Port-Hamiltonian formulations for the modeling, simulation and control of fluids

Authors

Flávio Luiz Cardoso-Ribeiro, Ghislain Haine, Yann Le Gorrec, Denis Matignon, Hector Ramirez

Abstract

This paper presents a state of the art on port-Hamiltonian formulations for the modeling and numerical simulation of open fluid systems. This literature review, with the help of more than one hundred classified references, highlights the main features, the positioning with respect to seminal works from the literature on this topic, and the advantages provided by such a framework. A focus is given on the shallow water equations and the incompressible Navier–Stokes equations in 2D, including numerical simulation results. It is also shown how it opens very stimulating and promising research lines towards thermodynamically consistent modeling and structure-preserving numerical methods for the simulation of complex fluid systems in interaction with their environment.

Keywords

boundary control, energy-based modeling, incompressible navier–stokes equations, port-hamiltonian systems, shallow water equations, structure-preserving discretization

Citation

Journal: Computers & Fluids
Year: 2024
Volume: 283
Issue:
Pages: 106407
Publisher: Elsevier BV
DOI: 10.1016/j.compfluid.2024.106407

BibTeX

@article{Cardoso_Ribeiro_2024,
  title={{Port-Hamiltonian formulations for the modeling, simulation and control of fluids}},
  volume={283},
  ISSN={0045-7930},
  DOI={10.1016/j.compfluid.2024.106407},
  journal={Computers \&amp; Fluids},
  publisher={Elsevier BV},
  author={Cardoso-Ribeiro, Flávio Luiz and Haine, Ghislain and Le Gorrec, Yann and Matignon, Denis and Ramirez, Hector},
  year={2024},
  pages={106407}
}

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References

Alonso AA, Ydstie BE, Banga JR (2002) From irreversible thermodynamics to a robust control theory for distributed process systems. Journal of Process Control 12(4):507–517. https://doi.org/10.1016/s0959-1524(01)00017- – 10.1016/s0959-1524(01)00017-8
Altmann R, Mehrmann V, Unger B (2021) Port-Hamiltonian formulations of poroelastic network models. Mathematical and Computer Modelling of Dynamical Systems 27(1):429–452. https://doi.org/10.1080/13873954.2021.197513 – 10.1080/13873954.2021.1975137
Altmann R, Schulze P (2017) A port-Hamiltonian formulation of the Navier–Stokes equations for reactive flows. Systems & Control Letters 100:51–55. https://doi.org/10.1016/j.sysconle.2016.12.00 – 10.1016/j.sysconle.2016.12.005
Aoues S, Cardoso-Ribeiro FL, Matignon D, Alazard D (2019) Modeling and Control of a Rotating Flexible Spacecraft: A Port-Hamiltonian Approach. IEEE Trans Contr Syst Technol 27(1):355–362. https://doi.org/10.1109/tcst.2017.277124 – 10.1109/tcst.2017.2771244
Arnold DN, Falk RS, Winther R (2006) Finite element exterior calculus, homological techniques, and applications. Acta Numerica 15:1–155. https://doi.org/10.1017/s096249290621001 – 10.1017/s0962492906210018
Arnold D, Falk R, Winther R (2010) Finite element exterior calculus: from Hodge theory to numerical stability. Bull Amer Math Soc 47(2):281–354. https://doi.org/10.1090/s0273-0979-10-01278- – 10.1090/s0273-0979-10-01278-4
Beattie C, Mehrmann V, Xu H, Zwart H (2018) Linear port-Hamiltonian descriptor systems. Math Control Signals Syst 30(4). https://doi.org/10.1007/s00498-018-0223- – 10.1007/s00498-018-0223-3
