Multiscale eigenelement method for periodical composites: A review

Based on the eigenvector expansion idea, the Multiscale Eigenelement Method (MEM) was proposed by the author and co-workers. MEM satisfies two equivalent conditions, one condition is the equivalence of strain energy, and the other is the deformation similarity. These two equivalent conditions character the structure-preserving property of a multiscale analysis method. The equivalence of strain energy is necessary for achieving accurate macro behaviors such as lower order frequencies, while the deformation similarity is essential for predicting accurate micro behaviors such as stresses. The MEM has become a powerful multiscale method for the analysis of composite structures because of its high accuracy and efficiency. In this paper, the research advances of MEM are reviewed and all types of eigenelement methods are compared, focusing on superiorities and deficiencies from practical viewpoint. It is concluded that the eigenelement methods with smooth shape functions are more suitable for the analysis of macro behaviors such as lower order frequencies, and the eigenelement methods with piecewise shape functions are suitable for the analysis of both macro and micro behaviors.