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Degree 2 Hellan–Herrmann–Johnson on a tetrahedron
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- \(R\) is the reference tetrahedron. The following numbering of the subentities of the reference cell is used:
- \(\mathcal{V}\) is spanned by: \(\left(\begin{array}{ccc}\displaystyle 1&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle 1&\displaystyle 0\\\displaystyle 1&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle 0&\displaystyle 1\\\displaystyle 0&\displaystyle 0&\displaystyle 0\\\displaystyle 1&\displaystyle 0&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 1&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle 1\\\displaystyle 0&\displaystyle 1&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle 1\end{array}\right)\), \(\left(\begin{array}{ccc}\displaystyle x&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle x&\displaystyle 0\\\displaystyle x&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle 0&\displaystyle x\\\displaystyle 0&\displaystyle 0&\displaystyle 0\\\displaystyle x&\displaystyle 0&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle x&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle x\\\displaystyle 0&\displaystyle x&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle x\end{array}\right)\), \(\left(\begin{array}{ccc}\displaystyle x^{2}&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle x^{2}&\displaystyle 0\\\displaystyle x^{2}&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle 0&\displaystyle x^{2}\\\displaystyle 0&\displaystyle 0&\displaystyle 0\\\displaystyle x^{2}&\displaystyle 0&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle x^{2}&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle x^{2}\\\displaystyle 0&\displaystyle x^{2}&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle x^{2}\end{array}\right)\), \(\left(\begin{array}{ccc}\displaystyle y&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle y&\displaystyle 0\\\displaystyle y&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle 0&\displaystyle y\\\displaystyle 0&\displaystyle 0&\displaystyle 0\\\displaystyle y&\displaystyle 0&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle y&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle y\\\displaystyle 0&\displaystyle y&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle y\end{array}\right)\), \(\left(\begin{array}{ccc}\displaystyle x y&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle x y&\displaystyle 0\\\displaystyle x y&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle 0&\displaystyle x y\\\displaystyle 0&\displaystyle 0&\displaystyle 0\\\displaystyle x y&\displaystyle 0&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle x y&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle x y\\\displaystyle 0&\displaystyle x y&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle x y\end{array}\right)\), \(\left(\begin{array}{ccc}\displaystyle y^{2}&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle y^{2}&\displaystyle 0\\\displaystyle y^{2}&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle 0&\displaystyle y^{2}\\\displaystyle 0&\displaystyle 0&\displaystyle 0\\\displaystyle y^{2}&\displaystyle 0&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle y^{2}&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle y^{2}\\\displaystyle 0&\displaystyle y^{2}&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle y^{2}\end{array}\right)\), \(\left(\begin{array}{ccc}\displaystyle z&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle z&\displaystyle 0\\\displaystyle z&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle 0&\displaystyle z\\\displaystyle 0&\displaystyle 0&\displaystyle 0\\\displaystyle z&\displaystyle 0&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle z&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle z\\\displaystyle 0&\displaystyle z&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle z\end{array}\right)\), \(\left(\begin{array}{ccc}\displaystyle x z&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle x z&\displaystyle 0\\\displaystyle x z&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle 0&\displaystyle x z\\\displaystyle 0&\displaystyle 0&\displaystyle 0\\\displaystyle x z&\displaystyle 0&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle x z&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle x z\\\displaystyle 0&\displaystyle x z&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle x z\end{array}\right)\), \(\left(\begin{array}{ccc}\displaystyle y z&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle y z&\displaystyle 0\\\displaystyle y z&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle 0&\displaystyle y z\\\displaystyle 0&\displaystyle 0&\displaystyle 0\\\displaystyle y z&\displaystyle 0&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle y z&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle y z\\\displaystyle 0&\displaystyle y z&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle y z\end{array}\right)\), \(\left(\begin{array}{ccc}\displaystyle z^{2}&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle z^{2}&\displaystyle 0\\\displaystyle z^{2}&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle 0&\displaystyle z^{2}\\\displaystyle 0&\displaystyle 0&\displaystyle 0\\\displaystyle z^{2}&\displaystyle 0&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle z^{2}&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle z^{2}\\\displaystyle 0&\displaystyle z^{2}&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle z^{2}\end{array}\right)\)
- \(\mathcal{L}=\{l_0,...,l_{59}\}\)
- Functionals and basis functions:
\(\displaystyle l_{0}:\mathbf{V}\mapsto\displaystyle\int_{f_{0}}(2 s_{0}^{2} + 4 s_{0} s_{1} - 3 s_{0} + 2 s_{1}^{2} - 3 s_{1} + 1)|{f_{0}}|\hat{\boldsymbol{n}}^{\text{t}}_{0}\mathbf{V}\hat{\boldsymbol{n}}_{0}\)
where \(f_{0}\) is the 0th face;
\(\hat{\boldsymbol{n}}_{0}\) is the normal to facet 0;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(f_{0}\).
\(\displaystyle \mathbf{\Phi}_{0} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle 30 x^{2} - 20 x + 2&\displaystyle 30 x^{2} - 20 x + 2\\\displaystyle 30 x^{2} - 20 x + 2&\displaystyle 0&\displaystyle 30 x^{2} - 20 x + 2\\\displaystyle 30 x^{2} - 20 x + 2&\displaystyle 30 x^{2} - 20 x + 2&\displaystyle 0\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
where \(f_{0}\) is the 0th face;
\(\hat{\boldsymbol{n}}_{0}\) is the normal to facet 0;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(f_{0}\).
\(\displaystyle \mathbf{\Phi}_{0} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle 30 x^{2} - 20 x + 2&\displaystyle 30 x^{2} - 20 x + 2\\\displaystyle 30 x^{2} - 20 x + 2&\displaystyle 0&\displaystyle 30 x^{2} - 20 x + 2\\\displaystyle 30 x^{2} - 20 x + 2&\displaystyle 30 x^{2} - 20 x + 2&\displaystyle 0\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{1}:\mathbf{V}\mapsto\displaystyle\int_{f_{0}}(s_{0} \left(2 s_{0} - 1\right))|{f_{0}}|\hat{\boldsymbol{n}}^{\text{t}}_{0}\mathbf{V}\hat{\boldsymbol{n}}_{0}\)
where \(f_{0}\) is the 0th face;
\(\hat{\boldsymbol{n}}_{0}\) is the normal to facet 0;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(f_{0}\).
\(\displaystyle \mathbf{\Phi}_{1} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle 30 y^{2} - 20 y + 2&\displaystyle 30 y^{2} - 20 y + 2\\\displaystyle 30 y^{2} - 20 y + 2&\displaystyle 0&\displaystyle 30 y^{2} - 20 y + 2\\\displaystyle 30 y^{2} - 20 y + 2&\displaystyle 30 y^{2} - 20 y + 2&\displaystyle 0\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
where \(f_{0}\) is the 0th face;
\(\hat{\boldsymbol{n}}_{0}\) is the normal to facet 0;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(f_{0}\).
\(\displaystyle \mathbf{\Phi}_{1} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle 30 y^{2} - 20 y + 2&\displaystyle 30 y^{2} - 20 y + 2\\\displaystyle 30 y^{2} - 20 y + 2&\displaystyle 0&\displaystyle 30 y^{2} - 20 y + 2\\\displaystyle 30 y^{2} - 20 y + 2&\displaystyle 30 y^{2} - 20 y + 2&\displaystyle 0\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{2}:\mathbf{V}\mapsto\displaystyle\int_{f_{0}}(s_{1} \left(2 s_{1} - 1\right))|{f_{0}}|\hat{\boldsymbol{n}}^{\text{t}}_{0}\mathbf{V}\hat{\boldsymbol{n}}_{0}\)
where \(f_{0}\) is the 0th face;
\(\hat{\boldsymbol{n}}_{0}\) is the normal to facet 0;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(f_{0}\).
\(\displaystyle \mathbf{\Phi}_{2} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle 30 z^{2} - 20 z + 2&\displaystyle 30 z^{2} - 20 z + 2\\\displaystyle 30 z^{2} - 20 z + 2&\displaystyle 0&\displaystyle 30 z^{2} - 20 z + 2\\\displaystyle 30 z^{2} - 20 z + 2&\displaystyle 30 z^{2} - 20 z + 2&\displaystyle 0\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
where \(f_{0}\) is the 0th face;
\(\hat{\boldsymbol{n}}_{0}\) is the normal to facet 0;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(f_{0}\).
\(\displaystyle \mathbf{\Phi}_{2} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle 30 z^{2} - 20 z + 2&\displaystyle 30 z^{2} - 20 z + 2\\\displaystyle 30 z^{2} - 20 z + 2&\displaystyle 0&\displaystyle 30 z^{2} - 20 z + 2\\\displaystyle 30 z^{2} - 20 z + 2&\displaystyle 30 z^{2} - 20 z + 2&\displaystyle 0\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{3}:\mathbf{V}\mapsto\displaystyle\int_{f_{0}}(4 s_{0} s_{1})|{f_{0}}|\hat{\boldsymbol{n}}^{\text{t}}_{0}\mathbf{V}\hat{\boldsymbol{n}}_{0}\)
where \(f_{0}\) is the 0th face;
\(\hat{\boldsymbol{n}}_{0}\) is the normal to facet 0;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(f_{0}\).
\(\displaystyle \mathbf{\Phi}_{3} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle \frac{15 y^{2}}{2} + 30 y z - 10 y + \frac{15 z^{2}}{2} - 10 z + 2&\displaystyle \frac{15 y^{2}}{2} + 30 y z - 10 y + \frac{15 z^{2}}{2} - 10 z + 2\\\displaystyle \frac{15 y^{2}}{2} + 30 y z - 10 y + \frac{15 z^{2}}{2} - 10 z + 2&\displaystyle 0&\displaystyle \frac{15 y^{2}}{2} + 30 y z - 10 y + \frac{15 z^{2}}{2} - 10 z + 2\\\displaystyle \frac{15 y^{2}}{2} + 30 y z - 10 y + \frac{15 z^{2}}{2} - 10 z + 2&\displaystyle \frac{15 y^{2}}{2} + 30 y z - 10 y + \frac{15 z^{2}}{2} - 10 z + 2&\displaystyle 0\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
where \(f_{0}\) is the 0th face;
\(\hat{\boldsymbol{n}}_{0}\) is the normal to facet 0;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(f_{0}\).
\(\displaystyle \mathbf{\Phi}_{3} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle \frac{15 y^{2}}{2} + 30 y z - 10 y + \frac{15 z^{2}}{2} - 10 z + 2&\displaystyle \frac{15 y^{2}}{2} + 30 y z - 10 y + \frac{15 z^{2}}{2} - 10 z + 2\\\displaystyle \frac{15 y^{2}}{2} + 30 y z - 10 y + \frac{15 z^{2}}{2} - 10 z + 2&\displaystyle 0&\displaystyle \frac{15 y^{2}}{2} + 30 y z - 10 y + \frac{15 z^{2}}{2} - 10 z + 2\\\displaystyle \frac{15 y^{2}}{2} + 30 y z - 10 y + \frac{15 z^{2}}{2} - 10 z + 2&\displaystyle \frac{15 y^{2}}{2} + 30 y z - 10 y + \frac{15 z^{2}}{2} - 10 z + 2&\displaystyle 0\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{4}:\mathbf{V}\mapsto\displaystyle\int_{f_{0}}(4 s_{1} \left(- s_{0} - s_{1} + 1\right))|{f_{0}}|\hat{\boldsymbol{n}}^{\text{t}}_{0}\mathbf{V}\hat{\boldsymbol{n}}_{0}\)
where \(f_{0}\) is the 0th face;
\(\hat{\boldsymbol{n}}_{0}\) is the normal to facet 0;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(f_{0}\).
\(\displaystyle \mathbf{\Phi}_{4} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle \frac{15 x^{2}}{2} + 30 x z - 10 x + \frac{15 z^{2}}{2} - 10 z + 2&\displaystyle \frac{15 x^{2}}{2} + 30 x z - 10 x + \frac{15 z^{2}}{2} - 10 z + 2\\\displaystyle \frac{15 x^{2}}{2} + 30 x z - 10 x + \frac{15 z^{2}}{2} - 10 z + 2&\displaystyle 0&\displaystyle \frac{15 x^{2}}{2} + 30 x z - 10 x + \frac{15 z^{2}}{2} - 10 z + 2\\\displaystyle \frac{15 x^{2}}{2} + 30 x z - 10 x + \frac{15 z^{2}}{2} - 10 z + 2&\displaystyle \frac{15 x^{2}}{2} + 30 x z - 10 x + \frac{15 z^{2}}{2} - 10 z + 2&\displaystyle 0\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
where \(f_{0}\) is the 0th face;
\(\hat{\boldsymbol{n}}_{0}\) is the normal to facet 0;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(f_{0}\).
\(\displaystyle \mathbf{\Phi}_{4} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle \frac{15 x^{2}}{2} + 30 x z - 10 x + \frac{15 z^{2}}{2} - 10 z + 2&\displaystyle \frac{15 x^{2}}{2} + 30 x z - 10 x + \frac{15 z^{2}}{2} - 10 z + 2\\\displaystyle \frac{15 x^{2}}{2} + 30 x z - 10 x + \frac{15 z^{2}}{2} - 10 z + 2&\displaystyle 0&\displaystyle \frac{15 x^{2}}{2} + 30 x z - 10 x + \frac{15 z^{2}}{2} - 10 z + 2\\\displaystyle \frac{15 x^{2}}{2} + 30 x z - 10 x + \frac{15 z^{2}}{2} - 10 z + 2&\displaystyle \frac{15 x^{2}}{2} + 30 x z - 10 x + \frac{15 z^{2}}{2} - 10 z + 2&\displaystyle 0\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{5}:\mathbf{V}\mapsto\displaystyle\int_{f_{0}}(4 s_{0} \left(- s_{0} - s_{1} + 1\right))|{f_{0}}|\hat{\boldsymbol{n}}^{\text{t}}_{0}\mathbf{V}\hat{\boldsymbol{n}}_{0}\)
where \(f_{0}\) is the 0th face;
\(\hat{\boldsymbol{n}}_{0}\) is the normal to facet 0;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(f_{0}\).
\(\displaystyle \mathbf{\Phi}_{5} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle \frac{15 x^{2}}{2} + 30 x y - 10 x + \frac{15 y^{2}}{2} - 10 y + 2&\displaystyle \frac{15 x^{2}}{2} + 30 x y - 10 x + \frac{15 y^{2}}{2} - 10 y + 2\\\displaystyle \frac{15 x^{2}}{2} + 30 x y - 10 x + \frac{15 y^{2}}{2} - 10 y + 2&\displaystyle 0&\displaystyle \frac{15 x^{2}}{2} + 30 x y - 10 x + \frac{15 y^{2}}{2} - 10 y + 2\\\displaystyle \frac{15 x^{2}}{2} + 30 x y - 10 x + \frac{15 y^{2}}{2} - 10 y + 2&\displaystyle \frac{15 x^{2}}{2} + 30 x y - 10 x + \frac{15 y^{2}}{2} - 10 y + 2&\displaystyle 0\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
where \(f_{0}\) is the 0th face;
\(\hat{\boldsymbol{n}}_{0}\) is the normal to facet 0;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(f_{0}\).
\(\displaystyle \mathbf{\Phi}_{5} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle \frac{15 x^{2}}{2} + 30 x y - 10 x + \frac{15 y^{2}}{2} - 10 y + 2&\displaystyle \frac{15 x^{2}}{2} + 30 x y - 10 x + \frac{15 y^{2}}{2} - 10 y + 2\\\displaystyle \frac{15 x^{2}}{2} + 30 x y - 10 x + \frac{15 y^{2}}{2} - 10 y + 2&\displaystyle 0&\displaystyle \frac{15 x^{2}}{2} + 30 x y - 10 x + \frac{15 y^{2}}{2} - 10 y + 2\\\displaystyle \frac{15 x^{2}}{2} + 30 x y - 10 x + \frac{15 y^{2}}{2} - 10 y + 2&\displaystyle \frac{15 x^{2}}{2} + 30 x y - 10 x + \frac{15 y^{2}}{2} - 10 y + 2&\displaystyle 0\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{6}:\mathbf{V}\mapsto\displaystyle\int_{f_{1}}(2 s_{0}^{2} + 4 s_{0} s_{1} - 3 s_{0} + 2 s_{1}^{2} - 3 s_{1} + 1)|{f_{1}}|\hat{\boldsymbol{n}}^{\text{t}}_{1}\mathbf{V}\hat{\boldsymbol{n}}_{1}\)
where \(f_{1}\) is the 1st face;
\(\hat{\boldsymbol{n}}_{1}\) is the normal to facet 1;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(f_{1}\).
\(\displaystyle \mathbf{\Phi}_{6} = \left(\begin{array}{ccc}\displaystyle 180 x^{2} + 360 x y + 360 x z - 240 x + 180 y^{2} + 360 y z - 240 y + 180 z^{2} - 240 z + 72&\displaystyle - 30 x^{2} - 60 x y - 60 x z + 40 x - 30 y^{2} - 60 y z + 40 y - 30 z^{2} + 40 z - 12&\displaystyle - 30 x^{2} - 60 x y - 60 x z + 40 x - 30 y^{2} - 60 y z + 40 y - 30 z^{2} + 40 z - 12\\\displaystyle - 30 x^{2} - 60 x y - 60 x z + 40 x - 30 y^{2} - 60 y z + 40 y - 30 z^{2} + 40 z - 12&\displaystyle 0&\displaystyle - 30 x^{2} - 60 x y - 60 x z + 40 x - 30 y^{2} - 60 y z + 40 y - 30 z^{2} + 40 z - 12\\\displaystyle - 30 x^{2} - 60 x y - 60 x z + 40 x - 30 y^{2} - 60 y z + 40 y - 30 z^{2} + 40 z - 12&\displaystyle - 30 x^{2} - 60 x y - 60 x z + 40 x - 30 y^{2} - 60 y z + 40 y - 30 z^{2} + 40 z - 12&\displaystyle 0\end{array}\right)\)
This DOF is associated with face 1 of the reference element.
where \(f_{1}\) is the 1st face;
\(\hat{\boldsymbol{n}}_{1}\) is the normal to facet 1;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(f_{1}\).
\(\displaystyle \mathbf{\Phi}_{6} = \left(\begin{array}{ccc}\displaystyle 180 x^{2} + 360 x y + 360 x z - 240 x + 180 y^{2} + 360 y z - 240 y + 180 z^{2} - 240 z + 72&\displaystyle - 30 x^{2} - 60 x y - 60 x z + 40 x - 30 y^{2} - 60 y z + 40 y - 30 z^{2} + 40 z - 12&\displaystyle - 30 x^{2} - 60 x y - 60 x z + 40 x - 30 y^{2} - 60 y z + 40 y - 30 z^{2} + 40 z - 12\\\displaystyle - 30 x^{2} - 60 x y - 60 x z + 40 x - 30 y^{2} - 60 y z + 40 y - 30 z^{2} + 40 z - 12&\displaystyle 0&\displaystyle - 30 x^{2} - 60 x y - 60 x z + 40 x - 30 y^{2} - 60 y z + 40 y - 30 z^{2} + 40 z - 12\\\displaystyle - 30 x^{2} - 60 x y - 60 x z + 40 x - 30 y^{2} - 60 y z + 40 y - 30 z^{2} + 40 z - 12&\displaystyle - 30 x^{2} - 60 x y - 60 x z + 40 x - 30 y^{2} - 60 y z + 40 y - 30 z^{2} + 40 z - 12&\displaystyle 0\end{array}\right)\)
This DOF is associated with face 1 of the reference element.
\(\displaystyle l_{7}:\mathbf{V}\mapsto\displaystyle\int_{f_{1}}(s_{0} \left(2 s_{0} - 1\right))|{f_{1}}|\hat{\boldsymbol{n}}^{\text{t}}_{1}\mathbf{V}\hat{\boldsymbol{n}}_{1}\)
where \(f_{1}\) is the 1st face;
\(\hat{\boldsymbol{n}}_{1}\) is the normal to facet 1;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(f_{1}\).
\(\displaystyle \mathbf{\Phi}_{7} = \left(\begin{array}{ccc}\displaystyle 180 y^{2} - 120 y + 12&\displaystyle - 30 y^{2} + 20 y - 2&\displaystyle - 30 y^{2} + 20 y - 2\\\displaystyle - 30 y^{2} + 20 y - 2&\displaystyle 0&\displaystyle - 30 y^{2} + 20 y - 2\\\displaystyle - 30 y^{2} + 20 y - 2&\displaystyle - 30 y^{2} + 20 y - 2&\displaystyle 0\end{array}\right)\)
This DOF is associated with face 1 of the reference element.
where \(f_{1}\) is the 1st face;
\(\hat{\boldsymbol{n}}_{1}\) is the normal to facet 1;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(f_{1}\).
