Weak form of Stokes–Dirac structures and geometric discretization of port-Hamiltonian systems

We present the mixed Galerkin discretization of distributed parameter port-Hamiltonian systems. On the prototypical example of hyperbolic systems of two conservation laws in arbitrary spatial dimension, we derive the main contributions: (i) A weak formulation of the underlying geometric (Stokes–Dira...

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Veröffentlicht in:Journal of computational physics 2018-05, Vol.361, p.442-476
Hauptverfasser: Kotyczka, Paul, Maschke, Bernhard, Lefèvre, Laurent
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Sprache:eng
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Zusammenfassung:We present the mixed Galerkin discretization of distributed parameter port-Hamiltonian systems. On the prototypical example of hyperbolic systems of two conservation laws in arbitrary spatial dimension, we derive the main contributions: (i) A weak formulation of the underlying geometric (Stokes–Dirac) structure with a segmented boundary according to the causality of the boundary ports. (ii) The geometric approximation of the Stokes–Dirac structure by a finite-dimensional Dirac structure is realized using a mixed Galerkin approach and power-preserving linear maps, which define minimal discrete power variables. (iii) With a consistent approximation of the Hamiltonian, we obtain finite-dimensional port-Hamiltonian state space models. By the degrees of freedom in the power-preserving maps, the resulting family of structure-preserving schemes allows for trade-offs between centered approximations and upwinding. We illustrate the method on the example of Whitney finite elements on a 2D simplicial triangulation and compare the eigenvalue approximation in 1D with a related approach. •Mixed Galerkin geometric discretization for open systems of conservation laws.•Weak form of the underlying Stokes–Dirac structure with boundary partitioning.•Power-preserving mappings to define Dirac structures on a minimal bond space.•Conservative discretization schemes with tunable approximation properties.•Explicit input–output state space models for interconnection and control.
ISSN:0021-9991
1090-2716
DOI:10.1016/j.jcp.2018.02.006