https://orcid.org/0000-0001-9925-9167
Abstract
A sparsity-based optimization approach is presented for determining the equations of motion of stochastically excited nonlinear structural systems. This is done by utilizing measured excitation-response realizations in the formulation of the related optimization problem, and by considering a library of candidate functions for representing the system governing dynamics. Note that a novel aspect of the approach relates to treating, also, systems endowed with fractional derivative elements. Clearly, this is of significant importance to a multitude of diverse applications in engineering mechanics taking into account the enhanced modeling capabilities of fractional calculus. Further, the fundamental theoretical and computational aspects of various representative, state-of-the-art, numerical schemes for solving the derived sparsity-based optimization problem are reviewed and discussed. A Bayesian compressive sampling approach that exhibits the additional advantage of quantifying the uncertainty of the estimates is considered as well. Furthermore, comparisons and a critical assessment of the employed numerical schemes are provided with respect to their efficacy in determining the nonlinear structural system equations of motion. In this regard, two illustrative numerical examples are considered pertaining to a nonlinear tuned mass-damper–inerter vibration control system and to a nonlinear electromechanical energy harvester, both endowed with fractional derivative elements.
Issue Section:
Special Papers
References
1.
Lin
, Y.-K. M.
, 1967
, Probabilistic Theory of Structural Dynamics
, McGraw-Hill
, New York.2.
Li
, J.
, and Chen
, J.
, 2009
, Stochastic Dynamics of Structures
, Wiley
, John Wiley & Sons (Asia), Singapore.3.
Kougioumtzoglou
, I. A.
, Psaros
, A. F.
, and Spanos
, P. D.
, 2024
, Path Integrals in Stochastic Engineering Dynamics
, Springer
, New York.4.
Wang
, W.-X.
, Lai
, Y.-C.
, and Grebogi
, C.
, 2016
, “Data Based Identification and Prediction of Nonlinear and Complex Dynamical Systems
,” Phys. Rep.
, 644
, pp. 1
–76
.10.1016/j.physrep.2016.06.0045.
Karniadakis
, G. E.
, Kevrekidis
, I. G.
, Lu
, L.
, Perdikaris
, P.
, Wang
, S.
, and Yang
, L.
, 2021
, “Physics-Informed Machine Learning
,” Nat. Rev. Phys.
, 3
(6
), pp. 422
–440
.10.1038/s42254-021-00314-56.
Brunton
, S.
, and Kutz
, J. N.
, 2022
, Data-Driven Science and Engineering: Machine Learning, Dynamical Systems, and Control
, Cambridge University Press
, Cambridge, UK.7.
Ghadami
, A.
, and Epureanu
, B. I.
, 2022
, “Data-Driven Prediction in Dynamical Systems: Recent Developments
,” Philos. Trans. R. Soc. A
, 380
(2229
), p. 20210213
.10.1098/rsta.2021.02138.
North
, J. S.
, Wikle
, C. K.
, and Schliep
, E. M.
, 2023
, “A Review of Data-Driven Discovery for Dynamic Systems
,” Int. Stat. Rev.
, 91
(3
), pp. 464
–492
.10.1111/insr.125549.
Clough
, R.
, and Penzien
, J.
, 1975
, Dynamics of Structures
, McGraw-Hill NY
, New York.10.
Kougioumtzoglou
, I. A.
, Petromichelakis
, I.
, and Psaros
, A. F.
, 2020
, “Sparse Representations and Compressive Sampling Approaches in Engineering Mechanics: A Review of Theoretical Concepts and Diverse Applications
,” Probab. Eng. Mech.
, 61
, p. 103082
.10.1016/j.probengmech.2020.10308211.
Lai
, Z.
, and Nagarajaiah
, S.
, 2019
, “Sparse Structural System Identification Method for Nonlinear Dynamic Systems With Hysteresis/Inelastic Behavior
,” Mech. Syst. Signal Process.
