  In this paper, we review some grid quality metrics Robinson1987, Lo1989,
Field2000, Knupp2001, Remacle2012 and define some new quality measures for
quadrilateral elements. Usually, a maximum value of a quality measure
corresponds to the minimum value of the energy density over the grid
Ivanenko2000.
  We also define new discrete functionals which are implemented as objective
functions in an optimization-based method for quadrilateral grid generation and
improvement. These functionals are linearly combined with a discrete functional
whose domain has an infinite barrier at the boundary of the set of unfolded
grids like $S_{\omega,\epsilon}(G)$, see Barrera2010, in order to preserve
convex grid cells in each step of the optimization process.
