Author(s)

Andrea Brugnoli¹, Ghislain Haine^2*, Anass Serhani³, Xavier Vasseur²

¹Robotics and Mechatronics Department, University of Twente, Enschede, Netherlands.
²ISAE-SUPAERO, Université de Toulouse, Toulouse, France.
³CERFACS, Toulouse, France.

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.

Port-Hamiltonian Systems, Partial Differential Equations, Boundary Control, Structure-Preserving Discretization, Finite Element Method

Brugnoli, A. , Haine, G. , Serhani, A. and Vasseur, X. (2021) Numerical Approximation of Port-Hamiltonian Systems for Hyperbolic or Parabolic PDEs with Boundary Control. Journal of Applied Mathematics and Physics, 9, 1278-1321. doi: 10.4236/jamp.2021.96088.