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

JOURNAL OF COMPUTATIONAL PHYSICS

卷:361

Article

Kotyczka, Paul^1,2  Maschke, Bernhard¹  Lefevre, Laurent³ 

[1] Univ Claude Bernard Lyon 1, Univ Lyon, CNRS, LAGEP UMR 5007, 43 Blvd 11 Novembre 1918, F-69622 Villeurbanne, France

[2] Tech Univ Munich, Dept Mech Engn, Chair Automat Control, Boltzmannstr 15, D-85748 Garching, Germany

[3] Univ Grenoble Alpes, LCIS, F-26902 Valence, France

关键词: Systems of conservation laws with boundary energy flows;    Port-Hamiltonian systems;    Mixed Galerkin methods;    Geometric spatial discretization;    Structure-preserving discretization;   

DOI  :  10.1016/j.jcp.2018.02.006

来源: Elsevier

【 摘 要 】

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. (C) 2018 Elsevier Inc. All rights reserved.

【 授权许可】

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