A partitioned finite element method for power-preserving discretization of open systems of conservation laws

Cardoso-Ribeiro, Flávio Luiz; Matignon, Denis; Lefèvre, Laurent

doi:10.1093/imamci/dnaa038

Cited by 48 publications

(28 citation statements)

References 43 publications

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“…Using the same function spaces as in Cardoso-Ribeiro's work [11], approximate flow and effort functions are introduced, f p , e p ∈ H 1 ( ) , f q , e q ∈ H div ( ) . (12) In the succeeding sections, we show that this choice of function spaces allows us to combine important features of Cardoso-Ribeiro's and Kotyczka's [22] methods.…”

Section: Weak Formmentioning

confidence: 99%

“…McDonald shows that multi-symplecticity preserves the travelling waves of hyperbolic equations [27]. The Partitioned Finite Element Method (PFEM) in the paper by Cardoso-Ribeiro et al [11] combines a finite element spatial discretisation with an SV time integration scheme in a way that conserves energy but requires Lagrange multipliers to implement mixed boundary conditions [21]. Brugnoli et al successfully apply this approach to Mindlin and Kirchhoff plate models [7,8].…”

Section: Introductionmentioning

confidence: 99%

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Theory and Implementation of Coupled Port-Hamiltonian Continuum and Lumped Parameter Models

Argus

Bradley

Hunter

2021

J Elast

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A continuous Galerkin finite element method that allows mixed boundary conditions without the need for Lagrange multipliers or user-defined parameters is developed. A mixed coupling of Lagrange and Raviart-Thomas basis functions are used. The method is proven to have a Hamiltonian-conserving spatial discretisation and a symplectic time discretisation. The energy residual is therefore guaranteed to be bounded for general problems and exactly conserved for linear problems. The linear 2D wave equation is discretised and modelled by making use of a port-Hamiltonian framework. This model is verified against an analytic solution and shown to have standard order of convergence for the temporal and spatial discretisation. The error growth over time is shown to grow linearly for this symplectic method, which agrees with theoretical results. A modal analysis is performed which verifies that the eigenvalues of the model accurately converge to the exact eigenvalues, as the mesh is refined. The port-Hamiltonian framework allows boundary coupling with bondgraph or, more generally, lumped parameter models, therefore unifying the two fields of lumped parameter modelling and continuum modelling of Hamiltonian systems. The wave domain discretisation is shown to be equivalent to a coupling of canonical port-Hamiltonian forms. This feature allows the model to have mixed boundary conditions as well as to have mixed causality interconnections with other port-Hamiltonian models. A model of the 2D wave equation is coupled, in a monolithic manner, with a lumped parameter model of an electromechanical linear actuator. The combined model is also verified to conserve energy exactly.

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Section: Weak Formmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Theory and Implementation of Coupled Port-Hamiltonian Continuum and Lumped Parameter Models

Argus

Bradley

Hunter

2021

J Elast

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“…The methodology detailed so far is certainly not limited to the previous three examples. Indeed higher-order differential [40], curl operator for Maxwell's equations [32], nonlinear system [41], 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 [9] [10].…”

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. 2021

JAMP

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We consider 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 novel general structure of infinite-dimensional port-Hamiltonian systems (pHs) for which the Partitioned Finite Element Method (PFEM) straightforwardly applies. The proposed strategy is applied 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 open-source 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 available.

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“…A finite-element based technique to obtain a finite-dimensional pH system is illustrated. This methodology relies on the results explained in [11] and ahead, used in [8,9]. The essential feature of this method is that it is structure-preserving.…”

Section: Discretization Proceduresmentioning

confidence: 99%

“…The problem is then written as a coupled system of ordinary and partial differential equations (ODEs and PDEs), extending the general definition of finite-dimensional port-Hamiltonian descriptor systems provided in [33]. A suitable structure-preserving discretization method, based on [11], is then used to obtain a finite-dimensional pH system. The modularity feature of pH systems makes the proposed approach analogous to a substructuring technique [26]: each individual component can be interconnected to the other bodies using standard interconnection of pH systems, as it is done in [31].…”

Section: Introductionmentioning

confidence: 99%

Port-Hamiltonian flexible multibody dynamics

et al. 2020

Self Cite

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A new formulation for the modular construction of flexible multibody systems is presented. By rearranging the equations for a flexible floating body and introducing the appropriate canonical momenta, the model is recast into a coupled system of ordinary and partial differential equations in port-Hamiltonian (pH) form. This approach relies on a floating frame description and is valid under the assumption of small deformations. This allows including mechanical models that cannot be easily formulated in terms of differential forms. Once a pH model is established, a finite element based method is then introduced to discretize the dynamics in a structure-preserving manner. Thanks to the features of the pH framework, complex multibody systems could be constructed in a modular way. Constraints are imposed at the velocity level, leading to an index 2 quasilinear differential-algebraic system. Numerical tests are carried out to assess the validity of the proposed approach.

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A partitioned finite element method for power-preserving discretization of open systems of conservation laws

Cited by 48 publications

References 43 publications

Theory and Implementation of Coupled Port-Hamiltonian Continuum and Lumped Parameter Models

Theory and Implementation of Coupled Port-Hamiltonian Continuum and Lumped Parameter Models

Numerical Approximation of Port-Hamiltonian Systems for Hyperbolic or Parabolic PDEs with Boundary Control

Port-Hamiltonian flexible multibody dynamics

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