Port-Hamiltonian FE models for filaments
Authors
Abstract
In this article, we present the port-Hamiltonian representation, the structure preserving discretization and the resulting finite-dimensional state space model of one-dimensional filaments based on a mixed finite element formulation. Due to the fact that the equations of motion of a filamentous body are based on the theory of geometrically nonlinear mechanical systems, the port-Hamiltonian formulation is expressed by means of its co-energy (effort) variables. The resulting port-Hamiltonian state space model features a quadratic Hamiltonian and the nonlinearity is reflected in the state dependence of its interconnection matrix. Numerical experiments generated with FEniCS illustrate the properties of the resulting finite element models.
Keywords
port-Hamiltonian systems; mixed finite elements; geometrically nonlinear mechanical systems; structure preserving discretization; filamentous bodies
Citation
- Journal: IFAC-PapersOnLine
- Year: 2022
- Volume: 55
- Issue: 30
- Pages: 353–358
- Publisher: Elsevier BV
- DOI: 10.1016/j.ifacol.2022.11.078
- Note: 25th International Symposium on Mathematical Theory of Networks and Systems MTNS 2022- Bayreuth, Germany, September 12-16, 2022
BibTeX
@article{Thoma_2022,
title={{Port-Hamiltonian FE models for filaments}},
volume={55},
ISSN={2405-8963},
DOI={10.1016/j.ifacol.2022.11.078},
number={30},
journal={IFAC-PapersOnLine},
publisher={Elsevier BV},
author={Thoma, Tobias and Kotyczka, Paul},
year={2022},
pages={353--358}
}
References
- Bonet, (2008)
- Brugnoli, A., Alazard, D., Pommier-Budinger, V. & Matignon, D. Port-Hamiltonian flexible multibody dynamics. Multibody System Dynamics vol. 51 343–375 (2020) – 10.1007/s11044-020-09758-6
- Brugnoli, A., Cardoso-Ribeiro, F. L., Haine, G. & Kotyczka, P. Partitioned finite element method for structured discretization with mixed boundary conditions. IFAC-PapersOnLine vol. 53 7557–7562 (2020) – 10.1016/j.ifacol.2020.12.1351
- Brugnoli, A port-Hamiltonian formulation for the full von-Kármán plate model. (2022)
- Brugnoli, A., Rashad, R., Califano, F., Stramigioli, S. & Matignon, D. Mixed finite elements for port-Hamiltonian models of von Kármán beams. IFAC-PapersOnLine vol. 54 186–191 (2021) – 10.1016/j.ifacol.2021.11.076
- Cardoso-Ribeiro, F. L., Matignon, D. & Lefèvre, L. A partitioned finite element method for power-preserving discretization of open systems of conservation laws. IMA Journal of Mathematical Control and Information vol. 38 493–533 (2020) – 10.1093/imamci/dnaa038
- Dresig, (2014)
- Duindam, (2009)
- Jacob, (2012)
- Kotyczka, (2019)
- Kuhn, A., Steiner, W., Zemann, J., Dinevski, D. & Troger, H. A comparison of various mathematical formulations and numerical solution methods for the large amplitude oscillations of a string pendulum. Applied Mathematics and Computation vol. 67 227–264 (1995) – 10.1016/0096-3003(94)00060-h
- Logg, (2012)
- Macchelli, A. & Melchiorri, C. Modeling and Control of the Timoshenko Beam. The Distributed Port Hamiltonian Approach. SIAM Journal on Control and Optimization vol. 43 743–767 (2004) – 10.1137/s0363012903429530
- Mankala, K. K. & Agrawal, S. K. Dynamic Modeling and Simulation of Satellite Tethered Systems. Journal of Vibration and Acoustics vol. 127 144–156 (2004) – 10.1115/1.1891811
- Rashad, R., Califano, F., van der Schaft, A. J. & Stramigioli, S. Twenty years of distributed port-Hamiltonian systems: a literature review. IMA Journal of Mathematical Control and Information vol. 37 1400–1422 (2020) – 10.1093/imamci/dnaa018
- Thoma, Explicit port-Hamiltonian FEM-models for linear mechanical systems with non-uniform boundary conditions. (2021)
- Thoma, Explicit port-Hamiltonian FEM models for geometrically nonlinear mechanical systems. (submitted to) Mathematical and Computer Modelling of Dynamical Systems (2022)
- Warsewa, A., Böhm, M., Sawodny, O. & Tarín, C. A port-Hamiltonian approach to modeling the structural dynamics of complex systems. Applied Mathematical Modelling vol. 89 1528–1546 (2021) – 10.1016/j.apm.2020.07.038
- Weiß, (2000)
- Zienkiewicz, (2005)