A discrete exterior approach to structure-preserving discretization of distributed-parameter port-Hamiltonian systems

This paper addresses the issue of structure-preserving discretization of open distributed-parameter systems with Hamiltonian dynamics. Employing the formalism of discrete exterior calculus, we introduce 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. This approach of discrete differential 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, we preserve a number of important topological and geometrical properties of the system.

[1]  Jerrold E. Marsden,et al.  Stokes-Dirac structures through reduction of infinite-dimensional Dirac structures , 2010, 49th IEEE Conference on Decision and Control (CDC).

[2]  R. Ho.  Algebraic Topology , 2022 .

[3]  A. Schaft,et al.  Discrete exterior geometry approach to structure-preserving discretization of distributed-parameter port-Hamiltonian systems , 2011, 1111.6403.

[4]  D. Arnold,et al.  Finite element exterior calculus: From hodge theory to numerical stability , 2009, 0906.4325.

[5]  Joris Vankerschaver,et al.  Discrete Lagrangian field theories on Lie groupoids , 2007 .

[6]  Anil N. Hirani,et al.  Discrete exterior calculus for variational problems in computer vision and graphics , 2003, 42nd IEEE International Conference on Decision and Control (IEEE Cat. No.03CH37475).

[7]  Thomas Rylander,et al.  Computational Electromagnetics , 2005, Electronics, Power Electronics, Optoelectronics, Microwaves, Electromagnetics, and Radar.

[8]  Alain Bossavit,et al.  Yee-like schemes on staggered cellular grids: a synthesis between FIT and FEM approaches , 2000 .

[9]  James C. Sexton,et al.  Geometric discretization scheme applied to the Abelian Chern-Simons theory , 2000, hep-th/0001030.

[10]  A. Schaft,et al.  Hamiltonian formulation of distributed-parameter systems with boundary energy flow , 2002 .

[11]  J. Marsden,et al.  Multisymplectic Geometry, Variational Integrators, and Nonlinear PDEs , 1998, math/9807080.

[12]  A. Schaft,et al.  On Representations and Integrability of Mathematical Structures in Energy-Conserving Physical Systems , 1999 .

[13]  Jacquelien M. A. Scherpen,et al.  Structure Preserving Spatial Discretization of a 1-D Piezoelectric Timoshenko Beam , 2011, Multiscale Model. Simul..

[14]  H. Whitney.  Geometric Integration Theory , 1957 .

[15]  Stefano Stramigioli,et al.  Port-Hamiltonian Systems , 2009 .

[16]  J. Marsden,et al.  Variational time integrators , 2004 .

[17]  Paul W. Gross,et al.  Electromagnetic Theory and Computation: A Topological Approach , 2004 .

[18]  J. Marsden,et al.  Asynchronous Variational Integrators , 2003 .

[19]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[20]  Yiying Tong,et al.  Discrete differential forms for computational modeling , 2005, SIGGRAPH Courses.

[21]  Ralf Hiptmair,et al.  Discrete Hodge operators , 2001, Numerische Mathematik.

[22]  Jonathan Richard Shewchuk,et al.  What is a Good Linear Element? Interpolation, Conditioning, and Quality Measures , 2002, IMR.

[23]  Irene Dorfman,et al.  Dirac Structures and Integrability of Nonlinear Evolution Equations , 1993 .

[24]  James R. Munkres,et al.  Elements of algebraic topology , 1984 .

[25]  Jerrold E. Marsden,et al.  Variational Methods, Multisymplectic Geometry and Continuum Mechanics , 2001 .

[26]  G. Golo,et al.  The wave equation as a Port-Hamiltonian system, and a finite-dimensional approximation , 2002 .

[27]  J. Marsden,et al.  Discrete mechanics and variational integrators , 2001, Acta Numerica.

[28]  Arjan van der Schaft,et al.  Hamiltonian discretization of boundary control systems , 2004, Autom..

[29]  R. Hiptmair.  Finite elements in computational electromagnetism , 2002, Acta Numerica.

[30]  W. Marsden.  I and J , 2012 .

[31]  Anil N. Hirani,et al.  Discrete exterior calculus , 2005, math/0508341.

[32]  A. Bossavit.  Computational Electromagnetism: Variational Formulations, Complementarity, Edge Elements , 1997 .

[33]  Arjan van der Schaft,et al.  Conservation Laws and Lumped System Dynamics , 2009 .

[34]  Alessandro Macchelli,et al.  An Algorithm to Discretize One-Dimensional Distributed Port Hamiltonian Systems , 2007 .

[35]  Michael J. Holst,et al.  Geometric Variational Crimes: Hilbert Complexes, Finite Element Exterior Calculus, and Problems on Hypersurfaces , 2010, Foundations of Computational Mathematics.

[36]  Thomas Voss,et al.  Port-Hamiltonian modeling and control of piezoelectric beams and plates: application to inflatable space structures , 2010 .