Diagonal Discrete Hodge Operators for Simplicial Meshes Using the Signed Dual Complex

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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“…with positive definite, diagonal matrices Q p , Q q that represent diagonal discrete Hodge operators [71].…”

Section: Discrete Constitutive Equationsmentioning

confidence: 99%

“…with positive definite, diagonal matrices Q p , Q q that represent diagonal discrete Hodge operators [71]. The discrete states p and q are constructed (as f p and f q ) as linear combinations of integral conserved quantities on the 2-and 1-simplices of the discretization grid.…”

Section: Discrete Constitutive Equationsmentioning

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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“…While all complementary and complementary-dual formulations may benefit from one stroke complementarity, the mixed-hybrid formulation is typically faster on tetrahedral meshes. An even more efficient formulation, still valid only for isotropic materials, is described in [15]. The bilateral bounds on global parameters obtained by the mixed-hybrid formulation together with vertex reconstruction are fairly symmetric with respect to the exact solution, thus usually there is an accuracy improvement by considering the mean of the values obtained by the two methods.…”

Section: Discussionmentioning

confidence: 99%

One Stroke Complementarity for Poisson-Like Problems

Specogna

2015

IEEE Trans. Magn.

Self Cite

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Taking electrokinetics as a paradigm problem for the sake of simplicity, complementarity originates when an irrotational electric field and a solenoidal current density satisfying boundary conditions are in hand. We first compare three formulations to obtain a solenoidal current density, both in terms of pure computational advantage and in the ability to pursue symmetric energy bounds with respect to the standard electric scalar potential formulation. For these formulations, we devise post-processing techniques that promise to provide bilateral bounds in one stroke, hence requiring the solution of just one linear system.

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“…Diese Wahl führt auf Approximationsräume unterschiedlicher Dimension. Lassen sich die Matrizen in (32) gemäß…”

Section: Gemischter Galerkin-ansatzunclassified

Zur Erhaltung von Struktur und Flachheit bei der torbasierten Ortsdiskretisierung

Kotyczka

2018

At - Automatisierungstechnik

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Zusammenfassung Mit einem kürzlich vorgestellten gemischten Galerkin-Ansatz lassen sich Port-Hamiltonsche Systeme hyperbolischer Erhaltungsgleichungen in beliebiger Ortsdimension strukturerhaltend diskretisieren. Der Ansatz ist ebenso für parabolische Systeme in strukturierter Darstellung geeignet. Dieser Beitrag fasst die Methode zusammen. Weiterhin werden die Struktur und Approximationsgüte der resultierenden endlich-dimensionalen Zustandsraummodelle in Abhängigkeit der Entwurfsfreiheitsgrade analysiert. Hierzu werden exemplarisch die eindimensionale lineare Wellen- und Wärmeleitungsgleichung betrachtet und zunächst Eigenwertlagen und Lösungen von Anfangswertaufgaben analysiert. Im Hinblick auf den Steuerungsentwurf wird die Erhaltung eines flachen Ausgangs für die Wärmeleitungsgleichung nachgewiesen und die Güte der angenäherten Randsteuerungen untersucht.

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Diagonal Discrete Hodge Operators for Simplicial Meshes Using the Signed Dual Complex

Cited by 8 publications

References 24 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

One Stroke Complementarity for Poisson-Like Problems

Zur Erhaltung von Struktur und Flachheit bei der torbasierten Ortsdiskretisierung

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