We present a mesh smoothing algorithm based on the MMPDE moving mesh method based on the direct geometric discretization of the underlying meshing functional on simplicial meshes. The nodal mesh velocities are given in a simple, analytical matrix form. We further combine the moving mesh smoothing with the lazy flip technique, a reversible edge removal algorithm to modify the mesh connectivity, and utilize radial basis function (RBF) surface reconstruction to improve tetrahedral meshes with curved boundary surfaces. Numerical comparison with some publicly available mesh improving software is provided. This work is a collaboration with Weizhang Huang (University of Kansas), Hang Si (Weierstrass Institute), Franco Dassi (Università degli Studi di Milano-Bicocca), and Patricio Farrell (Weierstrass Institute).