Publications on SSM-Theory
Contents
2024
SSMTool 2.5 allows for extracting forced response surfaces.
- M. Li, S. Jain & G. Haller, Fast computation and characterization of forced response surfaces via spectral submanifolds and parameter continuation Nonlinear Dyn 112 (2024) 7771–7797
- G. Haller, R. S. Kaundinya, Nonlinear model reduction to temporally aperiodic spectral submanifolds Chaos 34 (2024) 043152.
- Z. Xu, R. S. Kaundinya, S. Jain & G. Haller, Nonlinear model reduction to random spectral submanifolds in random vibrations arXiv:2407.03677 (2024)
Non-intrusive computation of SSMs
- M. Li, T. Thurnher, Z. Xu & S. Jain, Data-free Non-intrusive Model Reduction for Nonlinear Finite Element Models via Spectral Submanifolds https://arxiv.org/abs/2409.10126 (2024).
2023
SSMTool2.3, SSM theory is applied in for the analysis of dynamical systems with algebraic constraints.
- M. Li, S. Jain & G. Haller, Model reduction for constrained mechanical systems via spectral submanifolds. Nonlinear Dyn 111 (2023) 8881–8911.
SSM Theory is expanded to mixed-mode manifolds and expansions are developed for so called fractional spectral submanifolds. Results are mainly presented for the data driven setting but theoretical developments are valid for the analytical approach for computing SSMs as well.
- G. Haller, B. Kaszás, A. Liu & J. Axås, Nonlinear model reduction to fractional and mixed-mode spectral submanifolds. Chaos 33 (2023) 063138
SSMTool2.4, SSMs are used for the anlysis of systems subject to parametric excitation. An automated algorithm for the computation of the non-autonomous SSMs is developed and employed to obtain stability diagrams and FRCs directly from the reduced dynamics for such systems.
- T. Thurnher, G. Haller & S. Jain, Nonautonomous Spectral Submanifolds for Model Reduction of Nonlinear Mechanical Systems under Parametric Resonance Chaos 34 (2024) 073127
2022
SSMTool2.2, Systems with internal resonances which result in invariant tori are analysed in the following publications:
- M. Li, S. Jain & G. Haller, Nonlinear analysis of forced mechanical systems with internal resonance using spectral submanifolds, Part I: Periodic response and forced response curve. Nonlinear Dyn 110 (2022) 1005–1043
- M. Li & G. Haller, Nonlinear analysis of forced mechanical systems with internal resonance using spectral submanifolds, Part II: Bifurcation and quasi-periodic response. Nonlinear Dyn 110 (2022) 1045–1080
2021
SSMs are used for predicting an appropriate modal basis for creating projective ROMs. The curvature of the SSMs, which is and indicator for the strength of nonlinear couplings, is advocate to make a suitable modal selection.
- G. Buza, S. Jain & G. Haller, Using spectral submanifolds for optimal mode selection in model reduction. Proc. R. Soc. A 477 (2021) 20200725.
SSMTool 2.1, SSMs are now computed directly on physical coordinates, no modal transformation is required. This allows for the treatment of large scale dynamical systems (200k DOFs). Exact ROMs on the SSM are automatically computed and the capabilities of the toolbox are showcased with several peer reviewed examples.
- S. Jain & G. Haller, How to compute invariant manifolds and their reduced dynamics in high-dimensional finite-element models?. Nonlinear Dyn (2021).
2020
Improvements to the algorithmic implementation allow the treatment of larger dynamical systems with up to 10k DOFs. First exhibition of parallel computation for the FRC extraction over the desired frequency ranges.
- S. Ponsioen, S. Jain & G. Haller, Model reduction to spectral submanifolds and forced-response calculation in high-dimensional mechanical systems. Journal of Sound and Vibration 488 (2020) 115640.
2019
Analytic expressions for the extraction of forced response curves in 2D SSMs that allow for the direct extraction of isolated regions of forced response. Integration of these expressions with SSMTool.
- S. Ponsioen, T. Pedergnana & G. Haller, Analytic prediction of isolated forced response curves from spectral submanifolds. Nonlinear Dynamics 98, 4 (2019) 2755-2773
2018
SSMTool: an automated toolbox for the computation of autonomous SSMs on the modal space, the extraction of backbone curves and leading order approximations to forced response curves.
