Homogenization method of non-periodic multi-scale problem

In this paper, the basic idea and the finite element implementation of the homogenization theory applied in nonperiodic multi-scale problem were introduced. According to this, the mechanical properties are the macroscopic representation of particle behavior, such as atoms, molecules and grains. The importance of the arithmetic average is analyzed, and the homogenization method is proved to be the efficient method of solving multi-scale problems from algorithm analysis point of view. Furthermore, for heterogeneous materials with the same order of size of typical microscale structures, the calculation scale of finite element using the homogenization method is far less than that of the direct finite element method. The results show that the calculation accuracy of finite element using the homogenization method keeps consistent with that of the direct finite element method.