Bendimerad-Hohl A, Haine G, Lefèvre L, Matignon D (2023) Implicit port-Hamiltonian systems: structure-preserving discretization for the nonlocal vibrations in a viscoelastic nanorod, and for a seepage model. IFAC-PapersOnLine 56(2):6789–6795. https://doi.org/10.1016/j.ifacol.2023.10.38 – 10.1016/j.ifacol.2023.10.387
Bendimerad-Hohl, Structure-preserving discretization of the cahn-hilliard equations recast as a port-Hamiltonian system. (2023)
Bendimerad-Hohl A, Haine G, Matignon D, Maschke B (2022) Structure-preserving discretization of a coupled Allen-Cahn and heat equation system. IFAC-PapersOnLine 55(18):99–104. https://doi.org/10.1016/j.ifacol.2022.08.03 – 10.1016/j.ifacol.2022.08.037
Benner P, Goyal P, Van Dooren P (2020) Identification of port-Hamiltonian systems from frequency response data. Systems & Control Letters 143:104741. https://doi.org/10.1016/j.sysconle.2020.10474 – 10.1016/j.sysconle.2020.104741
Bird RB (2002) Transport phenomena. Applied Mechanics Reviews 55(1):R1–R4. https://doi.org/10.1115/1.142429 – 10.1115/1.1424298
Bochev, Principles of mimetic discretizations of differential operators. (2006)
Boffi, (2015)
Boyer, (2013)
Brayton RK, Moser JK (1964) A theory of nonlinear networks. I. Quart Appl Math 22(1):1–33. https://doi.org/10.1090/qam/16974 – 10.1090/qam/169746
Brugnoli A, Alazard D, Pommier-Budinger V, Matignon D (2019) Port-Hamiltonian formulation and symplectic discretization of plate models Part I: Mindlin model for thick plates. Applied Mathematical Modelling 75:940–960. https://doi.org/10.1016/j.apm.2019.04.03 – 10.1016/j.apm.2019.04.035
Brugnoli A, Alazard D, Pommier-Budinger V, Matignon D (2019) Port-Hamiltonian formulation and symplectic discretization of plate models Part II: Kirchhoff model for thin plates. Applied Mathematical Modelling 75:961–981. https://doi.org/10.1016/j.apm.2019.04.03 – 10.1016/j.apm.2019.04.036
Brugnoli A, Cardoso-Ribeiro FL, Haine G, Kotyczka P (2020) Partitioned finite element method for structured discretization with mixed boundary conditions. IFAC-PapersOnLine 53(2):7557–7562. https://doi.org/10.1016/j.ifacol.2020.12.135 – 10.1016/j.ifacol.2020.12.1351
Brugnoli A, Haine G, Matignon D (2022) Explicit structure-preserving discretization of port-Hamiltonian systems with mixed boundary control. IFAC-PapersOnLine 55(30):418–423. https://doi.org/10.1016/j.ifacol.2022.11.08 – 10.1016/j.ifacol.2022.11.089
Brugnoli A, Haine G, Matignon D (2023) Stokes-Dirac structures for distributed parameter port-Hamiltonian systems: An analytical viewpoint. CAM 15(3):362–387. https://doi.org/10.3934/cam.202301 – 10.3934/cam.2023018
Brugnoli A, Haine G, Serhani A, Vasseur X (2021) Numerical Approximation of Port-Hamiltonian Systems for Hyperbolic or Parabolic PDEs with Boundary Control. JAMP 09(06):1278–1321. https://doi.org/10.4236/jamp.2021.9608 – 10.4236/jamp.2021.96088
Brugnoli A, Rashad R, Stramigioli S (2022) Dual field structure-preserving discretization of port-Hamiltonian systems using finite element exterior calculus. Journal of Computational Physics 471:111601. https://doi.org/10.1016/j.jcp.2022.11160 – 10.1016/j.jcp.2022.111601
Califano F, Rashad R, Schuller FP, Stramigioli S (2021) Geometric and energy-aware decomposition of the Navier–Stokes equations: A port-Hamiltonian approach. Physics of Fluids 33(4). https://doi.org/10.1063/5.004835 – 10.1063/5.0048359
