, and de Langre
, E.
, Fluid-Structure Interactions: Cross-Flow-Induced Instabilities
( Cambridge University Press
, Cambridge
, 2010
).91.
Patel
, V.
, Parthasarathy
, S.
, Shinde
, V. J.
, and Gaitonde
, D. V.
, “ Machine learning based model reduction for fluid-structure interaction
,” AIAA Paper No. AIAA 2021-1747, 2021
.92.
Philipp
, F. M.
, Schaller
, M.
, Worthmann
, K.
, Peitz
, S.
, and Nüske
, F.
, “ Error bounds for kernel-based approximations of the Koopman operator
,” Appl. Comput. Harmonic Anal.
71
, 101657
(2024
).93.
Ping
, H.
, Zhu
, H.
, Zhang
, K.
, Zhou
, D.
, Bao
, Y.
, Xu
, Y.
, and Han
, Z.
, “ Dynamic mode decomposition based analysis of flow past a transversely oscillating cylinder
,” Phys. Fluids
33
, 033604
(2021
).94.
95.
Proctor
, J. L.
, Brunton
, S. L.
, and Kutz
, J. N.
, “ Dynamic mode decomposition with control
,” SIAM J. Appl. Dyn. Syst.
15
, 142
–161
(2016
).96.
Rahmani
, M.
and Redkar
, S.
, “ Fractional robust data-driven control of nonlinear MEMS gyroscope
,” Nonlinear Dyn.
111
, 19901
–19910
(2023
).97.
Rowley
, C. W.
and Dawson
, S. T. M.
, “ Model reduction for flow analysis and control
,” Annu. Rev. Fluid Mech.
49
, 387
–417
(2017
).98.
Rowley
, C. W.
, Mezić
, I.
, Bagheri
, S.
, Schlatter
, P.
, and Henningson
, D. S.
, “ Spectral analysis of nonlinear flows
,” J. Fluid Mech.
641
, 115
–127
(2009
).99.
Sarpkaya
, T.
, “ A critical review of the intrinsic nature of vortex-induced vibrations
,” J. Fluids Struct.
19
, 389
–447
(2004
).100.
Sarpkaya
, T.
, “ Vortex-induced oscillations: A selective review
,” J. Appl. Mech.
46
, 241
–258
(1979
).101.
Savoeurn
, N.
, Janya-Anurak
, C.
, and Uthaisangsuk
, V.
, “ Determination of dynamic characteristics of lattice structure using dynamic mode decomposition
,” J. Appl. Mech.
91
, 071003
(2024
).102.
Sayadi
, T.
, Schmid
, P. J.
, Richecoeur
, F.
, and Durox
, D.
, “ Parametrized data-driven decomposition for bifurcation analysis, with application to thermo-acoustically unstable systems
,” Phys. Fluids
27
, 37102
(2015
).103.
Schmid
, P. J.
, “ Dynamic mode decomposition and its variants
,” Annu. Rev. Fluid Mech.
54
, 225
–254
(2022
).104.
Schmid
, P. J.
, “ Dynamic mode decomposition of numerical and experimental data
,” J. Fluid Mech.
656
, 5
–28
(2010
).105.
Schmid
, P. J.
, Li
, L.
, Juniper
, M. P.
, and Pust
, O.
, “ Applications of the dynamic mode decomposition
,” Theor. Comput. Fluid Dyn.
25
, 249
–259
(2011
).106.
Schmid
, P. J.
and Sesterhenn
, J.
, “ Dynamic mode decomposition of numerical and experimental data
,” in 61st APS Meeting American Physical Society, San Antonio, Texas
(APS, 2008
), p. 208
.107.
Seenivasaharagavan
, G. S.
, Korda
, M.
, Arbabi
, H.
, and Mezić
, I.
, “ Mean subtraction and mode selection in dynamic mode decomposition
,” arXiv:2105.03607 (2021
).108.
Soleimani
, M.
, Irani
, F. N.
, Yadegar
, M.
, and Davoodi
, M.
, “ Multi-objective optimization of building HVAC operation: Advanced strategy using Koopman predictive control and deep learning
,” Build. Environ.
248
, 111073
(2024
).109.
Song
, H.
, Ba
, Y.
, Chen
, D.
, and Li
, Q.
, “ A model reduction method for parametric dynamical systems defined on complex geometries
,” J. Comput. Phys.
506
, 112923
(2024
).110.
Stankiewicz
, W.
, “ Recursive Dynamic Mode Decomposition for the flow around two square cylinders in tandem configuration
,” J. Fluids Struct.
110
, 103515
(2022
).111.
Taira
, K.
, Brunton
, S. L.
, Dawson
, S. T. M.
, Rowley
, C. W.
, Colonius
, T.
, McKeon
, B. J.
, Schmidt
, O. T.
, Gordeyev
, S.
, Theofilis
, V.
, and Ukeiley
, L. S.
