meshquality.f

In this routine the quality of each element is determined. To this end the ratio of the largest edge $L_i$ to the radius of the inscribed sphere is used. One can prove that the radius of the inscribed sphere of a linear tetrahedral is three times the volume $V_i$ divided by the sum of the area of its faces $(\sum S)_i$ [25]. Therefore, the quality $Q_i$ for element $i$ can be written as:

$$Q_i=\frac{\sqrt{6} }{12 } \frac{L_i (\sum S)_i }{3 \max(V_i, 10^{-30})}.$$ (715)

The factor $\frac{\sqrt{6} }{12 }$ is such that the quality of an equilateral tetrahedron is 1. For all other tetrahedra it exceeds 1. The larger the value, the worse the element. The cut-off of $10^{-30}$ was introduced to avoid dividing by zero or getting a negative value.