Port-Hamiltonian formulation and symplectic discretization of plate models. Part II : Kirchhoff model for thin plates

TL;DR

This paper models thin plates as port-Hamiltonian systems using a novel Kirchhoff-Love formulation with tensorial calculus, and develops a structure-preserving discretization method suitable for control applications.

Contribution

It introduces a port-Hamiltonian formulation of the Kirchhoff-Love thin plate model using tensor calculus and extends the Partitioned Finite Element Method for structure-preserving discretization.

Findings

Port-Hamiltonian formulation of thin plates with boundary control.

Tensorial calculus reveals structural similarities with Euler-Bernoulli beam.

Finite-dimensional port-Hamiltonian system obtained via discretization.

Abstract

The mechanical model of a thin plate with boundary control and observation is presented as a port-Hamiltonian system (pHs), both in vectorial and tensorial forms: the Kirchhoff-Love model of a plate is described by using a Stokes-Dirac structure and this represents a novelty with respect to the existing literature. This formulation is carried out both in vectorial and tensorial forms. Thanks to tensorial calculus, this model is found to mimic the interconnection structure of its one-dimensional counterpart, i.e. the Euler-Bernoulli beam. The Partitioned Finite Element Method (PFEM) is then extended to obtain a suitable, i.e. structure-preserving, weak form. The discretization procedure, performed on the vectorial formulation, leads to a finite-dimensional port-Hamiltonian system. This part II of the companion paper extends part I, dedicated to the Mindlin model for thick plates. The thin plate model comes along with additional difficulties, because of the higher order of the differential operator under consideration.