A hierarchical multilevel finite element method for mechanical analyses of periodical composite structures

Abstract

A hierarchical multilevel finite element method (HMM) is developed for simulating mechanical behaviors of periodical composite structures. This method is established based on the idea of constructing shape function numerically which is proposed in the multiscale finite element method (MsFEM). In order to improve the computational accuracy, a hierarchical shape function and a modal shape function are also constructed numerically on sub-grids domain in addition to the original shape function. Then totally three kinds of numerical shape function are introduced in the proposed HMM. Correspondingly, the macroscopic displacement of coarse element can be divided into three parts which are contributed by nodal (original), hierarchical and modal shape functions, respectively. By virtue of these functions, the equivalent mass matrix, stiffness matrix and external loading vector of one single coarse element can be determined easily. The overall macroscopic quantities of whole composite structure are obtained by assembling those of each coarse element. Finally the original problems can be solved at the macroscopic scale, which will save amount of computational cost. To verify the validation and efficiency of the developed HMM, the static analysis, generalized eigenvalue analysis and transient response analysis are investigated.