Port-Hamiltonian formulation and structure-preserving discretization of finite elasticity based on a mixed Hu-Washizu-type formulation

Authors

Moritz Hille, Peter Betsch, Marlon Franke

Abstract

We propose a port-Hamiltonian formulation and structure-preserving discretization of finite elasticity. The energy functional (or Hamiltonian) is based on a polyconvex representation of the stored energy and gives rise to three strain-type fields, which play the role of energy variables in the port-Hamiltonian formulation. We show that a Hu-Washizu-type extension of the variational principle of Livens can be used (i) to derive the continuous port-Hamiltonian formulation and (ii) to perform a structure-preserving spatial discretization. In particular, we show that the spatial finite element discretization of the underlying mixed formulation yields a discrete port-Hamiltonian system. Moreover, the temporal discretization of the underlying continuous formulation yields a new energy-momentum consistent framework, which accommodates alternative finite element formulations. The new framework, in particular, covers mixed finite elements that have been shown to be well suited for handling quasi-incompressible material behavior. Numerical examples are provided to evaluate the numerical performance and stability of the newly devised energy-momentum schemes.

Keywords

energy-momentum methods, hu-washizu principle, livens principle, mixed finite elements, nonlinear elastodynamics, port-hamiltonian formulation

Citation

• Journal: Computer Methods in Applied Mechanics and Engineering
• Year: 2026
• Volume: 458
• Issue:
• Pages: 118790
• Publisher: Elsevier BV
• DOI: 10.1016/j.cma.2026.118790

BibTeX

@article{Hille_2026,
  title={{Port-Hamiltonian formulation and structure-preserving discretization of finite elasticity based on a mixed Hu-Washizu-type formulation}},
  volume={458},
  ISSN={0045-7825},
  DOI={10.1016/j.cma.2026.118790},
  journal={Computer Methods in Applied Mechanics and Engineering},
  publisher={Elsevier BV},
  author={Hille, Moritz and Betsch, Peter and Franke, Marlon},
  year={2026},
  pages={118790}
}

