An eigenelement method of periodical composite structures

Eigenelement method is an eigenvector expansion based finite element method, which was proposed by the authors to solve the macro behaviors of composites with less computational cost. To improve the macroscopic accuracy of the classical eigenelement method (CEEM), a serendipity eigenelement method (SEEM) is proposed, which takes the geometry and elastic properties of different phases of composites into account to some extent. Moreover, the shape function and its construction method of a multiscale eigenelement method (MEM) are presented, and the results of SEEM and MEM are compared with that of CEEM and the mathematical homogenization method (MHM) whose physical interpretation is revealed for the first time. It is shown that MEM is the most accurate eigenelement, SEEM is more accurate than CEEM, and MEM satisfies the two essential homogenization conditions: the strain energy equivalence and the deformation similarity. The extensive numerical comparison is given for stresses, displacements and frequencies.