Authors

Cristobal Ponce, Yongxin Wu, Yann Le Gorrec, Hector Ramirez

Abstract

This article presents a systematic modeling methodology for deriving the infinite-dimensional port-Hamiltonian representation of geometrically nonlinear and hyperelastic systems, and a structure-preserving mixed FEM approach. The proposed methods provide a rigorous framework for obtaining the dynamic nonlinear partial differential equations governing these systems, ensuring that they are consistent with a Stokes–Dirac geometric structure. This structure is fundamental for modular multiphysics modeling and nonlinear passivity-based control. The modeling methodology is rooted in a total Lagrangian formulation, incorporating Green–Lagrange strains and second Piola–Kirchhoff stresses, where generalized displacements and strains define the interconnection structure. Using the generalized Hamilton’s principle, infinite-dimensional port-Hamiltonian systems are systematically derived. To preserve the structure upon spatial discretization, a three-field mixed finite element approach is proposed, in which displacements, strains, and stresses are explicitly treated as independent variables to retain the port-Hamiltonian structure. The effectiveness of the framework is demonstrated through model derivation and simulations, using a geometrically nonlinear planar beam with Saint Venant–Kirchhoff material, and a compressible nonlinear 2D elasticity problem with a Neo-Hookean material model, as illustrative examples.

Keywords

Port-Hamiltonian systems; Modeling; Structure-preserving discretization; Nonlinear elastodynamics; Hyperelasticity

Citation

  • Journal: Applied Mathematical Modelling
  • Year: 2026
  • Volume: 150
  • Issue:
  • Pages: 116403
  • Publisher: Elsevier BV
  • DOI: 10.1016/j.apm.2025.116403

BibTeX

@article{Ponce_2026,
  title={{A port-Hamiltonian framework for the modeling and FEM discretization of hyperelastic systems}},
  volume={150},
  ISSN={0307-904X},
  DOI={10.1016/j.apm.2025.116403},
  journal={Applied Mathematical Modelling},
  publisher={Elsevier BV},
  author={Ponce, Cristobal and Wu, Yongxin and Le Gorrec, Yann and Ramirez, Hector},
  year={2026},
  pages={116403}
}

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References