Structure-preserving discretization of port-Hamiltonian plate models

Authors

Andrea Brugnoli, Daniel Alazard, Valérie Pommier-Budinger, Denis Matignon

Abstract

Methods for discretizing port-Hamiltonian systems are of interest both for simulation and control purposes. Despite the large literature on mixed finite elements, no rigorous analysis of the connections between mixed elements and port-Hamiltonian systems has been carried out. In this paper we demonstrate how existing methods can be employed to discretize dynamical plate problems in a structure-preserving way. Based on convergence results of existing schemes, new error estimates are conjectured; numerical simulations confirm the expected behaviors.

Keywords

Port-Hamiltonian systems; Kirchhoff Plate; Mindlin-Reissner Plate; Mixed Finite Element Method; Numerical convergence

Citation

• Journal: IFAC-PapersOnLine
• Year: 2021
• Volume: 54
• Issue: 9
• Pages: 359–364
• Publisher: Elsevier BV
• DOI: 10.1016/j.ifacol.2021.06.094
• Note: 24th International Symposium on Mathematical Theory of Networks and Systems MTNS 2020- Cambridge, United Kingdom

BibTeX

@article{Brugnoli_2021,
  title={{Structure-preserving discretization of port-Hamiltonian plate models}},
  volume={54},
  ISSN={2405-8963},
  DOI={10.1016/j.ifacol.2021.06.094},
  number={9},
  journal={IFAC-PapersOnLine},
  publisher={Elsevier BV},
  author={Brugnoli, Andrea and Alazard, Daniel and Pommier-Budinger, Valérie and Matignon, Denis},
  year={2021},
  pages={359--364}
}

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References

• Arnold, D. N. & Lee, J. J. Mixed Methods for Elastodynamics with Weak Symmetry. SIAM Journal on Numerical Analysis vol. 52 2743–2769 (2014) – 10.1137/13095032x
• da Veiga, L. B., Mora, D. & Rodríguez, R. Numerical analysis of a locking‐free mixed finite element method for a bending moment formulation of Reissner‐Mindlin plate model. Numerical Methods for Partial Differential Equations vol. 29 40–63 (2012) – 10.1002/num.21698
• Blum, H. & Rannacher, R. On mixed finite element methods in plate bending analysis. Computational Mechanics vol. 6 221–236 (1990) – 10.1007/bf00350239
• Brugnoli, A., Alazard, D., Pommier-Budinger, V. & Matignon, D. Port-Hamiltonian formulation and symplectic discretization of plate models Part I: Mindlin model for thick plates. Applied Mathematical Modelling vol. 75 940–960 (2019) – 10.1016/j.apm.2019.04.035
• Brugnoli, A., Alazard, D., Pommier-Budinger, V. & Matignon, D. Port-Hamiltonian formulation and symplectic discretization of plate models Part II: Kirchhoff model for thin plates. Applied Mathematical Modelling vol. 75 961–981 (2019) – 10.1016/j.apm.2019.04.036
• Bécache, E., Joly, P. & Tsogka, C. An Analysis of New Mixed Finite Elements for the Approximation of Wave Propagation Problems. SIAM Journal on Numerical Analysis vol. 37 1053–1084 (2000) – 10.1137/s0036142998345499
• Bécache, E., Joly, P. & Tsogka, C. A New Family of Mixed Finite Elements for the Linear Elastodynamic Problem. SIAM Journal on Numerical Analysis vol. 39 2109–2132 (2002) – 10.1137/s0036142999359189
• Geveci, T. On the application of mixed finite element methods to the wave equations. ESAIM: Mathematical Modelling and Numerical Analysis vol. 22 243–250 (1988) – 10.1051/m2an/1988220202431
• Kirby, R. C. & Kieu, T. T. Symplectic-mixed finite element approximation of linear acoustic wave equations. Numerische Mathematik vol. 130 257–291 (2014) – 10.1007/s00211-014-0667-4
• 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
• McRae, A. T. T., Bercea, G.-T., Mitchell, L., Ham, D. A. & Cotter, C. J. Automated Generation and Symbolic Manipulation of Tensor Product Finite Elements. SIAM Journal on Scientific Computing vol. 38 S25–S47 (2016) – 10.1137/15m1021167
• Rafetseder, K. & Zulehner, W. A Decomposition Result for Kirchhoff Plate Bending Problems and a New Discretization Approach. SIAM Journal on Numerical Analysis vol. 56 1961–1986 (2018) – 10.1137/17m1118427
• Rathgeber, F. et al. Firedrake. ACM Transactions on Mathematical Software vol. 43 1–27 (2016) – 10.1145/2998441
• Timoshenko, (1959)
• 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