Multiscale asymptotic analysis and algorithm for the quadratic eigenvalue problem in composite materials

A novel second-order two-scale(SOTS) asymptotic analysis and computational approach are developed for solving the quadratic eigenvalue problem(QEP) in periodic composite domain. Two typical QEPs involving the velocity damping and the Rayleigh damping are considered and the asymptotic expansions for both the eigenfunctions and eigenvalues are performed. The first-order cell functions characterizing the detailed configuration of the presentative cell are formally defined and the homogenized QEPs are obtained with the macro effective coefficients. The second-order cell functions are further derived which are used to describe the rapid oscillation of the eigenfunctions more accurately. The nonlinear relationship between the original and the homogenized eigenvalues are established by introducing the auxiliary functions defined on the composite domain and the second-order expansions of the eigenvalues are obtained successively. Then, the error estimations of the expansions of eigenvalues are established. Finally, the finite element procedure is proposed, the homogenized QEPs are solved by the linearized method and the numerical examples demonstrating the accuracy and the efficiency of our proposed model and algorithm are reported. It is indicated that the SOTS method can effectively applied to this nonlinear eigenvalue problem and the second-order correctors play an important role for describing the local behavior of eigenfunctions and obtaining better approximation of the eigenvalues at lower cost.