This study proposes a new physics-constrained data-driven approach for sensitivity analysis and uncertainty quantification of large-scale chaotic Partial Differential Equations (PDEs). Unlike conventional sensitivity analysis, the proposed approach can manipulate the unsteady sensitivity function (i.e., tangent) for PDE-constrained optimizations. In this new approach, high-dimensional governing equations from physical space are transformed into an unphysical space (i.e., Hilbert space) to develop a closure model in the form of a Reduced-Order Model (ROM). This closure model is derived explicitly from the governing equations to set strong constraints on manifolds in Hilbert space. Afterward, a new data sampling method is proposed to build a data-driven approach for this framework. A series of least squares minimizations are set in the form of a novel auto-encoder system to solve this closure model. To compute sensitivities, least-squares shadowing minimization is applied to the ROM. It is shown that the proposed approach can capture sensitivities for large-scale chaotic dynamical systems, where finite difference approximations fail.
References
1.
F. D.
Witherden
, B. C.
Vermeire
, and P. E.
Vincent
, “Heterogeneous computing on mixed unstructured grids with PyFR
,” Comput. Fluids
120
, 173
–186
(2015
).2.
J.
Park
, F.
Witherden
, and P.
Vincent
, “High-order implicit large-eddy simulations of flow over a NACA 0021 aerofoil
,” AIAA J.
55
(7
), 2186
–2197
(2017
).3.
M.-F.
Tang
, G.-H.
Dong
, T.-J.
Xu
, C.
Wei Bi
, and S.
Wang
, “Large-eddy simulations of flow past cruciform circular cylinders in subcritical Reynolds numbers
,” Ocean Eng.
220
, 108484
(2021
).4.
G.
Zhu
, S.
Huang
, and Q.
Li
, “Large-eddy simulation of the inflow turbulence transport and aerodynamics of a rectangular 5:1 cylinder with high-order numerical methods
,” J. Wind Eng. Ind. Aerodyn.
207
, 104366
(2020
).5.
M.
Bovand
, S.
Rashidi
, M.
Dehghan
, J. A.
Esfahani
, and M.
Valipour
, “Control of wake and vortex shedding behind a porous circular obstacle by exerting an external magnetic field
,” J. Magn. Magn. Mater.
385
, 198
–206
(2015
).6.
H. R.
Karbasian
and K. C.
Kim
, “Numerical investigations on flow structure and behavior of vortices in the dynamic stall of an oscillating pitching hydrofoil
,” Ocean Eng.
127
, 200
–211
(2016
).7.
X.
Yang
, G.
He
, and X.
Zhang
, “Large-eddy simulation of flows past a flapping airfoil using immersed boundary method
,” Sci. China Phys., Mech. Astron.
53
(6
), 1101
–1108
(2010
).8.
P.
Benard
, A.
Vir
, V.
Moureau
, G.
Lartigue
, L.
Beaudet
, P.
Deglaire
, and L.
Bricteux
, “Large-Eddy simulation of wind turbines wakes including geometrical effects
,” Comput. Fluids
173
, 133
–139
(2018
).9.
J.
Esfahani
, E.
Barati
, and H.
Karbasian
, “Fluid structures of flapping airfoil with elliptical motion trajectory
,” Comput. Fluids
108
, 142
–155
(2015
).10.
H.
Karbasian
and J.
Esfahani
, “Enhancement of propulsive performance of flapping foil by fish-like motion pattern
,” Comput. Fluids
156
, 305
–316
(2017
).11.
H. R.
Karbasian
, J.
Esfahani
, and E.
Barati
, “Simulation of power extraction from tidal currents by flapping foil hydrokinetic turbines in tandem formation
,” Renewable Energy
81
, 816
–824
(2015
).12.
H.
Nakayama
, T.
Takemi
, and H.
Nagai
, “Large-eddy simulation of urban boundary-layer flows by generating turbulent inflows from mesoscale meteorological simulations
,” Atmos. Sci. Lett.
13
(3
), 180
–186
(2012
).13.
S. M.
Moreton
, D.
Ferreira
, M. J.
Roberts
, and H. T.
Hewitt
, “Evaluating surface eddy properties in coupled climate simulations with ‘eddy-present’and ‘eddy-rich’ ocean resolution
,” Ocean Modell.