\(\displaystyle \mathbf{\Phi}_{7} = \left(\begin{array}{ccc}\displaystyle 180 y^{2} - 120 y + 12&\displaystyle - 30 y^{2} + 20 y - 2&\displaystyle - 30 y^{2} + 20 y - 2\\\displaystyle - 30 y^{2} + 20 y - 2&\displaystyle 0&\displaystyle - 30 y^{2} + 20 y - 2\\\displaystyle - 30 y^{2} + 20 y - 2&\displaystyle - 30 y^{2} + 20 y - 2&\displaystyle 0\end{array}\right)\)
This DOF is associated with face 1 of the reference element.
\(\displaystyle l_{8}:\mathbf{V}\mapsto\displaystyle\int_{f_{1}}(s_{1} \left(2 s_{1} - 1\right))|{f_{1}}|\hat{\boldsymbol{n}}^{\text{t}}_{1}\mathbf{V}\hat{\boldsymbol{n}}_{1}\)
where \(f_{1}\) is the 1st face;
\(\hat{\boldsymbol{n}}_{1}\) is the normal to facet 1;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(f_{1}\).
\(\displaystyle \mathbf{\Phi}_{8} = \left(\begin{array}{ccc}\displaystyle 180 z^{2} - 120 z + 12&\displaystyle - 30 z^{2} + 20 z - 2&\displaystyle - 30 z^{2} + 20 z - 2\\\displaystyle - 30 z^{2} + 20 z - 2&\displaystyle 0&\displaystyle - 30 z^{2} + 20 z - 2\\\displaystyle - 30 z^{2} + 20 z - 2&\displaystyle - 30 z^{2} + 20 z - 2&\displaystyle 0\end{array}\right)\)
This DOF is associated with face 1 of the reference element.
where \(f_{1}\) is the 1st face;
\(\hat{\boldsymbol{n}}_{1}\) is the normal to facet 1;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(f_{1}\).
\(\displaystyle \mathbf{\Phi}_{8} = \left(\begin{array}{ccc}\displaystyle 180 z^{2} - 120 z + 12&\displaystyle - 30 z^{2} + 20 z - 2&\displaystyle - 30 z^{2} + 20 z - 2\\\displaystyle - 30 z^{2} + 20 z - 2&\displaystyle 0&\displaystyle - 30 z^{2} + 20 z - 2\\\displaystyle - 30 z^{2} + 20 z - 2&\displaystyle - 30 z^{2} + 20 z - 2&\displaystyle 0\end{array}\right)\)
This DOF is associated with face 1 of the reference element.
\(\displaystyle l_{9}:\mathbf{V}\mapsto\displaystyle\int_{f_{1}}(4 s_{0} s_{1})|{f_{1}}|\hat{\boldsymbol{n}}^{\text{t}}_{1}\mathbf{V}\hat{\boldsymbol{n}}_{1}\)
where \(f_{1}\) is the 1st face;
\(\hat{\boldsymbol{n}}_{1}\) is the normal to facet 1;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(f_{1}\).
\(\displaystyle \mathbf{\Phi}_{9} = \left(\begin{array}{ccc}\displaystyle 45 y^{2} + 180 y z - 60 y + 45 z^{2} - 60 z + 12&\displaystyle - \frac{15 y^{2}}{2} - 30 y z + 10 y - \frac{15 z^{2}}{2} + 10 z - 2&\displaystyle - \frac{15 y^{2}}{2} - 30 y z + 10 y - \frac{15 z^{2}}{2} + 10 z - 2\\\displaystyle - \frac{15 y^{2}}{2} - 30 y z + 10 y - \frac{15 z^{2}}{2} + 10 z - 2&\displaystyle 0&\displaystyle - \frac{15 y^{2}}{2} - 30 y z + 10 y - \frac{15 z^{2}}{2} + 10 z - 2\\\displaystyle - \frac{15 y^{2}}{2} - 30 y z + 10 y - \frac{15 z^{2}}{2} + 10 z - 2&\displaystyle - \frac{15 y^{2}}{2} - 30 y z + 10 y - \frac{15 z^{2}}{2} + 10 z - 2&\displaystyle 0\end{array}\right)\)
This DOF is associated with face 1 of the reference element.
where \(f_{1}\) is the 1st face;
\(\hat{\boldsymbol{n}}_{1}\) is the normal to facet 1;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(f_{1}\).
\(\displaystyle \mathbf{\Phi}_{9} = \left(\begin{array}{ccc}\displaystyle 45 y^{2} + 180 y z - 60 y + 45 z^{2} - 60 z + 12&\displaystyle - \frac{15 y^{2}}{2} - 30 y z + 10 y - \frac{15 z^{2}}{2} + 10 z - 2&\displaystyle - \frac{15 y^{2}}{2} - 30 y z + 10 y - \frac{15 z^{2}}{2} + 10 z - 2\\\displaystyle - \frac{15 y^{2}}{2} - 30 y z + 10 y - \frac{15 z^{2}}{2} + 10 z - 2&\displaystyle 0&\displaystyle - \frac{15 y^{2}}{2} - 30 y z + 10 y - \frac{15 z^{2}}{2} + 10 z - 2\\\displaystyle - \frac{15 y^{2}}{2} - 30 y z + 10 y - \frac{15 z^{2}}{2} + 10 z - 2&\displaystyle - \frac{15 y^{2}}{2} - 30 y z + 10 y - \frac{15 z^{2}}{2} + 10 z - 2&\displaystyle 0\end{array}\right)\)
This DOF is associated with face 1 of the reference element.
\(\displaystyle l_{10}:\mathbf{V}\mapsto\displaystyle\int_{f_{1}}(4 s_{1} \left(- s_{0} - s_{1} + 1\right))|{f_{1}}|\hat{\boldsymbol{n}}^{\text{t}}_{1}\mathbf{V}\hat{\boldsymbol{n}}_{1}\)
where \(f_{1}\) is the 1st face;
\(\hat{\boldsymbol{n}}_{1}\) is the normal to facet 1;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(f_{1}\).
\(\displaystyle \mathbf{\Phi}_{10} = \left(\begin{array}{ccc}\displaystyle 45 x^{2} + 90 x y - 90 x z - 30 x + 45 y^{2} - 90 y z - 30 y - 90 z^{2} + 90 z - 3&\displaystyle - \frac{15 x^{2}}{2} - 15 x y + 15 x z + 5 x - \frac{15 y^{2}}{2} + 15 y z + 5 y + 15 z^{2} - 15 z + \frac{1}{2}&\displaystyle - \frac{15 x^{2}}{2} - 15 x y + 15 x z + 5 x - \frac{15 y^{2}}{2} + 15 y z + 5 y + 15 z^{2} - 15 z + \frac{1}{2}\\\displaystyle - \frac{15 x^{2}}{2} - 15 x y + 15 x z + 5 x - \frac{15 y^{2}}{2} + 15 y z + 5 y + 15 z^{2} - 15 z + \frac{1}{2}&\displaystyle 0&\displaystyle - \frac{15 x^{2}}{2} - 15 x y + 15 x z + 5 x - \frac{15 y^{2}}{2} + 15 y z + 5 y + 15 z^{2} - 15 z + \frac{1}{2}\\\displaystyle - \frac{15 x^{2}}{2} - 15 x y + 15 x z + 5 x - \frac{15 y^{2}}{2} + 15 y z + 5 y + 15 z^{2} - 15 z + \frac{1}{2}&\displaystyle - \frac{15 x^{2}}{2} - 15 x y + 15 x z + 5 x - \frac{15 y^{2}}{2} + 15 y z + 5 y + 15 z^{2} - 15 z + \frac{1}{2}&\displaystyle 0\end{array}\right)\)
This DOF is associated with face 1 of the reference element.
where \(f_{1}\) is the 1st face;
\(\hat{\boldsymbol{n}}_{1}\) is the normal to facet 1;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(f_{1}\).
\(\displaystyle \mathbf{\Phi}_{10} = \left(\begin{array}{ccc}\displaystyle 45 x^{2} + 90 x y - 90 x z - 30 x + 45 y^{2} - 90 y z - 30 y - 90 z^{2} + 90 z - 3&\displaystyle - \frac{15 x^{2}}{2} - 15 x y + 15 x z + 5 x - \frac{15 y^{2}}{2} + 15 y z + 5 y + 15 z^{2} - 15 z + \frac{1}{2}&\displaystyle - \frac{15 x^{2}}{2} - 15 x y + 15 x z + 5 x - \frac{15 y^{2}}{2} + 15 y z + 5 y + 15 z^{2} - 15 z + \frac{1}{2}\\\displaystyle - \frac{15 x^{2}}{2} - 15 x y + 15 x z + 5 x - \frac{15 y^{2}}{2} + 15 y z + 5 y + 15 z^{2} - 15 z + \frac{1}{2}&\displaystyle 0&\displaystyle - \frac{15 x^{2}}{2} - 15 x y + 15 x z + 5 x - \frac{15 y^{2}}{2} + 15 y z + 5 y + 15 z^{2} - 15 z + \frac{1}{2}\\\displaystyle - \frac{15 x^{2}}{2} - 15 x y + 15 x z + 5 x - \frac{15 y^{2}}{2} + 15 y z + 5 y + 15 z^{2} - 15 z + \frac{1}{2}&\displaystyle - \frac{15 x^{2}}{2} - 15 x y + 15 x z + 5 x - \frac{15 y^{2}}{2} + 15 y z + 5 y + 15 z^{2} - 15 z + \frac{1}{2}&\displaystyle 0\end{array}\right)\)
This DOF is associated with face 1 of the reference element.
\(\displaystyle l_{11}:\mathbf{V}\mapsto\displaystyle\int_{f_{1}}(4 s_{0} \left(- s_{0} - s_{1} + 1\right))|{f_{1}}|\hat{\boldsymbol{n}}^{\text{t}}_{1}\mathbf{V}\hat{\boldsymbol{n}}_{1}\)
where \(f_{1}\) is the 1st face;
\(\hat{\boldsymbol{n}}_{1}\) is the normal to facet 1;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(f_{1}\).
\(\displaystyle \mathbf{\Phi}_{11} = \left(\begin{array}{ccc}\displaystyle 45 x^{2} - 90 x y + 90 x z - 30 x - 90 y^{2} - 90 y z + 90 y + 45 z^{2} - 30 z - 3&\displaystyle - \frac{15 x^{2}}{2} + 15 x y - 15 x z + 5 x + 15 y^{2} + 15 y z - 15 y - \frac{15 z^{2}}{2} + 5 z + \frac{1}{2}&\displaystyle - \frac{15 x^{2}}{2} + 15 x y - 15 x z + 5 x + 15 y^{2} + 15 y z - 15 y - \frac{15 z^{2}}{2} + 5 z + \frac{1}{2}\\\displaystyle - \frac{15 x^{2}}{2} + 15 x y - 15 x z + 5 x + 15 y^{2} + 15 y z - 15 y - \frac{15 z^{2}}{2} + 5 z + \frac{1}{2}&\displaystyle 0&\displaystyle - \frac{15 x^{2}}{2} + 15 x y - 15 x z + 5 x + 15 y^{2} + 15 y z - 15 y - \frac{15 z^{2}}{2} + 5 z + \frac{1}{2}\\\displaystyle - \frac{15 x^{2}}{2} + 15 x y - 15 x z + 5 x + 15 y^{2} + 15 y z - 15 y - \frac{15 z^{2}}{2} + 5 z + \frac{1}{2}&\displaystyle - \frac{15 x^{2}}{2} + 15 x y - 15 x z + 5 x + 15 y^{2} + 15 y z - 15 y - \frac{15 z^{2}}{2} + 5 z + \frac{1}{2}&\displaystyle 0\end{array}\right)\)
This DOF is associated with face 1 of the reference element.
where \(f_{1}\) is the 1st face;
\(\hat{\boldsymbol{n}}_{1}\) is the normal to facet 1;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(f_{1}\).
\(\displaystyle \mathbf{\Phi}_{11} = \left(\begin{array}{ccc}\displaystyle 45 x^{2} - 90 x y + 90 x z - 30 x - 90 y^{2} - 90 y z + 90 y + 45 z^{2} - 30 z - 3&\displaystyle - \frac{15 x^{2}}{2} + 15 x y - 15 x z + 5 x + 15 y^{2} + 15 y z - 15 y - \frac{15 z^{2}}{2} + 5 z + \frac{1}{2}&\displaystyle - \frac{15 x^{2}}{2} + 15 x y - 15 x z + 5 x + 15 y^{2} + 15 y z - 15 y - \frac{15 z^{2}}{2} + 5 z + \frac{1}{2}\\\displaystyle - \frac{15 x^{2}}{2} + 15 x y - 15 x z + 5 x + 15 y^{2} + 15 y z - 15 y - \frac{15 z^{2}}{2} + 5 z + \frac{1}{2}&\displaystyle 0&\displaystyle - \frac{15 x^{2}}{2} + 15 x y - 15 x z + 5 x + 15 y^{2} + 15 y z - 15 y - \frac{15 z^{2}}{2} + 5 z + \frac{1}{2}\\\displaystyle - \frac{15 x^{2}}{2} + 15 x y - 15 x z + 5 x + 15 y^{2} + 15 y z - 15 y - \frac{15 z^{2}}{2} + 5 z + \frac{1}{2}&\displaystyle - \frac{15 x^{2}}{2} + 15 x y - 15 x z + 5 x + 15 y^{2} + 15 y z - 15 y - \frac{15 z^{2}}{2} + 5 z + \frac{1}{2}&\displaystyle 0\end{array}\right)\)
This DOF is associated with face 1 of the reference element.
\(\displaystyle l_{12}:\mathbf{V}\mapsto\displaystyle\int_{f_{2}}(2 s_{0}^{2} + 4 s_{0} s_{1} - 3 s_{0} + 2 s_{1}^{2} - 3 s_{1} + 1)|{f_{2}}|\hat{\boldsymbol{n}}^{\text{t}}_{2}\mathbf{V}\hat{\boldsymbol{n}}_{2}\)
where \(f_{2}\) is the 2nd face;
\(\hat{\boldsymbol{n}}_{2}\) is the normal to facet 2;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(f_{2}\).
\(\displaystyle \mathbf{\Phi}_{12} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle - 30 x^{2} - 60 x y - 60 x z + 40 x - 30 y^{2} - 60 y z + 40 y - 30 z^{2} + 40 z - 12&\displaystyle - 30 x^{2} - 60 x y - 60 x z + 40 x - 30 y^{2} - 60 y z + 40 y - 30 z^{2} + 40 z - 12\\\displaystyle - 30 x^{2} - 60 x y - 60 x z + 40 x - 30 y^{2} - 60 y z + 40 y - 30 z^{2} + 40 z - 12&\displaystyle 180 x^{2} + 360 x y + 360 x z - 240 x + 180 y^{2} + 360 y z - 240 y + 180 z^{2} - 240 z + 72&\displaystyle - 30 x^{2} - 60 x y - 60 x z + 40 x - 30 y^{2} - 60 y z + 40 y - 30 z^{2} + 40 z - 12\\\displaystyle - 30 x^{2} - 60 x y - 60 x z + 40 x - 30 y^{2} - 60 y z + 40 y - 30 z^{2} + 40 z - 12&\displaystyle - 30 x^{2} - 60 x y - 60 x z + 40 x - 30 y^{2} - 60 y z + 40 y - 30 z^{2} + 40 z - 12&\displaystyle 0\end{array}\right)\)
This DOF is associated with face 2 of the reference element.
where \(f_{2}\) is the 2nd face;
\(\hat{\boldsymbol{n}}_{2}\) is the normal to facet 2;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(f_{2}\).
\(\displaystyle \mathbf{\Phi}_{12} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle - 30 x^{2} - 60 x y - 60 x z + 40 x - 30 y^{2} - 60 y z + 40 y - 30 z^{2} + 40 z - 12&\displaystyle - 30 x^{2} - 60 x y - 60 x z + 40 x - 30 y^{2} - 60 y z + 40 y - 30 z^{2} + 40 z - 12\\\displaystyle - 30 x^{2} - 60 x y - 60 x z + 40 x - 30 y^{2} - 60 y z + 40 y - 30 z^{2} + 40 z - 12&\displaystyle 180 x^{2} + 360 x y + 360 x z - 240 x + 180 y^{2} + 360 y z - 240 y + 180 z^{2} - 240 z + 72&\displaystyle - 30 x^{2} - 60 x y - 60 x z + 40 x - 30 y^{2} - 60 y z + 40 y - 30 z^{2} + 40 z - 12\\\displaystyle - 30 x^{2} - 60 x y - 60 x z + 40 x - 30 y^{2} - 60 y z + 40 y - 30 z^{2} + 40 z - 12&\displaystyle - 30 x^{2} - 60 x y - 60 x z + 40 x - 30 y^{2} - 60 y z + 40 y - 30 z^{2} + 40 z - 12&\displaystyle 0\end{array}\right)\)
This DOF is associated with face 2 of the reference element.
\(\displaystyle l_{13}:\mathbf{V}\mapsto\displaystyle\int_{f_{2}}(s_{0} \left(2 s_{0} - 1\right))|{f_{2}}|\hat{\boldsymbol{n}}^{\text{t}}_{2}\mathbf{V}\hat{\boldsymbol{n}}_{2}\)
where \(f_{2}\) is the 2nd face;
\(\hat{\boldsymbol{n}}_{2}\) is the normal to facet 2;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(f_{2}\).
\(\displaystyle \mathbf{\Phi}_{13} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle - 30 x^{2} + 20 x - 2&\displaystyle - 30 x^{2} + 20 x - 2\\\displaystyle - 30 x^{2} + 20 x - 2&\displaystyle 180 x^{2} - 120 x + 12&\displaystyle - 30 x^{2} + 20 x - 2\\\displaystyle - 30 x^{2} + 20 x - 2&\displaystyle - 30 x^{2} + 20 x - 2&\displaystyle 0\end{array}\right)\)
This DOF is associated with face 2 of the reference element.
where \(f_{2}\) is the 2nd face;
\(\hat{\boldsymbol{n}}_{2}\) is the normal to facet 2;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(f_{2}\).
\(\displaystyle \mathbf{\Phi}_{13} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle - 30 x^{2} + 20 x - 2&\displaystyle - 30 x^{2} + 20 x - 2\\\displaystyle - 30 x^{2} + 20 x - 2&\displaystyle 180 x^{2} - 120 x + 12&\displaystyle - 30 x^{2} + 20 x - 2\\\displaystyle - 30 x^{2} + 20 x - 2&\displaystyle - 30 x^{2} + 20 x - 2&\displaystyle 0\end{array}\right)\)
This DOF is associated with face 2 of the reference element.
\(\displaystyle l_{14}:\mathbf{V}\mapsto\displaystyle\int_{f_{2}}(s_{1} \left(2 s_{1} - 1\right))|{f_{2}}|\hat{\boldsymbol{n}}^{\text{t}}_{2}\mathbf{V}\hat{\boldsymbol{n}}_{2}\)
where \(f_{2}\) is the 2nd face;
\(\hat{\boldsymbol{n}}_{2}\) is the normal to facet 2;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(f_{2}\).
\(\displaystyle \mathbf{\Phi}_{14} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle - 30 z^{2} + 20 z - 2&\displaystyle - 30 z^{2} + 20 z - 2\\\displaystyle - 30 z^{2} + 20 z - 2&\displaystyle 180 z^{2} - 120 z + 12&\displaystyle - 30 z^{2} + 20 z - 2\\\displaystyle - 30 z^{2} + 20 z - 2&\displaystyle - 30 z^{2} + 20 z - 2&\displaystyle 0\end{array}\right)\)
This DOF is associated with face 2 of the reference element.
where \(f_{2}\) is the 2nd face;
\(\hat{\boldsymbol{n}}_{2}\) is the normal to facet 2;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(f_{2}\).
\(\displaystyle \mathbf{\Phi}_{14} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle - 30 z^{2} + 20 z - 2&\displaystyle - 30 z^{2} + 20 z - 2\\\displaystyle - 30 z^{2} + 20 z - 2&\displaystyle 180 z^{2} - 120 z + 12&\displaystyle - 30 z^{2} + 20 z - 2\\\displaystyle - 30 z^{2} + 20 z - 2&\displaystyle - 30 z^{2} + 20 z - 2&\displaystyle 0\end{array}\right)\)
This DOF is associated with face 2 of the reference element.
\(\displaystyle l_{15}:\mathbf{V}\mapsto\displaystyle\int_{f_{2}}(4 s_{0} s_{1})|{f_{2}}|\hat{\boldsymbol{n}}^{\text{t}}_{2}\mathbf{V}\hat{\boldsymbol{n}}_{2}\)
where \(f_{2}\) is the 2nd face;
\(\hat{\boldsymbol{n}}_{2}\) is the normal to facet 2;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(f_{2}\).
