Modelling and structure-preserving discretization of Maxwell’s equations as port-Hamiltonian system

Payen, Gabriel; Matignon, Denis; Haine, Ghislain

doi:10.1016/j.ifacol.2020.12.1355

Cited by 12 publications

(6 citation statements)

References 14 publications

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“…Beside the generalization of the classes related to the heat and wave equation to the three-dimensional case, we plan to propose in SCRIMP more advanced model problems based on the two-dimensional Shallow Water Equation (SWE) [19,17], the Kirchhoff model for thin plates [14] and Maxwell's equations [40]. Furthermore we will investigate both time integration methods that allow structure-preserving time discretization [31] of finite dimensional pHs and more accurate time integrators for nonlinear pHDAE.…”

Section: Discussionmentioning

confidence: 99%

“…The methodology detailed so far is certainly not limited to the previous three examples. Indeed higher-order differential [14], curl operator for Maxwell's equations [40], nonlinear system [19], and different Hamiltonian choices can be handled as well. For instance, in the case of the heat equation, the entropy or the classical L 2 -norm of the temperature can be alternatively considered as Hamiltonian functional [44,45].…”

Section: The Heat Equationmentioning

confidence: 99%

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Numerical approximation of port-Hamiltonian systems for hyperbolic or parabolic PDEs with boundary control

Brugnoli¹,

Haine²,

Serhani³

et al. 2020

Preprint

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The present manuscript concerns the design of structure-preserving discretization methods for the solution of systems of boundary controlled Partial Differential Equations (PDEs) thanks to the port-Hamiltonian formalism. We first provide a general structure of infinite-dimensional port-Hamiltonian systems (pHs) for which the Partitioned Finite Element Method (PFEM) straightforwardly applies. The proposed strategy is particularised to abstract multidimensional linear hyperbolic and parabolic systems of PDEs. Then we show that instructional model problems based on the wave equation, Mindlin equation and heat equation fit within this unified framework. Secondly we introduce the ongoing project SCRIMP (Simulation and ContRol of Interactions in Multi-Physics) developed for the numerical simulation of infinite-dimensional pHs. SCRIMP notably relies on the FEniCS opensource computing platform for the finite element spatial discretization. Finally, we illustrate how to solve the considered model problems within this framework by carefully explaining the methodology. As additional support, companion interactive Jupyter notebooks are provided.

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Section: Discussionmentioning

confidence: 99%

Section: The Heat Equationmentioning

confidence: 99%

Numerical approximation of port-Hamiltonian systems for hyperbolic or parabolic PDEs with boundary control

Brugnoli¹,

Haine²,

Serhani³

et al. 2020

Preprint

Self Cite

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“…The weak form of the dynamics accounts for the boundary conditions, that will be strongly enforced by considering test functions satisfying homogeneous boundary conditions in the adjoint system. Consider now the definition of the flows (28) and dual flows (47) and the associated weak formulation…”

Section: Overall Port-hamiltonian Dynamicsmentioning

confidence: 99%

“…This result follows by taking the exterior derivative of each line of system (136), under the assumption that the initial conditions respect these constraints. Mixed finite element strategies, like the ones proposed in [40,46,47], cannot satisfy both constraints as they do not employ a dual representation for each variable. Instead, the dual field formulation naturally capture this aspect as shown in Fig.…”

Section: The Maxwell Equations Equation In 3dmentioning

confidence: 99%

Dual field structure-preserving discretization of port-Hamiltonian systems using finite element exterior calculus

Brugnoli¹,

Rashad²

2022

Preprint

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In this paper we propose a novel approach to discretize linear port-Hamiltonian systems while preserving the underlying structure. We present a finite element exterior calculus formulation that is able to mimetically represent conservation laws and cope with mixed open boundary conditions using a single computational mesh. The possibility of including open boundary conditions allows for modular composition of complex multi-physical systems whereas the exterior calculus formulation provides a coordinate-free treatment. Our approach relies on a dual-field representation of the physical system that is redundant at the continuous level but eliminates the need of mimicking the Hodge star operator at the discrete level. By considering the Stokes-Dirac structure representing the system together with its adjoint, which embeds the metric information directly in the codifferential, the need for an explicit discrete Hodge star is avoided altogether. By imposing the boundary conditions in a strong manner, the power balance characterizing the Stokes-Dirac structure is then retrieved at the discrete level via symplectic Runge-Kutta integrators based on Gauss-Legendre collocation points. Numerical experiments validate the convergence of the method and the conservation properties in terms of energy balance both for the wave and Maxwell equations in a three dimensional domain. For the latter example, the magnetic and electric fields preserve their divergence free nature at the discrete level.

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“…It seems that the Partitioned Finite Element Method (PFEM), first proposed in [12] and since then widely studied (see e.g. [8,9,54,13,11,53] and references therein), is one of the most adapted scheme to construct a mimetic finite-dimensional Dirac structure. Furthermore, it only relies on the well-proven and robust finite element method: it gives rise to sparse matrices, it is easy to implement, and one can take advantage of the numerous existing softwares.…”

Section: Structure-preserving Discretization: State Of the Artmentioning

confidence: 99%

Numerical analysis of a structure-preserving space-discretization for an anisotropic and heterogeneous boundary controlled N-dimensional wave equation as port-Hamiltonian system

Haine¹,

Matignon²,

Serhani³

2020

Preprint

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The anisotropic and heterogeneous N -dimensional wave equation, controlled and observed at the boundary, is considered as a port-Hamiltonian system. The recent structure-preserving Partitioned Finite Element Method is applied, leading directly to a finite-dimensional port-Hamiltonian system, and its numerical analysis is done in a general framework, under usual assumptions for finite element. Compatibility conditions are then exhibited to reach the best trade off between the convergence rate and the number of degrees of freedom for both the state error and the Hamiltonian error. Numerical simulations in 2D are performed to illustrate the optimality of the main theorems among several choices of classical finite element families.

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Modelling and structure-preserving discretization of Maxwell’s equations as port-Hamiltonian system

Cited by 12 publications

References 14 publications

Numerical approximation of port-Hamiltonian systems for hyperbolic or parabolic PDEs with boundary control

Numerical approximation of port-Hamiltonian systems for hyperbolic or parabolic PDEs with boundary control

Dual field structure-preserving discretization of port-Hamiltonian systems using finite element exterior calculus

Numerical analysis of a structure-preserving space-discretization for an anisotropic and heterogeneous boundary controlled N-dimensional wave equation as port-Hamiltonian system

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