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

Antoine Bendimerad-Hohl, Ghislain Haine, Laurent Lefèvre, Denis Matignon

TL;DR

This work develops a structure-preserving space discretization framework for distributed port-Hamiltonian systems with implicit energy constitutive relations, by combining Stokes-Dirac and Stokes-Lagrange structures with a Partitioned Finite Element Method. It applies the approach to a 1D nonlocal nanorod, an implicit Euler–Bernoulli beam, and a 2D incompressible Navier–Stokes model in vorticity–stream function form, demonstrating exact-like preservation of power, energy, and, in the fluid case, enstrophy, at semi- and fully discrete levels. Latent-state formulations enable explicit Hamiltonians and consistent boundary-port handling, including Robin boundary conditions that contribute to energy accounting. The numerical results confirm robustness, convergence, and physical fidelity, highlighting the method’s potential for long-term, structure-preserving simulations in multiphysics settings and guiding future analytic and 3D extensions.

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.

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

TL;DR

This work develops a structure-preserving space discretization framework for distributed port-Hamiltonian systems with implicit energy constitutive relations, by combining Stokes-Dirac and Stokes-Lagrange structures with a Partitioned Finite Element Method. It applies the approach to a 1D nonlocal nanorod, an implicit Euler–Bernoulli beam, and a 2D incompressible Navier–Stokes model in vorticity–stream function form, demonstrating exact-like preservation of power, energy, and, in the fluid case, enstrophy, at semi- and fully discrete levels. Latent-state formulations enable explicit Hamiltonians and consistent boundary-port handling, including Robin boundary conditions that contribute to energy accounting. The numerical results confirm robustness, convergence, and physical fidelity, highlighting the method’s potential for long-term, structure-preserving simulations in multiphysics settings and guiding future analytic and 3D extensions.

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.

Paper Structure

This paper contains 54 sections, 22 theorems, 147 equations, 15 figures, 1 table.

Table of Contents

  1. Introduction
  2. Dirac and Lagrange structures
  3. Finite-dimensional case
  4. A toy model: a moving mass
  5. Infinite-dimensional case
  6. Continuous models
  7. Nanorod equation
  8. Nonlocal constitutive relations
  9. Implicit nanorod equation
  10. Implicit Euler-Bernoulli beam
  11. Incompressible Navier-Stokes equation
  12. Port-Hamiltonian representation
  13. Boundary conditions
  14. Enstrophy creation and kinetic energy dissipation
  15. Spatial Discretization
  16. ...and 39 more sections

Key Result

Lemma 2

Let $P$ and $S$ as in Definition def:linear-ph-state-repr. Let $\alpha,e \in \mathbb{R}^n$ and $y_L,u_L\in \mathbb{R}^{n_L}$, satisfying eqn:discr-linear-phs:LAGRANGE. Then, there exists a unique pair $(\lambda, \tilde{u}_L ) \in \mathbb{R}^{n} {\color{black} \times \mathbb{R}^{n_L}}$ such that: These variables are called latent state and latent control, respectively.

Figures (15)

  • Figure 1: Reduced complex of the Navier-Stokes equation on a 2D domain, with $\Lambda^k(\Omega)$ denoting the space of $k$-forms over $\Omega$.
  • Figure 2: Plots of the nanorod Hamiltonian and relative energy error for $\ell = 0,0.01,0.05$.
  • Figure 3: Phase velocity for $r=5~$cm (left) and $r=2.5~$cm (right), and a mesh size parameter $\textup{d} x = 5.0\times10^{-4}$ (implicit Euler-Bernoulli beam).
  • Figure 4: Snapshot of the implicit (left) and explicit (right) Euler-Bernoulli beam of radius $r=5$ cm with simply supported boundary conditions, at time $t=10$ ms. Parameters: $(h, D) = ( 7.85\times10^{-3}, 8.95\times10^{+3} )$, mesh size $\textup{d} x = 5.0\times10^{-4}$.
  • Figure 5: $L^2$ difference between the beam position computed with the explicit and implicit model over time.
  • ...and 10 more figures

Theorems & Definitions (69)

  • Definition 1
  • Remark 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Remark 2
  • Remark 3
  • Definition 4
  • Remark 4
  • ...and 59 more