SSMTool 2.5: Computation of invariant manifolds in high-dimensional mechanics problems

SSMTool 2.5: Computation of invariant manifolds in high-dimensional mechanics problems

Available here: https://zenodo.org/records/10018285

What's new in SSMTool 2.5

This package computes invariant manifolds in high-dimensional dynamical systems using the Parametrization Method with special attention to the computation of Spectral Submanifolds (SSM) and forced response curves in finite element models.

These invariant manifolds are computed in the physical coordinates using only the master modes resulting in efficient and feasible computations for high-dimensional finite-element problems. Additionally, the user has an option to choose among the graph or normal form style of parametrization. The computational methodology is described in the following article:

[1] Jain, S. & Haller, G. How to compute invariant manifolds and their reduced dynamics in high-dimensional finite element models. Nonlinear Dyn (2021). https://doi.org/10.1007/s11071-021-06957-4

The theoretical and computational aspects for analyzing systems with internal resonances via multi-dimensional SSMs are given in the following articles:

[2] Li, M., Jain, S. & Haller, G. Nonlinear analysis of forced mechanical systems with internal resonance using spectral submanifolds, Part I: Periodic response and forced response curve. Nonlinear Dyn 110, 1005–1043 (2022). https://doi.org/10.1007/s11071-022-07714-x

[3] Li, M & Haller, G. Nonlinear analysis of forced mechanical systems with internal resonance using spectral submanifolds, Part II: Bifurcation and quasi-periodic response. Nonlinear Dyn 110, 1045–1080 (2022). https://doi.org/10.1007/s11071-022-07476-6

How SSMs are extended to constrained mechanical systems are discussed in the following article:

[4] Li, M., Jain, S. & Haller, G. Model reduction for constrained mechanical systems via spectral submanifolds.Nonlinear Dyn 111(10): 8881-8911 (2023). https://doi.org/10.1007/s11071-023-08300-5

The treatment of systems subject to parametric resonance via higher-order approximations of nonautonomous SSMs is described in the following article:

[5] Thurnher, T., Haller, G. & Jain, S. Nonautonomous spectral submanifolds for model reduction of nonlinear mechanical systems under parametric resonances. Preprint (2023). Available on arXiv: https://doi.org/10.48550/arXiv.2307.10240

The use of SSM-based ROMs to extract forced response surfaces (FRSs) and their ridges and trenches via parameter continuation is discussed in the following article:

[6] Li, M., Jain, S. & Haller, G. Fast computation and characterization of forced response surface via spectral submanifolds and parameter continuation. Nonlinear Dyn 112, pages 7771–7797, (2024) https://doi.org/10.1007/s11071-024-09482-2

In this version, we demonstrate the computational methodology over the following small academic examples as well high-dimensional finite element problems using the FE package YetAnotherFECode

First-order examples:

Second-order examples:

Constrained mechanical systems [4]

Computation of stability diagrams and forced response curves in mechanical systems under parametric resonance:

Forced response surface and its ridges and trenches [6]:

This package uses the following external open-source packages:

  1. Continuation core (coco) https://sourceforge.net/projects/cocotools/
  2. Sandia tensor toolbox: https://gitlab.com/tensors/tensor_toolbox
  3. Combinator: https://www.mathworks.com/matlabcentral/fileexchange/24325-combinator-combinations-and-permutations
  4. YetAnotherFECode: Zenodo http://doi.org/10.5281/zenodo.4011281
  5. Tor: https://github.com/mingwu-li/torus_collocation

In order to install the program, simply run the install.m file in the main folder. The examples can be found in the examples directory. Note: When running the examples in the livescript files (workbooks), please ensure that the MATLAB 'Current Folder' is the directory of the specific example.

Please report any issues/bugs to Shobhit Jain ( shobhit.jain@tudelft.nl) and Mingwu Li (limw@sustech.edu.cn)