, 117
, pp. 813
–842
.10.1016/j.ymssp.2018.08.03312.
Zhang
, Y.
, Duan
, J.
, Jin
, Y.
, and Li
, Y.
, 2021
, “Discovering Governing Equation From Data for Multi-Stable Energy Harvester Under White Noise
,” Nonlinear Dyn.
, 106
(4
), pp. 2829
–2840
.10.1007/s11071-021-06960-913.
Liu
, Q.
, Cao
, J.
, Zhang
, Y.
, Zhao
, Z.
, Kerschen
, G.
, and Jing
, X.
, 2023
, “Interpretable Sparse Identification of a Bistable Nonlinear Energy Sink
,” Mech. Syst. Signal Process.
, 193
, p. 110254
.10.1016/j.ymssp.2023.11025414.
Tripura
, T.
, and Chakraborty
, S.
, 2023
, “A Sparse Bayesian Framework for Discovering Interpretable Nonlinear Stochastic Dynamical Systems With Gaussian White Noise
,” Mech. Syst. Signal Process.
, 187
, p. 109939
.10.1016/j.ymssp.2022.10993915.
Tripura
, T.
, Desai
, A. S.
, Adhikari
, S.
, and Chakraborty
, S.
, 2023
, “Probabilistic Machine Learning Based Predictive and Interpretable Digital Twin for Dynamical Systems
,” Comput. Struct.
, 281
, p. 107008
.10.1016/j.compstruc.2023.10700816.
Nayek
, R.
, Fuentes
, R.
, Worden
, K.
, and Cross
, E. J.
, 2021
, “On Spike-and-Slab Priors for Bayesian Equation Discovery of Nonlinear Dynamical Systems Via Sparse Linear Regression
,” Mech. Syst. Signal Process.
, 161
, p. 107986
.10.1016/j.ymssp.2021.10798617.
Fuentes
, R.
, Nayek
, R.
, Gardner
, P.
, Dervilis
, N.
, Rogers
, T.
, Worden
, K.
, and Cross
, E.
, 2021
, “Equation Discovery for Nonlinear Dynamical Systems: A Bayesian Viewpoint
,” Mech. Syst. Signal Process.
, 154
, p. 107528
.10.1016/j.ymssp.2020.10752818.
Oldham
, K.
, and Spanier
, J.
, 1974
, The Fractional Calculus Theory and Applications of Differentiation and Integration to Arbitrary Order
, Elsevier
, New York .19.
Sabatier
, J.
, Agrawal
, O. P.
, and Machado
, J. T.
, 2007
, Advances in Fractional Calculus
, Vol. 4, Springer
, Dordrecht, UK.20.
Rossikhin
, Y. A.
, and Shitikova
, M. V.
, 2010
, “Application of Fractional Calculus for Dynamic Problems of Solid Mechanics: Novel Trends and Recent Results
,” ASME Appl. Mech. Rev.
, 63
(1
), p. 010801
.10.1115/1.400056321.
Di Paola
, M.
, Failla
, G.
, Pirrotta
, A.
, Sofi
, A.
, and Zingales
, M.
, 2013
, “The Mechanically Based Non-Local Elasticity: An Overview of Main Results and Future Challenges
,” Philos. Trans. R. Soc. A: Math., Phys. Eng. Sci.
, 371
(1993
), p. 20120433
.10.1098/rsta.2012.043322.
Tarasov
, V. E.
, 2017
, “Fractional Mechanics of Elastic Solids: Continuum Aspects
,” J. Eng. Mech.
, 143
(5
), p. D4016001
.10.1061/(ASCE)EM.1943-7889.000107423.
Di Paola
, M.
, Pirrotta
, A.
, and Valenza
, A.
, 2011
, “Visco-Elastic Behavior Through Fractional Calculus: An Easier Method for Best Fitting Experimental Results
,” Mech. Mater.