- S. Ponsioen, T. Pedergnana & G. Haller, Automated computation of autonomous spectral submanifolds for nonlinear modal analysis. J. Sound Vib. 420 (2018) 269-295
SSMs are applied in combination with an initial slow-fast decomposition for the analysis of a finite element van Karman beam. An initial slow manifold is computed, further reduction to an SSM is consequently performed to obtain an exact reduced order model for the transverse dynamics of the beam.
- S. Jain, P. Tiso & G. Haller, Exact nonlinear model reduction for a von Karman beam: Slow-fast decomposition and spectral submanifolds. J. Sound Vib. 423 (2018) 195–211.
Development of analytical expressions for the analysis of FRCs and backbone curves in forced damped systems using SSM computations in modal coordinates.
- T. Breunung & G. Haller, Explicit backbone curves from spectral submanifolds of forced-damped nonlinear mechanical systems. Proc. R. Soc. A 474 (2018) 20180083.
2017
SSMs are computed for the extraction of backbone curves directly from the reduced order model provided by the autonomous reduced dynamics:
- R. Szalai, D. Ehrhardt & G. Haller, Nonlinear model identification and spectral submanifolds for multi-degree-of-freedom mechanical vibrations. Proc. Royal Soc. A 473 (2017) 20160759
Fundamentals
The foundations of SSM-Theory for autonomous and non-autonomous dynamical systems is established in
- G. Haller & S. Ponsioen, Nonlinear normal modes and spectral submanifolds: existence, uniqueness and use in model reduction. Nonlinear Dynamics 86 (2016) 1493-1534.
Data driven SSM Computation
SSMs can also be learned from trajectory data, rather than from analytical models. An open software toolbox for SSM computations from data is provided under SSMLearn.
The first publication which showcased how the behaviour of dynamical systems can be predicted from ROMs that are obtained from SSMs that have been learned from trajectory data is
- M. Cenedese, J. Axås, B. Bäuerlein, K. Avila & G. Haller, Data-driven modeling and prediction of non-linearizable dynamics via spectral submanifolds. Nat. Commun. 13 (2022) 872.
Further publications on data-driven SSMs are
- G. Haller, S. Jain & M. Cenedese Dynamics-based machine learning for nonlinearizable phenomena. Data-driven reduced models on spectral submanifolds. SIAM News 55/5 (2022) 1-4.
- M. Cenedese, J. Axås, H. Yang, M. Eriten & G. Haller, Data-driven nonlinear model reduction to spectral submanifolds in mechanical systems. Phil. Trans. R. Soc. A 380 (2022) 20210194 .
- J. Axås, M. Cenedese & G. Haller, <http://www.georgehaller.com/reprints/fastdatadriven_2023.pdf> Fast data-driven model reduction for nonlinear dynamical systems>. Nonlinear Dyn 111 (2023) 7941–7957.
- J. Axås & G. Haller, Model reduction for nonlinearizable dynamics via delay-embedded spectral submanifolds. Nonlinear Dyn (2023). (Published online)
- F. Mahlknecht, J.I. Alora, S. Jain, E. Schmerling, R. Bonalli, G. Haller & M. Pavone, Using spectral submanifolds for nonlinear periodic control arXiv:2209.06573 (2022).
- J.I. Alora, M. Cenedese, E. Schmerling, G. Haller & M. Pavone, Data-driven spectral submanifold reduction for nonlinear optimal control of high-dimensional robots. arXiv:2209.0571 (2022).
- J. I. Alora, M. Cenedese, E. Schmerling, G. Haller & M. Pavone, Practical Deployment of Spectral Submanifold Reduction for Optimal Control of High-Dimensional Systems. IFAC World Congress (2023). (Submitted)
- A. Liu, J. Axås & G. Haller, Data-driven modeling and forecasting of chaotic dynamics on inertial manifolds constructed as spectral submanifolds Chaos 34 (2024) 033140. [PDF]
- L. Bettini, M. Cenedese & G. Haller, Model reduction to spectral submanifolds in piecewise smooth dynamical systems International Journal of Non-Linear Mechanics 163 (2024) 1047
- A. Yang, J. Axås, F. Kádár, G. Stépán & G. Haller, Modeling nonlinear dynamics from videos arXiv:2406.08893 (2024)