Structure -preserving space discretization of differential and nonlocal constitutive relations for port-Hamiltonian systems

Authors

Antoine Bendimerad-Hohl, Ghislain Haine, Laurent Lefèvre, Denis Matignon

Abstract

We study the structure-preserving space discretization of port-Hamiltonian (pH) systems defined with differential constitutive relations. Using the concept of Stokes-Lagrange structure to describe these relations, these are reduced to a finite-dimensional Lagrange subspace of a pH system thanks to a structure-preserving Finite Element Method. To illustrate our results, the 1D nanorod case and the shear beam model are considered, which are given by differential and implicit constitutive relations for which a Stokes-Lagrange structure along with boundary energy ports naturally occur. Then, these results are extended to the nonlinear 2D incompressible Navier-Stokes equations written in a vorticity–stream function formulation. It is first recast as a pH system defined with a Stokes-Lagrange structure along with a modulated Stokes-Dirac structure. A careful structure-preserving space discretization is then performed, leading to a finite-dimensional pH system. Theoretical and numerical results show that both enstrophy and kinetic energy evolutions are preserved both at the semi-discrete and fully-discrete levels.

Keywords

euler-bernoulli beam, incompressible navier-stokes equation, nonlocal constitutive relations, port-hamiltonian systems, structure-preserving discretization

Citation

• Journal: Journal of Computational Physics
• Year: 2026
• Volume: 560
• Issue:
• Pages: 114951
• Publisher: Elsevier BV
• DOI: 10.1016/j.jcp.2026.114951

BibTeX

@article{Bendimerad_Hohl_2026,
  title={{Structure -preserving space discretization of differential and nonlocal constitutive relations for port-Hamiltonian systems}},
  volume={560},
  ISSN={0021-9991},
  DOI={10.1016/j.jcp.2026.114951},
  journal={Journal of Computational Physics},
  publisher={Elsevier BV},
  author={Bendimerad-Hohl, Antoine and Haine, Ghislain and Lefèvre, Laurent and Matignon, Denis},
  year={2026},
  pages={114951}
}

