Discrete gradient methods for port-Hamiltonian differential-algebraic equations

TL;DR

This paper develops structure-preserving discrete gradient methods tailored for port-Hamiltonian differential-algebraic equations, enabling accurate and stable numerical simulations of complex physical systems.

Contribution

It introduces a novel numerical scheme for semi-explicit DAE systems and extends discrete gradient methods to general port-Hamiltonian DAE frameworks.

Findings

Effective numerical scheme for semi-explicit DAEs

Application to multibody system dynamics

Demonstrated stability and structure preservation

Abstract

Discrete gradient methods are a powerful tool for the time discretization of dynamical systems, since they are structure-preserving regardless of the form of the total energy. In this work, we discuss the application of discrete gradient methods to the system class of nonlinear port-Hamiltonian differential-algebraic equations - as they emerge from the port- and energy-based modeling of physical systems in various domains. We introduce a novel numerical scheme tailored for semi-explicit differential-algebraic equations and further address more general settings using the concepts of discrete gradient pairs and Dirac-dissipative structures. Additionally, the behavior under system transformations is investigated and we demonstrate that under suitable assumptions port-Hamiltonian differential-algebraic equations admit a representation which consists of a parametrized port-Hamiltonian semi-explicit system and an unstructured equation. Finally, we present the application to multibody system dynamics and discuss numerical results to demonstrate the capabilities of our approach.