7.1.0.1 Definition:

The quality measurement for a tetrahedron $K$ in isotropic case is the following:

$$Q_K = \frac{h_s^3}{V_K},$$

where $h_s = \sqrt{\displaystyle{\sum_{i=1}^{6} h_i^2}}$, $h_i$ being the length of the edge $i$ of the tetrahedron $K$ and $V_{K}$ its volume.

When an anisotropic metric is defined on each mesh point, the quality is more difficult to define. In Mmg3d , we choose to calculate the tetrahedron quality as its quality measured on the euclidian space relative to an average metric defined on the tetrahedron. Lets $P_1$, $P_2$, $P_3$, $P_4$ the vertex of $K$. The average quality on $K$ is given by the following formula:

$${\cal M}_{moy} = \frac{1}{4} \left (\sum_{i=1}^4 {\cal M}_i^{-1}\right)^{-1},$$

where ${\cal M}_i$ is the metric at $P_i$.

The following expression gives the quality of $K$ on the metric ${\cal M}_{moy}$:

$$Q_K = \displaystyle{\frac{\left(\displaystyle{\sum_{1\leq i <... ...iP_j}\right)^\frac{3}{2}} {\sqrt{Det({\cal M}_{moy})} V_K}}$$

NB : Such a criterion allows to decide if a simplex is a good element. You can notice that the quality $Q_K$ varies in $[1 , +\infty[$ and that the smaller the volume of the tetrahedra is and the higher $Q_K$ is.