Meaning of "mesh quality" in anisotropic output result

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#1

Hi to all,

probably it is a trivial question, but I don’t understand how to interpret the “mesh quality” section in the output results after an anisotropic mesh adaptation.

For example:

BEST 0.999487
AVERAGE 0.723562
WORST 0.142121

95,94% > 0.5

What does it mean “best=0.999487” ?

Thank you for any help!

#2

Hi,

You will find a description of Mmg histograms on the Mmg webpage : Mmg histograms – Mmg Platform

Do not hesitate if you still have questions.

Best Regards,
Algiane

#3

yes i read this part, but it does not explain in which sense “best” is intended. How it is calculated? What is its relation with , for example, the volume of the tetrahedra or the stretching factors?

Can you explain me in more detail the idea behind this interesting output result?

Thanks

#4

Ok, I will try to add some formula and explanation on the webpage.

So, the quality in Mmg is a measure of the shape ratio in the metric space reconstructed from the size map.

A “perfect” element has quality 1. Quality 0 correspond to element with volume null.
For an isotropic metric an element of quality 1 is equilateral, for an anisotropic metric it is inscribed in the metric tensor.

More precisely, if I note e_i the i^{th} edges of an element, l_i their lengths, \bar{M} the mean metric over this element, V_{\bar{M}_i} the volume of this element in the mean metric:

  • quality formula for a triangle is:
Q_T = \alpha \frac{V_{\bar{M}}}{\sum_{e_i} l_i^2}
  • quality formula for a tetra is:
Q_T = \alpha \frac{V_{\bar{M}}}{(\sqrt{\sum_{e_i} l_i^2})^{\frac{3}{2}}}

with \alpha a normalization coefficient such as quality is 1 for an isotropic metric and and equilateral element.

Hope that it helps,

Best Regards,
Algiane

#5

ok now is much more clear. In other words, we can say that the quality quantify how well the metric is reproduced in the mesh, correct?

#6

Hi,

Yes, we can say that, while keeping in mind that it translates the deformation of the element (in the metric).

Best

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