\(\displaystyle \mathbf{\Phi}_{15} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle - \frac{15 x^{2}}{2} - 30 x z + 10 x - \frac{15 z^{2}}{2} + 10 z - 2&\displaystyle - \frac{15 x^{2}}{2} - 30 x z + 10 x - \frac{15 z^{2}}{2} + 10 z - 2\\\displaystyle - \frac{15 x^{2}}{2} - 30 x z + 10 x - \frac{15 z^{2}}{2} + 10 z - 2&\displaystyle 45 x^{2} + 180 x z - 60 x + 45 z^{2} - 60 z + 12&\displaystyle - \frac{15 x^{2}}{2} - 30 x z + 10 x - \frac{15 z^{2}}{2} + 10 z - 2\\\displaystyle - \frac{15 x^{2}}{2} - 30 x z + 10 x - \frac{15 z^{2}}{2} + 10 z - 2&\displaystyle - \frac{15 x^{2}}{2} - 30 x z + 10 x - \frac{15 z^{2}}{2} + 10 z - 2&\displaystyle 0\end{array}\right)\)
This DOF is associated with face 2 of the reference element.
where \(f_{2}\) is the 2nd face;
\(\hat{\boldsymbol{n}}_{2}\) is the normal to facet 2;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(f_{2}\).
\(\displaystyle \mathbf{\Phi}_{15} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle - \frac{15 x^{2}}{2} - 30 x z + 10 x - \frac{15 z^{2}}{2} + 10 z - 2&\displaystyle - \frac{15 x^{2}}{2} - 30 x z + 10 x - \frac{15 z^{2}}{2} + 10 z - 2\\\displaystyle - \frac{15 x^{2}}{2} - 30 x z + 10 x - \frac{15 z^{2}}{2} + 10 z - 2&\displaystyle 45 x^{2} + 180 x z - 60 x + 45 z^{2} - 60 z + 12&\displaystyle - \frac{15 x^{2}}{2} - 30 x z + 10 x - \frac{15 z^{2}}{2} + 10 z - 2\\\displaystyle - \frac{15 x^{2}}{2} - 30 x z + 10 x - \frac{15 z^{2}}{2} + 10 z - 2&\displaystyle - \frac{15 x^{2}}{2} - 30 x z + 10 x - \frac{15 z^{2}}{2} + 10 z - 2&\displaystyle 0\end{array}\right)\)
This DOF is associated with face 2 of the reference element.
\(\displaystyle l_{16}:\mathbf{V}\mapsto\displaystyle\int_{f_{2}}(4 s_{1} \left(- s_{0} - s_{1} + 1\right))|{f_{2}}|\hat{\boldsymbol{n}}^{\text{t}}_{2}\mathbf{V}\hat{\boldsymbol{n}}_{2}\)
where \(f_{2}\) is the 2nd face;
\(\hat{\boldsymbol{n}}_{2}\) is the normal to facet 2;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(f_{2}\).
\(\displaystyle \mathbf{\Phi}_{16} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle - \frac{15 x^{2}}{2} - 15 x y + 15 x z + 5 x - \frac{15 y^{2}}{2} + 15 y z + 5 y + 15 z^{2} - 15 z + \frac{1}{2}&\displaystyle - \frac{15 x^{2}}{2} - 15 x y + 15 x z + 5 x - \frac{15 y^{2}}{2} + 15 y z + 5 y + 15 z^{2} - 15 z + \frac{1}{2}\\\displaystyle - \frac{15 x^{2}}{2} - 15 x y + 15 x z + 5 x - \frac{15 y^{2}}{2} + 15 y z + 5 y + 15 z^{2} - 15 z + \frac{1}{2}&\displaystyle 45 x^{2} + 90 x y - 90 x z - 30 x + 45 y^{2} - 90 y z - 30 y - 90 z^{2} + 90 z - 3&\displaystyle - \frac{15 x^{2}}{2} - 15 x y + 15 x z + 5 x - \frac{15 y^{2}}{2} + 15 y z + 5 y + 15 z^{2} - 15 z + \frac{1}{2}\\\displaystyle - \frac{15 x^{2}}{2} - 15 x y + 15 x z + 5 x - \frac{15 y^{2}}{2} + 15 y z + 5 y + 15 z^{2} - 15 z + \frac{1}{2}&\displaystyle - \frac{15 x^{2}}{2} - 15 x y + 15 x z + 5 x - \frac{15 y^{2}}{2} + 15 y z + 5 y + 15 z^{2} - 15 z + \frac{1}{2}&\displaystyle 0\end{array}\right)\)
This DOF is associated with face 2 of the reference element.
where \(f_{2}\) is the 2nd face;
\(\hat{\boldsymbol{n}}_{2}\) is the normal to facet 2;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(f_{2}\).
\(\displaystyle \mathbf{\Phi}_{16} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle - \frac{15 x^{2}}{2} - 15 x y + 15 x z + 5 x - \frac{15 y^{2}}{2} + 15 y z + 5 y + 15 z^{2} - 15 z + \frac{1}{2}&\displaystyle - \frac{15 x^{2}}{2} - 15 x y + 15 x z + 5 x - \frac{15 y^{2}}{2} + 15 y z + 5 y + 15 z^{2} - 15 z + \frac{1}{2}\\\displaystyle - \frac{15 x^{2}}{2} - 15 x y + 15 x z + 5 x - \frac{15 y^{2}}{2} + 15 y z + 5 y + 15 z^{2} - 15 z + \frac{1}{2}&\displaystyle 45 x^{2} + 90 x y - 90 x z - 30 x + 45 y^{2} - 90 y z - 30 y - 90 z^{2} + 90 z - 3&\displaystyle - \frac{15 x^{2}}{2} - 15 x y + 15 x z + 5 x - \frac{15 y^{2}}{2} + 15 y z + 5 y + 15 z^{2} - 15 z + \frac{1}{2}\\\displaystyle - \frac{15 x^{2}}{2} - 15 x y + 15 x z + 5 x - \frac{15 y^{2}}{2} + 15 y z + 5 y + 15 z^{2} - 15 z + \frac{1}{2}&\displaystyle - \frac{15 x^{2}}{2} - 15 x y + 15 x z + 5 x - \frac{15 y^{2}}{2} + 15 y z + 5 y + 15 z^{2} - 15 z + \frac{1}{2}&\displaystyle 0\end{array}\right)\)
This DOF is associated with face 2 of the reference element.
\(\displaystyle l_{17}:\mathbf{V}\mapsto\displaystyle\int_{f_{2}}(4 s_{0} \left(- s_{0} - s_{1} + 1\right))|{f_{2}}|\hat{\boldsymbol{n}}^{\text{t}}_{2}\mathbf{V}\hat{\boldsymbol{n}}_{2}\)
where \(f_{2}\) is the 2nd face;
\(\hat{\boldsymbol{n}}_{2}\) is the normal to facet 2;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(f_{2}\).
\(\displaystyle \mathbf{\Phi}_{17} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle 15 x^{2} + 15 x y + 15 x z - 15 x - \frac{15 y^{2}}{2} - 15 y z + 5 y - \frac{15 z^{2}}{2} + 5 z + \frac{1}{2}&\displaystyle 15 x^{2} + 15 x y + 15 x z - 15 x - \frac{15 y^{2}}{2} - 15 y z + 5 y - \frac{15 z^{2}}{2} + 5 z + \frac{1}{2}\\\displaystyle 15 x^{2} + 15 x y + 15 x z - 15 x - \frac{15 y^{2}}{2} - 15 y z + 5 y - \frac{15 z^{2}}{2} + 5 z + \frac{1}{2}&\displaystyle - 90 x^{2} - 90 x y - 90 x z + 90 x + 45 y^{2} + 90 y z - 30 y + 45 z^{2} - 30 z - 3&\displaystyle 15 x^{2} + 15 x y + 15 x z - 15 x - \frac{15 y^{2}}{2} - 15 y z + 5 y - \frac{15 z^{2}}{2} + 5 z + \frac{1}{2}\\\displaystyle 15 x^{2} + 15 x y + 15 x z - 15 x - \frac{15 y^{2}}{2} - 15 y z + 5 y - \frac{15 z^{2}}{2} + 5 z + \frac{1}{2}&\displaystyle 15 x^{2} + 15 x y + 15 x z - 15 x - \frac{15 y^{2}}{2} - 15 y z + 5 y - \frac{15 z^{2}}{2} + 5 z + \frac{1}{2}&\displaystyle 0\end{array}\right)\)
This DOF is associated with face 2 of the reference element.
where \(f_{2}\) is the 2nd face;
\(\hat{\boldsymbol{n}}_{2}\) is the normal to facet 2;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(f_{2}\).
\(\displaystyle \mathbf{\Phi}_{17} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle 15 x^{2} + 15 x y + 15 x z - 15 x - \frac{15 y^{2}}{2} - 15 y z + 5 y - \frac{15 z^{2}}{2} + 5 z + \frac{1}{2}&\displaystyle 15 x^{2} + 15 x y + 15 x z - 15 x - \frac{15 y^{2}}{2} - 15 y z + 5 y - \frac{15 z^{2}}{2} + 5 z + \frac{1}{2}\\\displaystyle 15 x^{2} + 15 x y + 15 x z - 15 x - \frac{15 y^{2}}{2} - 15 y z + 5 y - \frac{15 z^{2}}{2} + 5 z + \frac{1}{2}&\displaystyle - 90 x^{2} - 90 x y - 90 x z + 90 x + 45 y^{2} + 90 y z - 30 y + 45 z^{2} - 30 z - 3&\displaystyle 15 x^{2} + 15 x y + 15 x z - 15 x - \frac{15 y^{2}}{2} - 15 y z + 5 y - \frac{15 z^{2}}{2} + 5 z + \frac{1}{2}\\\displaystyle 15 x^{2} + 15 x y + 15 x z - 15 x - \frac{15 y^{2}}{2} - 15 y z + 5 y - \frac{15 z^{2}}{2} + 5 z + \frac{1}{2}&\displaystyle 15 x^{2} + 15 x y + 15 x z - 15 x - \frac{15 y^{2}}{2} - 15 y z + 5 y - \frac{15 z^{2}}{2} + 5 z + \frac{1}{2}&\displaystyle 0\end{array}\right)\)
This DOF is associated with face 2 of the reference element.
\(\displaystyle l_{18}:\mathbf{V}\mapsto\displaystyle\int_{f_{3}}(2 s_{0}^{2} + 4 s_{0} s_{1} - 3 s_{0} + 2 s_{1}^{2} - 3 s_{1} + 1)|{f_{3}}|\hat{\boldsymbol{n}}^{\text{t}}_{3}\mathbf{V}\hat{\boldsymbol{n}}_{3}\)
where \(f_{3}\) is the 3rd face;
\(\hat{\boldsymbol{n}}_{3}\) is the normal to facet 3;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(f_{3}\).
\(\displaystyle \mathbf{\Phi}_{18} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle - 30 x^{2} - 60 x y - 60 x z + 40 x - 30 y^{2} - 60 y z + 40 y - 30 z^{2} + 40 z - 12&\displaystyle - 30 x^{2} - 60 x y - 60 x z + 40 x - 30 y^{2} - 60 y z + 40 y - 30 z^{2} + 40 z - 12\\\displaystyle - 30 x^{2} - 60 x y - 60 x z + 40 x - 30 y^{2} - 60 y z + 40 y - 30 z^{2} + 40 z - 12&\displaystyle 0&\displaystyle - 30 x^{2} - 60 x y - 60 x z + 40 x - 30 y^{2} - 60 y z + 40 y - 30 z^{2} + 40 z - 12\\\displaystyle - 30 x^{2} - 60 x y - 60 x z + 40 x - 30 y^{2} - 60 y z + 40 y - 30 z^{2} + 40 z - 12&\displaystyle - 30 x^{2} - 60 x y - 60 x z + 40 x - 30 y^{2} - 60 y z + 40 y - 30 z^{2} + 40 z - 12&\displaystyle 180 x^{2} + 360 x y + 360 x z - 240 x + 180 y^{2} + 360 y z - 240 y + 180 z^{2} - 240 z + 72\end{array}\right)\)
This DOF is associated with face 3 of the reference element.
where \(f_{3}\) is the 3rd face;
\(\hat{\boldsymbol{n}}_{3}\) is the normal to facet 3;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(f_{3}\).
\(\displaystyle \mathbf{\Phi}_{18} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle - 30 x^{2} - 60 x y - 60 x z + 40 x - 30 y^{2} - 60 y z + 40 y - 30 z^{2} + 40 z - 12&\displaystyle - 30 x^{2} - 60 x y - 60 x z + 40 x - 30 y^{2} - 60 y z + 40 y - 30 z^{2} + 40 z - 12\\\displaystyle - 30 x^{2} - 60 x y - 60 x z + 40 x - 30 y^{2} - 60 y z + 40 y - 30 z^{2} + 40 z - 12&\displaystyle 0&\displaystyle - 30 x^{2} - 60 x y - 60 x z + 40 x - 30 y^{2} - 60 y z + 40 y - 30 z^{2} + 40 z - 12\\\displaystyle - 30 x^{2} - 60 x y - 60 x z + 40 x - 30 y^{2} - 60 y z + 40 y - 30 z^{2} + 40 z - 12&\displaystyle - 30 x^{2} - 60 x y - 60 x z + 40 x - 30 y^{2} - 60 y z + 40 y - 30 z^{2} + 40 z - 12&\displaystyle 180 x^{2} + 360 x y + 360 x z - 240 x + 180 y^{2} + 360 y z - 240 y + 180 z^{2} - 240 z + 72\end{array}\right)\)
This DOF is associated with face 3 of the reference element.
\(\displaystyle l_{19}:\mathbf{V}\mapsto\displaystyle\int_{f_{3}}(s_{0} \left(2 s_{0} - 1\right))|{f_{3}}|\hat{\boldsymbol{n}}^{\text{t}}_{3}\mathbf{V}\hat{\boldsymbol{n}}_{3}\)
where \(f_{3}\) is the 3rd face;
\(\hat{\boldsymbol{n}}_{3}\) is the normal to facet 3;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(f_{3}\).
\(\displaystyle \mathbf{\Phi}_{19} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle - 30 x^{2} + 20 x - 2&\displaystyle - 30 x^{2} + 20 x - 2\\\displaystyle - 30 x^{2} + 20 x - 2&\displaystyle 0&\displaystyle - 30 x^{2} + 20 x - 2\\\displaystyle - 30 x^{2} + 20 x - 2&\displaystyle - 30 x^{2} + 20 x - 2&\displaystyle 180 x^{2} - 120 x + 12\end{array}\right)\)
This DOF is associated with face 3 of the reference element.
where \(f_{3}\) is the 3rd face;
\(\hat{\boldsymbol{n}}_{3}\) is the normal to facet 3;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(f_{3}\).
\(\displaystyle \mathbf{\Phi}_{19} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle - 30 x^{2} + 20 x - 2&\displaystyle - 30 x^{2} + 20 x - 2\\\displaystyle - 30 x^{2} + 20 x - 2&\displaystyle 0&\displaystyle - 30 x^{2} + 20 x - 2\\\displaystyle - 30 x^{2} + 20 x - 2&\displaystyle - 30 x^{2} + 20 x - 2&\displaystyle 180 x^{2} - 120 x + 12\end{array}\right)\)
This DOF is associated with face 3 of the reference element.
\(\displaystyle l_{20}:\mathbf{V}\mapsto\displaystyle\int_{f_{3}}(s_{1} \left(2 s_{1} - 1\right))|{f_{3}}|\hat{\boldsymbol{n}}^{\text{t}}_{3}\mathbf{V}\hat{\boldsymbol{n}}_{3}\)
where \(f_{3}\) is the 3rd face;
\(\hat{\boldsymbol{n}}_{3}\) is the normal to facet 3;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(f_{3}\).
\(\displaystyle \mathbf{\Phi}_{20} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle - 30 y^{2} + 20 y - 2&\displaystyle - 30 y^{2} + 20 y - 2\\\displaystyle - 30 y^{2} + 20 y - 2&\displaystyle 0&\displaystyle - 30 y^{2} + 20 y - 2\\\displaystyle - 30 y^{2} + 20 y - 2&\displaystyle - 30 y^{2} + 20 y - 2&\displaystyle 180 y^{2} - 120 y + 12\end{array}\right)\)
This DOF is associated with face 3 of the reference element.
where \(f_{3}\) is the 3rd face;
\(\hat{\boldsymbol{n}}_{3}\) is the normal to facet 3;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(f_{3}\).
\(\displaystyle \mathbf{\Phi}_{20} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle - 30 y^{2} + 20 y - 2&\displaystyle - 30 y^{2} + 20 y - 2\\\displaystyle - 30 y^{2} + 20 y - 2&\displaystyle 0&\displaystyle - 30 y^{2} + 20 y - 2\\\displaystyle - 30 y^{2} + 20 y - 2&\displaystyle - 30 y^{2} + 20 y - 2&\displaystyle 180 y^{2} - 120 y + 12\end{array}\right)\)
This DOF is associated with face 3 of the reference element.
\(\displaystyle l_{21}:\mathbf{V}\mapsto\displaystyle\int_{f_{3}}(4 s_{0} s_{1})|{f_{3}}|\hat{\boldsymbol{n}}^{\text{t}}_{3}\mathbf{V}\hat{\boldsymbol{n}}_{3}\)
where \(f_{3}\) is the 3rd face;
\(\hat{\boldsymbol{n}}_{3}\) is the normal to facet 3;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(f_{3}\).
\(\displaystyle \mathbf{\Phi}_{21} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle - \frac{15 x^{2}}{2} - 30 x y + 10 x - \frac{15 y^{2}}{2} + 10 y - 2&\displaystyle - \frac{15 x^{2}}{2} - 30 x y + 10 x - \frac{15 y^{2}}{2} + 10 y - 2\\\displaystyle - \frac{15 x^{2}}{2} - 30 x y + 10 x - \frac{15 y^{2}}{2} + 10 y - 2&\displaystyle 0&\displaystyle - \frac{15 x^{2}}{2} - 30 x y + 10 x - \frac{15 y^{2}}{2} + 10 y - 2\\\displaystyle - \frac{15 x^{2}}{2} - 30 x y + 10 x - \frac{15 y^{2}}{2} + 10 y - 2&\displaystyle - \frac{15 x^{2}}{2} - 30 x y + 10 x - \frac{15 y^{2}}{2} + 10 y - 2&\displaystyle 45 x^{2} + 180 x y - 60 x + 45 y^{2} - 60 y + 12\end{array}\right)\)
This DOF is associated with face 3 of the reference element.
where \(f_{3}\) is the 3rd face;
\(\hat{\boldsymbol{n}}_{3}\) is the normal to facet 3;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(f_{3}\).
\(\displaystyle \mathbf{\Phi}_{21} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle - \frac{15 x^{2}}{2} - 30 x y + 10 x - \frac{15 y^{2}}{2} + 10 y - 2&\displaystyle - \frac{15 x^{2}}{2} - 30 x y + 10 x - \frac{15 y^{2}}{2} + 10 y - 2\\\displaystyle - \frac{15 x^{2}}{2} - 30 x y + 10 x - \frac{15 y^{2}}{2} + 10 y - 2&\displaystyle 0&\displaystyle - \frac{15 x^{2}}{2} - 30 x y + 10 x - \frac{15 y^{2}}{2} + 10 y - 2\\\displaystyle - \frac{15 x^{2}}{2} - 30 x y + 10 x - \frac{15 y^{2}}{2} + 10 y - 2&\displaystyle - \frac{15 x^{2}}{2} - 30 x y + 10 x - \frac{15 y^{2}}{2} + 10 y - 2&\displaystyle 45 x^{2} + 180 x y - 60 x + 45 y^{2} - 60 y + 12\end{array}\right)\)
This DOF is associated with face 3 of the reference element.
\(\displaystyle l_{22}:\mathbf{V}\mapsto\displaystyle\int_{f_{3}}(4 s_{1} \left(- s_{0} - s_{1} + 1\right))|{f_{3}}|\hat{\boldsymbol{n}}^{\text{t}}_{3}\mathbf{V}\hat{\boldsymbol{n}}_{3}\)
where \(f_{3}\) is the 3rd face;
\(\hat{\boldsymbol{n}}_{3}\) is the normal to facet 3;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(f_{3}\).
