Explicit Structure-Preserving Discretization of Port-Hamiltonian Systems with Mixed Boundary Control

In this contribution, port-Hamiltonian systems with non-homogeneous mixed boundary conditions are discretized in a structure-preserving fashion by means of the Partitioned FEM. This means that the power balance and the port-Hamiltonian structure of the continuous equations is preserved at the discrete level. The general construction relies on a weak imposition of the boundary conditions by means of the Hellinger-Reissner variational principle, as recently proposed in [Thoma et al., 2021]. The case of linear hyperbolic wave-like systems, including the elastodynamic problem and the Maxwell equations in 3D, is then illustrated in detail. A numerical example is worked out on the case of the wave equation.