, 43
(12
), pp. 799
–806
.10.1016/j.mechmat.2011.08.01624.
Brunton
, S.
, Proctor
, J. L.
, and Kutz
, J. N.
, 2016
, “Discovering Governing Equations From Data by Sparse Identification of Nonlinear Dynamical Systems
,” Proc. Natl. Acad. Sci.
, 113
(15
), pp. 3932
–3937
.10.1073/pnas.151738411325.
Donoho
, D.
, 2006
, “Compressed Sensing
,” IEEE Trans. Inf. Theory
, 52
(4
), pp. 1289
–1306
.10.1109/TIT.2006.87158226.
Candès
, E. J.
, and Wakin
, M. B.
, 2008
, “An Introduction to Compressive Sampling
,” IEEE Signal Process. Mag.
, 25
(2
), pp. 21
–30
.10.1109/MSP.2007.91473127.
Champion
, K.
, Zheng
, P.
, Aravkin
, A. Y.
, Brunton
, S. L.
, and Kutz
, J. N.
, 2020
, “A Unified Sparse Optimization Framework to Learn Parsimonious Physics-Informed Models From Data
,” IEEE Access
, 8
, pp. 169259
–169271
.10.1109/ACCESS.2020.302362528.
Mangan
, N. M.
, Kutz
, J. N.
, Brunton
, S. L.
, and Proctor
, J. L.
, 2017
, “Model Selection for Dynamical Systems Via Sparse Regression and Information Criteria
,” Proc. R. Soc. A: Math., Phys. Eng. Sci.
, 473
(2204
), p. 20170009
.10.1098/rspa.2017.000929.
Ben-Israel
, A.
, and Greville
, T. N.
, 2006
, Generalized Inverses: Theory and Applications
, Springer Science & Business Media
, New York.30.
Zhang
, Y.
, Comerford
, L.
, Kougioumtzoglou
, I. A.
, and Beer
, M.
, 2018
, “Lp-Norm Minimization for Stochastic Process Power Spectrum Estimation Subject to Incomplete Data
,” Mech. Syst. Signal Process.
, 101
, pp. 361
–376
.10.1016/j.ymssp.2017.08.01731.
Zhang
, Y.
, Kougioumtzoglou
, I. A.
, and Kong
, F.
, 2022
, “Exploiting Expansion Basis Sparsity for Efficient Stochastic Response Determination of Nonlinear Systems Via the Wiener Path Integral Technique
,” Nonlinear Dyn.
, 107
(4
), pp. 3669
–3682
.10.1007/s11071-021-07153-032.
Champion
, K.
, Lusch
, B.
, Kutz
, J. N.
, and Brunton
, S. L.
, 2019
, “Data-Driven Discovery of Coordinates and Governing Equations
,” Proc. Natl. Acad. Sci.
, 116
(45
), pp. 22445
–22451
.10.1073/pnas.190699511633.
Cortiella
, A.
, Park
, K.-C.
, and Doostan
, A.
, 2021
, “Sparse Identification of Nonlinear Dynamical Systems Via Reweighted l1-Regularized Least Squares
,” Comput. Methods Appl. Mech. Eng.
, 376
, p. 113620
.10.1016/j.cma.2020.11362034.
Hokanson
, J. M.
, Iaccarino
, G.
, and Doostan
, A.
, 2023
, “Simultaneous Identification and Denoising of Dynamical Systems
,” SIAM J. Sci. Comput.
, 45
(4
), pp. A1413
–A1437
.10.1137/22M148630335.
Schaeffer
, H.
, Caflisch
, R.
, Hauck
, C. D.
, and Osher
, S.
, 2013
, “Sparse Dynamics for Partial Differential Equations
,” Proc. Natl. Acad. Sci.
, 110
(17
), pp. 6634
–6639
.10.1073/pnas.130275211036.
Eldar
, Y. C.
, and Kutyniok
, G.
