Structure Preserving Discretization of 1D Nonlinear Port-Hamiltonian Distributed Parameter Systems

TL;DR

This paper introduces a novel discretization method for nonlinear port-Hamiltonian distributed systems that preserves their geometric and energetic structure during spatial discretization, applicable to nonuniform meshes.

Contribution

It presents a new formal approach for structure-preserving discretization of nonlinear port-Hamiltonian systems on 1D manifolds, maintaining key physical properties.

Findings

Preserves Dirac structure in discretization

Ensures energy balance in the discretized model

Applicable to nonuniform spatial meshes

Abstract

This paper contributes with a new formal method of spatial discretization of a class of nonlinear distributed parameter systems that allow a port-Hamiltonian representation over a one dimensional manifold. A specific finite dimensional port-Hamiltonian element is defined that enables a structure preserving discretization of the infinite dimensional model that inherits the Dirac structure, the underlying energy balance and matches the Hamiltonian function on any, possibly nonuniform mesh of the spatial geometry.