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

TL;DR

This paper develops a structure-preserving finite element discretization method for port-Hamiltonian systems with differential and nonlocal constitutive relations, ensuring the preservation of physical quantities like energy.

Contribution

It introduces a novel discretization approach based on Stokes-Lagrange structures, extending to nonlinear 2D Navier-Stokes equations with proven energy conservation.

Findings

Finite element discretization preserves enstrophy and kinetic energy.

Method successfully applied to 1D nanorod and shear beam models.

Extended to nonlinear 2D Navier-Stokes equations with structure preservation.

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.