Explicit Port-Hamiltonian FEM-Models for Linear Mechanical Systems with Non-Uniform Boundary Conditions

TL;DR

This paper develops explicit port-Hamiltonian finite element models for linear mechanical systems with non-uniform boundary conditions, enabling better boundary condition handling and system analysis.

Contribution

It introduces a method to derive explicit port-Hamiltonian state space models using mixed finite element discretization with weak boundary conditions.

Findings

Explicit models reveal boundary degrees of freedom as inputs

Numerical experiments validate the approach on a 1D rod

Method preserves skew-adjointness after discretization

Abstract

In this contribution we present how to obtain explicit state space models in port-Hamiltonian form when a mixed finite element method is applied to a linear mechanical system with non-uniform boundary conditions. The key is to express the variational problem based on the principle of virtual power, with both the Dirichlet (velocity) and Neumann (stress) boundary conditions imposed in a weak sense. As a consequence, the formal skew-adjointness of the system operator becomes directly visible after integration by parts, and, after compatible FE discretization, the boundary degrees of freedom of both causalities appear as explicit inputs in the resulting state space model. The rationale behind our formulation is illustrated using a lumped parameter example, and numerical experiments on a one-dimensional rod show the properties of the approach in practice.