Discrete Geometry Approach to Structure-Preserving Discretization of Port-Hamiltonian Systems

Computers have emerged as essential tools in the modern scientific analysis and simulationbased 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. The thesis at hand offers a geometric framework for the discretization of a class of physical systems without destroying the underlying geometric structure of the original system. Hamiltonian systems are at the foundation of many current physical theories, including quantum and relativistic mechanics, electromagnetism, optics, solid and fluid mechanics. Geometry as the study of observable symmetries and dynamical invariants is de facto the lingua franca of the Hamiltonian theories. The prevailing paradigm in modeling of the complex largescale physical systems is network modeling. In many problems arising from modern science and engineering, such as multi-body systems, electrical networks and molecular dynamics, the port-based network modeling is a natural strategy of decomposing the overall system into subsystems, which are interconnected to each other through pairs of variables called ports and whose product is the power exchanged between the subsystems. The formalism that unifies the geometric Hamiltonian and the port-based network modeling is the port-Hamiltonian, which associates with interconnection structure of the network a geometric structure given by a Poisson, or more generally a Dirac structure. The generalized Hamiltonian dynamic is then defined with respect to this Poisson, or Dirac, structure by specifying the Hamiltonian representing the total stored energy, the energy-dissipating elements and the ports of the system. Apart from enunciating a remarkable structural unity, Poisson and Dirac geometry offers a mathematical framework that gives important insights into dynamical systems. Moreover, the geometric formalism transcends the finite-dimensional scenario and has been successfully applied to study of a number of distributed-parameter systems, systems described by a set of partial differential equations. In this work I provide a framework for structure-preserving discretization of open distributedparameter systems with generalized Hamiltonian dynamics. The underlying structure of distributed-parameter systems I address in the thesis is a StokesDirac structure, a type of infinite-dimensional Dirac structure, defined in terms of differential forms on a smooth finite-dimensional orientable, usually Riemannian, manifold with a boundary. The Stokes-Dirac structure generalizes the framework of the Poisson and symplectic structures by providing a theoretical account that permits the inclusion of varying boundary