\(\displaystyle \mathbf{\Phi}_{22} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle - \frac{15 x^{2}}{2} + 15 x y - 15 x z + 5 x + 15 y^{2} + 15 y z - 15 y - \frac{15 z^{2}}{2} + 5 z + \frac{1}{2}&\displaystyle - \frac{15 x^{2}}{2} + 15 x y - 15 x z + 5 x + 15 y^{2} + 15 y z - 15 y - \frac{15 z^{2}}{2} + 5 z + \frac{1}{2}\\\displaystyle - \frac{15 x^{2}}{2} + 15 x y - 15 x z + 5 x + 15 y^{2} + 15 y z - 15 y - \frac{15 z^{2}}{2} + 5 z + \frac{1}{2}&\displaystyle 0&\displaystyle - \frac{15 x^{2}}{2} + 15 x y - 15 x z + 5 x + 15 y^{2} + 15 y z - 15 y - \frac{15 z^{2}}{2} + 5 z + \frac{1}{2}\\\displaystyle - \frac{15 x^{2}}{2} + 15 x y - 15 x z + 5 x + 15 y^{2} + 15 y z - 15 y - \frac{15 z^{2}}{2} + 5 z + \frac{1}{2}&\displaystyle - \frac{15 x^{2}}{2} + 15 x y - 15 x z + 5 x + 15 y^{2} + 15 y z - 15 y - \frac{15 z^{2}}{2} + 5 z + \frac{1}{2}&\displaystyle 45 x^{2} - 90 x y + 90 x z - 30 x - 90 y^{2} - 90 y z + 90 y + 45 z^{2} - 30 z - 3\end{array}\right)\)
This DOF is associated with face 3 of the reference element.
where \(f_{3}\) is the 3rd face;
\(\hat{\boldsymbol{n}}_{3}\) is the normal to facet 3;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(f_{3}\).
\(\displaystyle \mathbf{\Phi}_{22} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle - \frac{15 x^{2}}{2} + 15 x y - 15 x z + 5 x + 15 y^{2} + 15 y z - 15 y - \frac{15 z^{2}}{2} + 5 z + \frac{1}{2}&\displaystyle - \frac{15 x^{2}}{2} + 15 x y - 15 x z + 5 x + 15 y^{2} + 15 y z - 15 y - \frac{15 z^{2}}{2} + 5 z + \frac{1}{2}\\\displaystyle - \frac{15 x^{2}}{2} + 15 x y - 15 x z + 5 x + 15 y^{2} + 15 y z - 15 y - \frac{15 z^{2}}{2} + 5 z + \frac{1}{2}&\displaystyle 0&\displaystyle - \frac{15 x^{2}}{2} + 15 x y - 15 x z + 5 x + 15 y^{2} + 15 y z - 15 y - \frac{15 z^{2}}{2} + 5 z + \frac{1}{2}\\\displaystyle - \frac{15 x^{2}}{2} + 15 x y - 15 x z + 5 x + 15 y^{2} + 15 y z - 15 y - \frac{15 z^{2}}{2} + 5 z + \frac{1}{2}&\displaystyle - \frac{15 x^{2}}{2} + 15 x y - 15 x z + 5 x + 15 y^{2} + 15 y z - 15 y - \frac{15 z^{2}}{2} + 5 z + \frac{1}{2}&\displaystyle 45 x^{2} - 90 x y + 90 x z - 30 x - 90 y^{2} - 90 y z + 90 y + 45 z^{2} - 30 z - 3\end{array}\right)\)
This DOF is associated with face 3 of the reference element.
\(\displaystyle l_{23}:\mathbf{V}\mapsto\displaystyle\int_{f_{3}}(4 s_{0} \left(- s_{0} - s_{1} + 1\right))|{f_{3}}|\hat{\boldsymbol{n}}^{\text{t}}_{3}\mathbf{V}\hat{\boldsymbol{n}}_{3}\)
where \(f_{3}\) is the 3rd face;
\(\hat{\boldsymbol{n}}_{3}\) is the normal to facet 3;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(f_{3}\).
\(\displaystyle \mathbf{\Phi}_{23} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle 15 x^{2} + 15 x y + 15 x z - 15 x - \frac{15 y^{2}}{2} - 15 y z + 5 y - \frac{15 z^{2}}{2} + 5 z + \frac{1}{2}&\displaystyle 15 x^{2} + 15 x y + 15 x z - 15 x - \frac{15 y^{2}}{2} - 15 y z + 5 y - \frac{15 z^{2}}{2} + 5 z + \frac{1}{2}\\\displaystyle 15 x^{2} + 15 x y + 15 x z - 15 x - \frac{15 y^{2}}{2} - 15 y z + 5 y - \frac{15 z^{2}}{2} + 5 z + \frac{1}{2}&\displaystyle 0&\displaystyle 15 x^{2} + 15 x y + 15 x z - 15 x - \frac{15 y^{2}}{2} - 15 y z + 5 y - \frac{15 z^{2}}{2} + 5 z + \frac{1}{2}\\\displaystyle 15 x^{2} + 15 x y + 15 x z - 15 x - \frac{15 y^{2}}{2} - 15 y z + 5 y - \frac{15 z^{2}}{2} + 5 z + \frac{1}{2}&\displaystyle 15 x^{2} + 15 x y + 15 x z - 15 x - \frac{15 y^{2}}{2} - 15 y z + 5 y - \frac{15 z^{2}}{2} + 5 z + \frac{1}{2}&\displaystyle - 90 x^{2} - 90 x y - 90 x z + 90 x + 45 y^{2} + 90 y z - 30 y + 45 z^{2} - 30 z - 3\end{array}\right)\)
This DOF is associated with face 3 of the reference element.
where \(f_{3}\) is the 3rd face;
\(\hat{\boldsymbol{n}}_{3}\) is the normal to facet 3;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(f_{3}\).
\(\displaystyle \mathbf{\Phi}_{23} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle 15 x^{2} + 15 x y + 15 x z - 15 x - \frac{15 y^{2}}{2} - 15 y z + 5 y - \frac{15 z^{2}}{2} + 5 z + \frac{1}{2}&\displaystyle 15 x^{2} + 15 x y + 15 x z - 15 x - \frac{15 y^{2}}{2} - 15 y z + 5 y - \frac{15 z^{2}}{2} + 5 z + \frac{1}{2}\\\displaystyle 15 x^{2} + 15 x y + 15 x z - 15 x - \frac{15 y^{2}}{2} - 15 y z + 5 y - \frac{15 z^{2}}{2} + 5 z + \frac{1}{2}&\displaystyle 0&\displaystyle 15 x^{2} + 15 x y + 15 x z - 15 x - \frac{15 y^{2}}{2} - 15 y z + 5 y - \frac{15 z^{2}}{2} + 5 z + \frac{1}{2}\\\displaystyle 15 x^{2} + 15 x y + 15 x z - 15 x - \frac{15 y^{2}}{2} - 15 y z + 5 y - \frac{15 z^{2}}{2} + 5 z + \frac{1}{2}&\displaystyle 15 x^{2} + 15 x y + 15 x z - 15 x - \frac{15 y^{2}}{2} - 15 y z + 5 y - \frac{15 z^{2}}{2} + 5 z + \frac{1}{2}&\displaystyle - 90 x^{2} - 90 x y - 90 x z + 90 x + 45 y^{2} + 90 y z - 30 y + 45 z^{2} - 30 z - 3\end{array}\right)\)
This DOF is associated with face 3 of the reference element.
\(\displaystyle l_{24}:v\mapsto\displaystyle\int_{R}(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle - s_{0} - s_{1} - s_{2} + 1&\displaystyle - s_{0} - s_{1} - s_{2} + 1\\\displaystyle - s_{0} - s_{1} - s_{2} + 1&\displaystyle 0&\displaystyle - s_{0} - s_{1} - s_{2} + 1\\\displaystyle - s_{0} - s_{1} - s_{2} + 1&\displaystyle - s_{0} - s_{1} - s_{2} + 1&\displaystyle 0\end{array}\right))v\)
where \(R\) is the reference element;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{24} = \left(\begin{array}{ccc}\displaystyle 20 x \left(- 6 x - 6 y - 6 z + 5\right)&\displaystyle 110 x^{2} + 220 x y + 220 x z - \frac{500 x}{3} + 110 y^{2} + 220 y z - \frac{500 y}{3} + 110 z^{2} - \frac{500 z}{3} + 60&\displaystyle 110 x^{2} + 220 x y + 220 x z - \frac{500 x}{3} + 110 y^{2} + 220 y z - \frac{500 y}{3} + 110 z^{2} - \frac{500 z}{3} + 60\\\displaystyle 110 x^{2} + 220 x y + 220 x z - \frac{500 x}{3} + 110 y^{2} + 220 y z - \frac{500 y}{3} + 110 z^{2} - \frac{500 z}{3} + 60&\displaystyle 20 y \left(- 6 x - 6 y - 6 z + 5\right)&\displaystyle 110 x^{2} + 220 x y + 220 x z - \frac{500 x}{3} + 110 y^{2} + 220 y z - \frac{500 y}{3} + 110 z^{2} - \frac{500 z}{3} + 60\\\displaystyle 110 x^{2} + 220 x y + 220 x z - \frac{500 x}{3} + 110 y^{2} + 220 y z - \frac{500 y}{3} + 110 z^{2} - \frac{500 z}{3} + 60&\displaystyle 110 x^{2} + 220 x y + 220 x z - \frac{500 x}{3} + 110 y^{2} + 220 y z - \frac{500 y}{3} + 110 z^{2} - \frac{500 z}{3} + 60&\displaystyle 20 z \left(- 6 x - 6 y - 6 z + 5\right)\end{array}\right)\)
This DOF is associated with volume 0 of the reference element.
where \(R\) is the reference element;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{24} = \left(\begin{array}{ccc}\displaystyle 20 x \left(- 6 x - 6 y - 6 z + 5\right)&\displaystyle 110 x^{2} + 220 x y + 220 x z - \frac{500 x}{3} + 110 y^{2} + 220 y z - \frac{500 y}{3} + 110 z^{2} - \frac{500 z}{3} + 60&\displaystyle 110 x^{2} + 220 x y + 220 x z - \frac{500 x}{3} + 110 y^{2} + 220 y z - \frac{500 y}{3} + 110 z^{2} - \frac{500 z}{3} + 60\\\displaystyle 110 x^{2} + 220 x y + 220 x z - \frac{500 x}{3} + 110 y^{2} + 220 y z - \frac{500 y}{3} + 110 z^{2} - \frac{500 z}{3} + 60&\displaystyle 20 y \left(- 6 x - 6 y - 6 z + 5\right)&\displaystyle 110 x^{2} + 220 x y + 220 x z - \frac{500 x}{3} + 110 y^{2} + 220 y z - \frac{500 y}{3} + 110 z^{2} - \frac{500 z}{3} + 60\\\displaystyle 110 x^{2} + 220 x y + 220 x z - \frac{500 x}{3} + 110 y^{2} + 220 y z - \frac{500 y}{3} + 110 z^{2} - \frac{500 z}{3} + 60&\displaystyle 110 x^{2} + 220 x y + 220 x z - \frac{500 x}{3} + 110 y^{2} + 220 y z - \frac{500 y}{3} + 110 z^{2} - \frac{500 z}{3} + 60&\displaystyle 20 z \left(- 6 x - 6 y - 6 z + 5\right)\end{array}\right)\)
This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{25}:v\mapsto\displaystyle\int_{R}(\left(\begin{array}{ccc}\displaystyle 6 s_{0} + 6 s_{1} + 6 s_{2} - 6&\displaystyle - s_{0} - s_{1} - s_{2} + 1&\displaystyle - s_{0} - s_{1} - s_{2} + 1\\\displaystyle - s_{0} - s_{1} - s_{2} + 1&\displaystyle 0&\displaystyle - s_{0} - s_{1} - s_{2} + 1\\\displaystyle - s_{0} - s_{1} - s_{2} + 1&\displaystyle - s_{0} - s_{1} - s_{2} + 1&\displaystyle 0\end{array}\right))v\)
where \(R\) is the reference element;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{25} = \left(\begin{array}{ccc}\displaystyle 20 x \left(6 x + 6 y + 6 z - 5\right)&\displaystyle - 30 x^{2} - 40 x y - 40 x z + \frac{100 x}{3} - 10 y^{2} - 20 y z + \frac{50 y}{3} - 10 z^{2} + \frac{50 z}{3} - \frac{20}{3}&\displaystyle - 30 x^{2} - 40 x y - 40 x z + \frac{100 x}{3} - 10 y^{2} - 20 y z + \frac{50 y}{3} - 10 z^{2} + \frac{50 z}{3} - \frac{20}{3}\\\displaystyle - 30 x^{2} - 40 x y - 40 x z + \frac{100 x}{3} - 10 y^{2} - 20 y z + \frac{50 y}{3} - 10 z^{2} + \frac{50 z}{3} - \frac{20}{3}&\displaystyle 0&\displaystyle - 30 x^{2} - 40 x y - 40 x z + \frac{100 x}{3} - 10 y^{2} - 20 y z + \frac{50 y}{3} - 10 z^{2} + \frac{50 z}{3} - \frac{20}{3}\\\displaystyle - 30 x^{2} - 40 x y - 40 x z + \frac{100 x}{3} - 10 y^{2} - 20 y z + \frac{50 y}{3} - 10 z^{2} + \frac{50 z}{3} - \frac{20}{3}&\displaystyle - 30 x^{2} - 40 x y - 40 x z + \frac{100 x}{3} - 10 y^{2} - 20 y z + \frac{50 y}{3} - 10 z^{2} + \frac{50 z}{3} - \frac{20}{3}&\displaystyle 0\end{array}\right)\)
This DOF is associated with volume 0 of the reference element.
where \(R\) is the reference element;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{25} = \left(\begin{array}{ccc}\displaystyle 20 x \left(6 x + 6 y + 6 z - 5\right)&\displaystyle - 30 x^{2} - 40 x y - 40 x z + \frac{100 x}{3} - 10 y^{2} - 20 y z + \frac{50 y}{3} - 10 z^{2} + \frac{50 z}{3} - \frac{20}{3}&\displaystyle - 30 x^{2} - 40 x y - 40 x z + \frac{100 x}{3} - 10 y^{2} - 20 y z + \frac{50 y}{3} - 10 z^{2} + \frac{50 z}{3} - \frac{20}{3}\\\displaystyle - 30 x^{2} - 40 x y - 40 x z + \frac{100 x}{3} - 10 y^{2} - 20 y z + \frac{50 y}{3} - 10 z^{2} + \frac{50 z}{3} - \frac{20}{3}&\displaystyle 0&\displaystyle - 30 x^{2} - 40 x y - 40 x z + \frac{100 x}{3} - 10 y^{2} - 20 y z + \frac{50 y}{3} - 10 z^{2} + \frac{50 z}{3} - \frac{20}{3}\\\displaystyle - 30 x^{2} - 40 x y - 40 x z + \frac{100 x}{3} - 10 y^{2} - 20 y z + \frac{50 y}{3} - 10 z^{2} + \frac{50 z}{3} - \frac{20}{3}&\displaystyle - 30 x^{2} - 40 x y - 40 x z + \frac{100 x}{3} - 10 y^{2} - 20 y z + \frac{50 y}{3} - 10 z^{2} + \frac{50 z}{3} - \frac{20}{3}&\displaystyle 0\end{array}\right)\)
This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{26}:v\mapsto\displaystyle\int_{R}(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle - s_{0} - s_{1} - s_{2} + 1&\displaystyle - s_{0} - s_{1} - s_{2} + 1\\\displaystyle - s_{0} - s_{1} - s_{2} + 1&\displaystyle 6 s_{0} + 6 s_{1} + 6 s_{2} - 6&\displaystyle - s_{0} - s_{1} - s_{2} + 1\\\displaystyle - s_{0} - s_{1} - s_{2} + 1&\displaystyle - s_{0} - s_{1} - s_{2} + 1&\displaystyle 0\end{array}\right))v\)
where \(R\) is the reference element;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{26} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle - 10 x^{2} - 40 x y - 20 x z + \frac{50 x}{3} - 30 y^{2} - 40 y z + \frac{100 y}{3} - 10 z^{2} + \frac{50 z}{3} - \frac{20}{3}&\displaystyle - 10 x^{2} - 40 x y - 20 x z + \frac{50 x}{3} - 30 y^{2} - 40 y z + \frac{100 y}{3} - 10 z^{2} + \frac{50 z}{3} - \frac{20}{3}\\\displaystyle - 10 x^{2} - 40 x y - 20 x z + \frac{50 x}{3} - 30 y^{2} - 40 y z + \frac{100 y}{3} - 10 z^{2} + \frac{50 z}{3} - \frac{20}{3}&\displaystyle 20 y \left(6 x + 6 y + 6 z - 5\right)&\displaystyle - 10 x^{2} - 40 x y - 20 x z + \frac{50 x}{3} - 30 y^{2} - 40 y z + \frac{100 y}{3} - 10 z^{2} + \frac{50 z}{3} - \frac{20}{3}\\\displaystyle - 10 x^{2} - 40 x y - 20 x z + \frac{50 x}{3} - 30 y^{2} - 40 y z + \frac{100 y}{3} - 10 z^{2} + \frac{50 z}{3} - \frac{20}{3}&\displaystyle - 10 x^{2} - 40 x y - 20 x z + \frac{50 x}{3} - 30 y^{2} - 40 y z + \frac{100 y}{3} - 10 z^{2} + \frac{50 z}{3} - \frac{20}{3}&\displaystyle 0\end{array}\right)\)
This DOF is associated with volume 0 of the reference element.
where \(R\) is the reference element;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{26} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle - 10 x^{2} - 40 x y - 20 x z + \frac{50 x}{3} - 30 y^{2} - 40 y z + \frac{100 y}{3} - 10 z^{2} + \frac{50 z}{3} - \frac{20}{3}&\displaystyle - 10 x^{2} - 40 x y - 20 x z + \frac{50 x}{3} - 30 y^{2} - 40 y z + \frac{100 y}{3} - 10 z^{2} + \frac{50 z}{3} - \frac{20}{3}\\\displaystyle - 10 x^{2} - 40 x y - 20 x z + \frac{50 x}{3} - 30 y^{2} - 40 y z + \frac{100 y}{3} - 10 z^{2} + \frac{50 z}{3} - \frac{20}{3}&\displaystyle 20 y \left(6 x + 6 y + 6 z - 5\right)&\displaystyle - 10 x^{2} - 40 x y - 20 x z + \frac{50 x}{3} - 30 y^{2} - 40 y z + \frac{100 y}{3} - 10 z^{2} + \frac{50 z}{3} - \frac{20}{3}\\\displaystyle - 10 x^{2} - 40 x y - 20 x z + \frac{50 x}{3} - 30 y^{2} - 40 y z + \frac{100 y}{3} - 10 z^{2} + \frac{50 z}{3} - \frac{20}{3}&\displaystyle - 10 x^{2} - 40 x y - 20 x z + \frac{50 x}{3} - 30 y^{2} - 40 y z + \frac{100 y}{3} - 10 z^{2} + \frac{50 z}{3} - \frac{20}{3}&\displaystyle 0\end{array}\right)\)
This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{27}:v\mapsto\displaystyle\int_{R}(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle - s_{0} - s_{1} - s_{2} + 1&\displaystyle - s_{0} - s_{1} - s_{2} + 1\\\displaystyle - s_{0} - s_{1} - s_{2} + 1&\displaystyle 0&\displaystyle - s_{0} - s_{1} - s_{2} + 1\\\displaystyle - s_{0} - s_{1} - s_{2} + 1&\displaystyle - s_{0} - s_{1} - s_{2} + 1&\displaystyle 6 s_{0} + 6 s_{1} + 6 s_{2} - 6\end{array}\right))v\)
where \(R\) is the reference element;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{27} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle - 10 x^{2} - 20 x y - 40 x z + \frac{50 x}{3} - 10 y^{2} - 40 y z + \frac{50 y}{3} - 30 z^{2} + \frac{100 z}{3} - \frac{20}{3}&\displaystyle - 10 x^{2} - 20 x y - 40 x z + \frac{50 x}{3} - 10 y^{2} - 40 y z + \frac{50 y}{3} - 30 z^{2} + \frac{100 z}{3} - \frac{20}{3}\\\displaystyle - 10 x^{2} - 20 x y - 40 x z + \frac{50 x}{3} - 10 y^{2} - 40 y z + \frac{50 y}{3} - 30 z^{2} + \frac{100 z}{3} - \frac{20}{3}&\displaystyle 0&\displaystyle - 10 x^{2} - 20 x y - 40 x z + \frac{50 x}{3} - 10 y^{2} - 40 y z + \frac{50 y}{3} - 30 z^{2} + \frac{100 z}{3} - \frac{20}{3}\\\displaystyle - 10 x^{2} - 20 x y - 40 x z + \frac{50 x}{3} - 10 y^{2} - 40 y z + \frac{50 y}{3} - 30 z^{2} + \frac{100 z}{3} - \frac{20}{3}&\displaystyle - 10 x^{2} - 20 x y - 40 x z + \frac{50 x}{3} - 10 y^{2} - 40 y z + \frac{50 y}{3} - 30 z^{2} + \frac{100 z}{3} - \frac{20}{3}&\displaystyle 20 z \left(6 x + 6 y + 6 z - 5\right)\end{array}\right)\)
This DOF is associated with volume 0 of the reference element.
