Theory and Implementation of Coupled Port-Hamiltonian Continuum and Lumped Parameter Models

Abstract

A continuous Galerkin finite element method that allows mixed boundary conditions without the need for Lagrange multipliers or user-defined parameters is developed. A mixed coupling of Lagrange and Raviart-Thomas basis functions are used. The method is proven to have a Hamiltonian-conserving spatial discretisation and a symplectic time discretisation. The energy residual is therefore guaranteed to be bounded for general problems and exactly conserved for linear problems. The linear 2D wave equation is discretised and modelled by making use of a port-Hamiltonian framework. This model is verified against an analytic solution and shown to have standard order of convergence for the temporal and spatial discretisation. The error growth over time is shown to grow linearly for this symplectic method, which agrees with theoretical results. A modal analysis is performed which verifies that the eigenvalues of the model accurately converge to the exact eigenvalues, as the mesh is refined. The port-Hamiltonian framework allows boundary coupling with bondgraph or, more generally, lumped parameter models, therefore unifying the two fields of lumped parameter modelling and continuum modelling of Hamiltonian systems. The wave domain discretisation is shown to be equivalent to a coupling of canonical port-Hamiltonian forms. This feature allows the model to have mixed boundary conditions as well as to have mixed causality interconnections with other port-Hamiltonian models. A model of the 2D wave equation is coupled, in a monolithic manner, with a lumped parameter model of an electromechanical linear actuator. The combined model is also verified to conserve energy exactly.

Energy-based advection modelling using bond graphs

“…This paper focuses on modelling advection in the one-dimensional case using lumped (zero-dimensional) approximations, but future work could be directed to including advection within the three-dimensional case. The Port-Hamiltonian approach [39] provides one way forward.We have assumed that the chemical domain has negligible effect on the hydraulic domain (§3). While this is in line with prior approaches [40], the validity of this assumption needs further investigation.This paper considers incompressible flow through rigid pipes; in the case of blood flow the pipes are not rigid, requiring extension to compartments with variable volumes; this will be the subject of future research.Hydraulic flow reversal leads to the switching function of equation (3.2).…”

“…This paper focuses on modelling advection in the one-dimensional case using lumped (zero-dimensional) approximations, but future work could be directed to including advection within the three-dimensional case. The Port-Hamiltonian approach [ 39 ] provides one way forward. We have assumed that the chemical domain has negligible effect on the hydraulic domain (§3).…”

Enablers and Tools for Agile Product Development

“…Within the pHs framework, different physical phenomena can be described and analyzed separately from another. There is an ongoing effort to apply structure-preserving model reduction techniques to pHs to solve distributed parameter systems keeping the properties of the pHs, i.e., the power conservation and composability (Argus et al 2021). In addition, a visual modeling language called bond-graphs exists, which is used to describe and reason about the simulations at an abstract level (Borutzky 2011).…”

“…The separation of concerns inherent in pHs modeling has been used to model the different physical phenomena occurring in a coupled thermo-elasticity problem (Brugnoli et al 2021;Argus et al 2021) separately (c.f. Fig.…”

Network thermodynamics of biological systems: A bond graph approach

“…Living systems have a significant spatial dimension including cell motility, blood flow and heterogenous organs [81]. The port-Hamiltonian is a spatial extension of the bond graph which has recently been applied to biology [82]. However, modelling biochemistry in this context remains the subject of further work.…”