Computation of piece-wise linear hat functions
First define one triangle and interpolations as follows:
$$
\begin{bmatrix} x \\ y \\ z \end{bmatrix} =
\begin{bmatrix} x_1 \\ y_1 \\ z_1 \end{bmatrix} +
\begin{bmatrix} x_2-x_1 \\ y_2-y_1 \\ z_2-z_1 \end{bmatrix} \xi +
\begin{bmatrix} x_3-x_1 \\ y_3-y_1 \\ z_3-z_1\end{bmatrix} \eta
\qquad \mbox{with:} \quad \begin{cases} \xi > 0 \\ \eta > 0 \\ \xi+\eta < 1 \end{cases}
$$
And quite in general:
$$
\phi = \phi_1 + (\phi_2-\phi_1)\,\xi + (\phi_3-\phi_1)\,\eta
$$
From this reference we infer that:
$$
\begin{cases}
\xi = [ (y_3 - y_1).(x - x_1) - (x_3 - x_1).(y - y_1) ] / \Delta \\
\eta = [ (x_2 - x_1).(y - y_1) - (y_2 - y_1).(x - x_1) ] / \Delta
\end{cases}
$$
Where $\Delta$ is twice the area of a triangle.
The Finite Element shape functions thus are, still for one triangle:
$$
N_1 = 1-\xi-\eta \quad ; \quad N_2 = \xi \quad ; \quad N_3 = \eta
$$ $$
x = N_1x_1+N_2x_2+N_3x_3 \\
y = N_1y_1+N_2y_2+N_3y_3 \\
z = N_1z_1+N_2z_2+N_3z_3 \\
\phi = N_1\phi_1+N_2\phi_2+N_3\phi_3
$$
Now specify for the 6 triangles in you mesh and you're done, for $\,\xi = \eta = 0\,$ eventually:
- $(1)\rightarrow(17) \;,\; (2)\rightarrow(11) \;,\; (3)\rightarrow(12)$
- $(1)\rightarrow(17) \;,\; (2)\rightarrow(12) \;,\; (3)\rightarrow(13)$
- $(1)\rightarrow(17) \;,\; (2)\rightarrow(13) \;,\; (3)\rightarrow(14)$
- $(1)\rightarrow(17) \;,\; (2)\rightarrow(14) \;,\; (3)\rightarrow(15)$
- $(1)\rightarrow(17) \;,\; (2)\rightarrow(15) \;,\; (3)\rightarrow(16)$
- $(1)\rightarrow(17) \;,\; (2)\rightarrow(16) \;,\; (3)\rightarrow(11)$