Distributed port-Hamiltonian modelling for irreversible processes

Zhou, Weijun; Hamroun, Boussad; Couenne, Françoise; Gorrec, Yann Le

doi:10.1080/13873954.2016.1237970

Cited by 14 publications

(7 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In this paper, we concentrate on canonical systems of two conservation laws in arbitrary spatial dimension. 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.…”

Section: Examplesmentioning

confidence: 99%

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

Kotyczka

Maschke²,

Lefèvre³

2018

Journal of Computational Physics

View full text Add to dashboard Cite

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...

show abstract

Section: Examplesmentioning

confidence: 99%

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

Kotyczka

Maschke²,

Lefèvre³

2018

Journal of Computational Physics

View full text Add to dashboard Cite

show abstract

“…To fix ideas and notations, a simple 1-D PDE model borrowed from [28] is first recalled: the lossy transmission line, on domain Ω = (0, ).…”

Section: A General Results Of Structure-preserving Discretization For mentioning

confidence: 99%

“…the choice of the finite element family remains quite open so far, but indeed, from first numerical experiments, some optimal choices can be observed in practice: the careful numerical analysis must still be investigated, structure-preserving model reduction can be carried out using methods presented in [11], for the time-domain discretization as last step procedure for numerical simulation, specific approaches should be followed, see e.g. [7], following [28], the introduction of entropy ports enables to transform dissipative systems into conservative systems, taking into account some thermodynamical laws; see [23,24] for the application to the heat equation. an alternative computational solution consists in making use of the transformation of the lossy system into a lossless one, and only then apply classical symplectic numerical schemes, see e.g.…”

Section: Constitutive Relation Are Approximated In Weak Formmentioning

confidence: 99%

A Partitioned Finite Element Method for the Structure-Preserving Discretization of Damped Infinite-Dimensional Port-Hamiltonian Systems with Boundary Control

Serhani

Matignon

Haine

2019

Lecture Notes in Computer Science

View full text Add to dashboard Cite

Many boundary controlled and observed Partial Differential Equations can be represented as port-Hamiltonian systems with dissipation, involving a Stokes-Dirac geometrical structure together with constitutive relations. The Partitioned Finite Element Method, introduced in Cardoso-Ribeiro et al. (2018), is a structure preserving numerical method which defines an underlying Dirac structure, and constitutive relations in weak form, leading to finite-dimensional port-Hamiltonian Differential Algebraic systems (pHDAE). Different types of dissipation are examined: internal damping, boundary damping and also diffusion models.

show abstract

“…Several studies have been also devoted for reaction-diffusion systems in which there is coupling between mass transport and chemical reactions. Different pH models can be found in Šešlija et al (2010, 2014b); Zhou et al (2017Zhou et al ( , 2012Zhou et al ( , 2015. A different way to model multi-scale systems stemming from the combination of hyperbolic and diffusive processes is studied in Le Gorrec & Matignon (2013), where fractional integrals and derivatives are used.…”

Section: Chemical Processesmentioning

confidence: 99%

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

Rashad

Califano

Schaft

2020

IMA Journal of Mathematical Control and Information

View full text Add to dashboard Cite

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.

show abstract

Distributed port-Hamiltonian modelling for irreversible processes

Cited by 14 publications

References 12 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

A Partitioned Finite Element Method for the Structure-Preserving Discretization of Damped Infinite-Dimensional Port-Hamiltonian Systems with Boundary Control

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

Contact Info

Product

Resources

About