Speaker
Description
In this work we study the structure-preserving space discretization of port-Hamiltonian (pH) systems defined with implicit or differential constitutive relations. Using the concept of Stokes-Lagrange structure to describe these relations, we show how these can be reduced to a finite-dimensional Lagrange subspace of a pH system thanks to a structure-preserving Finite Element Method. As an application, we first consider the linear nanorod case in 1-D, which is described by a nonlocal (implicit) constitutive relation for which a Stokes-Lagrange structure along with boundary energy ports naturally occurs. Then, these ideas are applied to a linear nonlocal example of a seepage model in 2-D, namely the Dzektser equation where distributed boundary energy ports have to be taken into account. In both cases we identify the underlying Stokes-Lagrange structure, that describes the Hamiltonian of the system; a focus is given on the boundary energy ports that appear because of the differential operator inside the Hamiltonian. This procedure yields two pH systems defined with Stokes-Dirac (power routing) and Stokes-Lagrange (energy definition) structures. Lastly, these structures are discretized into discrete Dirac and Lagrange structures along with their corresponding discrete control ports. Moreover, the computational burden of the implicit constitutive relation is shown to be better than its explicit counterpart, even if it requires solving a linear system.
Finally, we will extend our results to the nonlinear 2-D incompressible Navier-Stokes equation written in vorticity-stream function formulation. In this case, since a differential operator appears in the constitutive relations, it will be recast as a pH system defined with a Stokes-Lagrange structure and a Stokes-Dirac structure modulated by the energy variable.