147
, 101567
(2020
).14.
O.
Temel
, C. B.
Senel
, S.
Porchetta
, D.
Muñoz-Esparza
, M. A.
Mischna
, T.
Van Hoolst
, J.
van Beeck
, and Ö.
Karatekin
, “Large eddy simulations of the Martian convective boundary layer: Towards developing a new planetary boundary layer scheme
,” Atmos. Res.
250
, 105381
(2021
).15.
A. N.
Kolmogorov
, V.
Levin
, J. C. R.
Hunt
, O. M.
Phillips
, and D.
Williams
, “The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers
,” Proc. Roy. Soc. London, Ser. A
434
(1890
), 9
–13
(1991
).16.
A.
Jameson
, “Aerodynamic design via control theory
,” J. Sci. Comput.
3
(3
), 233
–260
(1988
).17.
T. R.
Bewley
, “Flow control: New challenges for a new renaissance
,” Prog. Aerosp. Sci.
37
(1
), 21
–58
(2001
).18.
T. R.
Bewley
, P.
Moin
, and R.
Temam
, “DNS-based predictive control of turbulence: An optimal benchmark for feedback algorithms
,” J. Fluid Mech.
447
(2
), 179
–225
(2001
).19.
B.
Mohseni-Gharyehsafa
, J. A.
Esfahani
, K. C.
Kim
, and H.
Ouerdane
, “Soft computing analysis of thermohydraulic enhancement using twisted tapes in a flat-plate solar collector: Sensitivity analysis and multi–objective optimization
,” J. Cleaner Prod.
314
, 127947
(2021
).20.
C. J.
Willmott
, C. M.
Rowe
, and W. D.
Philpot
, “Small-scale climate maps: A sensitivity analysis of some common assumptions associated with grid-point interpolation and contouring
,” Am. Cartogr.
12
(1
), 5
–16
(1985
).21.
D. S.
Cohan
, A.
Hakami
, Y.
Hu
, and A. G.
Russell
, “Nonlinear response of ozone to emissions: Source apportionment and sensitivity analysis
,” Environ. Sci. Technol.
39
(17
), 6739
–6748
(2005
).22.
L.
Hannah
, G. F.
Midgley
, and D.
Millar
, “Climate change-integrated conservation strategies
,” Global Ecol. Biogeogr.
11
(6
), 485
–495
(2002
).23.
D. J.
Lea
, M. R.
Allen
, and T. W.
Haine
, “Sensitivity analysis of the climate of a chaotic system
,” Tellus A
52
(5
), 523
–532
(2000
).24.
G.
Eyink
, T.
Haine
, and D.
Lea
, “Ruelle's linear response formula, ensemble adjoint schemes and Lévy flights
,” Nonlinearity
17
(5
), 1867
(2004
).25.
P. J.
Blonigan
, S. A.
Gomez
, and Q.
Wang
, “Least squares shadowing for sensitivity analysis of turbulent fluid flows
,” in 52nd Aerospace Sciences Meeting
(AIAA
, 2014
), p. 1426
.26.
J. R.
Martins
, J. J.
Alonso
, and J. J.
Reuther
, “A coupled-adjoint sensitivity analysis method for high-fidelity aero-structural design
,” Optim. Eng.
6
(1
), 33
–62
(2005
).27.
Y.
Shi
, C. A.
Mader
, S.
He
, G. L.
Halila
, and J. R.
Martins
, “Natural laminar-flow airfoil optimization design using a discrete adjoint approach
,” AIAA J.
58
(11
), 4702
–4722
(2020
).28.
D. W.
Zingg
, M.
Nemec
, and T. H.
Pulliam
, “A comparative evaluation of genetic and gradient-based algorithms applied to aerodynamic optimization
,” Eur. J. Comput. Mech.
17
(1–2
), 103
–126
(2008
).29.
B. C.
Vermeire
and P. E.
Vincent
, “On the properties of energy stable flux reconstruction schemes for implicit large eddy simulation
,” J. Comput. Phys.
327
, 368
–388
(2016
).30.
B. C.
Vermeire
, F. D.
Witherden
, and P. E.
Vincent
, “On the utility of GPU accelerated high-order methods for unsteady flow simulations: A comparison with industry-standard tools
,” J. Comput. Phys.
