Port-Hamiltonian formulation and structure-preserving discretization of hyperelastic strings

TL;DR

This paper develops a port-Hamiltonian framework for geometrically exact hyperelastic strings with nonlinear material behavior, including structure-preserving discretization and energy-consistent time-stepping for simulation and control.

Contribution

It introduces a novel port-Hamiltonian representation for nonlinear hyperelastic strings and develops a structure-preserving discretization method that retains the PH structure at the discrete level.

Findings

The semi-discrete model preserves the port-Hamiltonian structure.

Energy-consistent time-stepping method is developed.

Numerical experiments demonstrate the model's effectiveness.

Abstract

Port-Hamiltonian (PH) systems provide a framework for modeling, analysis and control of complex dynamical systems, where the complexity might result from multi-physical couplings, non-trivial domains and diverse nonlinearities. A major benefit of the PH representation is the explicit formulation of power interfaces, so-called ports, which allow for a power-preserving interconnection of subsystems to compose flexible multibody systems in a modular way. In this work, we present a PH representation of geometrically exact strings with nonlinear material behaviour. Furthermore, using structure-preserving discretization techniques a corresponding finite-dimensional PH state space model is developed. Applying mixed finite elements, the semi-discrete model retains the PH structure and the ports (pairs of velocities and forces) on the discrete level. Moreover, discrete derivatives are used in order to obtain an energy-consistent time-stepping method. The numerical properties of the newly devised model are investigated in a representative example. The developed PH state space model can be used for structure-preserving simulation and model order reduction as well as feedforward and feedback control design.