Califano F, Rashad R, Schuller FP, Stramigioli S (2022) Energetic decomposition of distributed systems with moving material domains: The port-Hamiltonian model of fluid-structure interaction. Journal of Geometry and Physics 175:104477. https://doi.org/10.1016/j.geomphys.2022.10447 – 10.1016/j.geomphys.2022.104477
Califano F, Rashad R, Stramigioli S (2022) A differential geometric description of thermodynamics in continuum mechanics with application to Fourier–Navier–Stokes fluids. Physics of Fluids 34(10). https://doi.org/10.1063/5.011951 – 10.1063/5.0119517
Cardoso-Ribeiro, Port-Hamiltonian modeling, discretization and feedback control of a circular water tank. (2019)
Cardoso-Ribeiro, (2023)
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/dnaa03 – 10.1093/imamci/dnaa038
Cardoso-Ribeiro FL, Matignon D, Lefèvre L (2021) Dissipative Shallow Water Equations: a port-Hamiltonian formulation. IFAC-PapersOnLine 54(19):167–172. https://doi.org/10.1016/j.ifacol.2021.11.07 – 10.1016/j.ifacol.2021.11.073
Cardoso-Ribeiro FL, Matignon D, Pommier-Budinger V (2017) A port-Hamiltonian model of liquid sloshing in moving containers and application to a fluid-structure system. Journal of Fluids and Structures 69:402–427. https://doi.org/10.1016/j.jfluidstructs.2016.12.00 – 10.1016/j.jfluidstructs.2016.12.007
Cardoso-Ribeiro FL, Matignon D, Pommier-Budinger V (2020) Port-Hamiltonian model of two-dimensional shallow water equations in moving containers. IMA Journal of Mathematical Control and Information 37(4):1348–1366. https://doi.org/10.1093/imamci/dnaa01 – 10.1093/imamci/dnaa016
Cervera J, van der Schaft AJ, Baños A (2007) Interconnection of port-Hamiltonian systems and composition of Dirac structures. Automatica 43(2):212–225. https://doi.org/10.1016/j.automatica.2006.08.01 – 10.1016/j.automatica.2006.08.014
Chorin, (1992)
Cisneros N, Rojas AJ, Ramirez H (2020) Port-Hamiltonian Modeling and Control of a Micro-Channel Experimental Plant. IEEE Access 8:176935–176946. https://doi.org/10.1109/access.2020.302665 – 10.1109/access.2020.3026653
Cotter CJ (2023) Compatible finite element methods for geophysical fluid dynamics. Acta Numerica 32:291–393. https://doi.org/10.1017/s096249292300002 – 10.1017/s0962492923000028
Courant TJ (1990) Dirac manifolds. Trans Amer Math Soc 319(2):631–661. https://doi.org/10.1090/s0002-9947-1990-0998124- – 10.1090/s0002-9947-1990-0998124-1
De Groot, (2013)
Diab, Splitting methods for linear circuit DAEs of index 1 in port-Hamiltonian form. (2021)
Domschke, (2021)
Dubljevic, Quo vadis advanced chemical process control. Can J Chem Eng (2022)
Duindam, (2009)
Eberard D, Maschke BM, van der Schaft AJ (2007) An extension of Hamiltonian systems to the thermodynamic phase space: Towards a geometry of nonreversible processes. Reports on Mathematical Physics 60(2):175–198. https://doi.org/10.1016/s0034-4877(07)00024- – 10.1016/s0034-4877(07)00024-9
Edwards BJ, Öttinger HC, Jongschaap RJJ (1997) On The Relationships Between Thermodynamic Formalisms For Complex Fluids. Journal of Non-Equilibrium Thermodynamics 22(4). https://doi.org/10.1515/jnet.1997.22.4.35 – 10.1515/jnet.1997.22.4.356
Egger H (2019) Structure preserving approximation of dissipative evolution problems. Numer Math 143(1):85–106. https://doi.org/10.1007/s00211-019-01050- – 10.1007/s00211-019-01050-w