, 2012
, Compressed Sensing: Theory and Applications
, Cambridge University Press
, Cambridge, UK.37.
Rish
, I.
, and Grabarnik
, G.
, 2014
, Sparse Modeling: Theory, Algorithms, and Applications
, CRC Press
, Boca Raton, FL.38.
Dumitrescu
, B.
, and Irofti
, P.
, 2018
, Dictionary Learning Algorithms and Applications
, Springer
, Berlin, Germany.39.
Pati
, Y. C.
, Rezaiifar
, R.
, and Krishnaprasad
, P. S.
, 1993
, “Orthogonal Matching Pursuit: Recursive Function Approximation With Applications to Wavelet Decomposition
,” Proceedings of 27th Asilomar Conference on Signals, Systems and Computers
, Pacific Grove, CA, Nov. 1–3
, pp. 40
–44
.10.1109/ACSSC.1993.34246540.
Adler
, J.
, Rao
, B. D.
, and Kreutz-Delgado
, K.
, 1996
, “Comparison of Basis Selection Methods
,” Conference Record of the 30th Asilomar Conference on Signals, Systems and Computers
, Vol. 1, Pacific Grove, CA, Nov. 3–6
, pp. 252
–257
.10.1109/ACSSC.1996.60086741.
Bloomfield
, P.
, and Steiger
, W. L.
, 1983
, Least Absolute Deviations: Theory, Applications, and Algorithms
, Springer
, Boston, MA.42.
Daubechies
, I.
, Defrise
, M.
, and De Mol
, C.
, 2004
, “An Iterative Thresholding Algorithm for Linear Inverse Problems With a Sparsity Constraint
,” Commun. Pure Appl. Math.: J. Issued Courant Inst. Math. Sci.
, 57
(11
), pp. 1413
–1457
.10.1002/cpa.2004243.
Candes
, E. J.
, Wakin
, M. B.
, and Boyd
, S. P.
, 2008
, “Enhancing Sparsity by Reweighted l1 Minimization
,” J. Fourier Anal. Appl.
, 14
(5–6
), pp. 877
–905
.10.1007/s00041-008-9045-x44.
Xu
, Z.
, Zhang
, H.
, Wang
, Y.
, Chang
, X.
, and Liang
, Y.
, 2010
, “L1/2 Regularization
,” Sci. Inf. Ser.
, 53
, pp. 1159
–1169
.45.
Rao
, B. D.
, and Kreutz-Delgado
, K.
, 1999
, “An Affine Scaling Methodology for Best Basis Selection
,” IEEE Trans. Signal Process.
, 47
(1
), pp. 187
–200
.10.1109/78.73825146.
Rao
, B. D.
, Engan
, K.
, Cotter
, S. F.
, Palmer
, J.
, and Kreutz-Delgado
, K.
, 2003
, “Subset Selection in Noise Based on Diversity Measure Minimization
,” IEEE Trans. Signal Process.
, 51
(3
), pp. 760
–770
.10.1109/TSP.2002.80807647.
Zhang
, S.
, and Lin
, G.
, 2018
, “Robust Data-Driven Discovery of Governing Physical Laws With Error Bars
,” Proc. R. Soc. A: Math., Phys. Eng. Sci.
, 474
(2217
), p. 20180305
.10.1098/rspa.2018.030548.
Yang
, Y.
, Aziz Bhouri
, M.
, and Perdikaris
, P.
, 2020
, “Bayesian Differential Programming for Robust Systems Identification Under Uncertainty
,” Proc. R. Soc. A
, 476
(2243
), p. 20200290
.10.1098/rspa.2020.029049.
Chatterjee
, T.
, Shaw
, A. D.
, Friswell
, M. I.
, and Khodaparast
, H. H.
, 2023
, “Sparse Bayesian Machine Learning for the Interpretable Identification of Nonlinear Structural Dynamics: Towards the Experimental Data-Driven Discovery of a Quasi Zero Stiffness Device
,” Mech. Syst. Signal Process.