334
, 497
–521
(2017
).31.
P. E.
Vincent
, P.
Castonguay
, and A.
Jameson
, “A new class of high-order energy stable flux reconstruction schemes
,” J. Sci. Comput.
47
(1
), 50
–72
(2011
).32.
W.
Trojak
, “Numerical analysis of flux reconstruction
,” Ph.D. thesis (University of Cambridge
, 2020
).33.
P. J.
Blonigan
, P.
Fernandez
, S. M.
Murman
, Q.
Wang
, G.
Rigas
, and L.
Magri
, “Toward a chaotic adjoint for LES
,” preprint arXiv:1702.06809 (2017
).34.
L.
Keefe
, P.
Moin
, and J.
Kim
, “The dimension of attractors underlying periodic turbulent Poiseuille flow
,” J. Fluid Mech.
242
, 1
–29
(1992
).35.
T.
Léonard
, L. Y.
Gicquel
, N.
Gourdain
, and F.
Duchaine
, “Steady/unsteady Reynolds-averaged Navier–Stokes and large eddy simulations of a turbine blade at high subsonic outlet Mach number
,” J. Turbomach.
137
(4
), 041001
(2015
).36.
H. R.
Karbasian
and B. C.
Vermeire
, “Gradient-free aerodynamic shape optimization using large eddy simulation
,” Comput. Fluids
232
, 105185
(2022
).37.
A.
Katok
and B.
Hasselblatt
, Introduction to the Modern Theory of Dynamical Systems
(Cambridge University Press
, 1997
), Vol. 54
.38.
Q.
Wang
, R.
Hu
, and P.
Blonigan
, “Least squares shadowing sensitivity analysis of chaotic limit cycle oscillations
,” J. Comput. Phys.
267
, 210
–224
(2014
).39.
P. J.
Blonigan
, Q.
Wang
, E. J.
Nielsen
, and B.
Diskin
, “Least-squares shadowing sensitivity analysis of chaotic flow around a two-dimensional airfoil
,” AIAA J.
56
(2
), 658
–672
(2018
).40.
P. J.
Blonigan
and Q.
Wang
, “Multiple shooting shadowing for sensitivity analysis of chaotic dynamical systems
,” J. Comput. Phys.
354
, 447
–475
(2018
).41.
A.
Ni
and Q.
Wang
, “Sensitivity analysis on chaotic dynamical systems by non-intrusive least squares shadowing (NILSS)
,” J. Comput. Phys.
347
, 56
–77
(2017
).42.
A.
Ni
, Q.
Wang
, P.
Fernandez
, and C.
Talnikar
, “Sensitivity analysis on chaotic dynamical systems by finite difference non-intrusive least squares shadowing (FD-NILSS)
,” J. Comput. Phys.
394
, 615
–631
(2019
).43.
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
(12
), 4013
–4041
(2017
).44.
K.
Taira
, M. S.
Hemati
, S. L.
Brunton
, Y.
Sun
, K.
Duraisamy
, S.
Bagheri
, S. T.
Dawson
, and C.-A.
Yeh
, “Modal analysis of fluid flows: Applications and outlook
,” AIAA J.
58
(3
), 998
–1022
(2020
).45.
G.
Rozza
and A.
Manzoni
, “Model order reduction by geometrical parametrization for shape optimization in computational fluid dynamics
,” in Proceedings of the ECCOMAS CFD 2010, V European Conference on Computational Fluid Dynamics
(ECCOMAS
, 2010
), pp. 1
–20
.46.
M. J.
Zahr
and C.
Farhat
, “Progressive construction of a parametric reduced-order model for PDE-constrained optimization
,” Int. J. Numer. Methods Eng.
102
(5
), 1111
–1135
(2015
).47.
M.
Bergmann
and L.
Cordier
, “Optimal control of the cylinder wake in the laminar regime by trust-region methods and POD reduced-order models
,” J. Comput. Phys.
227
(16
), 7813
–7840
(2008
).48.
K.
Kunisch
and S.
Volkwein
, “Proper orthogonal decomposition for optimality systems
,” ESAIM: Math. Modell. Numer. Anal.
42
(1
), 1
–23
(2008
).49.
T.
Bui-Thanh
, K.
Willcox
, and O.