Emmrich E, Mehrmann V (2013) Operator Differential-Algebraic Equations Arising in Fluid Dynamics. Computational Methods in Applied Mathematics 13(4):443–470. https://doi.org/10.1515/cmam-2013-001 – 10.1515/cmam-2013-0018
Erbay, (2024)
Erbay, (2024)
Farle O, Baltes R-B, Dyczij-Edlinger R (2014) Strukturerhaltende Diskretisierung verteilt-parametrischer Port-Hamiltonscher Systeme mittels finiter Elemente. at - Automatisierungstechnik 62(7):500–511. https://doi.org/10.1515/auto-2014-109 – 10.1515/auto-2014-1093
Farle, A port-Hamiltonian finite-element formulation for the maxwell equations. (2013)
Ferraro G, Fournié M, Haine G (2024) Simulation and control of interactions in multi-physics, a Python package for port-Hamiltonian systems. IFAC-PapersOnLine 58(6):119–124. https://doi.org/10.1016/j.ifacol.2024.08.26 – 10.1016/j.ifacol.2024.08.267
Gawlik ES, Gay-Balmaz F (2020) A Variational Finite Element Discretization of Compressible Flow. Found Comput Math 21(4):961–1001. https://doi.org/10.1007/s10208-020-09473- – 10.1007/s10208-020-09473-w
Gay-Balmaz, A variational formulation of nonequilibrium thermodynamics for discrete open systems with mass and heat transfer. Entropy (2018)
Gerbeau, Derivation of viscous saint-venant system for laminar shallow water; numerical validation. Discr Contin Dyn Syst - B (2001)
Geuzaine C, Remacle J (2009) Gmsh: A 3‐D finite element mesh generator with built‐in pre‐ and post‐processing facilities. Numerical Meth Engineering 79(11):1309–1331. https://doi.org/10.1002/nme.257 – 10.1002/nme.2579
Ghia U, Ghia KN, Shin CT (1982) High-Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method. Journal of Computational Physics 48(3):387–411. https://doi.org/10.1016/0021-9991(82)90058- – 10.1016/0021-9991(82)90058-4
Girault, (2011)
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.2003.12.01 – 10.1016/j.automatica.2003.12.017
Grmela M (2002) Lagrange hydrodynamics as extended Euler hydrodynamics: Hamiltonian and GENERIC structures. Physics Letters A 296(2–3):97–104. https://doi.org/10.1016/s0375-9601(02)00190- – 10.1016/s0375-9601(02)00190-1
Grmela M, Öttinger HC (1997) Dynamics and thermodynamics of complex fluids. I. Development of a general formalism. Phys Rev E 56(6):6620–6632. https://doi.org/10.1103/physreve.56.662 – 10.1103/physreve.56.6620
Gugercin S, Polyuga RV, Beattie C, van der Schaft A (2012) Structure-preserving tangential interpolation for model reduction of port-Hamiltonian systems. Automatica 48(9):1963–1974. https://doi.org/10.1016/j.automatica.2012.05.05 – 10.1016/j.automatica.2012.05.052
Haine G, Matignon D (2021) Incompressible Navier-Stokes Equation as port-Hamiltonian systems: velocity formulation versus vorticity formulation. IFAC-PapersOnLine 54(19):161–166. https://doi.org/10.1016/j.ifacol.2021.11.07 – 10.1016/j.ifacol.2021.11.072
Haine, Long-time behavior of a coupled heat-wave system using a structure-preserving finite element method. Math Rep (2022)
Haine G, Matignon D, Monteghetti F (2022) Structure-preserving discretization of Maxwell’s equations as a port-Hamiltonian system. IFAC-PapersOnLine 55(30):424–429. https://doi.org/10.1016/j.ifacol.2022.11.09 – 10.1016/j.ifacol.2022.11.090
