Vectorized computation of basis functions gradients
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

By construction, the gradients of basis functions are constants on each element :math:`T_k.` 
So, we denote, :math:`\forall \il\in\ENS{1}{4},` by :math:`\vecb{G}^\il` the :math:`3 \times \nme` array defined,
:math:`\forall k\in\ENS{1}{\nme},` by

.. math::

   \vecb{G}^\il(:,k)= \GRAD\BasisFunc_{\me(\il,k)}(\q),\ \forall \q\in T_k.

On :math:`T_k` tetrahedra
we set

.. math::
  \begin{array}{lcllcl}
  \vecb{D}^{12}&=&\q^{\me(1,k)}-\q^{\me(2,k)},\ & \vecb{D}^{23}&=&\q^{\me(2,k)}-\q^{\me(3,k)}\\
  \vecb{D}^{13}&=&\q^{\me(1,k)}-\q^{\me(3,k)},\ & \vecb{D}^{24}&=&\q^{\me(2,k)}-\q^{\me(4,k)}\\
  \vecb{D}^{14}&=&\q^{\me(1,k)}-\q^{\me(4,k)},\ & \vecb{D}^{34}&=&\q^{\me(3,k)}-\q^{\me(4,k)}
  \end{array}

Then, we have

.. math::
  \begin{array}{ll}
    \GRAD\BasisFunc_{1}^k(\q)=\frac{1}{6|T_k|}
    \begin{pmatrix}
    -\vecb{D}^{23}_y \vecb{D}^{24}_z + \vecb{D}^{23}_z \vecb{D}^{24}_y\\
          \vecb{D}^{23}_x \vecb{D}^{24}_z - \vecb{D}^{23}_z \vecb{D}^{24}_x\\
          -\vecb{D}^{23}_x \vecb{D}^{24}_y + \vecb{D}^{23}_y \vecb{D}^{24}_x
    \end{pmatrix},&\GRAD\BasisFunc_{2}^k(\q)=\frac{1}{6|T_k|}
    \begin{pmatrix} \vecb{D}^{13}_y \vecb{D}^{14}_z - \vecb{D}^{13}_z \vecb{D}^{14}_y\\
          -\vecb{D}^{13}_x \vecb{D}^{14}_z + \vecb{D}^{13}_z \vecb{D}^{14}_x\\
          \vecb{D}^{13}_x \vecb{D}^{14}_y - \vecb{D}^{13}_y \vecb{D}^{14}_x
    \end{pmatrix}\\
    \GRAD\BasisFunc_{3}^k(\q)=\frac{1}{6|T_k|}
    \begin{pmatrix} -\vecb{D}^{12}_y \vecb{D}^{14}_z + \vecb{D}^{12}_z \vecb{D}^{14}_y\\
          \vecb{D}^{12}_x \vecb{D}^{14}_z - \vecb{D}^{12}_z \vecb{D}^{14}_x\\
          -\vecb{D}^{12}_x \vecb{D}^{14}_y + \vecb{D}^{12}_y \vecb{D}^{14}_x
    \end{pmatrix},& 
    \GRAD\BasisFunc_{4}^k(\q)=\frac{1}{6|T_k|}
    \begin{pmatrix}     
    \vecb{D}^{12}_y \vecb{D}^{13}_z - \vecb{D}^{12}_z \vecb{D}^{13}_y\\
          -\vecb{D}^{12}_x \vecb{D}^{13}_z + \vecb{D}^{12}_z \vecb{D}^{13}_x\\
          \vecb{D}^{12}_x \vecb{D}^{13}_y - \vecb{D}^{12}_y \vecb{D}^{13}_x
    \end{pmatrix}
    \end{array}

    
With these formulas, we obtain the vectorized algorithm given in Algorithm :num:`gradientvec3dalgo`.

.. admonition:: Algorithm :num:`gradientvec3dalgo`

    .. _GradientVec3Dalgo:  
          
    .. figure::  images/GradientVec3D_algo.png
          :width: 600px
          :scale: 100%
          :align:   center
        
          Vectorized algorithm for computation of basis functions gradients in 3d