Dual field structure-preserving discretization of port-Hamiltonian systems using finite element exterior calculus

TL;DR

This paper introduces a structure-preserving finite element discretization method for port-Hamiltonian systems that employs dual-field representation and exterior calculus, ensuring conservation laws and boundary conditions are maintained numerically.

Contribution

It presents a novel dual-field, exterior calculus-based discretization that avoids explicit Hodge star operators and handles open boundary conditions for port-Hamiltonian systems.

Findings

The method preserves energy balance in wave and Maxwell equations.

Numerical experiments confirm convergence and conservation properties.

Magnetic and electric fields remain divergence-free at the discrete level.

Abstract

In this paper we propose a novel approach to discretize linear port-Hamiltonian systems while preserving the underlying structure. We present a finite element exterior calculus formulation that is able to mimetically represent conservation laws and cope with mixed open boundary conditions using a single computational mesh. The possibility of including open boundary conditions allows for modular composition of complex multi-physical systems whereas the exterior calculus formulation provides a coordinate-free treatment. Our approach relies on a dual-field representation of the physical system that is redundant at the continuous level but eliminates the need of mimicking the Hodge star operator at the discrete level. By considering the Stokes-Dirac structure representing the system together with its adjoint, which embeds the metric information directly in the codifferential, the need for an explicit discrete Hodge star is avoided altogether. By imposing the boundary conditions in a strong manner, the power balance characterizing the Stokes-Dirac structure is then retrieved at the discrete level via symplectic Runge-Kutta integrators based on Gauss-Legendre collocation points. Numerical experiments validate the convergence of the method and the conservation properties in terms of energy balance both for the wave and Maxwell equations in a three dimensional domain. For the latter example, the magnetic and electric fields preserve their divergence free nature at the discrete level.