Discrete geometry approach to structure-preserving discretization of port-Hamiltonian systems

PhD ceremony: Mr. M. Seslija, 14.30 uur, Academiegebouw, Broerstraat 5, Groningen

Disertation: Discrete geometry approach to structure-preserving discretization of port-Hamiltonian systems

Promotor(s): prof. J.M.A. Scherpen, prof. A.J. van der Schaft

Faculty: Mathematics and Natural Sciences

Computers have emerged as essential tools in the modern scientific analysis and simulation-based design of complex physical systems. The deeply-seated abstraction of continuity immanent to many physical systems inherently clashes with a digital computer's ability of storing and manipulating only finite sets of numbers. While there has been a number of computational techniques that proposed discretizations of differential equations, the geometric structures they model are often lost in the process. In his thesis Marko Seslija offers a geometric framework for the discretization of a class of physical systems (so-called port-Hamiltonian systems) without destroying the underlying geometric structure of the original system.

The most important consequences of this thesis are that many of the important results from differential geometry can be transferred into the discrete realm and thereby lead to numerically and physically faithful models, which later can be fed to computers and simulate crucial aspects of the physical reality.

In his thesis Seslija addresses the issue of structure-preserving discretization of open distributed-parameter systems with generalized Hamiltonian dynamics. Employing the formalism of discrete exterior calculus, he introduces 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. The spatial domain, in the continuous theory represented by a finite-dimensional smooth manifold with boundary, is replaced by a homological manifold-like simplicial complex and its circumcentric dual. The smooth differential forms, in the discrete setting, are mirrored by cochains on the primal and dual complexes, while the discrete exterior derivative is defined to be the coboundary operator.

This approach of discrete exterior 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, a number of important intrinsically topological and geometrical properties of the system are preserved.

Marko Seslija demonstrates general considerations on a number of physical examples, including reaction-diffusion systems, where the structure-preserving discretization recovers the standard compartmental model. Furthermore, he shows how a Poisson symmetry reduction of Dirac structures associated with infinite- and finite-dimensional models can be conducted in a unified fashion.