, 205
, p. 110858
.10.1016/j.ymssp.2023.11085850.
Champneys
, M. D.
, and Rogers
, T. J.
, 2024
, “Bindy–Bayesian Identification of Nonlinear Dynamics With Reversible-Jump Markov-Chain Monte Carlo
,” preprint arXiv:2408.08062.51.
Ji
, S.
, Xue
, Y.
, and Carin
, L.
, 2008
, “Bayesian Compressive Sensing
,” IEEE Trans. Signal Process.
, 56
(6
), pp. 2346
–2356
.10.1109/TSP.2007.91434552.
MacKay
, D. J.
, 1992
, “Bayesian Interpolation
,” Neural Comput.
, 4
(3
), pp. 415
–447
.10.1162/neco.1992.4.3.41553.
Smith
, M. C.
, 2020
, “The Inerter: A Retrospective
,” Annu. Rev. Control, Rob., Auton. Syst.
, 3
(1
), pp. 361
–391
.10.1146/annurev-control-053018-02391754.
Wagg
, D. J.
, 2021
, “A Review of the Mechanical Inerter: Historical Context, Physical Realisations and Nonlinear Applications
,” Nonlinear Dyn.
, 104
(1
), pp. 13
–34
.10.1007/s11071-021-06303-855.
Marian
, L.
, and Giaralis
, A.
, 2014
, “Optimal Design of a Novel Tuned Mass-Damper–Inerter (TMDI) Passive Vibration Control Configuration for Stochastically Support-Excited Structural Systems
,” Probab. Eng. Mech.
, 38
, pp. 156
–164
.10.1016/j.probengmech.2014.03.00756.
Mavromatis
, I. G.
, and Kougioumtzoglou
, I. A.
, 2023
, “A Reduced-Order Wiener Path Integral Formalism for Determining the Stochastic Response of Nonlinear Systems With Fractional Derivative Elements
,” ASCE-ASME J. Risk Uncertainty Eng. Syst., Part B: Mech. Eng.
, 9
(3
), p. 031201
.10.1115/1.405690257.
Shinozuka
, M.
, and Deodatis
, G.
, 1991
, “Simulation of Stochastic Processes by Spectral Representation
,” ASME Appl. Mech. Rev.
, 44
(4
), pp. 191
–204
.10.1115/1.311950158.
Koh
, C. G.
, and Kelly
, J. M.
, 1990
, “Application of Fractional Derivatives to Seismic Analysis of Base-Isolated Models
,” Earthquake Eng. Struct. Dyn.
, 19
(2
), pp. 229
–241
.10.1002/eqe.429019020759.
Daqaq
, M. F.
, Masana
, R.
, Erturk
, A.
, and Dane Quinn
, D.
, 2014
, “On the Role of Nonlinearities in Vibratory Energy Harvesting: A Critical Review and Discussion
,” ASME Appl. Mech. Rev.
, 66
(4
), p. 040801
.10.1115/1.402627860.
Petromichelakis
, I.
, Psaros
, A. F.
, and Kougioumtzoglou
, I. A.
, 2018
, “Stochastic Response Determination and Optimization of a Class of Nonlinear Electromechanical Energy Harvesters: A Wiener Path Integral Approach
,” Probab. Eng. Mech.
, 53
, pp. 116
–125
.10.1016/j.probengmech.2018.06.00461.
Petromichelakis
, I.
, Psaros
, A. F.
, and Kougioumtzoglou
, I. A.
, 2021
, “Stochastic Response Analysis and Reliability-Based Design Optimization of Nonlinear Electromechanical Energy Harvesters With Fractional Derivative Elements
,” ASCE-ASME J. Risk Uncertainty Eng. Syst., Part B: Mech. Eng.
, 7
(1
), p. 010901
.10.1115/1.4049232Copyright © 2025 by ASME
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