Haine G, Matignon D, Serhani A (2023) Numerical Analysis of a Structure-Preserving Space-Discretization for an Anisotropic and Heterogeneous Boundary Controlled $N$-Dimensional Wave Equation as a Port-Hamiltonian System. IJNAM 20(1):92–133. https://doi.org/10.4208/ijnam2023-100 – 10.4208/ijnam2023-1005
Hamroun B, Dimofte A, Lefèvre L, Mendes E (2010) Control by Interconnection and Energy-Shaping Methods of Port Hamiltonian Models. Application to the Shallow Water Equations. European Journal of Control 16(5):545–563. https://doi.org/10.3166/ejc.16.545-56 – 10.3166/ejc.16.545-563
Hamroun, Port-based modelling for open channel irrigation systems. Trans Fluid Mech (2006)
Heidari H, Zwart H (2019) Port-Hamiltonian modelling of nonlocal longitudinal vibrations in a viscoelastic nanorod. Mathematical and Computer Modelling of Dynamical Systems 25(5):447–462. https://doi.org/10.1080/13873954.2019.165937 – 10.1080/13873954.2019.1659374
Hiemstra RR, Toshniwal D, Huijsmans RHM, Gerritsma MI (2014) High order geometric methods with exact conservation properties. Journal of Computational Physics 257:1444–1471. https://doi.org/10.1016/j.jcp.2013.09.02 – 10.1016/j.jcp.2013.09.027
Jacob B, Morris K (2022) On Solvability of Dissipative Partial Differential-Algebraic Equations. IEEE Control Syst Lett 6:3188–3193. https://doi.org/10.1109/lcsys.2022.318347 – 10.1109/lcsys.2022.3183479
Jacob, (2012)
Jongschaap RJJ (2001) The matrix model, a driven state variables approach to non-equilibrium thermodynamics. Journal of Non-Newtonian Fluid Mechanics 96(1–2):63–76. https://doi.org/10.1016/s0377-0257(00)00136- – 10.1016/s0377-0257(00)00136-1
Jongschaap R, Öttinger HC (2004) The mathematical representation of driven thermodynamic systems. Journal of Non-Newtonian Fluid Mechanics 120(1–3):3–9. https://doi.org/10.1016/j.jnnfm.2003.11.00 – 10.1016/j.jnnfm.2003.11.008
Kotyczka, (2019)
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.2018.02.00 – 10.1016/j.jcp.2018.02.006
Kraus, Metriplectic integrators for dissipative fluids. (2021)
Kunkel P, Mehrmann V (2006) Differential-Algebraic Equations. EMS Textbooks in Mathematic – 10.4171/017
Kurula, Linear wave systems on n-d spatial domains. Internat J Control (2015)
Lagrée, (2021)
Lamour, (2013)
Le Gorrec Y, Zwart H, Maschke B (2005) Dirac structures and Boundary Control Systems associated with Skew-Symmetric Differential Operators. SIAM J Control Optim 44(5):1864–1892. https://doi.org/10.1137/04061167 – 10.1137/040611677
Lequeurre J, Munnier A (2020) Vorticity and Stream Function Formulations for the 2D Navier–Stokes Equations in a Bounded Domain. J Math Fluid Mech 22(2). https://doi.org/10.1007/s00021-019-0479- – 10.1007/s00021-019-0479-5
Logg, (2012)
Lohmayer M, Kotyczka P, Leyendecker S (2021) Exergetic port-Hamiltonian systems: modelling basics. Mathematical and Computer Modelling of Dynamical Systems 27(1):489–521. https://doi.org/10.1080/13873954.2021.197959 – 10.1080/13873954.2021.1979592
Lopes N, Hélie T (2016) Energy Balanced Model of a Jet Interacting With a Brass Player’s Lip. Acta Acustica united with Acustica 102(1):141–154. https://doi.org/10.3813/aaa.91893 – 10.3813/aaa.918931
Marche F (2007) Derivation of a new two-dimensional viscous shallow water model with varying topography, bottom friction and capillary effects. European Journal of Mechanics - B/Fluids 26(1):49–63. https://doi.org/10.1016/j.euromechflu.2006.04.00 – 10.1016/j.euromechflu.2006.04.007
