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

References

• Cardoso-Ribeiro FL, Matignon D, Lefèvre L (2018) A structure-preserving Partitioned Finite Element Method for the 2D wave equation ⁎ ⁎This work is supported by the project ANR-16-CE92-0028, entitled Interconnected Infinite-Dimensional systems for Heterogeneous Media, INFIDHEM, financed by the French National Research Agency (ANR). Further information is available at https://websites.isae-supaero.fr/infidhem/the-project/. IFAC-PapersOnLine 51(3):119–124. https://doi.org/10.1016/j.ifacol.2018.06.033 – 10.1016/j.ifacol.2018.06.033
• Cardoso-Ribeiro FL, Matignon D, Lefèvre L (2020) A partitioned finite element method for power-preserving discretization of open systems of conservation laws. IMA Journal of Mathematical Control and Information 38(2):493–533. https://doi.org/10.1093/imamci/dnaa038 – 10.1093/imamci/dnaa038
• Cowsar LC, Dupont TF, Wheeler MF (1996) A Priori Estimates for Mixed Finite Element Approximations of Second-Order Hyperbolic Equations with Absorbing Boundary Conditions. SIAM J Numer Anal 33(2):492–504. https://doi.org/10.1137/0733026 – 10.1137/0733026
• Duindam, (2009)
• Egger H, Kugler T, Liljegren-Sailer B, Marheineke N, Mehrmann V (2018) On Structure-Preserving Model Reduction for Damped Wave Propagation in Transport Networks. SIAM J Sci Comput 40(1):A331–A365. https://doi.org/10.1137/17m1125303 – 10.1137/17m1125303
• Engquist B, Majda A (1977) Absorbing boundary conditions for the numerical simulation of waves. Math Comp 31(139):629–651. https://doi.org/10.1090/s0025-5718-1977-0436612-4 – 10.1090/s0025-5718-1977-0436612-4
• Farle O, Klis D, Jochum M, Floch O, Dyczij-Edlinger R (2013) A port-hamiltonian finite-element formulation for the maxwell equations. In: 2013 International Conference on Electromagnetics in Advanced Applications (ICEAA). IEEE, pp 324–327 – 10.1109/iceaa.2013.6632246
• Golo G, Talasila V, van der Schaft A, Maschke B (2004) Hamiltonian discretization of boundary control systems. Automatica 40(5):757–771. https://doi.org/10.1016/j.automatica.2003.12.017 – 10.1016/j.automatica.2003.12.017
• Jacob, (2012)
• Kergomard J, Debut V, Matignon D (2006) Resonance modes in a one-dimensional medium with two purely resistive boundaries: Calculation methods, orthogonality, and completeness. The Journal of the Acoustical Society of America 119(3):1356–1367. https://doi.org/10.1121/1.2166709 – 10.1121/1.2166709
• Kotyczka P, Maschke B, Lefèvre L (2018) Weak form of Stokes–Dirac structures and geometric discretization of port-Hamiltonian systems. Journal of Computational Physics 361:442–476. https://doi.org/10.1016/j.jcp.2018.02.006 – 10.1016/j.jcp.2018.02.006
• Kurula, Linear wave systems on n-D spatial domains. International Journal of Control (2015)
• Le Gorrec Y, Matignon D (2013) Coupling between hyperbolic and diffusive systems: A port-Hamiltonian formulation. European Journal of Control 19(6):505–512. https://doi.org/10.1016/j.ejcon.2013.09.003 – 10.1016/j.ejcon.2013.09.003
• Maschke BM, van der Schaft AJ (1992) Port-Controlled Hamiltonian Systems: Modelling Origins and Systemtheoretic Properties. IFAC Proceedings Volumes 25(13):359–365. https://doi.org/10.1016/s1474-6670(17)52308-3 – 10.1016/s1474-6670(17)52308-3
• Matignon D, Hélie T (2013) A class of damping models preserving eigenspaces for linear conservative port-Hamiltonian systems. European Journal of Control 19(6):486–494. https://doi.org/10.1016/j.ejcon.2013.10.003 – 10.1016/j.ejcon.2013.10.003
• Monteghetti F, Matignon D, Piot E (2018) Energy analysis and discretization of nonlinear impedance boundary conditions for the time-domain linearized Euler equations. Journal of Computational Physics 375:393–426. https://doi.org/10.1016/j.jcp.2018.08.037 – 10.1016/j.jcp.2018.08.037
• Moulla R, Lefévre L, Maschke B (2012) Pseudo-spectral methods for the spatial symplectic reduction of open systems of conservation laws. Journal of Computational Physics 231(4):1272–1292. https://doi.org/10.1016/j.jcp.2011.10.008 – 10.1016/j.jcp.2011.10.008
• Serhani, A., Haine, G., and Matignon, D. (2019). Numerical analysis of a semi-discretization scheme for the n-D wave equation in port-Hamiltonian system formalism. Submitted.
• van der Schaft A, Jeltsema D (2014) Port-Hamiltonian Systems Theory: An Introductory Overview. now Publishers Inc – 10.1561/9781601987877
• van der Schaft AJ, Maschke BM (2002) Hamiltonian formulation of distributed-parameter systems with boundary energy flow. Journal of Geometry and Physics 42(1-2):166–194. https://doi.org/10.1016/s0393-0440(01)00083-3 – 10.1016/s0393-0440(01)00083-3
• Villegas JA, Zwart H, Le Gorrec Y, Maschke B (2009) Exponential Stability of a Class of Boundary Control Systems. IEEE Trans Automat Contr 54(1):142–147. https://doi.org/10.1109/tac.2008.2007176 – 10.1109/tac.2008.2007176