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

TL;DR

This paper develops a structure-preserving finite difference discretization method for 1D and 2D port-Hamiltonian systems on staggered grids, broadening applicability to complex systems like Timoshenko beams and Mindlin plates.

Contribution

It generalizes discretization techniques for port-Hamiltonian systems to include non-differential operators and introduces a 2D scheme requiring only two grids.

Findings

Effective discretization of Timoshenko beam equations.

Two-grid scheme for 2D port-Hamiltonian systems.

Applicable to systems with differential and non-differential operators.

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.