# New Quality Measures for Quadrilaterals and New Discrete Functionals for Grid Generation

^{*}

## Abstract

**:**

## 1. Introduction

**Clean-up**.- This consists of the elimination, insertion and reconnection of nodes to eliminate the worst elements. Some authors call that this procedure topological optimization in the sense that the connectivity of the nodes is removed to obtain an optimal configuration.
**Smoothing**.- This consists of node repositioning without changing the connectivity of the elements.

**Definition**

**1**

**.**We say that a real-valued function $\mu \left(Q\right)$ over a quadrilateral Q is a quality measure in the sense of Field-Oddy if it

- (1)
- Has the ability to detect degenerated elements;
- (2)
- Is bounded and continuous;
- (3)
- Is independent of scale;
- (4)
- Is normalized;
- (5)
- Is invariant under rigid transformations.

## 2. Background

## 3. New Quality Measures

#### 3.1. Quality Measure of Rectangles

**Example**

**1.**

#### 3.2. New Aspect Ratio

**Example**

**2.**

**Theorem**

**1.**

#### 3.3. Quality Measure of Parallelograms

_{1}and g

_{2}are equal $1/2$, ${g}_{3}={g}_{4}$ so that the other sides are parallel, see [4]. A quality measure to characterize parallelograms is

#### 3.4. Quality Measure of Squares

**Theorem**

**2.**

**Proof.**

## 4. Some Global Quality Metrics

- (1)
- A visual or exploratory or inspection;
- (2)
- Qualitative evaluation or shape parameters;
- (3)
- Statistical analysis.

- (1)
- How many elements are squares?
- (2)
- How many elements are rectangles with aspect ratio less than 4?
- (3)
- How many elements are parallelograms with aspect ratio less than 4?

## 5. Grid Quality Improvement

#### On Distortion of the Mesh

**Definition**

**2.**

## 6. New Quality Discrete Functionals

## 7. Examples

## 8. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**${A}^{\prime}BC{D}^{\prime}$ is the rectangle of minimum area for quadrilateral $ABCD$ and $RUTS$ is the rectangle associated by Robinson for Example 1.

**Figure 3.**$NBCD$ is the rectangle of minimum area for quadrilateral $ABCD$ and $JKLM$ is the rectangle associated by Robinson for the Example 2.

**Figure 5.**Level curves for (20) where (

**a**) Q has 3 fixed vertices $(0,0),(2,0)$ and $(0.5,2)$ and (

**b**) Q has 3 fixed vertices $(0,0),(2,0.5)$ and $(0.5,2)$.

**Figure 6.**Level curves for (22) where (

**a**) Q has 3 fixed vertices $(0,0),(2,0)$ and $(0.5,2)$ and (

**b**) Q has 3 fixed vertices $(0,0),(2,0.5)$ and $(0.5,2)$.

**Figure 7.**Level curves for (24) using (

**a**) ${\mu}_{1}\left(T\right)$ and (

**b**) radius ratio${\mu}_{2}\left(T\right)$.

**Figure 9.**Level curves for (

**a**) ${f}_{R}\left(Q\right)$ and (

**b**) ${f}_{r}\left(Q\right)$ where Q has 3 fixed vertices $(0,0),(2,0)$ and $(0,1)$.

**Figure 12.**Color map of (

**a**) rectangles quality measure and (

**b**) rectangle of minimum area quality measure.

**Figure 13.**Color map of (

**a**) rectangles quality measure and (

**b**) rectangle of minimum area quality measure.

**Table 1.**Summary of quadrangle quality measures for the mesh in Figure 8.

Shape | Name $\mathit{\mu}\left(\mathit{Q}\right)$ | Min | Max | MQ | MSE | SP |
---|---|---|---|---|---|---|

Parallelogram | AreaI | 0.0381 | 0.9998 | 0.9169 | 0.1081 | 0.9068 |

AreaII | 0.0117 | 0.9973 | 0.8615 | 0.1510 | 0.8284 | |

Rectangle | Lo1989 | 0.0160 | 0.9941 | 0.7597 | 0.1901 | 0.7241 |

ScaledJacobian | 0.2391 | 1.0000 | 0.9384 | 0.0975 | 0.9318 | |

ScaledJacobianM | 0.0457 | 0.9965 | 0.8639 | 0.1361 | 0.8478 | |

MinRect2015 | 0.0555 | 0.9942 | 0.7929 | 0.1837 | 0.7607 | |

Rectangles2015 | 0.0719 | 0.9983 | 0.8999 | 0.1170 | 0.8890 | |

Square | Lo1985 | 0.0339 | 0.9897 | 0.4792 | 0.2362 | 0.4162 |

Hua1995 | 0.0830 | 0.9996 | 0.5302 | 0.2573 | 0.4621 | |

Knupp2000 | 0.0365 | 0.9993 | 0.5113 | 0.2588 | 0.4402 | |

Pebay2002 | 0.0591 | 0.9814 | 0.6119 | 0.2208 | 0.5670 | |

Hmean2017E | 0.0797 | 0.9996 | 0.5292 | 0.2578 | 0.4607 | |

Hmean2017R | 0.0788 | 0.9996 | 0.5292 | 0.2578 | 0.4606 | |

Hmean2017r | 0.0297 | 0.9996 | 0.5351 | 0.2485 | 0.4713 |

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**MDPI and ACS Style**

González Flores, G.F.; Barrera Sánchez, P.
New Quality Measures for Quadrilaterals and New Discrete Functionals for Grid Generation. *Math. Comput. Appl.* **2023**, *28*, 95.
https://doi.org/10.3390/mca28050095

**AMA Style**

González Flores GF, Barrera Sánchez P.
New Quality Measures for Quadrilaterals and New Discrete Functionals for Grid Generation. *Mathematical and Computational Applications*. 2023; 28(5):95.
https://doi.org/10.3390/mca28050095

**Chicago/Turabian Style**

González Flores, Guilmer Ferdinand, and Pablo Barrera Sánchez.
2023. "New Quality Measures for Quadrilaterals and New Discrete Functionals for Grid Generation" *Mathematical and Computational Applications* 28, no. 5: 95.
https://doi.org/10.3390/mca28050095