where \(R\) is the reference element;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{27} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle - 10 x^{2} - 20 x y - 40 x z + \frac{50 x}{3} - 10 y^{2} - 40 y z + \frac{50 y}{3} - 30 z^{2} + \frac{100 z}{3} - \frac{20}{3}&\displaystyle - 10 x^{2} - 20 x y - 40 x z + \frac{50 x}{3} - 10 y^{2} - 40 y z + \frac{50 y}{3} - 30 z^{2} + \frac{100 z}{3} - \frac{20}{3}\\\displaystyle - 10 x^{2} - 20 x y - 40 x z + \frac{50 x}{3} - 10 y^{2} - 40 y z + \frac{50 y}{3} - 30 z^{2} + \frac{100 z}{3} - \frac{20}{3}&\displaystyle 0&\displaystyle - 10 x^{2} - 20 x y - 40 x z + \frac{50 x}{3} - 10 y^{2} - 40 y z + \frac{50 y}{3} - 30 z^{2} + \frac{100 z}{3} - \frac{20}{3}\\\displaystyle - 10 x^{2} - 20 x y - 40 x z + \frac{50 x}{3} - 10 y^{2} - 40 y z + \frac{50 y}{3} - 30 z^{2} + \frac{100 z}{3} - \frac{20}{3}&\displaystyle - 10 x^{2} - 20 x y - 40 x z + \frac{50 x}{3} - 10 y^{2} - 40 y z + \frac{50 y}{3} - 30 z^{2} + \frac{100 z}{3} - \frac{20}{3}&\displaystyle 20 z \left(6 x + 6 y + 6 z - 5\right)\end{array}\right)\)
This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{28}:v\mapsto\displaystyle\int_{R}(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle s_{0}&\displaystyle s_{0}\\\displaystyle s_{0}&\displaystyle 0&\displaystyle s_{0}\\\displaystyle s_{0}&\displaystyle s_{0}&\displaystyle 0\end{array}\right))v\)
where \(R\) is the reference element;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{28} = \left(\begin{array}{ccc}\displaystyle 20 x \left(3 x - 1\right)&\displaystyle - 190 x^{2} - 200 x y - 200 x z + \frac{640 x}{3} + \frac{100 y}{3} + \frac{100 z}{3} - 30&\displaystyle - 190 x^{2} - 200 x y - 200 x z + \frac{640 x}{3} + \frac{100 y}{3} + \frac{100 z}{3} - 30\\\displaystyle - 190 x^{2} - 200 x y - 200 x z + \frac{640 x}{3} + \frac{100 y}{3} + \frac{100 z}{3} - 30&\displaystyle 20 y \left(6 x - 1\right)&\displaystyle - 190 x^{2} - 200 x y - 200 x z + \frac{640 x}{3} + \frac{100 y}{3} + \frac{100 z}{3} - 30\\\displaystyle - 190 x^{2} - 200 x y - 200 x z + \frac{640 x}{3} + \frac{100 y}{3} + \frac{100 z}{3} - 30&\displaystyle - 190 x^{2} - 200 x y - 200 x z + \frac{640 x}{3} + \frac{100 y}{3} + \frac{100 z}{3} - 30&\displaystyle 20 z \left(6 x - 1\right)\end{array}\right)\)
This DOF is associated with volume 0 of the reference element.
where \(R\) is the reference element;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{28} = \left(\begin{array}{ccc}\displaystyle 20 x \left(3 x - 1\right)&\displaystyle - 190 x^{2} - 200 x y - 200 x z + \frac{640 x}{3} + \frac{100 y}{3} + \frac{100 z}{3} - 30&\displaystyle - 190 x^{2} - 200 x y - 200 x z + \frac{640 x}{3} + \frac{100 y}{3} + \frac{100 z}{3} - 30\\\displaystyle - 190 x^{2} - 200 x y - 200 x z + \frac{640 x}{3} + \frac{100 y}{3} + \frac{100 z}{3} - 30&\displaystyle 20 y \left(6 x - 1\right)&\displaystyle - 190 x^{2} - 200 x y - 200 x z + \frac{640 x}{3} + \frac{100 y}{3} + \frac{100 z}{3} - 30\\\displaystyle - 190 x^{2} - 200 x y - 200 x z + \frac{640 x}{3} + \frac{100 y}{3} + \frac{100 z}{3} - 30&\displaystyle - 190 x^{2} - 200 x y - 200 x z + \frac{640 x}{3} + \frac{100 y}{3} + \frac{100 z}{3} - 30&\displaystyle 20 z \left(6 x - 1\right)\end{array}\right)\)
This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{29}:v\mapsto\displaystyle\int_{R}(\left(\begin{array}{ccc}\displaystyle - 6 s_{0}&\displaystyle s_{0}&\displaystyle s_{0}\\\displaystyle s_{0}&\displaystyle 0&\displaystyle s_{0}\\\displaystyle s_{0}&\displaystyle s_{0}&\displaystyle 0\end{array}\right))v\)
where \(R\) is the reference element;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{29} = \left(\begin{array}{ccc}\displaystyle 20 x \left(1 - 3 x\right)&\displaystyle 30 x^{2} + 20 x y + 20 x z - \frac{80 x}{3} - \frac{10 y}{3} - \frac{10 z}{3} + \frac{10}{3}&\displaystyle 30 x^{2} + 20 x y + 20 x z - \frac{80 x}{3} - \frac{10 y}{3} - \frac{10 z}{3} + \frac{10}{3}\\\displaystyle 30 x^{2} + 20 x y + 20 x z - \frac{80 x}{3} - \frac{10 y}{3} - \frac{10 z}{3} + \frac{10}{3}&\displaystyle 0&\displaystyle 30 x^{2} + 20 x y + 20 x z - \frac{80 x}{3} - \frac{10 y}{3} - \frac{10 z}{3} + \frac{10}{3}\\\displaystyle 30 x^{2} + 20 x y + 20 x z - \frac{80 x}{3} - \frac{10 y}{3} - \frac{10 z}{3} + \frac{10}{3}&\displaystyle 30 x^{2} + 20 x y + 20 x z - \frac{80 x}{3} - \frac{10 y}{3} - \frac{10 z}{3} + \frac{10}{3}&\displaystyle 0\end{array}\right)\)
This DOF is associated with volume 0 of the reference element.
where \(R\) is the reference element;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{29} = \left(\begin{array}{ccc}\displaystyle 20 x \left(1 - 3 x\right)&\displaystyle 30 x^{2} + 20 x y + 20 x z - \frac{80 x}{3} - \frac{10 y}{3} - \frac{10 z}{3} + \frac{10}{3}&\displaystyle 30 x^{2} + 20 x y + 20 x z - \frac{80 x}{3} - \frac{10 y}{3} - \frac{10 z}{3} + \frac{10}{3}\\\displaystyle 30 x^{2} + 20 x y + 20 x z - \frac{80 x}{3} - \frac{10 y}{3} - \frac{10 z}{3} + \frac{10}{3}&\displaystyle 0&\displaystyle 30 x^{2} + 20 x y + 20 x z - \frac{80 x}{3} - \frac{10 y}{3} - \frac{10 z}{3} + \frac{10}{3}\\\displaystyle 30 x^{2} + 20 x y + 20 x z - \frac{80 x}{3} - \frac{10 y}{3} - \frac{10 z}{3} + \frac{10}{3}&\displaystyle 30 x^{2} + 20 x y + 20 x z - \frac{80 x}{3} - \frac{10 y}{3} - \frac{10 z}{3} + \frac{10}{3}&\displaystyle 0\end{array}\right)\)
This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{30}:v\mapsto\displaystyle\int_{R}(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle s_{0}&\displaystyle s_{0}\\\displaystyle s_{0}&\displaystyle - 6 s_{0}&\displaystyle s_{0}\\\displaystyle s_{0}&\displaystyle s_{0}&\displaystyle 0\end{array}\right))v\)
where \(R\) is the reference element;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{30} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle 20 x^{2} + 40 x y + 20 x z - \frac{70 x}{3} - \frac{20 y}{3} - \frac{10 z}{3} + \frac{10}{3}&\displaystyle 20 x^{2} + 40 x y + 20 x z - \frac{70 x}{3} - \frac{20 y}{3} - \frac{10 z}{3} + \frac{10}{3}\\\displaystyle 20 x^{2} + 40 x y + 20 x z - \frac{70 x}{3} - \frac{20 y}{3} - \frac{10 z}{3} + \frac{10}{3}&\displaystyle 20 y \left(1 - 6 x\right)&\displaystyle 20 x^{2} + 40 x y + 20 x z - \frac{70 x}{3} - \frac{20 y}{3} - \frac{10 z}{3} + \frac{10}{3}\\\displaystyle 20 x^{2} + 40 x y + 20 x z - \frac{70 x}{3} - \frac{20 y}{3} - \frac{10 z}{3} + \frac{10}{3}&\displaystyle 20 x^{2} + 40 x y + 20 x z - \frac{70 x}{3} - \frac{20 y}{3} - \frac{10 z}{3} + \frac{10}{3}&\displaystyle 0\end{array}\right)\)
This DOF is associated with volume 0 of the reference element.
where \(R\) is the reference element;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{30} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle 20 x^{2} + 40 x y + 20 x z - \frac{70 x}{3} - \frac{20 y}{3} - \frac{10 z}{3} + \frac{10}{3}&\displaystyle 20 x^{2} + 40 x y + 20 x z - \frac{70 x}{3} - \frac{20 y}{3} - \frac{10 z}{3} + \frac{10}{3}\\\displaystyle 20 x^{2} + 40 x y + 20 x z - \frac{70 x}{3} - \frac{20 y}{3} - \frac{10 z}{3} + \frac{10}{3}&\displaystyle 20 y \left(1 - 6 x\right)&\displaystyle 20 x^{2} + 40 x y + 20 x z - \frac{70 x}{3} - \frac{20 y}{3} - \frac{10 z}{3} + \frac{10}{3}\\\displaystyle 20 x^{2} + 40 x y + 20 x z - \frac{70 x}{3} - \frac{20 y}{3} - \frac{10 z}{3} + \frac{10}{3}&\displaystyle 20 x^{2} + 40 x y + 20 x z - \frac{70 x}{3} - \frac{20 y}{3} - \frac{10 z}{3} + \frac{10}{3}&\displaystyle 0\end{array}\right)\)
This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{31}:v\mapsto\displaystyle\int_{R}(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle s_{0}&\displaystyle s_{0}\\\displaystyle s_{0}&\displaystyle 0&\displaystyle s_{0}\\\displaystyle s_{0}&\displaystyle s_{0}&\displaystyle - 6 s_{0}\end{array}\right))v\)
where \(R\) is the reference element;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{31} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle 20 x^{2} + 20 x y + 40 x z - \frac{70 x}{3} - \frac{10 y}{3} - \frac{20 z}{3} + \frac{10}{3}&\displaystyle 20 x^{2} + 20 x y + 40 x z - \frac{70 x}{3} - \frac{10 y}{3} - \frac{20 z}{3} + \frac{10}{3}\\\displaystyle 20 x^{2} + 20 x y + 40 x z - \frac{70 x}{3} - \frac{10 y}{3} - \frac{20 z}{3} + \frac{10}{3}&\displaystyle 0&\displaystyle 20 x^{2} + 20 x y + 40 x z - \frac{70 x}{3} - \frac{10 y}{3} - \frac{20 z}{3} + \frac{10}{3}\\\displaystyle 20 x^{2} + 20 x y + 40 x z - \frac{70 x}{3} - \frac{10 y}{3} - \frac{20 z}{3} + \frac{10}{3}&\displaystyle 20 x^{2} + 20 x y + 40 x z - \frac{70 x}{3} - \frac{10 y}{3} - \frac{20 z}{3} + \frac{10}{3}&\displaystyle 20 z \left(1 - 6 x\right)\end{array}\right)\)
This DOF is associated with volume 0 of the reference element.
where \(R\) is the reference element;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{31} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle 20 x^{2} + 20 x y + 40 x z - \frac{70 x}{3} - \frac{10 y}{3} - \frac{20 z}{3} + \frac{10}{3}&\displaystyle 20 x^{2} + 20 x y + 40 x z - \frac{70 x}{3} - \frac{10 y}{3} - \frac{20 z}{3} + \frac{10}{3}\\\displaystyle 20 x^{2} + 20 x y + 40 x z - \frac{70 x}{3} - \frac{10 y}{3} - \frac{20 z}{3} + \frac{10}{3}&\displaystyle 0&\displaystyle 20 x^{2} + 20 x y + 40 x z - \frac{70 x}{3} - \frac{10 y}{3} - \frac{20 z}{3} + \frac{10}{3}\\\displaystyle 20 x^{2} + 20 x y + 40 x z - \frac{70 x}{3} - \frac{10 y}{3} - \frac{20 z}{3} + \frac{10}{3}&\displaystyle 20 x^{2} + 20 x y + 40 x z - \frac{70 x}{3} - \frac{10 y}{3} - \frac{20 z}{3} + \frac{10}{3}&\displaystyle 20 z \left(1 - 6 x\right)\end{array}\right)\)
This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{32}:v\mapsto\displaystyle\int_{R}(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle s_{1}&\displaystyle s_{1}\\\displaystyle s_{1}&\displaystyle 0&\displaystyle s_{1}\\\displaystyle s_{1}&\displaystyle s_{1}&\displaystyle 0\end{array}\right))v\)
where \(R\) is the reference element;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{32} = \left(\begin{array}{ccc}\displaystyle 20 x \left(6 y - 1\right)&\displaystyle - 200 x y + \frac{100 x}{3} - 190 y^{2} - 200 y z + \frac{640 y}{3} + \frac{100 z}{3} - 30&\displaystyle - 200 x y + \frac{100 x}{3} - 190 y^{2} - 200 y z + \frac{640 y}{3} + \frac{100 z}{3} - 30\\\displaystyle - 200 x y + \frac{100 x}{3} - 190 y^{2} - 200 y z + \frac{640 y}{3} + \frac{100 z}{3} - 30&\displaystyle 20 y \left(3 y - 1\right)&\displaystyle - 200 x y + \frac{100 x}{3} - 190 y^{2} - 200 y z + \frac{640 y}{3} + \frac{100 z}{3} - 30\\\displaystyle - 200 x y + \frac{100 x}{3} - 190 y^{2} - 200 y z + \frac{640 y}{3} + \frac{100 z}{3} - 30&\displaystyle - 200 x y + \frac{100 x}{3} - 190 y^{2} - 200 y z + \frac{640 y}{3} + \frac{100 z}{3} - 30&\displaystyle 20 z \left(6 y - 1\right)\end{array}\right)\)
This DOF is associated with volume 0 of the reference element.
where \(R\) is the reference element;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{32} = \left(\begin{array}{ccc}\displaystyle 20 x \left(6 y - 1\right)&\displaystyle - 200 x y + \frac{100 x}{3} - 190 y^{2} - 200 y z + \frac{640 y}{3} + \frac{100 z}{3} - 30&\displaystyle - 200 x y + \frac{100 x}{3} - 190 y^{2} - 200 y z + \frac{640 y}{3} + \frac{100 z}{3} - 30\\\displaystyle - 200 x y + \frac{100 x}{3} - 190 y^{2} - 200 y z + \frac{640 y}{3} + \frac{100 z}{3} - 30&\displaystyle 20 y \left(3 y - 1\right)&\displaystyle - 200 x y + \frac{100 x}{3} - 190 y^{2} - 200 y z + \frac{640 y}{3} + \frac{100 z}{3} - 30\\\displaystyle - 200 x y + \frac{100 x}{3} - 190 y^{2} - 200 y z + \frac{640 y}{3} + \frac{100 z}{3} - 30&\displaystyle - 200 x y + \frac{100 x}{3} - 190 y^{2} - 200 y z + \frac{640 y}{3} + \frac{100 z}{3} - 30&\displaystyle 20 z \left(6 y - 1\right)\end{array}\right)\)
This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{33}:v\mapsto\displaystyle\int_{R}(\left(\begin{array}{ccc}\displaystyle - 6 s_{1}&\displaystyle s_{1}&\displaystyle s_{1}\\\displaystyle s_{1}&\displaystyle 0&\displaystyle s_{1}\\\displaystyle s_{1}&\displaystyle s_{1}&\displaystyle 0\end{array}\right))v\)
where \(R\) is the reference element;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{33} = \left(\begin{array}{ccc}\displaystyle 20 x \left(1 - 6 y\right)&\displaystyle 40 x y - \frac{20 x}{3} + 20 y^{2} + 20 y z - \frac{70 y}{3} - \frac{10 z}{3} + \frac{10}{3}&\displaystyle 40 x y - \frac{20 x}{3} + 20 y^{2} + 20 y z - \frac{70 y}{3} - \frac{10 z}{3} + \frac{10}{3}\\\displaystyle 40 x y - \frac{20 x}{3} + 20 y^{2} + 20 y z - \frac{70 y}{3} - \frac{10 z}{3} + \frac{10}{3}&\displaystyle 0&\displaystyle 40 x y - \frac{20 x}{3} + 20 y^{2} + 20 y z - \frac{70 y}{3} - \frac{10 z}{3} + \frac{10}{3}\\\displaystyle 40 x y - \frac{20 x}{3} + 20 y^{2} + 20 y z - \frac{70 y}{3} - \frac{10 z}{3} + \frac{10}{3}&\displaystyle 40 x y - \frac{20 x}{3} + 20 y^{2} + 20 y z - \frac{70 y}{3} - \frac{10 z}{3} + \frac{10}{3}&\displaystyle 0\end{array}\right)\)
This DOF is associated with volume 0 of the reference element.
where \(R\) is the reference element;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{33} = \left(\begin{array}{ccc}\displaystyle 20 x \left(1 - 6 y\right)&\displaystyle 40 x y - \frac{20 x}{3} + 20 y^{2} + 20 y z - \frac{70 y}{3} - \frac{10 z}{3} + \frac{10}{3}&\displaystyle 40 x y - \frac{20 x}{3} + 20 y^{2} + 20 y z - \frac{70 y}{3} - \frac{10 z}{3} + \frac{10}{3}\\\displaystyle 40 x y - \frac{20 x}{3} + 20 y^{2} + 20 y z - \frac{70 y}{3} - \frac{10 z}{3} + \frac{10}{3}&\displaystyle 0&\displaystyle 40 x y - \frac{20 x}{3} + 20 y^{2} + 20 y z - \frac{70 y}{3} - \frac{10 z}{3} + \frac{10}{3}\\\displaystyle 40 x y - \frac{20 x}{3} + 20 y^{2} + 20 y z - \frac{70 y}{3} - \frac{10 z}{3} + \frac{10}{3}&\displaystyle 40 x y - \frac{20 x}{3} + 20 y^{2} + 20 y z - \frac{70 y}{3} - \frac{10 z}{3} + \frac{10}{3}&\displaystyle 0\end{array}\right)\)
This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{34}:v\mapsto\displaystyle\int_{R}(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle s_{1}&\displaystyle s_{1}\\\displaystyle s_{1}&\displaystyle - 6 s_{1}&\displaystyle s_{1}\\\displaystyle s_{1}&\displaystyle s_{1}&\displaystyle 0\end{array}\right))v\)
where \(R\) is the reference element;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{34} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle 20 x y - \frac{10 x}{3} + 30 y^{2} + 20 y z - \frac{80 y}{3} - \frac{10 z}{3} + \frac{10}{3}&\displaystyle 20 x y - \frac{10 x}{3} + 30 y^{2} + 20 y z - \frac{80 y}{3} - \frac{10 z}{3} + \frac{10}{3}\\\displaystyle 20 x y - \frac{10 x}{3} + 30 y^{2} + 20 y z - \frac{80 y}{3} - \frac{10 z}{3} + \frac{10}{3}&\displaystyle 20 y \left(1 - 3 y\right)&\displaystyle 20 x y - \frac{10 x}{3} + 30 y^{2} + 20 y z - \frac{80 y}{3} - \frac{10 z}{3} + \frac{10}{3}\\\displaystyle 20 x y - \frac{10 x}{3} + 30 y^{2} + 20 y z - \frac{80 y}{3} - \frac{10 z}{3} + \frac{10}{3}&\displaystyle 20 x y - \frac{10 x}{3} + 30 y^{2} + 20 y z - \frac{80 y}{3} - \frac{10 z}{3} + \frac{10}{3}&\displaystyle 0\end{array}\right)\)
This DOF is associated with volume 0 of the reference element.
