126 research outputs found
Explicit Simplicial Discretization of Distributed-Parameter Port-Hamiltonian Systems
Simplicial Dirac structures as finite analogues of the canonical Stokes-Dirac
structure, capturing the topological laws of the system, are defined on
simplicial manifolds in terms of primal and dual cochains related by the
coboundary operators. These finite-dimensional Dirac structures offer a
framework for the formulation of standard input-output finite-dimensional
port-Hamiltonian systems that emulate the behavior of distributed-parameter
port-Hamiltonian systems. This paper elaborates on the matrix representations
of simplicial Dirac structures and the resulting port-Hamiltonian systems on
simplicial manifolds. Employing these representations, we consider the
existence of structural invariants and demonstrate how they pertain to the
energy shaping of port-Hamiltonian systems on simplicial manifolds
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-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.Comment: Copyright 2018. This manuscript version is made available under the
CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0
Reaction-Diffusion Systems as Complex Networks
The spatially distributed reaction networks are indispensable for the
understanding of many important phenomena concerning the development of
organisms, coordinated cell behavior, and pattern formation. The purpose of
this brief discussion paper is to point out some open problems in the theory of
PDE and compartmental ODE models of balanced reaction-diffusion networks.Comment: A discussion paper for the 1st IFAC Workshop on Control of Systems
Governed by Partial Differential Equation
Structure Preserving Discretization of 1D Nonlinear Port-Hamiltonian Distributed Parameter Systems
This paper contributes with a new formal method of spatial discretization of a class of nonlinear distributed parameter systems that allow a port-Hamiltonian representation over a one dimensional manifold. A specific finite dimensional port-Hamiltonian element is defined that enables a structure preserving discretization of the infinite dimensional model that inherits the Dirac structure, the underlying energy balance and matches the Hamiltonian function on any, possibly nonuniform mesh of the spatial geometry
Twenty years of distributed port-Hamiltonian systems:A literature review
The port-Hamiltonian (pH) theory for distributed parameter systems has developed greatly in the past two decades. The theory has been successfully extended from finite-dimensional to infinite-dimensional systems through a lot of research efforts. This article collects the different research studies carried out for distributed pH systems. We classify over a hundred and fifty studies based on different research focuses ranging from modeling, discretization, control and theoretical foundations. This literature review highlights the wide applicability of the pH systems theory to complex systems with multi-physical domains using the same tools and language. We also supplement this article with a bibliographical database including all papers reviewed in this paper classified in their respective groups
Energy preserving model order reduction of the nonlinear Schr\"odinger equation
An energy preserving reduced order model is developed for two dimensional
nonlinear Schr\"odinger equation (NLSE) with plane wave solutions and with an
external potential. The NLSE is discretized in space by the symmetric interior
penalty discontinuous Galerkin (SIPG) method. The resulting system of
Hamiltonian ordinary differential equations are integrated in time by the
energy preserving average vector field (AVF) method. The mass and energy
preserving reduced order model (ROM) is constructed by proper orthogonal
decomposition (POD) Galerkin projection. The nonlinearities are computed for
the ROM efficiently by discrete empirical interpolation method (DEIM) and
dynamic mode decomposition (DMD). Preservation of the semi-discrete energy and
mass are shown for the full order model (FOM) and for the ROM which ensures the
long term stability of the solutions. Numerical simulations illustrate the
preservation of the energy and mass in the reduced order model for the two
dimensional NLSE with and without the external potential. The POD-DMD makes a
remarkable improvement in computational speed-up over the POD-DEIM. Both
methods approximate accurately the FOM, whereas POD-DEIM is more accurate than
the POD-DMD
Numerical analysis of a structure-preserving space-discretization for an anisotropic and heterogeneous boundary controlled N-dimensional wave equation as port-Hamiltonian system
The anisotropic and heterogeneous N-dimensional wave equation, controlled and
observed at the boundary, is considered as a port-Hamiltonian system. The
recent structure-preserving Partitioned Finite Element Method is applied,
leading directly to a finite-dimensional port-Hamiltonian system, and its
numerical analysis is done in a general framework, under usual assumptions for
finite element. Compatibility conditions are then exhibited to reach the best
trade off between the convergence rate and the number of degrees of freedom for
both the state error and the Hamiltonian error. Numerical simulations in 2D are
performed to illustrate the optimality of the main theorems among several
choices of classical finite element families.Comment: 36 pages, 1 figure, submitte
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