Accuracy of multiscale asymptotic expansion method

The multiscale asymptotic expansion method (MsAEM) is generally implemented by finite element method. The calculating accuracy of MsAEM depends completely on the order of asymptotic expansion and the order of finite element. First, the necessary number of expansion term is decided in a mechanical view from the pseudo loads used for solving influence functions. Next for different order of load cases, the analytical solutions of the static problems of the periodical composite rod are obtained using different order of MsAEM and finite elements. In those solutions, the element order for solving analytical macro displacements depends on the external loads whereas the element orders for solving analytical influence functions are determined from the governing differential equations of influence functions. Then, two dimensional (2D) periodical composite are explored similarly. Finally, the potential energy functional is used to evaluate the accuracy of MsAEM, and numerical comparisons validate the conclusions.