A Symplectic Numerical Power Flow Framework Based on Wave Finite-Element Method for Assembled Structural Systems

Identifying the propagation paths of dominant wave modes in complex assembled structure is critical for implementing wave-based vibration and noise control strategies, such as phononic band gaps. This paper presents a symplectic numerical framework to compute the wave-mode power flow in engineering assembled structures based on wave finite element method (WFEM). The power orthogonality among wave modes is explicitly formulated through the symplectic orthogonality (SO) and its adjoint form (SAO), and this formulation is further extended to the Zhong-Williams and lambda(phi) symplectic schemes. The generalized symplectic adjoint orthogonality (GSAO) and phi_SAO are subsequently proposed, providing a physically consistent basis for modal diagonalization and coherent wave propagation within the generalized symplectic eigenspace. These developments enable direct computation of the forced response and power flow entirely within the symplectic space, without reverting to the wave space. Six power-flow formulations are systematically compared and shown to yield consistent results on both beam and cylindrical shell structures. An electric motor housing is used as a case study, in which the proposed approach establishes a wave-mode power flow network. It is noted that the power-flow formulation relies on symplectic orthogonality defined for conservative WFEM systems and therefore cannot be directly applied to non-Hermitian systems.