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

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.

To cite this paper:
Bendimerad-Hohl Antoine, Haine Ghislain, Lefèvre Laurent, Matignon Denis (2026) Structure-preserving space discretization of differential and nonlocal constitutive relations for port-Hamiltonian systems. Journal of Computational Physics; X(X):PP–PP

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