Theory and Implementation of Coupled Port-Hamiltonian Continuum and Lumped Parameter Models
Authors
Finbar J. Argus, Chris P. Bradley, Peter J. Hunter
Abstract
A continuous Galerkin finite element method that allows mixed boundary conditions without the need for Lagrange multipliers or user-defined parameters is developed. A mixed coupling of Lagrange and Raviart-Thomas basis functions are used. The method is proven to have a Hamiltonian-conserving spatial discretisation and a symplectic time discretisation. The energy residual is therefore guaranteed to be bounded for general problems and exactly conserved for linear problems. The linear 2D wave equation is discretised and modelled by making use of a port-Hamiltonian framework. This model is verified against an analytic solution and shown to have standard order of convergence for the temporal and spatial discretisation. The error growth over time is shown to grow linearly for this symplectic method, which agrees with theoretical results. A modal analysis is performed which verifies that the eigenvalues of the model accurately converge to the exact eigenvalues, as the mesh is refined. The port-Hamiltonian framework allows boundary coupling with bond-graph or, more generally, lumped parameter models, therefore unifying the two fields of lumped parameter modelling and continuum modelling of Hamiltonian systems. The wave domain discretisation is shown to be equivalent to a coupling of canonical port-Hamiltonian forms. This feature allows the model to have mixed boundary conditions as well as to have mixed causality interconnections with other port-Hamiltonian models. A model of the 2D wave equation is coupled, in a monolithic manner, with a lumped parameter model of an electromechanical linear actuator. The combined model is also verified to conserve energy exactly.
Keywords
Port-Hamiltonian; Modelling; PDE; FEniCS; Hamiltonian; Finite element; Galerkin; Symplectic; Lumped parameter; Continuum; Monolithic; Bond graph; 65P10; 35L05, 74F99, 65M60
Citation
- Journal: Journal of Elasticity
- Year: 2021
- Volume: 145
- Issue: 1-2
- Pages: 339–382
- Publisher: Springer Science and Business Media LLC
- DOI: 10.1007/s10659-021-09846-4
BibTeX
@article{Argus_2021,
title={{Theory and Implementation of Coupled Port-Hamiltonian Continuum and Lumped Parameter Models}},
volume={145},
ISSN={1573-2681},
DOI={10.1007/s10659-021-09846-4},
number={1–2},
journal={Journal of Elasticity},
publisher={Springer Science and Business Media LLC},
author={Argus, Finbar J. and Bradley, Chris P. and Hunter, Peter J.},
year={2021},
pages={339--382}
}
References
- Alnæs, M. et al. The FEniCS Project Version 1.5. <p>Archive of Numerical Software Vol 3, Starting Point and Frequency: Year: 2013</p> (2015) – 10.11588/ans.2015.100.20553
- Bauer, W. & Cotter, C. J. Energy–enstrophy conserving compatible finite element schemes for the rotating shallow water equations with slip boundary conditions. Journal of Computational Physics vol. 373 171–187 (2018) – 10.1016/j.jcp.2018.06.071
- Bazilevs, Y. & Hughes, T. J. R. Weak imposition of Dirichlet boundary conditions in fluid mechanics. Computers & Fluids vol. 36 12–26 (2007) – 10.1016/j.compfluid.2005.07.012
- Falk, R. S. Mixed and Hybrid Finite Element Methods (Franco Brezzi and Michel Fortin). SIAM Review vol. 35 514–517 (1993) – 10.1137/1035113
- Brugnano, L., Frasca Caccia, G. & Iavernaro, F. Energy conservation issues in the numerical solution of the semilinear wave equation. Applied Mathematics and Computation vol. 270 842–870 (2015) – 10.1016/j.amc.2015.08.078
- Brugnoli, A., Alazard, D., Pommier-Budinger, V. & Matignon, D. Interconnection of the Kirchhoff plate within the port-Hamiltonian framework. 2019 IEEE 58th Conference on Decision and Control (CDC) 6857–6862 (2019) doi:10.1109/cdc40024.2019.9029487 – 10.1109/cdc40024.2019.9029487
- 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
- A. Brugnoli. Brugnoli, A., Cardoso-Ribeiro, F.L., Haine, G., Kotyczka, P.: Partitioned finite element method for power-preserving structured discretization with mixed boundary conditions. In: Proceedings of the 21st IFAC. World Congress, ??? (2020) (2020)
- 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
- Cockburn, B., Gopalakrishnan, J. & Lazarov, R. Unified Hybridization of Discontinuous Galerkin, Mixed, and Continuous Galerkin Methods for Second Order Elliptic Problems. SIAM Journal on Numerical Analysis vol. 47 1319–1365 (2009) – 10.1137/070706616
- Duck, F. Tissue non-linearity. Proceedings of the Institution of Mechanical Engineers, Part H: Journal of Engineering in Medicine vol. 224 155–170 (2009) – 10.1243/09544119jeim574
- Duindam, V., Macchelli, A., Stramigioli, S. & Bruyninckx, H. Modeling and Control of Complex Physical Systems. (Springer Berlin Heidelberg, 2009). doi:10.1007/978-3-642-03196-0 – 10.1007/978-3-642-03196-0
- Eldred, C., Dubos, T. & Kritsikis, E. A quasi-Hamiltonian discretization of the thermal shallow water equations. Journal of Computational Physics vol. 379 1–31 (2019) – 10.1016/j.jcp.2018.10.038
