Port-Hamiltonian formulation and structure-preserving discretization of finite elasticity based on a mixed Hu-Washizu-type formulation

We propose a port-Hamiltonian formulation and structure-preserving discretization of finite elasticity. The energy functional (or Hamiltonian) is based on a polyconvex representation of the stored energy and gives rise to three strain-type fields, which play the role of energy variables in the port-Hamiltonian formulation. We show that a Hu-Washizu-type extension of the variational principle of Livens can be used (i) to derive the continuous port-Hamiltonian formulation and (ii) to perform a structure-preserving spatial discretization. In particular, we show that the spatial finite element discretization of the underlying mixed formulation yields a discrete port-Hamiltonian system. Moreover, the temporal discretization of the underlying continuous formulation yields a new energy-momentum consistent framework, which accommodates alternative finite element formulations. The new framework, in particular, covers mixed finite elements that have been shown to be well suited for handling quasi-incompressible material behavior. Numerical examples are provided to evaluate the numerical performance and stability of the newly devised energy-momentum schemes.