Partitioned Finite Element Method for port-Hamiltonian systems with Boundary Damping: Anisotropic Heterogeneous 2D wave equations

Authors

Anass Serhani, Denis Matignon, Ghislain Haine

Abstract

A 2D wave equation with boundary damping of impedance type can be recast into an infinite-dimensional port-Hamiltonian system (pHs) with an appropriate feedback law, where the structure operator J is formally skew-symmetric. It is known that the underlying semigroup proves dissipative, even though no dissipation operator R is to be found in the pHs model. The Partitioned Finite Element Method (PFEM) introduced in Cardoso-Ribeiro et al. (2018), is structure-preserving and provides a natural way to discretize such systems. It gives rise to a non null symmetric matrix R. Moreover, since this matrix accounts for boundary damping, its rank is very low: only the basis functions at the boundary have an influence. Lastly, this matrix can be factorized out when considering the boundary condition as a feedback law for the pHs, involving the impedance parameter. Note that pHs - as open system - is used here as a tool to accurately discretize the wave equation with boundary damping as a closed system. In the worked-out numerical examples in 2D, the isotropic and homogeneous case is presented and the influence of the impedance is assessed; then, an anisotropic and heterogeneous wave equation with space-varying impedance at the boundary is investigated.

Keywords

Port-Hamiltonian systems (pHs); distributed-parameter system (DPS); structure preserving discretization; partitioned finite element method (PFEM); boundary damping

Citation

Journal: IFAC-PapersOnLine
Year: 2019
Volume: 52
Issue: 2
Pages: 96–101
Publisher: Elsevier BV
DOI: 10.1016/j.ifacol.2019.08.017
Note: 3rd IFAC Workshop on Control of Systems Governed by Partial Differential Equations CPDE 2019- Oaxaca, Mexico, 20–24 May 2019

BibTeX

@article{Serhani_2019,
  title={{Partitioned Finite Element Method for port-Hamiltonian systems with Boundary Damping: Anisotropic Heterogeneous 2D wave equations}},
  volume={52},
  ISSN={2405-8963},
  DOI={10.1016/j.ifacol.2019.08.017},
  number={2},
  journal={IFAC-PapersOnLine},
  publisher={Elsevier BV},
  author={Serhani, Anass and Matignon, Denis and Haine, Ghislain},
  year={2019},
  pages={96--101}
}

Download the bib file

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