Explicit Port-Hamiltonian FEM-Models for Linear Mechanical Systems with Non-Uniform Boundary Conditions

The paper’s Theorem 2 states that the mixed Galerkin discretization of the weak elastodynamics formulation with weak Neumann/Dirichlet boundary conditions yields an explicit port-Hamiltonian ODE M ė = J e + G u with y = G^T e, where M = diag(Mv, Mσ), J = [0 −K; K^T 0], and matrices Mv, Mσ, K, Gv, Gσ are given by (25)–(29) , starting from the weak form (18a)–(18b) and the by-parts step (30)–(32) that exposes the skew-adjoint coupling leading to J’s skew-symmetry and the energy balance (34) . The candidate solution reproduces the same semi-discrete system by inserting the same Galerkin expansions, using one Green/IBP identity to identify the same K and boundary input/output matrices, and then verifies J’s skew-symmetry, M’s invertibility/positive definiteness, and the discrete power balance—all matching the paper’s result. Differences are not substantive (integration-by-parts placement and transposes due to basis orientation), so both are correct and essentially the same derivation.