Download the bib file

References

• Modeling and Control of Complex Physical Systems. The Port-Hamiltonian Approach. (2009)
• 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
• 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- – 10.1016/s1474-6670(17)52308-3
• 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
• 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
• 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
• Kinon PL, Thoma T, Betsch P, Kotyczka P (2024) Generalized Maxwell viscoelasticity for geometrically exact strings: Nonlinear port-Hamiltonian formulation and structure-preserving discretization. IFAC-PapersOnLine 58(6):101–106. https://doi.org/10.1016/j.ifacol.2024.08.26 – 10.1016/j.ifacol.2024.08.264
• Cardoso-Ribeiro FL, Matignon D, Pommier-Budinger V (2016) Piezoelectric beam with distributed control ports: a power-preserving discretization using weak formulation.**The contribution of the authors has been done within the context of the French National Research Agency sponsored project HAMECMOPSYS. Further information is available at http://www.hamecmopsys.ens2m.fr/. IFAC-PapersOnLine 49(8):290–297. https://doi.org/10.1016/j.ifacol.2016.07.45 – 10.1016/j.ifacol.2016.07.456
• 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
• Ponce C, Ramirez H, Le Gorrec Y, Wu Y (2025) Constrained port-Hamiltonian modeling and structure-preserving discretization of the Rayleigh beam. IFAC-PapersOnLine 59(8):108–113. https://doi.org/10.1016/j.ifacol.2025.08.07 – 10.1016/j.ifacol.2025.08.075
• Kinon PL, Betsch P, Eugster SR (2025) Energy-momentum-consistent simulation of planar geometrically exact beams in a port-Hamiltonian framework. Multibody Syst Dyn. https://doi.org/10.1007/s11044-025-10087- – 10.1007/s11044-025-10087-9
• Brugnoli A, Alazard D, Pommier-Budinger V, Matignon D (2021) Structure-preserving discretization of port-Hamiltonian plate models. IFAC-PapersOnLine 54(9):359–364. https://doi.org/10.1016/j.ifacol.2021.06.09 – 10.1016/j.ifacol.2021.06.094
• 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.2018.06.03 – 10.1016/j.ifacol.2018.06.033
• 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
• 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
• Thoma T, Kotyczka P, Egger H (2024) On the velocity-stress formulation for geometrically nonlinear elastodynamics and its structure-preserving discretization. Mathematical and Computer Modelling of Dynamical Systems 30(1):701–720. https://doi.org/10.1080/13873954.2024.239748 – 10.1080/13873954.2024.2397486
• Brugnoli A, Matignon D, Morlier J (2025) A linearly-implicit energy-momentum preserving scheme for geometrically nonlinear mechanics based on non-canonical Hamiltonian formulations. Nonlinear Dyn 113(20):27539–27566. https://doi.org/10.1007/s11071-025-11601- – 10.1007/s11071-025-11601-6
• Ponce, A port-hamiltonian framework for the modeling and FEM discretization of hyperelastic systems. Appl. Math. Model. (2025)
• Simo JC, Tarnow N (1992) The discrete energy-momentum method. Conserving algorithms for nonlinear elastodynamics. Z angew Math Phys 43(5):757–792. https://doi.org/10.1007/bf0091340 – 10.1007/bf00913408
• Gonzalez O (2000) Exact energy and momentum conserving algorithms for general models in nonlinear elasticity. Computer Methods in Applied Mechanics and Engineering 190(13–14):1763–1783. https://doi.org/10.1016/s0045-7825(00)00189- – 10.1016/s0045-7825(00)00189-4
• Romero I (2012) An analysis of the stress formula for energy-momentum methods in nonlinear elastodynamics. Comput Mech 50(5):603–610. https://doi.org/10.1007/s00466-012-0693- – 10.1007/s00466-012-0693-y
• Betsch P, Steinmann P (2001) Conservation properties of a time FE method—part II: Time‐stepping schemes for non‐linear elastodynamics. Numerical Meth Engineering 50(8):1931–1955. https://doi.org/10.1002/nme.10 – 10.1002/nme.103
• Groß M, Betsch P, Steinmann P (2005) Conservation properties of a time FE method. Part IV: Higher order energy and momentum conserving schemes. Int J Numer Meth Engng 63(13):1849–1897. https://doi.org/10.1002/nme.133 – 10.1002/nme.1339
• Betsch P, Janz A, Hesch C (2018) A mixed variational framework for the design of energy–momentum schemes inspired by the structure of polyconvex stored energy functions. Computer Methods in Applied Mechanics and Engineering 335:660–696. https://doi.org/10.1016/j.cma.2018.01.01 – 10.1016/j.cma.2018.01.013
• Schröder J, Wriggers P, Balzani D (2011) A new mixed finite element based on different approximations of the minors of deformation tensors. Computer Methods in Applied Mechanics and Engineering 200(49–52):3583–3600. https://doi.org/10.1016/j.cma.2011.08.00 – 10.1016/j.cma.2011.08.009
• Kraus A, Wriggers P, Viebahn N, Schröder J (2019) Low‐order locking‐free mixed finite element formulation with approximation of the minors of the deformation gradient. Numerical Meth Engineering 120(8):1011–1026. https://doi.org/10.1002/nme.616 – 10.1002/nme.6168
• de Boer, (1982)
• Bonet J, Gil AJ, Ortigosa R (2016) On a tensor cross product based formulation of large strain solid mechanics. International Journal of Solids and Structures 84:49–63. https://doi.org/10.1016/j.ijsolstr.2015.12.03 – 10.1016/j.ijsolstr.2015.12.030
• Kinon PL, Betsch P, Schneider S (2023) Structure-preserving integrators based on a new variational principle for constrained mechanical systems. Nonlinear Dyn 111(15):14231–14261. https://doi.org/10.1007/s11071-023-08522- – 10.1007/s11071-023-08522-7
• Washizu, (1975)
• Mehrmann, Structure-preserving discretization for port-hamiltonian descriptor systems. (2019)
• 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
• 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
• Greenspan D (1984) Conservative numerical methods for. Journal of Computational Physics 56(1):28–41. https://doi.org/10.1016/0021-9991(84)90081- – 10.1016/0021-9991(84)90081-0
• Wriggers, (2008)
• Franke M, Klein DK, Weeger O, Betsch P (2023) Advanced discretization techniques for hyperelastic physics-augmented neural networks. Computer Methods in Applied Mechanics and Engineering 416:116333. https://doi.org/10.1016/j.cma.2023.11633 – 10.1016/j.cma.2023.116333
• Franke M, Janz A, Schiebl M, Betsch P (2018) An energy momentum consistent integration scheme using a polyconvexity‐based framework for nonlinear thermo‐elastodynamics. Numerical Meth Engineering 115(5):549–577. https://doi.org/10.1002/nme.581 – 10.1002/nme.5816
• Gil AJ, Lee CH, Bonet J, Aguirre M (2014) A stabilised Petrov–Galerkin formulation for linear tetrahedral elements in compressible, nearly incompressible and truly incompressible fast dynamics. Computer Methods in Applied Mechanics and Engineering 276:659–690. https://doi.org/10.1016/j.cma.2014.04.00 – 10.1016/j.cma.2014.04.006
• Scovazzi G, Carnes B, Zeng X, Rossi S (2015) A simple, stable, and accurate linear tetrahedral finite element for transient, nearly, and fully incompressible solid dynamics: a dynamic variational multiscale approach. Numerical Meth Engineering 106(10):799–839. https://doi.org/10.1002/nme.513 – 10.1002/nme.5138
• Bonet J, Gil AJ, Lee CH, Aguirre M, Ortigosa R (2015) A first order hyperbolic framework for large strain computational solid dynamics. Part I: Total Lagrangian isothermal elasticity. Computer Methods in Applied Mechanics and Engineering 283:689–732. https://doi.org/10.1016/j.cma.2014.09.02 – 10.1016/j.cma.2014.09.024