Numerics for Physics-Based PDEs with Boundary Control: the Partitioned Finite Element Method for Port-Hamiltonian Systems

Abstract

The numerical simulation of complex open multiphysics systems in Computational Science and Engineering is a challenging topic. Based on energy exchanges, the port-Hamiltonian formalism aims at describing physics in a structured manner. One of the major interests of this approach is its versatility, allowing for coupling and interconnection that preserve this structure. We propose a Finite Element based technique for the structure-preserving discretization of a large class of port-Hamiltonian systems. Assuming a partitioned structure of the system associated to an integration-by-parts formula, it is possible to derive a consistent weak-formulation sharing the main features of the original boundary-controlled PDE. This allows using Galerkin approximations to obtain finite-dimensional systems that mimic the properties of the original distributed ones: the Partitioned Finite Element Method producing sparse matrices enables the use of dedicated algorithms in scientific computing. Indeed, this method can be easily implemented using well-established and robust libraries. This strategy is illustrated by means of physically motivated PDEs: acoustic waves, Mindlin and Kirchhoff plates, heat equation, Maxwell's equation. Interactive Jupyter notebooks are available, relying on the FEniCS open-source software. Advanced applications include multiphysics problems, e.g. fluid-structure interactions, thermoelasticity, and modular modelling of complex systems, e.g. multibody dynamics.