The Lyapunov–Schmidt–Koiter asymptotic method using SymPy
1 Welcome!
These are my notes on the LSK method for the analysis of the stability and bifurcation(s) of a conservative system. These notes are based on several references: the initial PhD thesis of Warner Tjardus Koiter (1945) as well as some graphical illustrations from his lecture notes (W. T. Koiter and Heijden 2009). I enjoyed the concise presentation of Nguyen (2000) as well as the lecture notes of Triantafyllidis (2017). Finally, the chapter by Potier-Ferry (1987) helped me clear some issues. I used the direct approach presented in Appendix A of the paper by Chakrabarti et al. (2018) to derive the bifurcation equations. \(\phantom{x}\)
When the system under consideration exhibits multiple simultaneous buckling modes, some derivations become a bit tedious. I therefore used the SymPy computer algebra system as much as possible. This required a few “tricks”, and I believe that SymPy is now fully part of this work, as reflected by the new title.
I hope the reader will find these notes useful, even though there are still a few points which I do not fully understand (they are clearly indicated in the text). Should you want to report an error or suggest improvements, please do so by creating an issue on the Github repository.
These notes are available in two forms
- as an online website at https://sbrisard.github.io/LSK/
- as a PDF document at ….
The Quarto sources of these notes are available on the Github repo https://github.com/sbrisard/LSK. In particular, all SymPy simulations are available as Jupyter notebooks.
These notes by Sébastien Brisard are licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.