Finite differences on staggered grids preserving the port-Hamiltonian structure with application to an acoustic duct

Authors

Vincent Trenchant, Hector Ramirez, Yann Le Gorrec, Paul Kotyczka

Abstract

A finite-difference spatial discretization scheme that preserves the port-Hamiltonian structure of infinite dimensional systems governed by the wave equation is proposed. The scheme is based on the use of staggered grids for the discretization of different variables of the system. The discretization is given in 2D for rectilinear and regular triangular meshes. The proposed method is completed with the midpoint rule for time integration and numerical results are provided, including considerations for interconnection and closed loop behaviors and isotropy comparison between the proposed meshes.

Keywords

Distributed port-Hamiltonian systems; Wave equation; Staggered grids; Finite-difference method; Midpoint rule; Structured mesh

Citation

• Journal: Journal of Computational Physics
• Year: 2018
• Volume: 373
• Issue:
• Pages: 673–697
• Publisher: Elsevier BV
• DOI: 10.1016/j.jcp.2018.06.051

BibTeX

@article{Trenchant_2018,
  title={{Finite differences on staggered grids preserving the port-Hamiltonian structure with application to an acoustic duct}},
  volume={373},
  ISSN={0021-9991},
  DOI={10.1016/j.jcp.2018.06.051},
  journal={Journal of Computational Physics},
  publisher={Elsevier BV},
  author={Trenchant, Vincent and Ramirez, Hector and Le Gorrec, Yann and Kotyczka, Paul},
  year={2018},
  pages={673--697}
}

