Weak Form of the Stokes-Dirac Structure and Geometric Discretization of Port-Hamiltonian Systems

P Kotyczka (1) , B. Maschke (2) , Laurent Lefèvre (3)

1 Institute of Automation and Information Systems [ Munich]
2 LAGEP - Laboratoire d'automatique et de génie des procédés
3 LCIS - Laboratoire de Conception et d'Intégration des Systèmes

We present the mixed Galerkin discretization of distributed-parameter port-Hamiltonian systems. Due to the inherent definition of (boundary) interconnection ports, this system representation is particularly useful for the modeling of interconnected multi-physics systems and control. At the prototypical example of a system of two conservation laws in arbitrary spatial dimension, we derive the main contributions: (i) A weak formulation of the underlying geometric (Stokes-Dirac) structure, (ii) its geometric approximation by a finite-dimensional Dirac structure using a mixed Galerkin approach and power-preserving maps on the space of discrete power variables and (iii) the approximation of the Hamiltonian to obtain finite-dimensional port-Hamiltonian state space models. The power-preserving maps on the discrete bond space offer design degrees of freedom for the discretization, which is illustrated at the example Whitney finite elements on a 2D simplicial triangulation. The resulting schemes can be considered as trade-offs between centered approximations and upwinding.

Mots clés: Open systems of conservation laws; port-Hamiltonian systems; mixed Galerkin meth- ods; structure-preserving discretization; geometric discretization

Domaines: Automatique / Robotique; Analyse numérique [math.NA]

P Kotyczka, B. Maschke, Laurent Lefèvre. Weak Form of the Stokes-Dirac Structure and Geometric Discretization of Port-Hamiltonian Systems. Journal of Computational Physics, 2018, 361, pp.442-476. ⟨10.1016/j.jcp.2018.02.006⟩. ⟨hal-01831948⟩