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

TL;DR

This paper introduces new quality metrics and discrete functionals for quadrilateral grid generation, improving grid quality assessment and optimization-based grid improvement methods.

Contribution

It proposes novel quality measures for quadrilaterals and develops new discrete functionals for optimization-based grid generation and enhancement.

Findings

New quadrilateral quality measures defined.

Discrete functionals implemented for grid optimization.

Enhanced grid generation methods with preserved convexity.

Abstract

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.