- Bond-graph modeling. IEEE Control Systems vol. 27 24–45 (2007) – 10.1109/mcs.2007.338279
- 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
- Hairer, E. & Lubich, C. Symmetric multistep methods over long times. Numerische Mathematik vol. 97 699–723 (2004) – 10.1007/s00211-004-0520-2
- Hairer, E., Lubich, C. & Wanner, G. Geometric numerical integration illustrated by the Störmer–Verlet method. Acta Numerica vol. 12 399–450 (2003) – 10.1017/s0962492902000144
- Geometric Numerical Integration. Springer Series in Computational Mathematics (Springer-Verlag, 2006). doi:10.1007/3-540-30666-8 – 10.1007/3-540-30666-8
- Joly, P. Variational Methods for Time-Dependent Wave Propagation Problems. Lecture Notes in Computational Science and Engineering 201–264 (2003) doi:10.1007/978-3-642-55483-4_6 – 10.1007/978-3-642-55483-4_6
- Kotyczka, P. Numerical Methods for Distributed Parameter Port-Hamiltonian Systems. (Technical University of Munich, 2019). doi:10.14459/2019MD1510230 – 10.14459/2019md1510230
- Kotyczka, P. & Lefèvre, L. Discrete-time port-Hamiltonian systems: A definition based on symplectic integration. Systems & Control Letters vol. 133 104530 (2019) – 10.1016/j.sysconle.2019.104530
- 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
- General Boundary Conditions. Boundary Conditions in Electromagnetics 101–141 (2019) doi:10.1002/9781119632429.ch5 – 10.1002/9781119632429.ch5
- Automated Solution of Differential Equations by the Finite Element Method. Lecture Notes in Computational Science and Engineering (Springer Berlin Heidelberg, 2012). doi:10.1007/978-3-642-23099-8 – 10.1007/978-3-642-23099-8
- McDonald, F., McLachlan, R. I., Moore, B. E. & Quispel, G. R. W. Travelling wave solutions of multisymplectic discretizations of semi-linear wave equations. Journal of Difference Equations and Applications vol. 22 913–940 (2016) – 10.1080/10236198.2016.1162161
- McLachlan, R. Symplectic integration of Hamiltonian wave equations. Numerische Mathematik vol. 66 465–492 (1993) – 10.1007/bf01385708
- McLachlan, R. I. & Stern, A. Multisymplecticity of Hybridizable Discontinuous Galerkin Methods. Foundations of Computational Mathematics vol. 20 35–69 (2019) – 10.1007/s10208-019-09415-1
- Nitsche, J. Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg vol. 36 9–15 (1971) – 10.1007/bf02995904
- Ophir, J., Céspedes, I., Ponnekanti, H., Yazdi, Y. & Li, X. Elastography: A Quantitative Method for Imaging the Elasticity of Biological Tissues. Ultrasonic Imaging vol. 13 111–134 (1991) – 10.1177/016173469101300201
- Handbook of linear partial differential equations for engineers and scientists. Choice Reviews Online vol. 40 40-0964-40–0964 (2002) – 10.5860/choice.40-0964
- Raviart, P. A. & Thomas, J. M. A mixed finite element method for 2-nd order elliptic problems. Lecture Notes in Mathematics 292–315 (1977) doi:10.1007/bfb0064470 – 10.1007/bfb0064470
- Reich, S. Multi-Symplectic Runge–Kutta Collocation Methods for Hamiltonian Wave Equations. Journal of Computational Physics vol. 157 473–499 (2000) – 10.1006/jcph.1999.6372
- Sadjina, S., Kyllingstad, L. T., Skjong, S. & Pedersen, E. Energy conservation and power bonds in co-simulations: non-iterative adaptive step size control and error estimation. Engineering with Computers vol. 33 607–620 (2016) – 10.1007/s00366-016-0492-8
- Sánchez, M. A., Ciuca, C., Nguyen, N. C., Peraire, J. & Cockburn, B. Symplectic Hamiltonian HDG methods for wave propagation phenomena. Journal of Computational Physics vol. 350 951–973 (2017) – 10.1016/j.jcp.2017.09.010
- Scovazzi, G. & Carnes, B. Weak boundary conditions for wave propagation problems in confined domains: Formulation and implementation using a variational multiscale method. Computer Methods in Applied Mechanics and Engineering vols 221–222 117–131 (2012) – 10.1016/j.cma.2012.01.018
- 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
- Süli, E. & Mayers, D. F. An Introduction to Numerical Analysis. (2003) doi:10.1017/cbo9780511801181 – 10.1017/cbo9780511801181
- Trenchant, V., Fares, Y., Ramirez, H. & Le Gorrec, Y. A port-Hamiltonian formulation of a 2D boundary controlled acoustic system. IFAC-PapersOnLine vol. 48 235–240 (2015) – 10.1016/j.ifacol.2015.10.245
- Trenchant, V., Ramirez, H., Le Gorrec, Y. & Kotyczka, P. Structure preserving spatial discretization of 2D hyperbolic systems using staggered grids finite difference. 2017 American Control Conference (ACC) 2491–2496 (2017) doi:10.23919/acc.2017.7963327 – 10.23919/acc.2017.7963327
- van der Schaft, A. & Jeltsema, D. Port-Hamiltonian Systems Theory: An Introductory Overview. Foundations and Trends® in Systems and Control vol. 1 173–378 (2014) – 10.1561/2600000002
- 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
- Vu, N. M. T., Lefèvre, L., Nouailletas, R. & Brémond, S. Symplectic spatial integration schemes for systems of balance equations. Journal of Process Control vol. 51 1–17 (2017) – 10.1016/j.jprocont.2016.12.005
- Zhai, Z. & Chen, Q. (Yan). Solution characters of iterative coupling between energy simulation and CFD programs. Energy and Buildings vol. 35 493–505 (2003) – 10.1016/s0378-7788(02)00156-1