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Structure preserving discretization method for 1D and 2D port-Hamiltonian systems using finite differences on staggered grids

Ignacio Diaz Alastuey, Yann Le Gorrec, Yongxin Wu

TL;DR

The article develops a structure-preserving finite-difference discretization on staggered grids for port-Hamiltonian systems in 1D and extends it to 2D on rectangular domains. It generalizes prior wave-equation discretizations to a broader class of interconnection operators, including non-differential terms, by partitioning the state into two staggered sets and constructing a discrete Hamiltonian that preserves energy. The authors demonstrate energy-preserving PH-ODE formulations for both 1D (e.g., Timoshenko beam) and 2D (e.g., Mindlin plate) systems and validate the approach with numerical examples showing energy conservation. They also discuss limitations to rectangular geometries and outline future work on nonlinearities, triangular grids, and more general interconnections across ports.

Abstract

This paper extends previous work on finitedifference schemes over staggered grids for infinite-dimensional port-Hamiltonian systems. In the one-dimensional setting, it generalizes the discretization approach originally developed for the wave equation to a broader class of systems characterized by interconnection operators that include both differential and non-differential terms, such as the Timoshenko beam equation. The paper then introduces a discretization strategy for the two-dimensional case that requires only two grids, thereby accommodating a wider range of systems, including those whose interconnection operators contain non-differential components, such as the Mindlin plate model.

Structure preserving discretization method for 1D and 2D port-Hamiltonian systems using finite differences on staggered grids

TL;DR

The article develops a structure-preserving finite-difference discretization on staggered grids for port-Hamiltonian systems in 1D and extends it to 2D on rectangular domains. It generalizes prior wave-equation discretizations to a broader class of interconnection operators, including non-differential terms, by partitioning the state into two staggered sets and constructing a discrete Hamiltonian that preserves energy. The authors demonstrate energy-preserving PH-ODE formulations for both 1D (e.g., Timoshenko beam) and 2D (e.g., Mindlin plate) systems and validate the approach with numerical examples showing energy conservation. They also discuss limitations to rectangular geometries and outline future work on nonlinearities, triangular grids, and more general interconnections across ports.

Abstract

This paper extends previous work on finitedifference schemes over staggered grids for infinite-dimensional port-Hamiltonian systems. In the one-dimensional setting, it generalizes the discretization approach originally developed for the wave equation to a broader class of systems characterized by interconnection operators that include both differential and non-differential terms, such as the Timoshenko beam equation. The paper then introduces a discretization strategy for the two-dimensional case that requires only two grids, thereby accommodating a wider range of systems, including those whose interconnection operators contain non-differential components, such as the Mindlin plate model.

Paper Structure

This paper contains 12 sections, 48 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Discretization of a class of 1D port-Hamiltonian systems
  3. Class of systems
  4. Discretization
  5. Energy and Structure preserving interconnection
  6. Extension to 2D port-Hamiltonian systems
  7. Class of systems
  8. Discretization
  9. Numerical Examples
  10. Example: Timoshenko beam application
  11. Example: Mindlin plate application
  12. Conclusions

Figures (4)

  • Figure 1: Example of discretization scheme.
  • Figure 2: Example of discretization scheme.
  • Figure 3: Timoshenko beam simulation.
  • Figure 4: Hamiltonian over time for 1D and 2D simulations.