Solution methods for two key problems in multiscale asymptotic expansion method

The accuracy of decoupled multiscale asymptotic expansion method (MsAEM) depends completely on the accuracy of influence functions and the accuracy of homogenized displacement derivatives. Unit cell problem must be solved first for obtaining influence functions whose calculational accuracy depends largely on the boundary conditions of the unit cell problem; and homogenized problem must be solved for obtaining homogenized displacement derivatives whose accuracy depends largely on the order of used finite elements and meshes. In this work, as for two-dimensional (2D) periodical composite structures, super unit cell approach is proposed to solve for accurate influence functions, and quasi potential energy functional corresponding to influence functions (called quasi displacements in this paper), is constructed to evaluate the accuracy of influence functions or boundary conditions of unit cell problem, and it follows that clamp boundary condition is not always suitable for solving influence functions of different orders although it is exact for unit cell problem of one-dimensional (1D) rod. Finally, the differential quadrature finite element method is employed to improve the computational accuracy of homogenized displacement derivatives.