A Hamiltonian vorticity–dilatation formulation of the compressible Euler equations

Polner, Mónika; Vegt, J.J.W. van der

doi:10.1016/j.na.2014.07.005

Cited by 11 publications

(7 citation statements)

References 14 publications

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“…Beyond this basic class of PH systems (which however covers different linear and nonlinear physical phenomena), there exists a growing number of PH models for different physical phenomena, see e. g. [54] for the modeling of the plasma in a fusion reactor, [21] for the reactive Navier-Stokes flow or [55] for irreversible thermodynamic systems to mention only a few interesting examples. In [56], a PH formulation of the compressible Euler equations in terms of density, weighted vorticity and dilatation is presented. The PH representation is not unique.…”

Section: Examplesmentioning

confidence: 99%

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

Kotyczka

Maschke²,

Lefèvre³

2018

Journal of Computational Physics

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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 powerpreserving 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.Keywords: Systems of conservation laws with boundary energy flows, port-Hamiltonian systems, mixed Galerkin methods, geometric spatial discretization, structure-preserving discretization. * Accepted version of P. Kotyczka et al., Weak form of Stokes-Dirac structures and geometric discretization of port-Hamiltonian systems, J. Comput. Phys. 361 (2018) 442-476, https://doi.• The power-preserving maps for the discrete power variables offer design degrees of freedom to parametrize the resulting finite-dimensional PH state space models. They can be used to realize upwinding.• Mapping the flow variables instead of the efforts avoids a structural artificial feedthrough, which is not desirable for the approximation of hyperbolic systems.We consider as the prototypical example of distributed parameter PH systems, an open system of two hyperbolic conservation laws in canonical form, as presented in [1]. We use the language of differential forms, see e. g. [23], which highlights the geometric nature of each variable and allows for a unifying representation independent from the dimension of the spatial domain.An important reason for expressing the spatial discretization of PH systems based on the weak form is to make the link with modern geometric discretization methods. Bossavit's work in computational electromagnetism [24], [25] and Tonti's cell method [26] keep track of the geometric nature of the system variables which allows for a direct interpretation of the discrete variables in terms of integral system quantities. This integral point of view is also adopted in discrete exterior calculus [11]. Finite element exterior calculus [27] gives a theoretical frame to describe functional spaces of differential forms and their compatible approximations, which includes the construction of higher order approximation bases that generalize the famous Whitney forms [28], see also [29]. We refer also to the recent article [30] which proposes conforming polynomial ap...

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Section: Examplesmentioning

confidence: 99%

“…The vectors of discrete flows and efforts f p/q , e p/q that satisfy (56), together with the discrete boundary ports of different causality, define a subset of the bond space…”

Section: Power Balance On the Discrete Bond Spacementioning

confidence: 99%

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

Kotyczka

Maschke²,

Lefèvre³

2018

Journal of Computational Physics

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“…The SDS describing the system is nonlinear; however, it was also shown that for irrotational flow, the underlying SDS becomes the canonical one. An extended model describing the case of non-isentropic fluids is presented in Polner & van der Vegt (2014) in terms of the vorticity-dilatation variables. Moreover, for the spatial one-dimensional case, different pH models of fluid-thermal equations can be found in Lopezlena & Scherpen (2004b), a dpH of Navier-Stokes equations for reactive flows can be found in Altmann & Schulze (2017), while jet-bundle formulations of the Kortweg-de Vries and Boussinesq equations can be found in Maschke & van der Schaft (2013).…”

Section: Fluid Mechanicsmentioning

confidence: 99%

Twenty years of distributed port-Hamiltonian systems: a literature review

Rashad

Califano

Schaft

2020

IMA Journal of Mathematical Control and Information

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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.

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“…The procedure to construct the port-Hamiltonian model we are aiming for relies greatly on understanding the underlying geometric structure of the state space of each energetic subsystem. This geometric formulation, pioneered by [8] and [9], will allow a systematic derivation of the underlying Hamiltonian dynamical equations and Dirac structures, usually postulated a priori in the literature [10,6,11,12]. Furthermore, it will allow for the boundary terms, which are always absent in the traditional Hamiltonian picture, to be easily identified and transformed into power ports which can be used for energy exchange through the boundary of the spatial domain.…”

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confidence: 99%

Port-Hamiltonian Modeling of Ideal Fluid Flow: Part I. Foundations and Kinetic Energy

Rashad,

Califano,

Schuller

et al. 2020

Preprint

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In this two-parts paper, we present a systematic procedure to extend the known Hamiltonian model of ideal inviscid fluid flow on Riemannian manifolds in terms of Lie-Poisson structures to a port-Hamiltonian model in terms of Stokes-Dirac structures. The first novelty of the presented model is the inclusion of non-zero energy exchange through, and within, the spatial boundaries of the domain containing the fluid. The second novelty is that the port-Hamiltonian model is constructed as the interconnection of a small set of building blocks of open energetic subsystems. Depending only on the choice of subsystems one composes and their energy-aware interconnection, the geometric description of a wide range of fluid dynamical systems can be achieved. The constructed port-Hamiltonian models include a number of inviscid fluid dynamical systems with variable boundary conditions. Namely, compressible isentropic flow, compressible adiabatic flow, and incompressible flow. Furthermore, all the derived fluid flow models are valid covariantly and globally on n-dimensional Riemannian manifolds using differential geometric tools of exterior calculus.

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A Hamiltonian vorticity–dilatation formulation of the compressible Euler equations

Cited by 11 publications

References 14 publications

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

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

Twenty years of distributed port-Hamiltonian systems: a literature review

Port-Hamiltonian Modeling of Ideal Fluid Flow: Part I. Foundations and Kinetic Energy

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