High-efficient complex eigen-solution algorithms for transcendental dynamic stiffness formulations of plate built-up structures with frequency-dependent viscoelastic models

Two highly accurate and reliable eigen-solution techniques, the new homotopy perturbation method and the extended argument principle method, are proposed for analysing orthotropic viscoelastic plate built-up structures. These techniques are formulated to solve transcendental eigenvalue problems in modal analysis based on the analytical damped dynamic stiffness formulations. The homotopy perturbation method uses undamped real-valued eigenvalues and eigenvectors computed by the Wittrick-Williams algorithm as the exact initial solutions. The internal damping coefficient and external damping coefficient are set as convergence control parameters by using the homotopy method, and the initial solutions are updated through inverse iteration to efficiently obtain the final complex eigenvalues. Conversely, the extended argument principle method utilizes the dichotomy of mode count in the complex domain, based on the denominators of elemental dynamic stiffness matrices, to pinpoint complex eigenvalues. Validation against finite element solutions from COMSOL shows that while the extended argument principle method offers benchmark solutions, it is computationally intensive. In contrast, the proposed homotopy perturbation method presents a valuable tool in engineering applications due to its exceptional balance of accuracy and computational efficiency. This method facilitates rapid analyses and design parameter optimization within the context of viscoelastic plate structures.