Periodic microstructural composites have gained considerable attention in material science and engineering attributable to their excellent flexibility in tailoring various desirable physical properties. Conventionally, the finite element technique has been widely used in implementing the homogenization. However, the standard finite element method (FEM) leads to an overly stiff model which sometimes gives unsatisfactory accuracy especially using triangular elements in 2D or tetrahedral elements in 3D with coarse mesh. In this paper, different forms of smoothed finite element method (SFEM) are presented to develop new asymptotic homogenization techniques for analyzing various effective physical properties of periodic microstructural composite materials. A range of multifunctional material examples, including elastic modulus with multiphase composites, conductivity of thermal and electrical composites, and diffusivity/permeability of 3D tissue scaffold, has exemplified herein to demonstrate that SFEM is able to provide more accurate results using the same set of mesh compared with the standard FEM. In addition, the computational efficiency of SFEM is also higher than that of the standard FEM counterpart.