A port-Hamiltonian framework for the modeling and FEM discretization of hyperelastic systems

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.