Anisotropic heterogeneous n-D heat equation with boundary control and observation: II. Structure-preserving discretization

Authors

Anass Serhani, Ghislain Haine, Denis Matignon

Abstract

The heat equation with boundary control and observation can be described by means of three different Hamiltonians, the internal energy, the entropy, or a classical Lyapunov functional, as shown in the companion paper (Serhani et al. (2019a)). The aim of this work is to apply the partitioned finite element method (PFEM) proposed in Cardoso-Ribeiro et al. (2018) to the three associated port-Hamiltonian systems. Differential Algebraic Equations are obtained. The strategy proves very efficient to mimic the continuous Stokes-Dirac structure at the discrete level, and especially preserving the associated power balance. Anisotropic and heterogeneous 2D simulations are finally performed on the Lyapunov formulation to provide numerical evidence that this strategy proves very efficient for the accurate simulation of a boundary controlled and observed infinite-dimensional system.

Keywords

Port-Hamiltonian Differential Algebraic System; Heat Equation; Structure Preserving Discretization; Partitionned Finite Element Method (PFEM); Boundary Control

Citation

• Journal: IFAC-PapersOnLine
• Year: 2019
• Volume: 52
• Issue: 7
• Pages: 57–62
• Publisher: Elsevier BV
• DOI: 10.1016/j.ifacol.2019.07.010
• Note: 3rd IFAC Workshop on Thermodynamic Foundations for a Mathematical Systems Theory TFMST 2019- Louvain-la-Neuve, Belgium, 3–5 July 2019

BibTeX

@article{Serhani_2019,
  title={{Anisotropic heterogeneous n-D heat equation with boundary control and observation: II. Structure-preserving discretization}},
  volume={52},
  ISSN={2405-8963},
  DOI={10.1016/j.ifacol.2019.07.010},
  number={7},
  journal={IFAC-PapersOnLine},
  publisher={Elsevier BV},
  author={Serhani, Anass and Haine, Ghislain and Matignon, Denis},
  year={2019},
  pages={57--62}
}

Download the bib file

References

• Alnæs, The FEniCS Project Version 1.5. Archive of Numerical Software (2015)
• Brugnoli, A., Alazard, D., Pommier-Budinger, V. & Matignon, D. Partitioned Finite Element Method for the Mindlin Plate as a Port-Hamiltonian system. IFAC-PapersOnLine vol. 52 88–95 (2019) – 10.1016/j.ifacol.2019.08.016
• 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
• Cardoso-Ribeiro, F. L., Matignon, D. & Lefèvre, L. A structure-preserving Partitioned Finite Element Method for the 2D wave equation. IFAC-PapersOnLine vol. 51 119–124 (2018) – 10.1016/j.ifacol.2018.06.033
• 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
• Egger, H., Kugler, T., Liljegren-Sailer, B., Marheineke, N. & Mehrmann, V. On Structure-Preserving Model Reduction for Damped Wave Propagation in Transport Networks. SIAM Journal on Scientific Computing vol. 40 A331–A365 (2018) – 10.1137/17m1125303
• Golo, Hamiltonian discretization of boundary control systems. Auto-matica (2004)
• Hairer, (2006)
• Kotyczka, P. Finite Volume Structure-Preserving Discretization of 1D Distributed-Parameter Port-Hamiltonian Systems. IFAC-PapersOnLine vol. 49 298–303 (2016) – 10.1016/j.ifacol.2016.07.457
• 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
• Kunkel, P. & Mehrmann, V. Differential-Algebraic Equations. EMS Textbooks in Mathematics (2006) doi:10.4171/017 – 10.4171/017
• Leimkuhler, (2004)
• Serhani, A., Haine, G. & Matignon, D. Anisotropic heterogeneous n-D heat equation with boundary control and observation: I. Modeling as port-Hamiltonian system. IFAC-PapersOnLine vol. 52 51–56 (2019) – 10.1016/j.ifacol.2019.07.009
• Serhani, A partitioned finite element method (PFEM) for the structure-preserving discretization of damped infinite-dimensional port-Hamiltonian systems with boundary control. (2019)
• Serhani, A., Matignon, D. & Haine, G. Structure-Preserving Finite Volume Method for 2D Linear and Non-Linear Port-Hamiltonian Systems. IFAC-PapersOnLine vol. 51 131–136 (2018) – 10.1016/j.ifacol.2018.06.037
• Serhani, A., Matignon, D. & Haine, G. Partitioned Finite Element Method for port-Hamiltonian systems with Boundary Damping: Anisotropic Heterogeneous 2D wave equations. IFAC-PapersOnLine vol. 52 96–101 (2019) – 10.1016/j.ifacol.2019.08.017
• Trenchant, V., Ramirez, H., Le Gorrec, Y. & Kotyczka, P. Finite differences on staggered grids preserving the port-Hamiltonian structure with application to an acoustic duct. Journal of Computational Physics vol. 373 673–697 (2018) – 10.1016/j.jcp.2018.06.051