A discrete exterior approach to structure-preserving discretization of distributed-parameter port-Hamiltonian systems

Abstract

A discrete exterior approach to structure-preserving discretization of distributed-parameter port-Hamiltonian systems Seslija, Marko; Scherpen, Jacquelien M.A.; van der Schaft, Arjan Take-down policy If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons the number of authors shown on this cover page is limited to 10 maximum. Download date: 02-04-2019A discrete exterior approach to structure-preserving discretization of distributed-parameter port-Hamiltonian systems MarkoŠešlija, Jacquelien M.A. Scherpen and Arjan van der SchaftAbstract-This paper addresses the issue of structurepreserving discretization of open distributed-parameter systems with Hamiltonian dynamics. Employing the formalism of discrete exterior calculus, we introduce simplicial Dirac structures as discrete analogues of the Stokes-Dirac structure and demonstrate that they provide a natural framework for deriving finite-dimensional port-Hamiltonian systems that emulate their infinite-dimensional counterparts. This approach of discrete differential geometry, rather than discretizing the partial differential equations, allows to first discretize the underlying Stokes-Dirac structure and then to impose the corresponding finite-dimensional port-Hamiltonian dynamics. In this manner, we preserve a number of important topological and geometrical properties of the system.

Port-Hamiltonian Modeling of Ideal Fluid Flow: Part II. Compressible and Incompressible Flow

“…This promises an avenue to ensure compatibility of our energy-aware decomposition with correspondingly designed structure-preserving numerical schemes, e.g. [9,10] that choose to discretize some parts of a fluid's domain in a more refined fashion than others, as is often needed.…”

“…where T ds corresponds to the heat exchanged per unit mass, and pdν corresponds to the mechanical work done by the fluid system due to pressure forces. Note that from (9) we have that…”

Structure Preserving Discretization of 1D Nonlinear Port-Hamiltonian Distributed Parameter Systems

“…In the following, we define a power balance for the expansions ( 24) and ( 27) that matches (22) and the inclusion (23). For this, first observe that, by (19) and (22), (e p ab , e q ab , e b ab ) | ( f p ab , f q ab , f b ab…”

“…Specific for this model class, discrete differential geometries were introduced in [21,22] to define discrete Dirac structures and discrete port-Hamiltonian dynamics on discrete manifolds. Collocation based methods and simplicial Dirac structures are employed in [23,24,3,25,26] to derive a discrete exterior calculus in which numerical integration preserves Dirac structures, dynamics and constitutive equations. The authors of [27,28] consider discretizations of damped and controlled systems.…”