A Port-Hamiltonian Approach to Modeling the Structural Dynamics of Complex Systems

Part A: Both the paper and the candidate give the identical ISO port-Hamiltonian realization of the second-order FE model with x = [Mfe qdot; q], J = [[0,-I],[I,0]], R = [[Dfe,0],[0,0]], Q = diag(Mfe^{-1}, Kfe), G = [[I],[0]], and y = qdot; this is exactly Eq. (3) in the paper. Part B: The paper’s PFEM path yields a DAE, eliminates constraints, reduces to 2 nDOF states, and via two explicit transformations Tu and Tp maps to the standard FE form (3) using the same bases; the candidate’s argument mirrors this, phrasing the reduction as a linear bijection from energy coordinates to (p, q) with p = Mfe qdot and strains = discrete gradients of q. Aside from minor differences in emphasis (e.g., candidate names a discrete gradient operator), the proofs are materially the same.