Discrete exterior geometry approach to structure-preserving discretization of distributed-parameter port-Hamiltonian systems

Seslija, Marko; Schaft, A.J. van der; Scherpen, Jacquelien M. A.

doi:10.1016/j.geomphys.2012.02.006

Cited by 36 publications

(22 citation statements)

References 32 publications

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“…In this paragraph, we explicitly indicate the arguments (z, t), for the Hamiltonian can depend on z as in the case of the shallow water equations with variable bed profile. In the sequel, we will omit the arguments 9. See e. g [2],.…”

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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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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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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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“…(Desbrun, Hirani, Leok, and Marsden 2005;Hirani 2003). Since its conception, DEC has found many applications and has been extended in various directions, including general relativity (Frauendiener 2006), electrodynamics (Stern, Tong, Desbrun, and Marsden 2007), linear elasticity (Yavari 2008), computational modeling (Desbrun, Kanso, and Tong 2008), port-Hamiltonian systems (Seslija, Schaft, and Scherpen 2012), digital geometry processing (Crane, de Goes, Desbrun, and Schroder 2013), Darcy flow (Hirani, Nakshatrala, and Chaudhry 2015), and the Navier-Stokes equations (Mohamed, Hirani, and Samtaney 2016). However, a rigorous convergence analysis of DEC has always been lacking; As far as we are aware, the only convergence proof of DEC so far appeared is for the scalar Poisson problem in two dimensions, and it Date: June 5, 2018. is based on reinterpreting the discretization as a finite element method, cf.…”

Section: Introductionmentioning

confidence: 99%

Convergence of Discrete Exterior Calculus Approximations for Poisson Problems

Schulz

Tsogtgerel

2019

Discrete Comput Geom

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Discrete exterior calculus (DEC) is a framework for constructing discrete versions of exterior differential calculus objects, and is widely used in computer graphics, computational topology, and discretizations of the Hodge-Laplace operator and other related partial differential equations. However, a rigorous convergence analysis of DEC has always been lacking; as far as we are aware, the only convergence proof of DEC so far appeared is for the scalar Poisson problem in two dimensions, and it is based on reinterpreting the discretization as a finite element method. Moreover, even in two dimensions, there have been some puzzling numerical experiments reported in the literature, apparently suggesting that there is convergence without consistency.In this paper, we develop a general independent framework for analyzing issues such as convergence of DEC without relying on theories of other discretization methods, and demonstrate its usefulness by establishing convergence results for DEC beyond the Poisson problem in two dimensions. Namely, we prove that DEC solutions to the scalar Poisson problem in arbitrary dimensions converge pointwise to the exact solution at least linearly with respect to the mesh size. We illustrate the findings by various numerical experiments, which show that the convergence is in fact of second order when the solution is sufficiently regular. The problems of explaining the second order convergence, and of proving convergence for general p-forms remain open.

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“…More recent works in electromagnetism include [106][107][108][109]. In the field of mechanics, DEC has been employed to solve Darcy, Euler and Navier-Stokes equations in some basic configurations [110][111][112], to geometrizes elasticity problems [113,114] or to solve port-Hamiltonian systems [115]. DEC belongs to the family of cochain-based mimetic discretization methods described in [116].…”

Section: Introductionmentioning

confidence: 99%

A review of some geometric integrators

Razafindralandy

Hamdouni

Chhay

2018

Adv. Model. and Simul. in Eng. Sci.

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Some of the most important geometric integrators for both ordinary and partial differential equations are reviewed and illustrated with examples in mechanics. The class of Hamiltonian differential systems is recalled and its symplectic structure is highlighted. The associated natural geometric integrators, known as symplectic integrators, are then presented. In particular, their ability to numerically reproduce first integrals with a bounded error over a long time interval is shown. The extension to partial differential Hamiltonian systems and to multisymplectic integrators is presented afterwards. Next, the class of Lagrangian systems is described. It is highlighted that the variational structure carries both the dynamics (Euler-Lagrange equations) and the conservation laws (Noether's theorem). Integrators preserving the variational structure are constructed by mimicking the calculus of variation at the discrete level. We show that this approach leads to numerical schemes which preserve exactly the energy of the system. After that, the Lie group of local symmetries of partial differential equations is recalled. A construction of Lie-symmetry-preserving numerical scheme is then exposed. This is done via the moving frame method. Applications to Burgers equation are shown. The last part is devoted to the Discrete Exterior Calculus, which is a structure-preserving integrator based on differential geometry and exterior calculus. The efficiency of the approach is demonstrated on fluid flow problems with a passive scalar advection.

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Discrete exterior geometry approach to structure-preserving discretization of distributed-parameter port-Hamiltonian systems

Cited by 36 publications

References 32 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

Convergence of Discrete Exterior Calculus Approximations for Poisson Problems

A review of some geometric integrators

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