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In this paper, we review some grid quality metrics Robinson1987, Lo1989,Field2000, Knupp2001, Remacle2012 and define some new quality measures forquadrilateral elements. Usually, a maximum value of a quality measurecorresponds to the minimum value of the energy density over the gridIvanenko2000.  We also define new discrete functionals which are implemented as objectivefunctions in an optimization-based method for quadrilateral grid generation andimprovement. These functionals are linearly combined with a discrete functionalwhose domain has an infinite barrier at the boundary of the set of unfoldedgrids like $S_{\omega,\epsilon}(G)$, see Barrera2010, in order to preserveconvex grid cells in each step of the optimization process.

New quality measures for quadrilaterals and new discrete functionals for grid generation