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References

• 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
• Duindam, (2009)
• 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
• 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
• 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
• 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
• 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
• 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
• Ponce C, Wu Y, Le Gorrec Y, Ramirez H (2024) A systematic methodology for port-Hamiltonian modeling of multidimensional flexible linear mechanical systems. Applied Mathematical Modelling 134:434–451. https://doi.org/10.1016/j.apm.2024.05.04 – 10.1016/j.apm.2024.05.040
• Cardoso-Ribeiro FL, Haine G, Le Gorrec Y, Matignon D, Ramirez H (2024) Port-Hamiltonian formulations for the modeling, simulation and control of fluids. Computers & Fluids 283:106407. https://doi.org/10.1016/j.compfluid.2024.10640 – 10.1016/j.compfluid.2024.106407
• Farle, A port-Hamiltonian finite-element formulation for the Maxwell equations. (2013)
• 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
• Vincent B, Couenne F, Lefèvre L, Maschke B (2020) Port Hamiltonian systems with moving interface: a phase field approach. IFAC-PapersOnLine 53(2):7569–7574. https://doi.org/10.1016/j.ifacol.2020.12.135 – 10.1016/j.ifacol.2020.12.1353
• Bendimerad-Hohl, Structure-preserving discretization of the Cahn-Hilliard equations recast as a port-Hamiltonian system. (2023)
• 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 A, Maschke B (2020) Dirac and Lagrange Algebraic Constraints in Nonlinear Port-Hamiltonian Systems. Vietnam J Math 48(4):929–939. https://doi.org/10.1007/s10013-020-00419- – 10.1007/s10013-020-00419-x
• Krhač K, Maschke B, van der Schaft A (2024) Port-Hamiltonian systems with energy and power ports. IFAC-PapersOnLine 58(6):280–285. https://doi.org/10.1016/j.ifacol.2024.08.29 – 10.1016/j.ifacol.2024.08.294
• van der Schaft A, Mehrmann V (2022) Linear Port-Hamiltonian DAE Systems Revisited. SSRN Journal. https://doi.org/10.2139/ssrn.429299 – 10.2139/ssrn.4292998
• Bendimerad-Hohl A, Haine G, Lefèvre L, Matignon D (2025) Stokes-Lagrange and Stokes-Dirac representations of $ N $-dimensional port-Hamiltonian systems for modeling and control. CAM 17(2):474–519. https://doi.org/10.3934/cam.202502 – 10.3934/cam.2025020
• Yaghi M, Couenne F, Galfré A, Lefèvre L, Maschke B (2022) Port Hamiltonian formulation of the solidification process for a pure substance: A phase field approach*. IFAC-PapersOnLine 55(18):93–98. https://doi.org/10.1016/j.ifacol.2022.08.03 – 10.1016/j.ifacol.2022.08.036
• Dzektser, Generalization of the equation of motion of ground waters with free surface. Dokl. Akad. Nauk SSSR (1972)
• 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
• 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
• Heidari H, Zwart H (2022) Nonlocal longitudinal vibration in a nanorod, a system theoretic analysis. Math Model Nat Phenom 17:24. https://doi.org/10.1051/mmnp/202202 – 10.1051/mmnp/2022028
• McLachlan RI, Quispel GRW, Robidoux N (1999) Geometric integration using discrete gradients. Philosophical Transactions of the Royal Society of London Series A: Mathematical, Physical and Engineering Sciences 357(1754):1021–1045. https://doi.org/10.1098/rsta.1999.036 – 10.1098/rsta.1999.0363
• Gonzalez O (1996) Time integration and discrete Hamiltonian systems. J Nonlinear Sci 6(5):449–467. https://doi.org/10.1007/bf0244016 – 10.1007/bf02440162
• Aoues S, Di Loreto M, Eberard D, Marquis-Favre W (2017) Hamiltonian systems discrete-time approximation: Losslessness, passivity and composability. Systems & Control Letters 110:9–14. https://doi.org/10.1016/j.sysconle.2017.10.00 – 10.1016/j.sysconle.2017.10.003
• Kotyczka P, Lefèvre L (2019) Discrete-time port-Hamiltonian systems: A definition based on symplectic integration. Systems & Control Letters 133:104530. https://doi.org/10.1016/j.sysconle.2019.10453 – 10.1016/j.sysconle.2019.104530
• Mehrmann, Structure-preserving discretization for port-Hamiltonian descriptor systems. (2019)
• Celledoni E, Jackaman J (2021) Discrete conservation laws for finite element discretisations of multisymplectic PDEs. Journal of Computational Physics 444:110520. https://doi.org/10.1016/j.jcp.2021.11052 – 10.1016/j.jcp.2021.110520
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• Boffi, (2013)
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• Brugnoli A, Rashad R, Califano F, Stramigioli S, Matignon D (2021) Mixed finite elements for port-Hamiltonian models of von Kármán beams. IFAC-PapersOnLine 54(19):186–191. https://doi.org/10.1016/j.ifacol.2021.11.07 – 10.1016/j.ifacol.2021.11.076
• Brugnoli A, Alazard D, Pommier-Budinger V, Matignon D (2021) A Port-Hamiltonian formulation of linear thermoelasticity and its mixed finite element discretization. Journal of Thermal Stresses 44(6):643–661. https://doi.org/10.1080/01495739.2021.191732 – 10.1080/01495739.2021.1917322
• 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
