Hi Algiane,

I thank you for you answer, it clarifies a lot of a ideas.

I am currently solving PDEs on parallelepipeds, so that the Hausdorff parameters should be always respected, at least far from the domain edges, doesn’t it?

Regarding the gradation option I have always set it to 0.0 in order to avoid its use, so it has been wrong all this time? If 0.0 is considered a non-disabling value, then mmg provides adapted meshes varying very little, right? (and this would explain why my adapted meshes have low gradation rate, at most 1.2, but still varying in accordance with my error estimator, considering also the [0.6, 1.3] feature).

Apart from this, does mmg limits the gradation also with respect to the orientation (in anisotropic adapt)? E.g., suppose I have two adjacent tetras, with same diagonal values (same shape), but totally different orientation, say they are orthogonal, does mmg adjust them?

What is the justification of 0.6 and 1.3? Is it possible to tune them without modifying source code? If not, where are they located? I suppose that the computational cost of the algorithm rises if the interval [0.6, 1.3] is restricted, there have been studies for this problem?

Up to now I focused on the fact that the product of the eigenvalues of the metric provides the future cell volume. So that, integrating this value and using the domain measure an a priori estimation for the number of elements is provided. This is done on the metric on the old mesh before adaptation. Anyway, this estimate jumps a lot and it is not reliable for this purpose. This estimator assumes that the future mesh is constructed on a not graded metric, and that all the new edges are perfectly unitary. To be more precise, this estimator captures the mean behavior, but it has too much variability (I performed some preliminary statistical exploration).

So that, a good idea could be: disable hgrad and focus on the distribution of the edges with respect to the future mesh. I think that if many edges are in [0.6, 1] w.r.t. the metric in the new mesh, then the estimator overestimates, on the contrary if many edges are in [1, 1.3] then the estimator overestimates. Is it right? If so, it would be great to obtain in advance some sort of measures on how many/ how much edges will miss the value of 1.0.

Anyway, I will go back to the isotropic metric and follow your advises. Thank you!