The multiscale eigenelement method in dynamic analyses of periodical composite structures

Multiscale eigenelement method (MEM), which was presented by the author and co-authors, is based on the idea of eigenvector expansion, and the shape functions in MEM bridge the information between macro and micro scales. In previous works, the shape functions in the classical MEM are solved from the self-equilibrium equation of unit cell, while in the improved MEM they are from the self-equilibrium equation as well as the eigenvalue problem of clamped unit cell. In this work, MEM is compared with substructure method for static problem, two effective correction terms are suggested to improve accuracy if necessary. Another contribution of the present work is, the dynamic governing differential equation of MEM is derived for periodical composite structures according to the generalized Hamilton variational principle, and the details about efficient and accurate use of mode superposition method and Newmark integration method are presented. The numerical comparisons with the standard FEM using fine meshes are conducted for free and forced vibration analyses of one-dimensional and two-dimensional problems. And it is concluded that the improved MEM is suitable for the analyses of both static and dynamic problems, and the classical MEM is suitable for the static problems, especially the one with correction term.