Vectorized computation of basis functions gradients¶

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

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

On $T_k$ tetrahedra we set

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

Then, we have

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

With these formulas, we obtain the vectorized algorithm given in Algorithm 26.

Algorithm 26

Figure 26: Vectorized algorithm for computation of basis functions gradients in 3d

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