where \(R\) is the reference element;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{34} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle 20 x y - \frac{10 x}{3} + 30 y^{2} + 20 y z - \frac{80 y}{3} - \frac{10 z}{3} + \frac{10}{3}&\displaystyle 20 x y - \frac{10 x}{3} + 30 y^{2} + 20 y z - \frac{80 y}{3} - \frac{10 z}{3} + \frac{10}{3}\\\displaystyle 20 x y - \frac{10 x}{3} + 30 y^{2} + 20 y z - \frac{80 y}{3} - \frac{10 z}{3} + \frac{10}{3}&\displaystyle 20 y \left(1 - 3 y\right)&\displaystyle 20 x y - \frac{10 x}{3} + 30 y^{2} + 20 y z - \frac{80 y}{3} - \frac{10 z}{3} + \frac{10}{3}\\\displaystyle 20 x y - \frac{10 x}{3} + 30 y^{2} + 20 y z - \frac{80 y}{3} - \frac{10 z}{3} + \frac{10}{3}&\displaystyle 20 x y - \frac{10 x}{3} + 30 y^{2} + 20 y z - \frac{80 y}{3} - \frac{10 z}{3} + \frac{10}{3}&\displaystyle 0\end{array}\right)\)
This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{35}:v\mapsto\displaystyle\int_{R}(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle s_{1}&\displaystyle s_{1}\\\displaystyle s_{1}&\displaystyle 0&\displaystyle s_{1}\\\displaystyle s_{1}&\displaystyle s_{1}&\displaystyle - 6 s_{1}\end{array}\right))v\)
where \(R\) is the reference element;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{35} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle 20 x y - \frac{10 x}{3} + 20 y^{2} + 40 y z - \frac{70 y}{3} - \frac{20 z}{3} + \frac{10}{3}&\displaystyle 20 x y - \frac{10 x}{3} + 20 y^{2} + 40 y z - \frac{70 y}{3} - \frac{20 z}{3} + \frac{10}{3}\\\displaystyle 20 x y - \frac{10 x}{3} + 20 y^{2} + 40 y z - \frac{70 y}{3} - \frac{20 z}{3} + \frac{10}{3}&\displaystyle 0&\displaystyle 20 x y - \frac{10 x}{3} + 20 y^{2} + 40 y z - \frac{70 y}{3} - \frac{20 z}{3} + \frac{10}{3}\\\displaystyle 20 x y - \frac{10 x}{3} + 20 y^{2} + 40 y z - \frac{70 y}{3} - \frac{20 z}{3} + \frac{10}{3}&\displaystyle 20 x y - \frac{10 x}{3} + 20 y^{2} + 40 y z - \frac{70 y}{3} - \frac{20 z}{3} + \frac{10}{3}&\displaystyle 20 z \left(1 - 6 y\right)\end{array}\right)\)
This DOF is associated with volume 0 of the reference element.
where \(R\) is the reference element;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{35} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle 20 x y - \frac{10 x}{3} + 20 y^{2} + 40 y z - \frac{70 y}{3} - \frac{20 z}{3} + \frac{10}{3}&\displaystyle 20 x y - \frac{10 x}{3} + 20 y^{2} + 40 y z - \frac{70 y}{3} - \frac{20 z}{3} + \frac{10}{3}\\\displaystyle 20 x y - \frac{10 x}{3} + 20 y^{2} + 40 y z - \frac{70 y}{3} - \frac{20 z}{3} + \frac{10}{3}&\displaystyle 0&\displaystyle 20 x y - \frac{10 x}{3} + 20 y^{2} + 40 y z - \frac{70 y}{3} - \frac{20 z}{3} + \frac{10}{3}\\\displaystyle 20 x y - \frac{10 x}{3} + 20 y^{2} + 40 y z - \frac{70 y}{3} - \frac{20 z}{3} + \frac{10}{3}&\displaystyle 20 x y - \frac{10 x}{3} + 20 y^{2} + 40 y z - \frac{70 y}{3} - \frac{20 z}{3} + \frac{10}{3}&\displaystyle 20 z \left(1 - 6 y\right)\end{array}\right)\)
This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{36}:v\mapsto\displaystyle\int_{R}(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle s_{2}&\displaystyle s_{2}\\\displaystyle s_{2}&\displaystyle 0&\displaystyle s_{2}\\\displaystyle s_{2}&\displaystyle s_{2}&\displaystyle 0\end{array}\right))v\)
where \(R\) is the reference element;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{36} = \left(\begin{array}{ccc}\displaystyle 20 x \left(6 z - 1\right)&\displaystyle - 200 x z + \frac{100 x}{3} - 200 y z + \frac{100 y}{3} - 190 z^{2} + \frac{640 z}{3} - 30&\displaystyle - 200 x z + \frac{100 x}{3} - 200 y z + \frac{100 y}{3} - 190 z^{2} + \frac{640 z}{3} - 30\\\displaystyle - 200 x z + \frac{100 x}{3} - 200 y z + \frac{100 y}{3} - 190 z^{2} + \frac{640 z}{3} - 30&\displaystyle 20 y \left(6 z - 1\right)&\displaystyle - 200 x z + \frac{100 x}{3} - 200 y z + \frac{100 y}{3} - 190 z^{2} + \frac{640 z}{3} - 30\\\displaystyle - 200 x z + \frac{100 x}{3} - 200 y z + \frac{100 y}{3} - 190 z^{2} + \frac{640 z}{3} - 30&\displaystyle - 200 x z + \frac{100 x}{3} - 200 y z + \frac{100 y}{3} - 190 z^{2} + \frac{640 z}{3} - 30&\displaystyle 20 z \left(3 z - 1\right)\end{array}\right)\)
This DOF is associated with volume 0 of the reference element.
where \(R\) is the reference element;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{36} = \left(\begin{array}{ccc}\displaystyle 20 x \left(6 z - 1\right)&\displaystyle - 200 x z + \frac{100 x}{3} - 200 y z + \frac{100 y}{3} - 190 z^{2} + \frac{640 z}{3} - 30&\displaystyle - 200 x z + \frac{100 x}{3} - 200 y z + \frac{100 y}{3} - 190 z^{2} + \frac{640 z}{3} - 30\\\displaystyle - 200 x z + \frac{100 x}{3} - 200 y z + \frac{100 y}{3} - 190 z^{2} + \frac{640 z}{3} - 30&\displaystyle 20 y \left(6 z - 1\right)&\displaystyle - 200 x z + \frac{100 x}{3} - 200 y z + \frac{100 y}{3} - 190 z^{2} + \frac{640 z}{3} - 30\\\displaystyle - 200 x z + \frac{100 x}{3} - 200 y z + \frac{100 y}{3} - 190 z^{2} + \frac{640 z}{3} - 30&\displaystyle - 200 x z + \frac{100 x}{3} - 200 y z + \frac{100 y}{3} - 190 z^{2} + \frac{640 z}{3} - 30&\displaystyle 20 z \left(3 z - 1\right)\end{array}\right)\)
This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{37}:v\mapsto\displaystyle\int_{R}(\left(\begin{array}{ccc}\displaystyle - 6 s_{2}&\displaystyle s_{2}&\displaystyle s_{2}\\\displaystyle s_{2}&\displaystyle 0&\displaystyle s_{2}\\\displaystyle s_{2}&\displaystyle s_{2}&\displaystyle 0\end{array}\right))v\)
where \(R\) is the reference element;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{37} = \left(\begin{array}{ccc}\displaystyle 20 x \left(1 - 6 z\right)&\displaystyle 40 x z - \frac{20 x}{3} + 20 y z - \frac{10 y}{3} + 20 z^{2} - \frac{70 z}{3} + \frac{10}{3}&\displaystyle 40 x z - \frac{20 x}{3} + 20 y z - \frac{10 y}{3} + 20 z^{2} - \frac{70 z}{3} + \frac{10}{3}\\\displaystyle 40 x z - \frac{20 x}{3} + 20 y z - \frac{10 y}{3} + 20 z^{2} - \frac{70 z}{3} + \frac{10}{3}&\displaystyle 0&\displaystyle 40 x z - \frac{20 x}{3} + 20 y z - \frac{10 y}{3} + 20 z^{2} - \frac{70 z}{3} + \frac{10}{3}\\\displaystyle 40 x z - \frac{20 x}{3} + 20 y z - \frac{10 y}{3} + 20 z^{2} - \frac{70 z}{3} + \frac{10}{3}&\displaystyle 40 x z - \frac{20 x}{3} + 20 y z - \frac{10 y}{3} + 20 z^{2} - \frac{70 z}{3} + \frac{10}{3}&\displaystyle 0\end{array}\right)\)
This DOF is associated with volume 0 of the reference element.
where \(R\) is the reference element;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{37} = \left(\begin{array}{ccc}\displaystyle 20 x \left(1 - 6 z\right)&\displaystyle 40 x z - \frac{20 x}{3} + 20 y z - \frac{10 y}{3} + 20 z^{2} - \frac{70 z}{3} + \frac{10}{3}&\displaystyle 40 x z - \frac{20 x}{3} + 20 y z - \frac{10 y}{3} + 20 z^{2} - \frac{70 z}{3} + \frac{10}{3}\\\displaystyle 40 x z - \frac{20 x}{3} + 20 y z - \frac{10 y}{3} + 20 z^{2} - \frac{70 z}{3} + \frac{10}{3}&\displaystyle 0&\displaystyle 40 x z - \frac{20 x}{3} + 20 y z - \frac{10 y}{3} + 20 z^{2} - \frac{70 z}{3} + \frac{10}{3}\\\displaystyle 40 x z - \frac{20 x}{3} + 20 y z - \frac{10 y}{3} + 20 z^{2} - \frac{70 z}{3} + \frac{10}{3}&\displaystyle 40 x z - \frac{20 x}{3} + 20 y z - \frac{10 y}{3} + 20 z^{2} - \frac{70 z}{3} + \frac{10}{3}&\displaystyle 0\end{array}\right)\)
This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{38}:v\mapsto\displaystyle\int_{R}(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle s_{2}&\displaystyle s_{2}\\\displaystyle s_{2}&\displaystyle - 6 s_{2}&\displaystyle s_{2}\\\displaystyle s_{2}&\displaystyle s_{2}&\displaystyle 0\end{array}\right))v\)
where \(R\) is the reference element;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{38} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle 20 x z - \frac{10 x}{3} + 40 y z - \frac{20 y}{3} + 20 z^{2} - \frac{70 z}{3} + \frac{10}{3}&\displaystyle 20 x z - \frac{10 x}{3} + 40 y z - \frac{20 y}{3} + 20 z^{2} - \frac{70 z}{3} + \frac{10}{3}\\\displaystyle 20 x z - \frac{10 x}{3} + 40 y z - \frac{20 y}{3} + 20 z^{2} - \frac{70 z}{3} + \frac{10}{3}&\displaystyle 20 y \left(1 - 6 z\right)&\displaystyle 20 x z - \frac{10 x}{3} + 40 y z - \frac{20 y}{3} + 20 z^{2} - \frac{70 z}{3} + \frac{10}{3}\\\displaystyle 20 x z - \frac{10 x}{3} + 40 y z - \frac{20 y}{3} + 20 z^{2} - \frac{70 z}{3} + \frac{10}{3}&\displaystyle 20 x z - \frac{10 x}{3} + 40 y z - \frac{20 y}{3} + 20 z^{2} - \frac{70 z}{3} + \frac{10}{3}&\displaystyle 0\end{array}\right)\)
This DOF is associated with volume 0 of the reference element.
where \(R\) is the reference element;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{38} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle 20 x z - \frac{10 x}{3} + 40 y z - \frac{20 y}{3} + 20 z^{2} - \frac{70 z}{3} + \frac{10}{3}&\displaystyle 20 x z - \frac{10 x}{3} + 40 y z - \frac{20 y}{3} + 20 z^{2} - \frac{70 z}{3} + \frac{10}{3}\\\displaystyle 20 x z - \frac{10 x}{3} + 40 y z - \frac{20 y}{3} + 20 z^{2} - \frac{70 z}{3} + \frac{10}{3}&\displaystyle 20 y \left(1 - 6 z\right)&\displaystyle 20 x z - \frac{10 x}{3} + 40 y z - \frac{20 y}{3} + 20 z^{2} - \frac{70 z}{3} + \frac{10}{3}\\\displaystyle 20 x z - \frac{10 x}{3} + 40 y z - \frac{20 y}{3} + 20 z^{2} - \frac{70 z}{3} + \frac{10}{3}&\displaystyle 20 x z - \frac{10 x}{3} + 40 y z - \frac{20 y}{3} + 20 z^{2} - \frac{70 z}{3} + \frac{10}{3}&\displaystyle 0\end{array}\right)\)
This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{39}:v\mapsto\displaystyle\int_{R}(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle s_{2}&\displaystyle s_{2}\\\displaystyle s_{2}&\displaystyle 0&\displaystyle s_{2}\\\displaystyle s_{2}&\displaystyle s_{2}&\displaystyle - 6 s_{2}\end{array}\right))v\)
where \(R\) is the reference element;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{39} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle 20 x z - \frac{10 x}{3} + 20 y z - \frac{10 y}{3} + 30 z^{2} - \frac{80 z}{3} + \frac{10}{3}&\displaystyle 20 x z - \frac{10 x}{3} + 20 y z - \frac{10 y}{3} + 30 z^{2} - \frac{80 z}{3} + \frac{10}{3}\\\displaystyle 20 x z - \frac{10 x}{3} + 20 y z - \frac{10 y}{3} + 30 z^{2} - \frac{80 z}{3} + \frac{10}{3}&\displaystyle 0&\displaystyle 20 x z - \frac{10 x}{3} + 20 y z - \frac{10 y}{3} + 30 z^{2} - \frac{80 z}{3} + \frac{10}{3}\\\displaystyle 20 x z - \frac{10 x}{3} + 20 y z - \frac{10 y}{3} + 30 z^{2} - \frac{80 z}{3} + \frac{10}{3}&\displaystyle 20 x z - \frac{10 x}{3} + 20 y z - \frac{10 y}{3} + 30 z^{2} - \frac{80 z}{3} + \frac{10}{3}&\displaystyle 20 z \left(1 - 3 z\right)\end{array}\right)\)
This DOF is associated with volume 0 of the reference element.
where \(R\) is the reference element;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{39} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle 20 x z - \frac{10 x}{3} + 20 y z - \frac{10 y}{3} + 30 z^{2} - \frac{80 z}{3} + \frac{10}{3}&\displaystyle 20 x z - \frac{10 x}{3} + 20 y z - \frac{10 y}{3} + 30 z^{2} - \frac{80 z}{3} + \frac{10}{3}\\\displaystyle 20 x z - \frac{10 x}{3} + 20 y z - \frac{10 y}{3} + 30 z^{2} - \frac{80 z}{3} + \frac{10}{3}&\displaystyle 0&\displaystyle 20 x z - \frac{10 x}{3} + 20 y z - \frac{10 y}{3} + 30 z^{2} - \frac{80 z}{3} + \frac{10}{3}\\\displaystyle 20 x z - \frac{10 x}{3} + 20 y z - \frac{10 y}{3} + 30 z^{2} - \frac{80 z}{3} + \frac{10}{3}&\displaystyle 20 x z - \frac{10 x}{3} + 20 y z - \frac{10 y}{3} + 30 z^{2} - \frac{80 z}{3} + \frac{10}{3}&\displaystyle 20 z \left(1 - 3 z\right)\end{array}\right)\)
This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{40}:v\mapsto\displaystyle\int_{R}(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle 0&\displaystyle - 2 s_{0}^{2} - 4 s_{0} s_{1} - 4 s_{0} s_{2} + 3 s_{0} - 2 s_{1}^{2} - 4 s_{1} s_{2} + 3 s_{1} - 2 s_{2}^{2} + 3 s_{2} - 1\\\displaystyle 0&\displaystyle 0&\displaystyle 2 s_{0}^{2} + 4 s_{0} s_{1} + 4 s_{0} s_{2} - 3 s_{0} + 2 s_{1}^{2} + 4 s_{1} s_{2} - 3 s_{1} + 2 s_{2}^{2} - 3 s_{2} + 1\\\displaystyle - 2 s_{0}^{2} - 4 s_{0} s_{1} - 4 s_{0} s_{2} + 3 s_{0} - 2 s_{1}^{2} - 4 s_{1} s_{2} + 3 s_{1} - 2 s_{2}^{2} + 3 s_{2} - 1&\displaystyle 2 s_{0}^{2} + 4 s_{0} s_{1} + 4 s_{0} s_{2} - 3 s_{0} + 2 s_{1}^{2} + 4 s_{1} s_{2} - 3 s_{1} + 2 s_{2}^{2} - 3 s_{2} + 1&\displaystyle 0\end{array}\right))v\)
where \(R\) is the reference element;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{40} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle 210 x^{2} + 420 x y + 420 x z - 300 x + 210 y^{2} + 420 y z - 300 y + 210 z^{2} - 300 z + 100&\displaystyle - 420 x^{2} - 840 x y - 840 x z + 600 x - 420 y^{2} - 840 y z + 600 y - 420 z^{2} + 600 z - 200\\\displaystyle 210 x^{2} + 420 x y + 420 x z - 300 x + 210 y^{2} + 420 y z - 300 y + 210 z^{2} - 300 z + 100&\displaystyle 0&\displaystyle 210 x^{2} + 420 x y + 420 x z - 300 x + 210 y^{2} + 420 y z - 300 y + 210 z^{2} - 300 z + 100\\\displaystyle - 420 x^{2} - 840 x y - 840 x z + 600 x - 420 y^{2} - 840 y z + 600 y - 420 z^{2} + 600 z - 200&\displaystyle 210 x^{2} + 420 x y + 420 x z - 300 x + 210 y^{2} + 420 y z - 300 y + 210 z^{2} - 300 z + 100&\displaystyle 0\end{array}\right)\)
This DOF is associated with volume 0 of the reference element.
where \(R\) is the reference element;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{40} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle 210 x^{2} + 420 x y + 420 x z - 300 x + 210 y^{2} + 420 y z - 300 y + 210 z^{2} - 300 z + 100&\displaystyle - 420 x^{2} - 840 x y - 840 x z + 600 x - 420 y^{2} - 840 y z + 600 y - 420 z^{2} + 600 z - 200\\\displaystyle 210 x^{2} + 420 x y + 420 x z - 300 x + 210 y^{2} + 420 y z - 300 y + 210 z^{2} - 300 z + 100&\displaystyle 0&\displaystyle 210 x^{2} + 420 x y + 420 x z - 300 x + 210 y^{2} + 420 y z - 300 y + 210 z^{2} - 300 z + 100\\\displaystyle - 420 x^{2} - 840 x y - 840 x z + 600 x - 420 y^{2} - 840 y z + 600 y - 420 z^{2} + 600 z - 200&\displaystyle 210 x^{2} + 420 x y + 420 x z - 300 x + 210 y^{2} + 420 y z - 300 y + 210 z^{2} - 300 z + 100&\displaystyle 0\end{array}\right)\)
This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{41}:v\mapsto\displaystyle\int_{R}(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle - 2 s_{0}^{2} - 4 s_{0} s_{1} - 4 s_{0} s_{2} + 3 s_{0} - 2 s_{1}^{2} - 4 s_{1} s_{2} + 3 s_{1} - 2 s_{2}^{2} + 3 s_{2} - 1&\displaystyle 0\\\displaystyle - 2 s_{0}^{2} - 4 s_{0} s_{1} - 4 s_{0} s_{2} + 3 s_{0} - 2 s_{1}^{2} - 4 s_{1} s_{2} + 3 s_{1} - 2 s_{2}^{2} + 3 s_{2} - 1&\displaystyle 0&\displaystyle 2 s_{0}^{2} + 4 s_{0} s_{1} + 4 s_{0} s_{2} - 3 s_{0} + 2 s_{1}^{2} + 4 s_{1} s_{2} - 3 s_{1} + 2 s_{2}^{2} - 3 s_{2} + 1\\\displaystyle 0&\displaystyle 2 s_{0}^{2} + 4 s_{0} s_{1} + 4 s_{0} s_{2} - 3 s_{0} + 2 s_{1}^{2} + 4 s_{1} s_{2} - 3 s_{1} + 2 s_{2}^{2} - 3 s_{2} + 1&\displaystyle 0\end{array}\right))v\)
where \(R\) is the reference element;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{41} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle - 420 x^{2} - 840 x y - 840 x z + 600 x - 420 y^{2} - 840 y z + 600 y - 420 z^{2} + 600 z - 200&\displaystyle 210 x^{2} + 420 x y + 420 x z - 300 x + 210 y^{2} + 420 y z - 300 y + 210 z^{2} - 300 z + 100\\\displaystyle - 420 x^{2} - 840 x y - 840 x z + 600 x - 420 y^{2} - 840 y z + 600 y - 420 z^{2} + 600 z - 200&\displaystyle 0&\displaystyle 210 x^{2} + 420 x y + 420 x z - 300 x + 210 y^{2} + 420 y z - 300 y + 210 z^{2} - 300 z + 100\\\displaystyle 210 x^{2} + 420 x y + 420 x z - 300 x + 210 y^{2} + 420 y z - 300 y + 210 z^{2} - 300 z + 100&\displaystyle 210 x^{2} + 420 x y + 420 x z - 300 x + 210 y^{2} + 420 y z - 300 y + 210 z^{2} - 300 z + 100&\displaystyle 0\end{array}\right)\)
This DOF is associated with volume 0 of the reference element.
