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arXiv · 2507.06869

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

Antoine Bendimerad-Hohl·Ghislain Haine·Laurent Lefèvre·Denis Matignon
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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.

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Antoine Bendimerad-Hohl, Ghislain Haine, Laurent Lefèvre, Denis Matignon. 2025-07-09. Structure-preserving space discretization of differential and nonlocal constitutive relations for port-Hamiltonian systems. https://arxiv.org/abs/2507.06869

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