In order to determine the validity and the quality of meshes, mesh optimization methods have been formulated with quality measures. The basic idea of mesh optimization is to relocate the vertices to obtain a valid mesh (untangling) or improve the mesh quality (smoothing), or both. We will look at a new algebraic way of calculating quality measure on quadrilateral meshes, based on triangular meshes in 2D as well as new optimization methods for simultaneous untangling and smoothing for severely deformed meshes. An innovative anisotropic diffusion method will be introduced for consideration of inner boundary deformation movements for quadrangle meshes in 2D.