Maschke B, Schaft A van der (2020) Linear Boundary Port Hamiltonian Systems defined on Lagrangian submanifolds. IFAC-PapersOnLine 53(2):7734–7739. https://doi.org/10.1016/j.ifacol.2020.12.152 – 10.1016/j.ifacol.2020.12.1526
Mehrmann V, Unger B (2023) Control of port-Hamiltonian differential-algebraic systems and applications. Acta Numerica 32:395–515. https://doi.org/10.1017/s096249292200008 – 10.1017/s0962492922000083
Mehrmann V, van der Schaft A (2023) Differential–algebraic systems with dissipative Hamiltonian structure. Math Control Signals Syst 35(3):541–584. https://doi.org/10.1007/s00498-023-00349- – 10.1007/s00498-023-00349-2
Mehrmann, (2024)
Merker J, Krüger M (2012) On a variational principle in thermodynamics. Continuum Mech Thermodyn 25(6):779–793. https://doi.org/10.1007/s00161-012-0277- – 10.1007/s00161-012-0277-2
Mohamed MS, Hirani AN, Samtaney R (2016) Discrete exterior calculus discretization of incompressible Navier–Stokes equations over surface simplicial meshes. Journal of Computational Physics 312:175–191. https://doi.org/10.1016/j.jcp.2016.02.02 – 10.1016/j.jcp.2016.02.028
Mora LA, Gorrec YL, Matignon D, Ramirez H (2023) Irreversible port-Hamiltonian modelling of 3D compressible fluids. IFAC-PapersOnLine 56(2):6394–6399. https://doi.org/10.1016/j.ifacol.2023.10.83 – 10.1016/j.ifacol.2023.10.836
Mora LA, Le Gorrec Y, Matignon D, Ramirez H, Yuz JI (2021) On port-Hamiltonian formulations of 3-dimensional compressible Newtonian fluids. Physics of Fluids 33(11). https://doi.org/10.1063/5.006778 – 10.1063/5.0067784
Mora LA, Ramirez H, Yuz JI, Le Gorec Y, Zañartu M (2020) Energy-based fluid–structure model of the vocal folds. IMA Journal of Mathematical Control and Information 38(2):466–492. https://doi.org/10.1093/imamci/dnaa03 – 10.1093/imamci/dnaa031
Mora LA, Yann LG, Ramirez H, Yuz J (2020) Fluid-Structure Port-Hamiltonian Model for Incompressible Flows in Tubes with Time Varying Geometries. Mathematical and Computer Modelling of Dynamical Systems 26(5):409–433. https://doi.org/10.1080/13873954.2020.178684 – 10.1080/13873954.2020.1786841
Morandin, (2024)
Morrison PJ (1984) Bracket formulation for irreversible classical fields. Physics Letters A 100(8):423–427. https://doi.org/10.1016/0375-9601(84)90635- – 10.1016/0375-9601(84)90635-2
Morrison PJ (1998) Hamiltonian description of the ideal fluid. Rev Mod Phys 70(2):467–521. https://doi.org/10.1103/revmodphys.70.46 – 10.1103/revmodphys.70.467
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.2011.10.00 – 10.1016/j.jcp.2011.10.008
Mrugala R, Nulton JD, Christian Schön J, Salamon P (1991) Contact structure in thermodynamic theory. Reports on Mathematical Physics 29(1):109–121. https://doi.org/10.1016/0034-4877(91)90017- – 10.1016/0034-4877(91)90017-h
Vu NMT, Lefèvre L, Maschke B (2016) A structured control model for the thermo-magneto-hydrodynamics of plasmas in tokamaks. Mathematical and Computer Modelling of Dynamical Systems 22(3):181–206. https://doi.org/10.1080/13873954.2016.115487 – 10.1080/13873954.2016.1154874
Olver, (1993)
Öttinger HC (2018) GENERIC Integrators: Structure Preserving Time Integration for Thermodynamic Systems. Journal of Non-Equilibrium Thermodynamics 43(2):89–100. https://doi.org/10.1515/jnet-2017-003 – 10.1515/jnet-2017-0034
