Explicit Port-Hamiltonian FEM-Models for Linear Mechanical Systems with Non-Uniform Boundary Conditions
Authors
Abstract
In this contribution, we present how to obtain explicit state space models in port-Hamiltonian form when a mixed finite element method is applied to a linear mechanical system with non-uniform boundary conditions. The key is to express the variational problem based on the principle of virtual power, with both the Dirichlet (velocity) and Neumann (stress) boundary conditions imposed in a weak sense. As a consequence, the formal skew-adjointness of the system operator becomes directly visible after integration by parts, and, after compatible FE discretization, the boundary degrees of freedom of both causalities appear as explicit inputs in the resulting state space model. The rationale behind our formulation is illustrated using a lumped parameter example, and numerical experiments on a one-dimensional rod show the properties of the approach in practice.
Keywords
port-Hamiltonian systems; elastodynamics; non-uniform boundary conditions; structure-preserving discretization; mixed finite elements; weak form; principle of virtual power
Citation
- Journal: IFAC-PapersOnLine
- Year: 2022
- Volume: 55
- Issue: 20
- Pages: 499–504
- Publisher: Elsevier BV
- DOI: 10.1016/j.ifacol.2022.09.144
- Note: 10th Vienna International Conference on Mathematical Modelling MATHMOD 2022- Vienna Austria, 27–29 July 2022
BibTeX
@article{Thoma_2022,
title={{Explicit Port-Hamiltonian FEM-Models for Linear Mechanical Systems with Non-Uniform Boundary Conditions}},
volume={55},
ISSN={2405-8963},
DOI={10.1016/j.ifacol.2022.09.144},
number={20},
journal={IFAC-PapersOnLine},
publisher={Elsevier BV},
author={Thoma, Tobias and Kotyczka, Paul},
year={2022},
pages={499--504}
}
References
- Brugnoli, Port-Hamiltonian flexible multi-body dynamics. Multibody System Dynamics (2020)
- 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
- Cardoso-Ribeiro, A partitioned finite element method for power-preserving discretization of open systems of conservation laws. IMA Journal of Mathematical Control and Information (2020)
- Farle, A port-Hamiltonian finite-element formulation for the Maxwell equations. (2013)
- Jacob, (2012)
- Kotyczka, (2019)
- Kotyczka, P., Maschke, B. & Lefèvre, L. Weak form of Stokes–Dirac structures and geometric discretization of port-Hamiltonian systems. Journal of Computational Physics vol. 361 442–476 (2018) – 10.1016/j.jcp.2018.02.006
- Logg, (2012)
- Lu, K., Augarde, C. E., Coombs, W. M. & Hu, Z. Weak impositions of Dirichlet boundary conditions in solid mechanics: A critique of current approaches and extension to partially prescribed boundaries. Computer Methods in Applied Mechanics and Engineering vol. 348 632–659 (2019) – 10.1016/j.cma.2019.01.035
- 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
- van der Schaft, A. J. & Maschke, B. M. Hamiltonian formulation of distributed-parameter systems with boundary energy flow. Journal of Geometry and Physics vol. 42 166–194 (2002) – 10.1016/s0393-0440(01)00083-3
- Villegas, (2007)
- 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
- Zienkiewicz, (2005)