Strain smoothing operation has been recently adopted to soften the stiffness of the model created using tetrahedron mesh, such as the Face-based Smoothed Finite Element Method (FS-FEM), with the aim to improve solution accuracy and the applicability of low order tetrahedral elements. In this paper, a new method with strain smoothing operation based on the edge of four-node tetrahedron mesh is proposed, and the edge-based smoothing domain of tetrahedron mesh is serving as the assembly unit for computing the 3D stiffness matrix. Numerical results demonstrate that the proposed method possesses a close-to-exact stiffness of the continuous system and gives better results than both the FEM and FS-FEM using tetrahedron mesh or even the FEM using hexahedral mesh in the static and dynamic analysis. In addition, a novel domain-based selective scheme is proposed leading to a combined ES-T-/NS-FEM model that is immune from volumetric locking and hence works well for nearly incompressible materials. The proposed method is an innovative and unique numerical method with its distinct features, which possesses strong potentials in the successful applications for static and dynamics problems.