Öttinger HC, Grmela M (1997) Dynamics and thermodynamics of complex fluids. II. Illustrations of a general formalism. Phys Rev E 56(6):6633–6655. https://doi.org/10.1103/physreve.56.663 – 10.1103/physreve.56.6633
Pasumarthy, Port-Hamiltonian formulation of shallow water equations with coriolis force and topography. (2008)
Pasumarthy R, Ambati VR, van der Schaft AJ (2012) Port-Hamiltonian discretization for open channel flows. Systems & Control Letters 61(9):950–958. https://doi.org/10.1016/j.sysconle.2012.05.00 – 10.1016/j.sysconle.2012.05.003
Payen G, Matignon D, Haine G (2020) Modelling and structure-preserving discretization of Maxwell’s equations as port-Hamiltonian system. IFAC-PapersOnLine 53(2):7581–7586. https://doi.org/10.1016/j.ifacol.2020.12.135 – 10.1016/j.ifacol.2020.12.1355
Philipp, (2023)
Ramirez H, Gorrec YL (2016) An irreversible port-Hamiltonian formulation of distributed diffusion processes. IFAC-PapersOnLine 49(24):46–51. https://doi.org/10.1016/j.ifacol.2016.10.75 – 10.1016/j.ifacol.2016.10.752
Ramirez, An overview on irreversible port-Hamiltonian systems. Entropy (2022)
Ramirez H, Gorrec YL, Maschke B (2022) Boundary controlled irreversible port-Hamiltonian systems. Chemical Engineering Science 248:117107. https://doi.org/10.1016/j.ces.2021.11710 – 10.1016/j.ces.2021.117107
Ramirez H, Maschke B, Sbarbaro D (2013) Irreversible port-Hamiltonian systems: A general formulation of irreversible processes with application to the CSTR. Chemical Engineering Science 89:223–234. https://doi.org/10.1016/j.ces.2012.12.00 – 10.1016/j.ces.2012.12.002
Rashad, Port-Hamiltonian modeling of ideal fluid flow: Part I. Foundations and kinetic energy. J Geom Phys (2021)
Rashad, Port-Hamiltonian modeling of ideal fluid flow: Part II. Compressible and incompressible flow. J Geom Phys (2021)
Rashad R, Califano F, van der Schaft AJ, Stramigioli S (2020) Twenty years of distributed port-Hamiltonian systems: a literature review. IMA Journal of Mathematical Control and Information 37(4):1400–1422. https://doi.org/10.1093/imamci/dnaa01 – 10.1093/imamci/dnaa018
Reis, (2023)
Rettberg J, Wittwar D, Buchfink P, Brauchler A, Ziegler P, Fehr J, Haasdonk B (2023) Port-Hamiltonian fluid–structure interaction modelling and structure-preserving model order reduction of a classical guitar. Mathematical and Computer Modelling of Dynamical Systems 29(1):116–148. https://doi.org/10.1080/13873954.2023.217323 – 10.1080/13873954.2023.2173238
Schiebl M, Betsch P (2021) Structure‐preserving space‐time discretization of large‐strain thermo‐viscoelasticity in the framework of GENERIC. Numerical Meth Engineering 122(14):3448–3488. https://doi.org/10.1002/nme.667 – 10.1002/nme.6670
Serhani A, Matignon D, Haine G (2019) A Partitioned Finite Element Method for the Structure-Preserving Discretization of Damped Infinite-Dimensional Port-Hamiltonian Systems with Boundary Control. Lecture Notes in Computer Science 549–55 – 10.1007/978-3-030-26980-7_57
Serhani A, Haine G, Matignon D (2019) Anisotropic heterogeneous n-D heat equation with boundary control and observation: II. Structure-preserving discretization. IFAC-PapersOnLine 52(7):57–62. https://doi.org/10.1016/j.ifacol.2019.07.01 – 10.1016/j.ifacol.2019.07.010
Serhani A, Matignon D, Haine G (2019) Partitioned Finite Element Method for port-Hamiltonian systems with Boundary Damping: Anisotropic Heterogeneous 2D wave equations. IFAC-PapersOnLine 52(2):96–101. https://doi.org/10.1016/j.ifacol.2019.08.01 – 10.1016/j.ifacol.2019.08.017
