Accuracy analysis of mathematical homogenization method for several periodical composite structure

The mathematical homogenization method (MHM) is generally implemented by finite element method (FEM) and its calculating accuracy depends completely on the order of perturbation and finite element, the perturbed displacements in uncoupled form are defined as the multiplications of influence functions and the derivatives of homogenized displacements. The order of elements depends on the accuracy requirements of influence function and homogenized displacements while the order of perturbations depends mainly on the accuracy of different-order derivatives of homogenized displacements and the properties of quasi loads. For the static problems of periodical composite rod, the exact solutions can be obtained using correct order of MHM and finite element for the static problem of periodic composite rod subjected to different order of loads. Then two-dimensional (2D) periodical composite structures are explored similarly, and the clamped boundary condition of periodical unite cell and the derivatives of homogenized displacements have great effects on the calculation accuracy of MHM. The effect of second-order perturbations is stressed for the accuracy of MHM. Finally, the potential energy functional is used to evaluate the accuracy of MHM and numerical comparisons validate the conclusions.