1,384 research outputs found
Model reduction of port-Hamiltonian systems based on reduction of Dirac structures
The geometric formulation of general port-Hamiltonian systems is used in order to obtain two structure preserving reduction methods. The main idea is to construct a reduced-order Dirac structure corresponding to zero power flow in some of the energy-storage ports. This can be performed in two canonical ways, called the effort- and the flow-constraint methods. We show how the effort-constraint method can be regarded as a projection-based model reduction method
Symplectic Model Reduction of Hamiltonian Systems
In this paper, a symplectic model reduction technique, proper symplectic
decomposition (PSD) with symplectic Galerkin projection, is proposed to save
the computational cost for the simplification of large-scale Hamiltonian
systems while preserving the symplectic structure. As an analogy to the
classical proper orthogonal decomposition (POD)-Galerkin approach, PSD is
designed to build a symplectic subspace to fit empirical data, while the
symplectic Galerkin projection constructs a reduced Hamiltonian system on the
symplectic subspace. For practical use, we introduce three algorithms for PSD,
which are based upon: the cotangent lift, complex singular value decomposition,
and nonlinear programming. The proposed technique has been proven to preserve
system energy and stability. Moreover, PSD can be combined with the discrete
empirical interpolation method to reduce the computational cost for nonlinear
Hamiltonian systems. Owing to these properties, the proposed technique is
better suited than the classical POD-Galerkin approach for model reduction of
Hamiltonian systems, especially when long-time integration is required. The
stability, accuracy, and efficiency of the proposed technique are illustrated
through numerical simulations of linear and nonlinear wave equations.Comment: 25 pages, 13 figure
Port-Hamiltonian Modeling of Ideal Fluid Flow: Part II. Compressible and Incompressible Flow
Part I of this paper presented a systematic derivation of the Stokes Dirac
structure underlying the port-Hamiltonian model of ideal fluid flow on
Riemannian manifolds. Starting from the group of diffeomorphisms as a
configuration space for the fluid, the Stokes Dirac structure is derived by
Poisson reduction and then augmented by boundary ports and distributed ports.
The additional boundary ports have been shown to appear naturally as surface
terms in the pairings of dual maps, always neglected in standard Hamiltonian
theory. The port-Hamiltonian model presented in Part I corresponded only to the
kinetic energy of the fluid and how its energy variables evolve such that the
energy is conserved.
In Part II, we utilize the distributed port of the kinetic energy
port-Hamiltonian system for representing a number of fluid-dynamical systems.
By adding internal energy we model compressible flow, both adiabatic and
isentropic, and by adding constraint forces we model incompressible flow. The
key tools used are the interconnection maps relating the dynamics of fluid
motion to the dynamics of advected quantities.Comment: This is a prevprint submitted to the journal of Geometry and Physics.
Please DO NOT CITE this version, but only the published manuscrip
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
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