%---------------------------Distortion---------------------------
\section{Distortion} 

Given a set of Gauss points $G=\{g_k\}$ for a hexahedron, let
\[
|J| = \min_{g_k}\left\{\det\left(J_{g_k}\right)\right\}
\]
be the minimum determinant of the Jacobian when evaluated at each Gauss point $g_k$.
Then the distortion is
\[
q = \frac{|J| V_m}{V}  
\]
where $V_m = 8$ is the volume of a ``master'' hexahedron defined by the vertices
\[
\begin{array}{lcrcrcrl}
  \vec P_0 &= (&-1&,&-1&,& -1&)\\
  \vec P_1 &= (& 1&,&-1&,& -1&)\\
  \vec P_2 &= (& 1&,& 1&,& -1&)\\
  \vec P_3 &= (&-1&,& 1&,& -1&)
\end{array}\rule{5em}{0pt}
\begin{array}{lcrcrcrl}
  \vec P_4 &= (&-1&,&-1&,&  1&)\\
  \vec P_5 &= (& 1&,&-1&,&  1&)\\
  \vec P_6 &= (& 1&,& 1&,&  1&)\\
  \vec P_7 &= (&-1&,& 1&,&  1&)
\end{array}
\]
and $V$ is the volume of the hexahedron being evaluated.
See \S\ref{s:hex-volume} for details on computing the hex volume $V$.

\hexmetrictable{distortion}%
{$L^3$}%                                      Dimension
{$[0.5,1]$}%                                  Acceptable range
{$[0,1]$}%                                    Normal range
{$[-DBL\_MAX,DBL\_MAX]$}%                     Full range
{$1$}%                                        Cube
{Adapted from \cite{ideas:xx}}%               Citation
{v\_hex\_distortion}%                         Verdict function name