Seslija M, van der Schaft A, Scherpen JMA (2012) Discrete exterior geometry approach to structure-preserving discretization of distributed-parameter port-Hamiltonian systems. Journal of Geometry and Physics 62(6):1509–1531. https://doi.org/10.1016/j.geomphys.2012.02.00 – 10.1016/j.geomphys.2012.02.006
Skrepek N (2021) Well-posedness of linear first order port-Hamiltonian Systems on multidimensional spatial domains. EECT 10(4):965–1006. https://doi.org/10.3934/eect.202009 – 10.3934/eect.2020098
Temam, (2001)
Thuburn J, Cotter CJ (2015) A primal–dual mimetic finite element scheme for the rotating shallow water equations on polygonal spherical meshes. Journal of Computational Physics 290:274–297. https://doi.org/10.1016/j.jcp.2015.02.04 – 10.1016/j.jcp.2015.02.045
Trenchant V, Ramirez H, Le Gorrec Y, Kotyczka P (2018) Finite differences on staggered grids preserving the port-Hamiltonian structure with application to an acoustic duct. Journal of Computational Physics 373:673–697. https://doi.org/10.1016/j.jcp.2018.06.05 – 10.1016/j.jcp.2018.06.051
van der Schaft, (2013)
van der Schaft A (2021) Classical Thermodynamics Revisited: A Systems and Control Perspective. IEEE Control Syst 41(5):32–60. https://doi.org/10.1109/mcs.2021.309280 – 10.1109/mcs.2021.3092809
van der Schaft A, Jeltsema D (2014) Port-Hamiltonian Systems Theory: An Introductory Overview. Foundations and Trends® in Systems and Control 1(2–3):173–378. https://doi.org/10.1561/260000000 – 10.1561/2600000002
Van Der Schaft AJ, Maschke BM (1994) On the Hamiltonian formulation of nonholonomic mechanical systems. Reports on Mathematical Physics 34(2):225–233. https://doi.org/10.1016/0034-4877(94)90038- – 10.1016/0034-4877(94)90038-8
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- – 10.1016/s0393-0440(01)00083-3
van der Schaft A, Maschke B (2018) Generalized port-Hamiltonian DAE systems. Systems & Control Letters 121:31–37. https://doi.org/10.1016/j.sysconle.2018.09.00 – 10.1016/j.sysconle.2018.09.008
van der Schaft, Geometry of thermodynamic processes. Entropy (2018)
van der Schaft A, Mehrmann V (2023) Linear port-Hamiltonian DAE systems revisited. Systems & Control Letters 177:105564. https://doi.org/10.1016/j.sysconle.2023.10556 – 10.1016/j.sysconle.2023.105564
Vu NMT, Lefèvre L, Maschke B (2019) Geometric spatial reduction for port-Hamiltonian systems. Systems & Control Letters 125:1–8. https://doi.org/10.1016/j.sysconle.2019.01.00 – 10.1016/j.sysconle.2019.01.002
Vu NMT, Lefèvre L, Nouailletas R, Brémond S (2017) Symplectic spatial integration schemes for systems of balance equations. Journal of Process Control 51:1–17. https://doi.org/10.1016/j.jprocont.2016.12.00 – 10.1016/j.jprocont.2016.12.005
Warsewa A, Böhm M, Sawodny O, Tarín C (2021) A port-Hamiltonian approach to modeling the structural dynamics of complex systems. Applied Mathematical Modelling 89:1528–1546. https://doi.org/10.1016/j.apm.2020.07.03 – 10.1016/j.apm.2020.07.038
Zhang Y, Palha A, Gerritsma M, Rebholz LG (2022) A mass-, kinetic energy- and helicity-conserving mimetic dual-field discretization for three-dimensional incompressible Navier-Stokes equations, part I: Periodic domains. Journal of Computational Physics 451:110868. https://doi.org/10.1016/j.jcp.2021.11086 – 10.1016/j.jcp.2021.110868