Ghattas
, “Parametric reduced-order models for probabilistic analysis of unsteady aerodynamic applications
,” AIAA J.
46
(10
), 2520
–2529
(2008
).50.
M. J.
Zahr
and P.-O.
Persson
, “An adjoint method for a high-order discretization of deforming domain conservation laws for optimization of flow problems
,” J. Comput. Phys.
326
, 516
–543
(2016
).51.
K.
Carlberg
, C.
Bou-Mosleh
, and C.
Farhat
, “Efficient non-linear model reduction via a least-squares Petrov–Galerkin projection and compressive tensor approximations
,” Int. J. Numer. Methods Eng.
86
(2
), 155
–181
(2011
).52.
K.
Carlberg
, M.
Barone
, and H.
Antil
, “Galerkin v. least-squares Petrov–Galerkin projection in nonlinear model reduction
,” J. Comput. Phys.
330
, 693
–734
(2017
).53.
E. J.
Parish
, C.
Wentland
, and K.
Duraisamy
, “The adjoint Petrov-Galerkin method for non-linear model reduction
,” arXiv:1810.03455 (2018
).54.
H. F.
Lui
and W. R.
Wolf
, “Construction of reduced order models for fluid flows using deep feedforward neural networks
,” preprint arXiv:1903.05206 (2019
).55.
S. L.
Brunton
and J. N.
Kutz
, Data-Driven Science and Engineering: Machine Learning, Dynamical Systems, and Control
(Cambridge University Press
, 2019
).56.
S. L.
Brunton
, B. R.
Noack
, and P.
Koumoutsakos
, “Machine learning for fluid mechanics
,” Annu. Rev. Fluid Mech.
52
, 477
–508
(2020
).57.
T.
Murata
, K.
Fukami
, and K.
Fukagata
, “Nonlinear mode decomposition with convolutional neural networks for fluid dynamics
,” J. Fluid Mech.
882
, A13
(2020
).58.
B.
Epureanu
, “A parametric analysis of reduced order models of viscous flows in turbomachinery
,” J. Fluids Struct.
17
(7
), 971
–982
(2003
).59.
A.
Hay
, J. T.
Borggaard
, and D.
Pelletier
, “Local improvements to reduced-order models using sensitivity analysis of the proper orthogonal decomposition
,” J. Fluid Mech.
629
, 41
–72
(2009
).60.
D.
Amsallem
and C.
Farhat
, “Interpolation method for adapting reduced-order models and application to aeroelasticity
,” AIAA J.
46
(7
), 1803
–1813
(2008
).61.
M.
Hairer
and J. C.
Mattingly
, “Ergodicity of the 2D Navier–Stokes equations with degenerate stochastic forcing
,” Ann. Math.
164
, 993
–1032
(2006
).62.
F.
Flandoli
and B.
Maslowski
, “Ergodicity of the 2-D Navier–Stokes equation under random perturbations
,” Commun. Math. Phys.
172
(1
), 119
–141
(1995
).63.
F.
Ginelli
, P.
Poggi
, A.
Turchi
, H.
Chaté
, R.
Livi
, and A.
Politi
, “Characterizing dynamics with covariant Lyapunov vectors
,” Phys. Rev. Lett.
99
(13
), 130601
(2007
).64.
V. I.
Oseledets
, “A multiplicative ergodic theorem: Characteristic Lyapunov exponents of dynamical systems
,” Trudy Mosk. Mat. O-va.
19
, 179
–210
(1968
).65.
P.
Fernandez
and Q.
Wang
, “Lyapunov spectrum of the separated flow around the NACA 0012 airfoil and its dependence on numerical discretization
,” J. Comput. Phys.
350
, 453
–469
(2017
).66.
D. A.
Nelson
and G. B.
Jacobs
, “High-order visualization of three-dimensional Lagrangian coherent structures with DG-FTLE
,” Comput. Fluids
139
, 197
–215
(2016
).67.
M.
Sudharsan
, S. L.
Brunton
, and J. J.
Riley
, “Lagrangian coherent structures and inertial particle dynamics
,” Phys. Rev. E
93
(3
), 033108
(2016
).68.
L.
Kourentis
and E.
Konstantinidis
, “Uncovering large-scale coherent structures in natural and forced turbulent wakes by combining PIV, POD, and FTLE
,” Exp. Fluids
52
(3
), 749
–763
(2012
).69.
