Abstract
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.
Highlights
The efficient numerical simulation of complex multiphysics systems is ubiquitous in Computational Science and Engineering
The driving forces of Partitioned Finite Element Method (PFEM) are threefold: first, PFEM takes collocated boundary controls and observations into account in a simple manner; secondly, PFEM is structure-preserving, meaning in particular that the discrete power balance perfectly mimics the continuous one; thirdly, the implementation of PFEM only relies on existing finite element libraries, such as FEniCS [23], selected in the ongoing project SCRIMP for its robustness and efficiency
Its core components notably include the Unified Form Language (UFL) [44], the FEniCS Form Compiler (FFC) [45] and the finite element library DOLFIN [46], which contains various types of conforming finite element methods, e.g., nodal Lagrangian finite elements for grad-conforming approximations or non non-nodal finite elements (e.g., Raviart-Thomas spaces for div-conforming approximations) as well
Summary
The efficient numerical simulation of complex multiphysics systems is ubiquitous in Computational Science and Engineering. Infinite-dimensional port-Hamiltonian systems (pHs) have been first introduced in [1] using the language of differential geometry They provide a powerful tool to model complex multiphysics open systems (whether or not being linear) for control purpose. The existence of an underlying common structure for many pHs is highlighted Obtaining such a general scheme for infinite-dimensional pHs is of major importance for control purposes [11], and for coupling atomic elements into a more complex system with the guarantee of well-preserved energy exchanges between subsystems [12]. The first proposed structure-preserving scheme for pHs dates back to [13], where the authors proposed a mixed finite element spatial discretization for hyperbolic systems of conservation laws. Thanks to [8] it has become clear that there exists a deep relation between structure-preserving discretization of pHs and Mixed Finite Element Method
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