Port-Hamiltonian formulation and symplectic discretization of Plate models. Part I : Mindlin model for thick plates

TL;DR

This paper develops a port-Hamiltonian tensorial formulation for the Mindlin plate model, extending structure-preserving discretization methods and validating them through numerical examples.

Contribution

It introduces a tensorial port-Hamiltonian formulation for thick plates and extends the Partitioned Finite Element Method to preserve system structure.

Findings

Tensorial formulation mimics interconnection structure of 1D models

Structure-preserving discretization maintains properties of the original system

Numerical validation confirms effectiveness of the approach

Abstract

The port-Hamiltonian formulation is a powerful method for modeling and interconnecting systems of different natures. In this paper, the port-Hamiltonian formulation in tensorial form of a thick plate described by the Mindlin-Reissner model is presented. Boundary control and observation are taken into account. Thanks to tensorial calculus, it can be seen that the Mindlin plate model mimics the interconnection structure of its one-dimensional counterpart, i.e. the Timoshenko beam. The Partitioned Finite Element Method (PFEM) is then extended to both the vectorial and tensorial formulations in order to obtain a suitable, i.e. structure-preserving, finite-dimensional port-Hamiltonian system (PHs), which preserves the structure and properties of the original distributed parameter system. Mixed boundary conditions are finally handled by introducing some algebraic constraints. Numerical examples are finally presented to validate this approach.