, “ Modal analysis of fluid flows: An overview
,” AIAA J.
55
, 4013
–4041
(2017
).112.
Tu
, J. H.
, Rowley
, C. W.
, Luchtenburg
, D. M.
, Brunton
, S. L.
, and Kutz
, J. N.
, “ On dynamic mode decomposition: Theory and applications
,” J. Comput. Dyn.
1
, 391
–421
(2014
).113.
Unal
, M. F.
and Rockwell
, D.
, “ On vortex formation from a cylinder. Part 1. The initial instability
,” J. Fluid Mech.
190
, 491
–512
(1988
).114.
Wahba
, N.
, Rismanchi
, B.
, Pu
, Y.
, and Aye
, L.
, “ Efficient HVAC system identification using Koopman operator and machine learning for thermal comfort optimisation
,” Build. Environ.
242
, 110567
(2023
).115.
Wang
, J.
, Xu
, B.
, Lai
, J.
, Wang
, Y.
, Hu
, C.
, Li
, H.
, and Song
, A.
, “ An improved Koopman-MPC framework for data-driven modeling and control of soft actuators
,” IEEE Rob. Autom. Lett.
8
, 616
–623
(2023
).116.
Wang
, M.
, Lou
, X.
, Wu
, W.
, and Cui
, B.
, “ Koopman-based MPC with learned dynamics: Hierarchical neural network approach
,” IEEE Trans. Neural Networks Learn. Syst.
35
, 3630
–3639
(2024
).117.
Williams
, M. O.
, Kevrekidis
, I. G.
, and Rowley
, C. W.
, “ A data–driven approximation of the Koopman operator: Extending dynamic mode decomposition
,” J. Nonlinear Sci.
25
, 1307
–1346
(2015
).118.
Wu
, J.
, Sheridan
, J.
, Hourigan
, K.
, and Soria
, J.
, “ Shear layer vortices and longitudinal vortices in the near wake of a circular cylinder
,” Exp. Therm. Fluid Sci.
12
, 169
–174
(1996
).119.
Wu
, W.
, Meneveau
, C.
, and Mittal
, R.
, “ Spatio-temporal dynamics of turbulent separation bubbles
,” J. Fluid Mech.
883
, A45
(2020
).120.
Wu
, Z.
, Laurence
, D.
, Utyuzhnikov
, S.
, and Afgan
, I.
, “ Proper orthogonal decomposition and dynamic mode decomposition of jet in channel crossflow
,” Nucl. Eng. Des.
344
, 54
–68
(2019
).121.
Yeomans
, J. M.
, “ Fluid flows on many scales
,” Nat. Phys.
17
, 756
(2021
).122.
Zhang
, H.
, Xin
, D.
, and Ou
, J.
, “ Wake control using spanwise-varying vortex generators on bridge decks: A computational study
,” J. Wind Eng. Ind. Aerodyn.
184
, 185
–197
(2019
).123.
Zhang
, Q.
, Liu
, Y.
, and Wang
, S.
, “ The identification of coherent structures using proper orthogonal decomposition and dynamic mode decomposition
,” J. Fluids Struct.
49
, 53
–72
(2014
).124.
Zhang
, X.
, Weerasuriya
, A. U.
, Wang
, J.
, Li
, C. Y.
, Chen
, Z.
, Tse
, K. T.
, and Hang
, J.
, “ Cross-ventilation of a generic building with various configurations of external and internal openings
,” Build. Environ.
207
, 108447
(2022
).125.
Zhang
, X.
, Weerasuriya
, A. U.
, Zhang
, X.
, Tse
, K. T.
, Lu
, B.
, Li
, C. Y.
, and Liu
, C.-H.
, “ Pedestrian wind comfort near a super-tall building with various configurations in an urban-like setting
,” Build. Simul.
13
, 1385
–1408
(2020
).126.
Zhao
, J.
, Leontini
, J. S.
, Lo Jacono
, D.
, and Sheridan
, J.
, “ Fluid-structure interaction of a square cylinder at different angles of attack
,” J. Fluid Mech.
747
, 688
–721
(2014
).127.
Zhao
, M.
and Jiang
, L.
, “ Data-driven probability density forecast for stochastic dynamical systems
,” J. Comput. Phys.
492
, 112422
(2023
).128.
Zhou
, L.
, Tse
, K. T.
, Hu
, G.
, and Li
, C. Y.
, “ Mode interpretation of interference effects between tall buildings in tandem and side-by-side arrangement with POD and ICA
,” Eng. Struct.
243
, 112616
(2021a
).129.
Zhou
, L.
, Tse
, K. T.
, Hu
, G.
, and Li
, Y.
, “ Higher order dynamic mode decomposition of wind pressures on square buildings
,” J. Wind Eng. Ind. Aerodyn.
211
, 104545
(2021b
).© 2024 Author(s). Published under an exclusive license by AIP Publishing.
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