• {“status”:”error” – 10.1016/j.ifacol.2021.11.072
• Cardoso-Ribeiro FL, Haine G, Lefèvre L, Matignon D (2024) Rotational shallow water equations with viscous damping and boundary control: structure-preserving spatial discretization. Math Control Signals Syst 37(2):361–394. https://doi.org/10.1007/s00498-024-00404- – 10.1007/s00498-024-00404-6
• 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
• 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
• 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
• Eringen AC (1983) On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. Journal of Applied Physics 54(9):4703–4710. https://doi.org/10.1063/1.33280 – 10.1063/1.332803
• Ducceschi M, Bilbao S (2019) Conservative finite difference time domain schemes for the prestressed Timoshenko, shear and Euler–Bernoulli beam equations. Wave Motion 89:142–165. https://doi.org/10.1016/j.wavemoti.2019.03.00 – 10.1016/j.wavemoti.2019.03.006
• Ducceschi M, Bilbao S (2016) Linear stiff string vibrations in musical acoustics: Assessment and comparison of models. The Journal of the Acoustical Society of America 140(4):2445–2454. https://doi.org/10.1121/1.496255 – 10.1121/1.4962553
• Boyer, (2012)
• Clercx HJH, Bruneau C-H (2006) The normal and oblique collision of a dipole with a no-slip boundary. Computers & Fluids 35(3):245–279. https://doi.org/10.1016/j.compfluid.2004.11.00 – 10.1016/j.compfluid.2004.11.009
• de Diego GG, Palha A, Gerritsma M (2019) Inclusion of no-slip boundary conditions in the MEEVC scheme. Journal of Computational Physics 378:615–633. https://doi.org/10.1016/j.jcp.2018.11.02 – 10.1016/j.jcp.2018.11.025
• Gernandt, On the equivalence of geometric and descriptor representations of linear port-Hamiltonian systems. (2022)
• Gangi AF (2000) Constitutive equations and reciprocity. Geophysical Journal International 143(2):311–318. https://doi.org/10.1046/j.1365-246x.2000.01259. – 10.1046/j.1365-246x.2000.01259.x
• Jacob, (2012)
• Kurula M, Zwart H, van der Schaft A, Behrndt J (2010) Dirac structures and their composition on Hilbert spaces. Journal of Mathematical Analysis and Applications 372(2):402–422. https://doi.org/10.1016/j.jmaa.2010.07.00 – 10.1016/j.jmaa.2010.07.004
• Gernandt H, Philipp FM, Preuster T, Schaller M (2026) Extension Theory Via Boundary Triplets for Infinite-Dimensional Implicit Port-Hamiltonian Systems. Integr Equ Oper Theory 98(2). https://doi.org/10.1007/s00020-026-02841- – 10.1007/s00020-026-02841-1
• Bendimerad-Hohl A, Matignon D, Haine G, Lefèvre L (2024) On Stokes-Lagrange and Stokes-Dirac representations for 1D distributed port-Hamiltonian systems. IFAC-PapersOnLine 58(17):238–243. https://doi.org/10.1016/j.ifacol.2024.10.17 – 10.1016/j.ifacol.2024.10.174
• Brugnoli A, Mehrmann V (2024) On the discrete equivalence of Lagrangian, Hamiltonian and mixed finite element formulations for linear wave phenomena. IFAC-PapersOnLine 58(6):95–100. https://doi.org/10.1016/j.ifacol.2024.08.26 – 10.1016/j.ifacol.2024.08.263
• Mirouze I, Weaver AT (2010) Representation of correlation functions in variational assimilation using an implicit diffusion operator. Quart J Royal Meteoro Soc 136(651):1421–1443. https://doi.org/10.1002/qj.64 – 10.1002/qj.643
• Guillet O, Weaver AT, Vasseur X, Michel Y, Gratton S, Gürol S (2019) Modelling spatially correlated observation errors in variational data assimilation using a diffusion operator on an unstructured mesh. Quart J Royal Meteoro Soc 145(722):1947–1967. https://doi.org/10.1002/qj.353 – 10.1002/qj.3537
• Naylor, (1982)
• Romano, Nonlocal elasticity in nanobeams: the stress-driven integral model. Int. J. Eng. Sci. (2017)
• Assous, (2018)
• Arnold, (2018)
• Palha A, Gerritsma M (2017) A mass, energy, enstrophy and vorticity conserving (MEEVC) mimetic spectral element discretization for the 2D incompressible Navier–Stokes equations. Journal of Computational Physics 328:200–220. https://doi.org/10.1016/j.jcp.2016.10.00 – 10.1016/j.jcp.2016.10.009
• Zhang Y, Palha A, Gerritsma M, Yao Q (2024) A MEEVC discretization for two-dimensional incompressible Navier-Stokes equations with general boundary conditions. Journal of Computational Physics 510:113080. https://doi.org/10.1016/j.jcp.2024.11308 – 10.1016/j.jcp.2024.113080
• Roze D, Raibaud M, Geoffroy T (2024) Passive-guaranteed modeling and simulation of a finite element nonlinear string model. IFAC-PapersOnLine 58(6):226–231. https://doi.org/10.1016/j.ifacol.2024.08.28 – 10.1016/j.ifacol.2024.08.285
• 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
• Rempfer D (2006) On Boundary Conditions for Incompressible Navier-Stokes Problems. Applied Mechanics Reviews 59(3):107–125. https://doi.org/10.1115/1.217768 – 10.1115/1.2177683
• E W, Liu J-G (1996) Vorticity Boundary Condition and Related Issues for Finite Difference Schemes. Journal of Computational Physics 124(2):368–382. https://doi.org/10.1006/jcph.1996.006 – 10.1006/jcph.1996.0066
• Renard Y, Poulios K (2020) GetFEM. ACM Trans Math Softw 47(1):1–31. https://doi.org/10.1145/341284 – 10.1145/3412849
• Lagnese, Boundary stabilization of thin plates. (1989)