where \(R\) is the reference element;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{41} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle - 420 x^{2} - 840 x y - 840 x z + 600 x - 420 y^{2} - 840 y z + 600 y - 420 z^{2} + 600 z - 200&\displaystyle 210 x^{2} + 420 x y + 420 x z - 300 x + 210 y^{2} + 420 y z - 300 y + 210 z^{2} - 300 z + 100\\\displaystyle - 420 x^{2} - 840 x y - 840 x z + 600 x - 420 y^{2} - 840 y z + 600 y - 420 z^{2} + 600 z - 200&\displaystyle 0&\displaystyle 210 x^{2} + 420 x y + 420 x z - 300 x + 210 y^{2} + 420 y z - 300 y + 210 z^{2} - 300 z + 100\\\displaystyle 210 x^{2} + 420 x y + 420 x z - 300 x + 210 y^{2} + 420 y z - 300 y + 210 z^{2} - 300 z + 100&\displaystyle 210 x^{2} + 420 x y + 420 x z - 300 x + 210 y^{2} + 420 y z - 300 y + 210 z^{2} - 300 z + 100&\displaystyle 0\end{array}\right)\)
This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{42}:v\mapsto\displaystyle\int_{R}(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle 0&\displaystyle s_{0} \left(1 - 2 s_{0}\right)\\\displaystyle 0&\displaystyle 0&\displaystyle s_{0} \left(2 s_{0} - 1\right)\\\displaystyle s_{0} \left(1 - 2 s_{0}\right)&\displaystyle s_{0} \left(2 s_{0} - 1\right)&\displaystyle 0\end{array}\right))v\)
where \(R\) is the reference element;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{42} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle 210 x^{2} - 120 x + 10&\displaystyle - 420 x^{2} + 240 x - 20\\\displaystyle 210 x^{2} - 120 x + 10&\displaystyle 0&\displaystyle 210 x^{2} - 120 x + 10\\\displaystyle - 420 x^{2} + 240 x - 20&\displaystyle 210 x^{2} - 120 x + 10&\displaystyle 0\end{array}\right)\)
This DOF is associated with volume 0 of the reference element.
where \(R\) is the reference element;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{42} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle 210 x^{2} - 120 x + 10&\displaystyle - 420 x^{2} + 240 x - 20\\\displaystyle 210 x^{2} - 120 x + 10&\displaystyle 0&\displaystyle 210 x^{2} - 120 x + 10\\\displaystyle - 420 x^{2} + 240 x - 20&\displaystyle 210 x^{2} - 120 x + 10&\displaystyle 0\end{array}\right)\)
This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{43}:v\mapsto\displaystyle\int_{R}(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle s_{0} \left(1 - 2 s_{0}\right)&\displaystyle 0\\\displaystyle s_{0} \left(1 - 2 s_{0}\right)&\displaystyle 0&\displaystyle s_{0} \left(2 s_{0} - 1\right)\\\displaystyle 0&\displaystyle s_{0} \left(2 s_{0} - 1\right)&\displaystyle 0\end{array}\right))v\)
where \(R\) is the reference element;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{43} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle - 420 x^{2} + 240 x - 20&\displaystyle 210 x^{2} - 120 x + 10\\\displaystyle - 420 x^{2} + 240 x - 20&\displaystyle 0&\displaystyle 210 x^{2} - 120 x + 10\\\displaystyle 210 x^{2} - 120 x + 10&\displaystyle 210 x^{2} - 120 x + 10&\displaystyle 0\end{array}\right)\)
This DOF is associated with volume 0 of the reference element.
where \(R\) is the reference element;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{43} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle - 420 x^{2} + 240 x - 20&\displaystyle 210 x^{2} - 120 x + 10\\\displaystyle - 420 x^{2} + 240 x - 20&\displaystyle 0&\displaystyle 210 x^{2} - 120 x + 10\\\displaystyle 210 x^{2} - 120 x + 10&\displaystyle 210 x^{2} - 120 x + 10&\displaystyle 0\end{array}\right)\)
This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{44}:v\mapsto\displaystyle\int_{R}(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle 0&\displaystyle s_{1} \left(1 - 2 s_{1}\right)\\\displaystyle 0&\displaystyle 0&\displaystyle s_{1} \left(2 s_{1} - 1\right)\\\displaystyle s_{1} \left(1 - 2 s_{1}\right)&\displaystyle s_{1} \left(2 s_{1} - 1\right)&\displaystyle 0\end{array}\right))v\)
where \(R\) is the reference element;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{44} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle 210 y^{2} - 120 y + 10&\displaystyle - 420 y^{2} + 240 y - 20\\\displaystyle 210 y^{2} - 120 y + 10&\displaystyle 0&\displaystyle 210 y^{2} - 120 y + 10\\\displaystyle - 420 y^{2} + 240 y - 20&\displaystyle 210 y^{2} - 120 y + 10&\displaystyle 0\end{array}\right)\)
This DOF is associated with volume 0 of the reference element.
where \(R\) is the reference element;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{44} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle 210 y^{2} - 120 y + 10&\displaystyle - 420 y^{2} + 240 y - 20\\\displaystyle 210 y^{2} - 120 y + 10&\displaystyle 0&\displaystyle 210 y^{2} - 120 y + 10\\\displaystyle - 420 y^{2} + 240 y - 20&\displaystyle 210 y^{2} - 120 y + 10&\displaystyle 0\end{array}\right)\)
This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{45}:v\mapsto\displaystyle\int_{R}(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle s_{1} \left(1 - 2 s_{1}\right)&\displaystyle 0\\\displaystyle s_{1} \left(1 - 2 s_{1}\right)&\displaystyle 0&\displaystyle s_{1} \left(2 s_{1} - 1\right)\\\displaystyle 0&\displaystyle s_{1} \left(2 s_{1} - 1\right)&\displaystyle 0\end{array}\right))v\)
where \(R\) is the reference element;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{45} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle - 420 y^{2} + 240 y - 20&\displaystyle 210 y^{2} - 120 y + 10\\\displaystyle - 420 y^{2} + 240 y - 20&\displaystyle 0&\displaystyle 210 y^{2} - 120 y + 10\\\displaystyle 210 y^{2} - 120 y + 10&\displaystyle 210 y^{2} - 120 y + 10&\displaystyle 0\end{array}\right)\)
This DOF is associated with volume 0 of the reference element.
where \(R\) is the reference element;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{45} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle - 420 y^{2} + 240 y - 20&\displaystyle 210 y^{2} - 120 y + 10\\\displaystyle - 420 y^{2} + 240 y - 20&\displaystyle 0&\displaystyle 210 y^{2} - 120 y + 10\\\displaystyle 210 y^{2} - 120 y + 10&\displaystyle 210 y^{2} - 120 y + 10&\displaystyle 0\end{array}\right)\)
This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{46}:v\mapsto\displaystyle\int_{R}(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle 0&\displaystyle s_{2} \left(1 - 2 s_{2}\right)\\\displaystyle 0&\displaystyle 0&\displaystyle s_{2} \left(2 s_{2} - 1\right)\\\displaystyle s_{2} \left(1 - 2 s_{2}\right)&\displaystyle s_{2} \left(2 s_{2} - 1\right)&\displaystyle 0\end{array}\right))v\)
where \(R\) is the reference element;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{46} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle 210 z^{2} - 120 z + 10&\displaystyle - 420 z^{2} + 240 z - 20\\\displaystyle 210 z^{2} - 120 z + 10&\displaystyle 0&\displaystyle 210 z^{2} - 120 z + 10\\\displaystyle - 420 z^{2} + 240 z - 20&\displaystyle 210 z^{2} - 120 z + 10&\displaystyle 0\end{array}\right)\)
This DOF is associated with volume 0 of the reference element.
where \(R\) is the reference element;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{46} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle 210 z^{2} - 120 z + 10&\displaystyle - 420 z^{2} + 240 z - 20\\\displaystyle 210 z^{2} - 120 z + 10&\displaystyle 0&\displaystyle 210 z^{2} - 120 z + 10\\\displaystyle - 420 z^{2} + 240 z - 20&\displaystyle 210 z^{2} - 120 z + 10&\displaystyle 0\end{array}\right)\)
This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{47}:v\mapsto\displaystyle\int_{R}(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle s_{2} \left(1 - 2 s_{2}\right)&\displaystyle 0\\\displaystyle s_{2} \left(1 - 2 s_{2}\right)&\displaystyle 0&\displaystyle s_{2} \left(2 s_{2} - 1\right)\\\displaystyle 0&\displaystyle s_{2} \left(2 s_{2} - 1\right)&\displaystyle 0\end{array}\right))v\)
where \(R\) is the reference element;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{47} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle - 420 z^{2} + 240 z - 20&\displaystyle 210 z^{2} - 120 z + 10\\\displaystyle - 420 z^{2} + 240 z - 20&\displaystyle 0&\displaystyle 210 z^{2} - 120 z + 10\\\displaystyle 210 z^{2} - 120 z + 10&\displaystyle 210 z^{2} - 120 z + 10&\displaystyle 0\end{array}\right)\)
This DOF is associated with volume 0 of the reference element.
where \(R\) is the reference element;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{47} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle - 420 z^{2} + 240 z - 20&\displaystyle 210 z^{2} - 120 z + 10\\\displaystyle - 420 z^{2} + 240 z - 20&\displaystyle 0&\displaystyle 210 z^{2} - 120 z + 10\\\displaystyle 210 z^{2} - 120 z + 10&\displaystyle 210 z^{2} - 120 z + 10&\displaystyle 0\end{array}\right)\)
This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{48}:v\mapsto\displaystyle\int_{R}(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle 0&\displaystyle - 4 s_{1} s_{2}\\\displaystyle 0&\displaystyle 0&\displaystyle 4 s_{1} s_{2}\\\displaystyle - 4 s_{1} s_{2}&\displaystyle 4 s_{1} s_{2}&\displaystyle 0\end{array}\right))v\)
where \(R\) is the reference element;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{48} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle \frac{105 y^{2}}{2} + 210 y z - 60 y + \frac{105 z^{2}}{2} - 60 z + 10&\displaystyle - 105 y^{2} - 420 y z + 120 y - 105 z^{2} + 120 z - 20\\\displaystyle \frac{105 y^{2}}{2} + 210 y z - 60 y + \frac{105 z^{2}}{2} - 60 z + 10&\displaystyle 0&\displaystyle \frac{105 y^{2}}{2} + 210 y z - 60 y + \frac{105 z^{2}}{2} - 60 z + 10\\\displaystyle - 105 y^{2} - 420 y z + 120 y - 105 z^{2} + 120 z - 20&\displaystyle \frac{105 y^{2}}{2} + 210 y z - 60 y + \frac{105 z^{2}}{2} - 60 z + 10&\displaystyle 0\end{array}\right)\)
This DOF is associated with volume 0 of the reference element.
where \(R\) is the reference element;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{48} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle \frac{105 y^{2}}{2} + 210 y z - 60 y + \frac{105 z^{2}}{2} - 60 z + 10&\displaystyle - 105 y^{2} - 420 y z + 120 y - 105 z^{2} + 120 z - 20\\\displaystyle \frac{105 y^{2}}{2} + 210 y z - 60 y + \frac{105 z^{2}}{2} - 60 z + 10&\displaystyle 0&\displaystyle \frac{105 y^{2}}{2} + 210 y z - 60 y + \frac{105 z^{2}}{2} - 60 z + 10\\\displaystyle - 105 y^{2} - 420 y z + 120 y - 105 z^{2} + 120 z - 20&\displaystyle \frac{105 y^{2}}{2} + 210 y z - 60 y + \frac{105 z^{2}}{2} - 60 z + 10&\displaystyle 0\end{array}\right)\)
This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{49}:v\mapsto\displaystyle\int_{R}(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle - 4 s_{1} s_{2}&\displaystyle 0\\\displaystyle - 4 s_{1} s_{2}&\displaystyle 0&\displaystyle 4 s_{1} s_{2}\\\displaystyle 0&\displaystyle 4 s_{1} s_{2}&\displaystyle 0\end{array}\right))v\)
where \(R\) is the reference element;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{49} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle - 105 y^{2} - 420 y z + 120 y - 105 z^{2} + 120 z - 20&\displaystyle \frac{105 y^{2}}{2} + 210 y z - 60 y + \frac{105 z^{2}}{2} - 60 z + 10\\\displaystyle - 105 y^{2} - 420 y z + 120 y - 105 z^{2} + 120 z - 20&\displaystyle 0&\displaystyle \frac{105 y^{2}}{2} + 210 y z - 60 y + \frac{105 z^{2}}{2} - 60 z + 10\\\displaystyle \frac{105 y^{2}}{2} + 210 y z - 60 y + \frac{105 z^{2}}{2} - 60 z + 10&\displaystyle \frac{105 y^{2}}{2} + 210 y z - 60 y + \frac{105 z^{2}}{2} - 60 z + 10&\displaystyle 0\end{array}\right)\)
This DOF is associated with volume 0 of the reference element.
where \(R\) is the reference element;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{49} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle - 105 y^{2} - 420 y z + 120 y - 105 z^{2} + 120 z - 20&\displaystyle \frac{105 y^{2}}{2} + 210 y z - 60 y + \frac{105 z^{2}}{2} - 60 z + 10\\\displaystyle - 105 y^{2} - 420 y z + 120 y - 105 z^{2} + 120 z - 20&\displaystyle 0&\displaystyle \frac{105 y^{2}}{2} + 210 y z - 60 y + \frac{105 z^{2}}{2} - 60 z + 10\\\displaystyle \frac{105 y^{2}}{2} + 210 y z - 60 y + \frac{105 z^{2}}{2} - 60 z + 10&\displaystyle \frac{105 y^{2}}{2} + 210 y z - 60 y + \frac{105 z^{2}}{2} - 60 z + 10&\displaystyle 0\end{array}\right)\)
This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{50}:v\mapsto\displaystyle\int_{R}(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle 0&\displaystyle - 4 s_{0} s_{2}\\\displaystyle 0&\displaystyle 0&\displaystyle 4 s_{0} s_{2}\\\displaystyle - 4 s_{0} s_{2}&\displaystyle 4 s_{0} s_{2}&\displaystyle 0\end{array}\right))v\)
where \(R\) is the reference element;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{50} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle \frac{105 x^{2}}{2} + 210 x z - 60 x + \frac{105 z^{2}}{2} - 60 z + 10&\displaystyle - 105 x^{2} - 420 x z + 120 x - 105 z^{2} + 120 z - 20\\\displaystyle \frac{105 x^{2}}{2} + 210 x z - 60 x + \frac{105 z^{2}}{2} - 60 z + 10&\displaystyle 0&\displaystyle \frac{105 x^{2}}{2} + 210 x z - 60 x + \frac{105 z^{2}}{2} - 60 z + 10\\\displaystyle - 105 x^{2} - 420 x z + 120 x - 105 z^{2} + 120 z - 20&\displaystyle \frac{105 x^{2}}{2} + 210 x z - 60 x + \frac{105 z^{2}}{2} - 60 z + 10&\displaystyle 0\end{array}\right)\)
This DOF is associated with volume 0 of the reference element.
where \(R\) is the reference element;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{50} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle \frac{105 x^{2}}{2} + 210 x z - 60 x + \frac{105 z^{2}}{2} - 60 z + 10&\displaystyle - 105 x^{2} - 420 x z + 120 x - 105 z^{2} + 120 z - 20\\\displaystyle \frac{105 x^{2}}{2} + 210 x z - 60 x + \frac{105 z^{2}}{2} - 60 z + 10&\displaystyle 0&\displaystyle \frac{105 x^{2}}{2} + 210 x z - 60 x + \frac{105 z^{2}}{2} - 60 z + 10\\\displaystyle - 105 x^{2} - 420 x z + 120 x - 105 z^{2} + 120 z - 20&\displaystyle \frac{105 x^{2}}{2} + 210 x z - 60 x + \frac{105 z^{2}}{2} - 60 z + 10&\displaystyle 0\end{array}\right)\)
This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{51}:v\mapsto\displaystyle\int_{R}(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle - 4 s_{0} s_{2}&\displaystyle 0\\\displaystyle - 4 s_{0} s_{2}&\displaystyle 0&\displaystyle 4 s_{0} s_{2}\\\displaystyle 0&\displaystyle 4 s_{0} s_{2}&\displaystyle 0\end{array}\right))v\)
where \(R\) is the reference element;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{51} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle - 105 x^{2} - 420 x z + 120 x - 105 z^{2} + 120 z - 20&\displaystyle \frac{105 x^{2}}{2} + 210 x z - 60 x + \frac{105 z^{2}}{2} - 60 z + 10\\\displaystyle - 105 x^{2} - 420 x z + 120 x - 105 z^{2} + 120 z - 20&\displaystyle 0&\displaystyle \frac{105 x^{2}}{2} + 210 x z - 60 x + \frac{105 z^{2}}{2} - 60 z + 10\\\displaystyle \frac{105 x^{2}}{2} + 210 x z - 60 x + \frac{105 z^{2}}{2} - 60 z + 10&\displaystyle \frac{105 x^{2}}{2} + 210 x z - 60 x + \frac{105 z^{2}}{2} - 60 z + 10&\displaystyle 0\end{array}\right)\)
This DOF is associated with volume 0 of the reference element.
where \(R\) is the reference element;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{51} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle - 105 x^{2} - 420 x z + 120 x - 105 z^{2} + 120 z - 20&\displaystyle \frac{105 x^{2}}{2} + 210 x z - 60 x + \frac{105 z^{2}}{2} - 60 z + 10\\\displaystyle - 105 x^{2} - 420 x z + 120 x - 105 z^{2} + 120 z - 20&\displaystyle 0&\displaystyle \frac{105 x^{2}}{2} + 210 x z - 60 x + \frac{105 z^{2}}{2} - 60 z + 10\\\displaystyle \frac{105 x^{2}}{2} + 210 x z - 60 x + \frac{105 z^{2}}{2} - 60 z + 10&\displaystyle \frac{105 x^{2}}{2} + 210 x z - 60 x + \frac{105 z^{2}}{2} - 60 z + 10&\displaystyle 0\end{array}\right)\)
This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{52}:v\mapsto\displaystyle\int_{R}(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle 0&\displaystyle - 4 s_{0} s_{1}\\\displaystyle 0&\displaystyle 0&\displaystyle 4 s_{0} s_{1}\\\displaystyle - 4 s_{0} s_{1}&\displaystyle 4 s_{0} s_{1}&\displaystyle 0\end{array}\right))v\)
where \(R\) is the reference element;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{52} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle \frac{105 x^{2}}{2} + 210 x y - 60 x + \frac{105 y^{2}}{2} - 60 y + 10&\displaystyle - 105 x^{2} - 420 x y + 120 x - 105 y^{2} + 120 y - 20\\\displaystyle \frac{105 x^{2}}{2} + 210 x y - 60 x + \frac{105 y^{2}}{2} - 60 y + 10&\displaystyle 0&\displaystyle \frac{105 x^{2}}{2} + 210 x y - 60 x + \frac{105 y^{2}}{2} - 60 y + 10\\\displaystyle - 105 x^{2} - 420 x y + 120 x - 105 y^{2} + 120 y - 20&\displaystyle \frac{105 x^{2}}{2} + 210 x y - 60 x + \frac{105 y^{2}}{2} - 60 y + 10&\displaystyle 0\end{array}\right)\)
This DOF is associated with volume 0 of the reference element.
where \(R\) is the reference element;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{52} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle \frac{105 x^{2}}{2} + 210 x y - 60 x + \frac{105 y^{2}}{2} - 60 y + 10&\displaystyle - 105 x^{2} - 420 x y + 120 x - 105 y^{2} + 120 y - 20\\\displaystyle \frac{105 x^{2}}{2} + 210 x y - 60 x + \frac{105 y^{2}}{2} - 60 y + 10&\displaystyle 0&\displaystyle \frac{105 x^{2}}{2} + 210 x y - 60 x + \frac{105 y^{2}}{2} - 60 y + 10\\\displaystyle - 105 x^{2} - 420 x y + 120 x - 105 y^{2} + 120 y - 20&\displaystyle \frac{105 x^{2}}{2} + 210 x y - 60 x + \frac{105 y^{2}}{2} - 60 y + 10&\displaystyle 0\end{array}\right)\)
This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{53}:v\mapsto\displaystyle\int_{R}(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle - 4 s_{0} s_{1}&\displaystyle 0\\\displaystyle - 4 s_{0} s_{1}&\displaystyle 0&\displaystyle 4 s_{0} s_{1}\\\displaystyle 0&\displaystyle 4 s_{0} s_{1}&\displaystyle 0\end{array}\right))v\)
where \(R\) is the reference element;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{53} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle - 105 x^{2} - 420 x y + 120 x - 105 y^{2} + 120 y - 20&\displaystyle \frac{105 x^{2}}{2} + 210 x y - 60 x + \frac{105 y^{2}}{2} - 60 y + 10\\\displaystyle - 105 x^{2} - 420 x y + 120 x - 105 y^{2} + 120 y - 20&\displaystyle 0&\displaystyle \frac{105 x^{2}}{2} + 210 x y - 60 x + \frac{105 y^{2}}{2} - 60 y + 10\\\displaystyle \frac{105 x^{2}}{2} + 210 x y - 60 x + \frac{105 y^{2}}{2} - 60 y + 10&\displaystyle \frac{105 x^{2}}{2} + 210 x y - 60 x + \frac{105 y^{2}}{2} - 60 y + 10&\displaystyle 0\end{array}\right)\)
This DOF is associated with volume 0 of the reference element.