G.
Benettin
, L.
Galgani
, A.
Giorgilli
, and J.-M.
Strelcyn
, “Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems; a method for computing all of them. Part 1: Theory
,” Meccanica
15
(1
), 9
–20
(1980
).70.
P. J.
Blonigan
, “Least squares shadowing for sensitivity analysis of large chaotic systems and fluid flows
,” Ph.D. thesis (Massachusetts Institute of Technology
, 2016
).71.
D.
Huynh
and A.
Patera
, “Reduced basis approximation and a posteriori error estimation for stress intensity factors
,” Int. J. Numer. Methods Eng.
72
(10
), 1219
–1259
(2007
).72.
K.
Carlberg
and C.
Farhat
, “A compact proper orthogonal decomposition basis for optimization-oriented reduced-order models
,” in 12th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference
(AIAA
, 2008
), p. 5964
.73.
C. A.
Pereira
and B. C.
Vermeire
, “Spectral properties of high-order element types for implicit large eddy simulation
,” J. Sci. Comput.
85
(2
), 48
(2020
).74.
H. T.
Huynh
, “A flux reconstruction approach to high-order schemes including discontinuous galerkin methods
,” in 18th AIAA Computational Fluid Dynamics Conference
(AIAA
, 2007
), p. 4079
.75.
V.
Hernandez
, J. E.
Roman
, and V.
Vidal
, “SLEPc: A scalable and flexible toolkit for the solution of eigenvalue problems
,” ACM Trans. Math. Software
31
(3
), 351
–362
(2005
).76.
L. D.
Dalcin
, R. R.
Paz
, P. A.
Kler
, and A.
Cosimo
, “Parallel distributed computing using Python
,” Adv. Water Resources
34
(9
), 1124
–1139
(2011
).77.
S.
Balay
, S.
Abhyankar
, M. F.
Adams
, J.
Brown
, P.
Brune
, K.
Buschelman
, L.
Dalcin
, A.
Dener
, V.
Eijkhout
, W. D.
Gropp
, D.
Karpeyev
, D.
Kaushik
, M. G.
Knepley
, D. A.
May
, L. C.
McInnes
, R. T.
Mills
, T.
Munson
, K.
Rupp
, P.
Sanan
, B. F.
Smith
, S.
Zampini
, H.
Zhang
, and H.
Zhang
, “PETSc users manual
,” Technical Report No. ANL-95/11—Revision 3.14
(Argonne National Laboratory
, 2020
).78.
L.
Dalcín
, R.
Paz
, and M.
Storti
, “MPI for Python
,” J. Parallel Distributed Comput.
65
(9
), 1108
–1115
(2005
).79.
O.
Inoue
and N.
Hatakeyama
, “Sound generation by a two-dimensional circular cylinder in a uniform flow
,” J. Fluid Mech.
471
, 285
(2002
).80.
J.
Cagnone
, B. C.
Vermeire
, and S.
Nadarajah
, “A p-adaptive LCP formulation for the compressible Navier–Stokes equations
,” J. Comput. Phys.
233
, 324
–338
(2013
).81.
S.
Yarusevych
, P. E.
Sullivan
, and J. G.
Kawall
, “On vortex shedding from an airfoil in low-Reynolds-number flows
,” J. Fluid Mech.
632
, 245
(2009
).82.
R. F.
Huang
and C. L.
Lin
, “Vortex shedding and shear-layer instability of wing at low-Reynolds numbers
,” AIAA J.
33
(8
), 1398
–1403
(1995
).83.
B. C.
Vermeire
and S. H.
Nasab
, “Accelerated implicit-explicit Runge-Kutta schemes for locally stiff systems
,” J. Comput. Phys.
429
, 110022
(2021
).84.
M. E.
Rosti
, M.
Omidyeganeh
, and A.
Pinelli
, “Direct numerical simulation of the flow around an aerofoil in ramp-up motion
,” Phys. Fluids
28
(2
), 025106
(2016
).85.
See https://www.tensorflow.org/ for “
TensorFlow: Large-scale machine learning on heterogeneous systems
” (2015
).© 2022 Author(s). Published under an exclusive license by AIP Publishing.
2022
Author(s)
You do not currently have access to this content.