Download the bib file

References

• van der Schaft, (2017)
• (2009)
• Maschke, Port controlled Hamiltonian systems: modeling origins and system theoretic properties. (1992)
• Ortega, R., van der Schaft, A., Maschke, B. & Escobar, G. Interconnection and damping assignment passivity-based control of port-controlled Hamiltonian systems. Automatica vol. 38 585–596 (2002) – 10.1016/s0005-1098(01)00278-3
• 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
• Le Gorrec, Y., Zwart, H. & Maschke, B. Dirac structures and Boundary Control Systems associated with Skew-Symmetric Differential Operators. SIAM Journal on Control and Optimization vol. 44 1864–1892 (2005) – 10.1137/040611677
• Villegas, J. A., Zwart, H., Le Gorrec, Y. & Maschke, B. Exponential Stability of a Class of Boundary Control Systems. IEEE Transactions on Automatic Control vol. 54 142–147 (2009) – 10.1109/tac.2008.2007176
• Jacob, Linear Port-Hamiltonian Systems on Infinite-dimensional Spaces. (2012)
• Ramirez, H., Le Gorrec, Y., Macchelli, A. & Zwart, H. Exponential Stabilization of Boundary Controlled Port-Hamiltonian Systems With Dynamic Feedback. IEEE Transactions on Automatic Control vol. 59 2849–2855 (2014) – 10.1109/tac.2014.2315754
• Macchelli, A., Le Gorrec, Y., Ramirez, H. & Zwart, H. On the Synthesis of Boundary Control Laws for Distributed Port-Hamiltonian Systems. IEEE Transactions on Automatic Control vol. 62 1700–1713 (2017) – 10.1109/tac.2016.2595263
• Seslija, M., van der Schaft, A. & Scherpen, J. M. A. Discrete exterior geometry approach to structure-preserving discretization of distributed-parameter port-Hamiltonian systems. Journal of Geometry and Physics vol. 62 1509–1531 (2012) – 10.1016/j.geomphys.2012.02.006
• Golo, G., Talasila, V., van der Schaft, A. & Maschke, B. Hamiltonian discretization of boundary control systems. Automatica vol. 40 757–771 (2004) – 10.1016/j.automatica.2003.12.017
• Hamroun, Port-based modelling and geometric reduction for open channel irrigation systems. (2007)
• Moulla, R., Lefévre, L. & Maschke, B. Pseudo-spectral methods for the spatial symplectic reduction of open systems of conservation laws. Journal of Computational Physics vol. 231 1272–1292 (2012) – 10.1016/j.jcp.2011.10.008
• Kotyczka, P. & Maschke, B. Discrete port-Hamiltonian formulation and numerical approximation for systems of two conservation laws. at - Automatisierungstechnik vol. 65 308–322 (2017) – 10.1515/auto-2016-0098
• Bassi, An algorithm to discretize one-dimensional distributed port Hamiltonian systems. (2007)
• Macchelli, A. Energy shaping of distributed parameter port-Hamiltonian systems based on finite element approximation. Systems & Control Letters vol. 60 579–589 (2011) – 10.1016/j.sysconle.2011.04.016
• Wu, Power preserving model reduction of 2D vibro-acoustic system: a port Hamiltonian approach. (2015)
• Le Gorrec, Systèmes hamiltoniens à ports de dimension infinie: réduction et propriétés spectrales. J. Eur. Syst. Autom. (2011)
• Kotyczka, Finite volume structure-preserving discretization of 1D distributed-parameter port-Hamiltonian systems. (2016)
• 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
• Mazumder, Numerical methods for partial differential equations. (2015)
• Strikwerda, (2004)
• ISERLES, A. Generalized Leapfrog Methods. IMA Journal of Numerical Analysis vol. 6 381–392 (1986) – 10.1093/imanum/6.4.381
• Fornberg, B. High-Order Finite Differences and the Pseudospectral Method on Staggered Grids. SIAM Journal on Numerical Analysis vol. 27 904–918 (1990) – 10.1137/0727052
• Zeng, Y. Q. & Liu, Q. H. A staggered-grid finite-difference method with perfectly matched layers for poroelastic wave equations. The Journal of the Acoustical Society of America vol. 109 2571–2580 (2001) – 10.1121/1.1369783
• Hustedt, B., Operto, S. & Virieux, J. Mixed-grid and staggered-grid finite-difference methods for frequency-domain acoustic wave modelling. Geophysical Journal International vol. 157 1269–1296 (2004) – 10.1111/j.1365-246x.2004.02289.x
• Štekl, I. & Pratt, R. G. Accurate viscoelastic modeling by frequency‐domain finite differences using rotated operators. GEOPHYSICS vol. 63 1779–1794 (1998) – 10.1190/1.1444472
• Shin, C. & Sohn, H. A frequency‐space 2-D scalar wave extrapolator using extended 25-point finite‐difference operator. GEOPHYSICS vol. 63 289–296 (1998) – 10.1190/1.1444323
• Graves, R. W. Simulating seismic wave propagation in 3D elastic media using staggered-grid finite differences. Bulletin of the Seismological Society of America vol. 86 1091–1106 (1996) – 10.1785/bssa0860041091
• Hamilton, B. & Bilbao, S. Hexagonal vs. rectilinear grids for explicit finite difference schemes for the two-dimensional wave equation. Proceedings of Meetings on Acoustics 015120–015120 (2013) doi:10.1121/1.4800308 – 10.1121/1.4800308
• Bilbao, S. Modeling of Complex Geometries and Boundary Conditions in Finite Difference/Finite Volume Time Domain Room Acoustics Simulation. IEEE Transactions on Audio, Speech, and Language Processing vol. 21 1524–1533 (2013) – 10.1109/tasl.2013.2256897
• Zingg, D. W. & Lomax, H. Finite-Difference Schemes on Regular Triangular Grids. Journal of Computational Physics vol. 108 306–313 (1993) – 10.1006/jcph.1993.1184
• Kantorovich, (1958)
• Hairer, (2006)
• Laila, D. S. & Astolfi, A. Construction of discrete-time models for port-controlled Hamiltonian systems with applications. Systems & Control Letters vol. 55 673–680 (2006) – 10.1016/j.sysconle.2005.09.012
• Falaize, A. & Hélie, T. Passive simulation of the nonlinear port-Hamiltonian modeling of a Rhodes Piano. Journal of Sound and Vibration vol. 390 289–309 (2017) – 10.1016/j.jsv.2016.11.008
• Trenchant, Structure preserving spatial discretization of 2D hyperbolic systems using staggered grids finite difference. (2017)
• Collet, Semi-active optimization of 2D wave’s dispersion into shunted piezocomposite systems for controlling acoustic interaction. (2011)
• Collet, M., David, P. & Berthillier, M. Active acoustical impedance using distributed electrodynamical transducers. The Journal of the Acoustical Society of America vol. 125 882–894 (2009) – 10.1121/1.3026329
• Trenchant, A port-Hamiltonian formulation of a 2D boundary controlled acoustic system. (2015)
• Hagood, N. W. & von Flotow, A. Damping of structural vibrations with piezoelectric materials and passive electrical networks. Journal of Sound and Vibration vol. 146 243–268 (1991) – 10.1016/0022-460x(91)90762-9
• Hansen, (2012)
• Thiel, W. & Katehi, L. P. B. Some aspects of stability and numerical dissipation of the finite-difference time-domain (FDTD) technique including passive and active lumped elements. IEEE Transactions on Microwave Theory and Techniques vol. 50 2159–2165 (2002) – 10.1109/tmtt.2002.802330
• Nijmeijer, (1990)
• Hamilton, (2014)
• Falaize, (2016)
• Krstic, (2008)
• Brogliato, B., Maschke, B., Lozano, R. & Egeland, O. Dissipative Systems Analysis and Control. Communications and Control Engineering (Springer London, 2007). doi:10.1007/978-1-84628-517-2 – 10.1007/978-1-84628-517-2