where \(R\) is the reference element;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{53} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle - 105 x^{2} - 420 x y + 120 x - 105 y^{2} + 120 y - 20&\displaystyle \frac{105 x^{2}}{2} + 210 x y - 60 x + \frac{105 y^{2}}{2} - 60 y + 10\\\displaystyle - 105 x^{2} - 420 x y + 120 x - 105 y^{2} + 120 y - 20&\displaystyle 0&\displaystyle \frac{105 x^{2}}{2} + 210 x y - 60 x + \frac{105 y^{2}}{2} - 60 y + 10\\\displaystyle \frac{105 x^{2}}{2} + 210 x y - 60 x + \frac{105 y^{2}}{2} - 60 y + 10&\displaystyle \frac{105 x^{2}}{2} + 210 x y - 60 x + \frac{105 y^{2}}{2} - 60 y + 10&\displaystyle 0\end{array}\right)\)
This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{54}:v\mapsto\displaystyle\int_{R}(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle 0&\displaystyle 4 s_{2} \left(s_{0} + s_{1} + s_{2} - 1\right)\\\displaystyle 0&\displaystyle 0&\displaystyle 4 s_{2} \left(- s_{0} - s_{1} - s_{2} + 1\right)\\\displaystyle 4 s_{2} \left(s_{0} + s_{1} + s_{2} - 1\right)&\displaystyle 4 s_{2} \left(- s_{0} - s_{1} - s_{2} + 1\right)&\displaystyle 0\end{array}\right))v\)
where \(R\) is the reference element;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{54} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle \frac{105 x^{2}}{2} + 105 x y - 105 x z - 45 x + \frac{105 y^{2}}{2} - 105 y z - 45 y - 105 z^{2} + 105 z + \frac{5}{2}&\displaystyle - 105 x^{2} - 210 x y + 210 x z + 90 x - 105 y^{2} + 210 y z + 90 y + 210 z^{2} - 210 z - 5\\\displaystyle \frac{105 x^{2}}{2} + 105 x y - 105 x z - 45 x + \frac{105 y^{2}}{2} - 105 y z - 45 y - 105 z^{2} + 105 z + \frac{5}{2}&\displaystyle 0&\displaystyle \frac{105 x^{2}}{2} + 105 x y - 105 x z - 45 x + \frac{105 y^{2}}{2} - 105 y z - 45 y - 105 z^{2} + 105 z + \frac{5}{2}\\\displaystyle - 105 x^{2} - 210 x y + 210 x z + 90 x - 105 y^{2} + 210 y z + 90 y + 210 z^{2} - 210 z - 5&\displaystyle \frac{105 x^{2}}{2} + 105 x y - 105 x z - 45 x + \frac{105 y^{2}}{2} - 105 y z - 45 y - 105 z^{2} + 105 z + \frac{5}{2}&\displaystyle 0\end{array}\right)\)
This DOF is associated with volume 0 of the reference element.
where \(R\) is the reference element;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{54} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle \frac{105 x^{2}}{2} + 105 x y - 105 x z - 45 x + \frac{105 y^{2}}{2} - 105 y z - 45 y - 105 z^{2} + 105 z + \frac{5}{2}&\displaystyle - 105 x^{2} - 210 x y + 210 x z + 90 x - 105 y^{2} + 210 y z + 90 y + 210 z^{2} - 210 z - 5\\\displaystyle \frac{105 x^{2}}{2} + 105 x y - 105 x z - 45 x + \frac{105 y^{2}}{2} - 105 y z - 45 y - 105 z^{2} + 105 z + \frac{5}{2}&\displaystyle 0&\displaystyle \frac{105 x^{2}}{2} + 105 x y - 105 x z - 45 x + \frac{105 y^{2}}{2} - 105 y z - 45 y - 105 z^{2} + 105 z + \frac{5}{2}\\\displaystyle - 105 x^{2} - 210 x y + 210 x z + 90 x - 105 y^{2} + 210 y z + 90 y + 210 z^{2} - 210 z - 5&\displaystyle \frac{105 x^{2}}{2} + 105 x y - 105 x z - 45 x + \frac{105 y^{2}}{2} - 105 y z - 45 y - 105 z^{2} + 105 z + \frac{5}{2}&\displaystyle 0\end{array}\right)\)
This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{55}:v\mapsto\displaystyle\int_{R}(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle 4 s_{2} \left(s_{0} + s_{1} + s_{2} - 1\right)&\displaystyle 0\\\displaystyle 4 s_{2} \left(s_{0} + s_{1} + s_{2} - 1\right)&\displaystyle 0&\displaystyle 4 s_{2} \left(- s_{0} - s_{1} - s_{2} + 1\right)\\\displaystyle 0&\displaystyle 4 s_{2} \left(- s_{0} - s_{1} - s_{2} + 1\right)&\displaystyle 0\end{array}\right))v\)
where \(R\) is the reference element;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{55} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle - 105 x^{2} - 210 x y + 210 x z + 90 x - 105 y^{2} + 210 y z + 90 y + 210 z^{2} - 210 z - 5&\displaystyle \frac{105 x^{2}}{2} + 105 x y - 105 x z - 45 x + \frac{105 y^{2}}{2} - 105 y z - 45 y - 105 z^{2} + 105 z + \frac{5}{2}\\\displaystyle - 105 x^{2} - 210 x y + 210 x z + 90 x - 105 y^{2} + 210 y z + 90 y + 210 z^{2} - 210 z - 5&\displaystyle 0&\displaystyle \frac{105 x^{2}}{2} + 105 x y - 105 x z - 45 x + \frac{105 y^{2}}{2} - 105 y z - 45 y - 105 z^{2} + 105 z + \frac{5}{2}\\\displaystyle \frac{105 x^{2}}{2} + 105 x y - 105 x z - 45 x + \frac{105 y^{2}}{2} - 105 y z - 45 y - 105 z^{2} + 105 z + \frac{5}{2}&\displaystyle \frac{105 x^{2}}{2} + 105 x y - 105 x z - 45 x + \frac{105 y^{2}}{2} - 105 y z - 45 y - 105 z^{2} + 105 z + \frac{5}{2}&\displaystyle 0\end{array}\right)\)
This DOF is associated with volume 0 of the reference element.
where \(R\) is the reference element;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{55} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle - 105 x^{2} - 210 x y + 210 x z + 90 x - 105 y^{2} + 210 y z + 90 y + 210 z^{2} - 210 z - 5&\displaystyle \frac{105 x^{2}}{2} + 105 x y - 105 x z - 45 x + \frac{105 y^{2}}{2} - 105 y z - 45 y - 105 z^{2} + 105 z + \frac{5}{2}\\\displaystyle - 105 x^{2} - 210 x y + 210 x z + 90 x - 105 y^{2} + 210 y z + 90 y + 210 z^{2} - 210 z - 5&\displaystyle 0&\displaystyle \frac{105 x^{2}}{2} + 105 x y - 105 x z - 45 x + \frac{105 y^{2}}{2} - 105 y z - 45 y - 105 z^{2} + 105 z + \frac{5}{2}\\\displaystyle \frac{105 x^{2}}{2} + 105 x y - 105 x z - 45 x + \frac{105 y^{2}}{2} - 105 y z - 45 y - 105 z^{2} + 105 z + \frac{5}{2}&\displaystyle \frac{105 x^{2}}{2} + 105 x y - 105 x z - 45 x + \frac{105 y^{2}}{2} - 105 y z - 45 y - 105 z^{2} + 105 z + \frac{5}{2}&\displaystyle 0\end{array}\right)\)
This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{56}:v\mapsto\displaystyle\int_{R}(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle 0&\displaystyle 4 s_{1} \left(s_{0} + s_{1} + s_{2} - 1\right)\\\displaystyle 0&\displaystyle 0&\displaystyle 4 s_{1} \left(- s_{0} - s_{1} - s_{2} + 1\right)\\\displaystyle 4 s_{1} \left(s_{0} + s_{1} + s_{2} - 1\right)&\displaystyle 4 s_{1} \left(- s_{0} - s_{1} - s_{2} + 1\right)&\displaystyle 0\end{array}\right))v\)
where \(R\) is the reference element;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{56} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle \frac{105 x^{2}}{2} - 105 x y + 105 x z - 45 x - 105 y^{2} - 105 y z + 105 y + \frac{105 z^{2}}{2} - 45 z + \frac{5}{2}&\displaystyle - 105 x^{2} + 210 x y - 210 x z + 90 x + 210 y^{2} + 210 y z - 210 y - 105 z^{2} + 90 z - 5\\\displaystyle \frac{105 x^{2}}{2} - 105 x y + 105 x z - 45 x - 105 y^{2} - 105 y z + 105 y + \frac{105 z^{2}}{2} - 45 z + \frac{5}{2}&\displaystyle 0&\displaystyle \frac{105 x^{2}}{2} - 105 x y + 105 x z - 45 x - 105 y^{2} - 105 y z + 105 y + \frac{105 z^{2}}{2} - 45 z + \frac{5}{2}\\\displaystyle - 105 x^{2} + 210 x y - 210 x z + 90 x + 210 y^{2} + 210 y z - 210 y - 105 z^{2} + 90 z - 5&\displaystyle \frac{105 x^{2}}{2} - 105 x y + 105 x z - 45 x - 105 y^{2} - 105 y z + 105 y + \frac{105 z^{2}}{2} - 45 z + \frac{5}{2}&\displaystyle 0\end{array}\right)\)
This DOF is associated with volume 0 of the reference element.
where \(R\) is the reference element;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{56} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle \frac{105 x^{2}}{2} - 105 x y + 105 x z - 45 x - 105 y^{2} - 105 y z + 105 y + \frac{105 z^{2}}{2} - 45 z + \frac{5}{2}&\displaystyle - 105 x^{2} + 210 x y - 210 x z + 90 x + 210 y^{2} + 210 y z - 210 y - 105 z^{2} + 90 z - 5\\\displaystyle \frac{105 x^{2}}{2} - 105 x y + 105 x z - 45 x - 105 y^{2} - 105 y z + 105 y + \frac{105 z^{2}}{2} - 45 z + \frac{5}{2}&\displaystyle 0&\displaystyle \frac{105 x^{2}}{2} - 105 x y + 105 x z - 45 x - 105 y^{2} - 105 y z + 105 y + \frac{105 z^{2}}{2} - 45 z + \frac{5}{2}\\\displaystyle - 105 x^{2} + 210 x y - 210 x z + 90 x + 210 y^{2} + 210 y z - 210 y - 105 z^{2} + 90 z - 5&\displaystyle \frac{105 x^{2}}{2} - 105 x y + 105 x z - 45 x - 105 y^{2} - 105 y z + 105 y + \frac{105 z^{2}}{2} - 45 z + \frac{5}{2}&\displaystyle 0\end{array}\right)\)
This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{57}:v\mapsto\displaystyle\int_{R}(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle 4 s_{1} \left(s_{0} + s_{1} + s_{2} - 1\right)&\displaystyle 0\\\displaystyle 4 s_{1} \left(s_{0} + s_{1} + s_{2} - 1\right)&\displaystyle 0&\displaystyle 4 s_{1} \left(- s_{0} - s_{1} - s_{2} + 1\right)\\\displaystyle 0&\displaystyle 4 s_{1} \left(- s_{0} - s_{1} - s_{2} + 1\right)&\displaystyle 0\end{array}\right))v\)
where \(R\) is the reference element;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{57} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle - 105 x^{2} + 210 x y - 210 x z + 90 x + 210 y^{2} + 210 y z - 210 y - 105 z^{2} + 90 z - 5&\displaystyle \frac{105 x^{2}}{2} - 105 x y + 105 x z - 45 x - 105 y^{2} - 105 y z + 105 y + \frac{105 z^{2}}{2} - 45 z + \frac{5}{2}\\\displaystyle - 105 x^{2} + 210 x y - 210 x z + 90 x + 210 y^{2} + 210 y z - 210 y - 105 z^{2} + 90 z - 5&\displaystyle 0&\displaystyle \frac{105 x^{2}}{2} - 105 x y + 105 x z - 45 x - 105 y^{2} - 105 y z + 105 y + \frac{105 z^{2}}{2} - 45 z + \frac{5}{2}\\\displaystyle \frac{105 x^{2}}{2} - 105 x y + 105 x z - 45 x - 105 y^{2} - 105 y z + 105 y + \frac{105 z^{2}}{2} - 45 z + \frac{5}{2}&\displaystyle \frac{105 x^{2}}{2} - 105 x y + 105 x z - 45 x - 105 y^{2} - 105 y z + 105 y + \frac{105 z^{2}}{2} - 45 z + \frac{5}{2}&\displaystyle 0\end{array}\right)\)
This DOF is associated with volume 0 of the reference element.
where \(R\) is the reference element;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{57} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle - 105 x^{2} + 210 x y - 210 x z + 90 x + 210 y^{2} + 210 y z - 210 y - 105 z^{2} + 90 z - 5&\displaystyle \frac{105 x^{2}}{2} - 105 x y + 105 x z - 45 x - 105 y^{2} - 105 y z + 105 y + \frac{105 z^{2}}{2} - 45 z + \frac{5}{2}\\\displaystyle - 105 x^{2} + 210 x y - 210 x z + 90 x + 210 y^{2} + 210 y z - 210 y - 105 z^{2} + 90 z - 5&\displaystyle 0&\displaystyle \frac{105 x^{2}}{2} - 105 x y + 105 x z - 45 x - 105 y^{2} - 105 y z + 105 y + \frac{105 z^{2}}{2} - 45 z + \frac{5}{2}\\\displaystyle \frac{105 x^{2}}{2} - 105 x y + 105 x z - 45 x - 105 y^{2} - 105 y z + 105 y + \frac{105 z^{2}}{2} - 45 z + \frac{5}{2}&\displaystyle \frac{105 x^{2}}{2} - 105 x y + 105 x z - 45 x - 105 y^{2} - 105 y z + 105 y + \frac{105 z^{2}}{2} - 45 z + \frac{5}{2}&\displaystyle 0\end{array}\right)\)
This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{58}:v\mapsto\displaystyle\int_{R}(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle 0&\displaystyle 4 s_{0} \left(s_{0} + s_{1} + s_{2} - 1\right)\\\displaystyle 0&\displaystyle 0&\displaystyle 4 s_{0} \left(- s_{0} - s_{1} - s_{2} + 1\right)\\\displaystyle 4 s_{0} \left(s_{0} + s_{1} + s_{2} - 1\right)&\displaystyle 4 s_{0} \left(- s_{0} - s_{1} - s_{2} + 1\right)&\displaystyle 0\end{array}\right))v\)
where \(R\) is the reference element;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{58} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle - 105 x^{2} - 105 x y - 105 x z + 105 x + \frac{105 y^{2}}{2} + 105 y z - 45 y + \frac{105 z^{2}}{2} - 45 z + \frac{5}{2}&\displaystyle 210 x^{2} + 210 x y + 210 x z - 210 x - 105 y^{2} - 210 y z + 90 y - 105 z^{2} + 90 z - 5\\\displaystyle - 105 x^{2} - 105 x y - 105 x z + 105 x + \frac{105 y^{2}}{2} + 105 y z - 45 y + \frac{105 z^{2}}{2} - 45 z + \frac{5}{2}&\displaystyle 0&\displaystyle - 105 x^{2} - 105 x y - 105 x z + 105 x + \frac{105 y^{2}}{2} + 105 y z - 45 y + \frac{105 z^{2}}{2} - 45 z + \frac{5}{2}\\\displaystyle 210 x^{2} + 210 x y + 210 x z - 210 x - 105 y^{2} - 210 y z + 90 y - 105 z^{2} + 90 z - 5&\displaystyle - 105 x^{2} - 105 x y - 105 x z + 105 x + \frac{105 y^{2}}{2} + 105 y z - 45 y + \frac{105 z^{2}}{2} - 45 z + \frac{5}{2}&\displaystyle 0\end{array}\right)\)
This DOF is associated with volume 0 of the reference element.
where \(R\) is the reference element;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{58} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle - 105 x^{2} - 105 x y - 105 x z + 105 x + \frac{105 y^{2}}{2} + 105 y z - 45 y + \frac{105 z^{2}}{2} - 45 z + \frac{5}{2}&\displaystyle 210 x^{2} + 210 x y + 210 x z - 210 x - 105 y^{2} - 210 y z + 90 y - 105 z^{2} + 90 z - 5\\\displaystyle - 105 x^{2} - 105 x y - 105 x z + 105 x + \frac{105 y^{2}}{2} + 105 y z - 45 y + \frac{105 z^{2}}{2} - 45 z + \frac{5}{2}&\displaystyle 0&\displaystyle - 105 x^{2} - 105 x y - 105 x z + 105 x + \frac{105 y^{2}}{2} + 105 y z - 45 y + \frac{105 z^{2}}{2} - 45 z + \frac{5}{2}\\\displaystyle 210 x^{2} + 210 x y + 210 x z - 210 x - 105 y^{2} - 210 y z + 90 y - 105 z^{2} + 90 z - 5&\displaystyle - 105 x^{2} - 105 x y - 105 x z + 105 x + \frac{105 y^{2}}{2} + 105 y z - 45 y + \frac{105 z^{2}}{2} - 45 z + \frac{5}{2}&\displaystyle 0\end{array}\right)\)
This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{59}:v\mapsto\displaystyle\int_{R}(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle 4 s_{0} \left(s_{0} + s_{1} + s_{2} - 1\right)&\displaystyle 0\\\displaystyle 4 s_{0} \left(s_{0} + s_{1} + s_{2} - 1\right)&\displaystyle 0&\displaystyle 4 s_{0} \left(- s_{0} - s_{1} - s_{2} + 1\right)\\\displaystyle 0&\displaystyle 4 s_{0} \left(- s_{0} - s_{1} - s_{2} + 1\right)&\displaystyle 0\end{array}\right))v\)
where \(R\) is the reference element;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{59} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle 210 x^{2} + 210 x y + 210 x z - 210 x - 105 y^{2} - 210 y z + 90 y - 105 z^{2} + 90 z - 5&\displaystyle - 105 x^{2} - 105 x y - 105 x z + 105 x + \frac{105 y^{2}}{2} + 105 y z - 45 y + \frac{105 z^{2}}{2} - 45 z + \frac{5}{2}\\\displaystyle 210 x^{2} + 210 x y + 210 x z - 210 x - 105 y^{2} - 210 y z + 90 y - 105 z^{2} + 90 z - 5&\displaystyle 0&\displaystyle - 105 x^{2} - 105 x y - 105 x z + 105 x + \frac{105 y^{2}}{2} + 105 y z - 45 y + \frac{105 z^{2}}{2} - 45 z + \frac{5}{2}\\\displaystyle - 105 x^{2} - 105 x y - 105 x z + 105 x + \frac{105 y^{2}}{2} + 105 y z - 45 y + \frac{105 z^{2}}{2} - 45 z + \frac{5}{2}&\displaystyle - 105 x^{2} - 105 x y - 105 x z + 105 x + \frac{105 y^{2}}{2} + 105 y z - 45 y + \frac{105 z^{2}}{2} - 45 z + \frac{5}{2}&\displaystyle 0\end{array}\right)\)
This DOF is associated with volume 0 of the reference element.
where \(R\) is the reference element;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{59} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle 210 x^{2} + 210 x y + 210 x z - 210 x - 105 y^{2} - 210 y z + 90 y - 105 z^{2} + 90 z - 5&\displaystyle - 105 x^{2} - 105 x y - 105 x z + 105 x + \frac{105 y^{2}}{2} + 105 y z - 45 y + \frac{105 z^{2}}{2} - 45 z + \frac{5}{2}\\\displaystyle 210 x^{2} + 210 x y + 210 x z - 210 x - 105 y^{2} - 210 y z + 90 y - 105 z^{2} + 90 z - 5&\displaystyle 0&\displaystyle - 105 x^{2} - 105 x y - 105 x z + 105 x + \frac{105 y^{2}}{2} + 105 y z - 45 y + \frac{105 z^{2}}{2} - 45 z + \frac{5}{2}\\\displaystyle - 105 x^{2} - 105 x y - 105 x z + 105 x + \frac{105 y^{2}}{2} + 105 y z - 45 y + \frac{105 z^{2}}{2} - 45 z + \frac{5}{2}&\displaystyle - 105 x^{2} - 105 x y - 105 x z + 105 x + \frac{105 y^{2}}{2} + 105 y z - 45 y + \frac{105 z^{2}}{2} - 45 z + \frac{5}{2}&\displaystyle 0\end{array}\right)\)
This DOF is associated with volume 0 of the reference element.