an encyclopedia of finite element definitions
Degree 1 Hellan–Herrmann–Johnson on a tetrahedron
◀ Back to Hellan–Herrmann–Johnson definition page
- \(R\) is the reference tetrahedron. The following numbering of the subentities of the reference cell is used:
- \(\mathcal{V}\) is spanned by: \(\left(\begin{array}{ccc}\displaystyle 1&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle 1&\displaystyle 0\\\displaystyle 1&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle 0&\displaystyle 1\\\displaystyle 0&\displaystyle 0&\displaystyle 0\\\displaystyle 1&\displaystyle 0&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 1&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle 1\\\displaystyle 0&\displaystyle 1&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle 1\end{array}\right)\), \(\left(\begin{array}{ccc}\displaystyle x&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle x&\displaystyle 0\\\displaystyle x&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle 0&\displaystyle x\\\displaystyle 0&\displaystyle 0&\displaystyle 0\\\displaystyle x&\displaystyle 0&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle x&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle x\\\displaystyle 0&\displaystyle x&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle x\end{array}\right)\), \(\left(\begin{array}{ccc}\displaystyle y&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle y&\displaystyle 0\\\displaystyle y&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle 0&\displaystyle y\\\displaystyle 0&\displaystyle 0&\displaystyle 0\\\displaystyle y&\displaystyle 0&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle y&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle y\\\displaystyle 0&\displaystyle y&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle y\end{array}\right)\), \(\left(\begin{array}{ccc}\displaystyle z&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle z&\displaystyle 0\\\displaystyle z&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle 0&\displaystyle z\\\displaystyle 0&\displaystyle 0&\displaystyle 0\\\displaystyle z&\displaystyle 0&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle z&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle z\\\displaystyle 0&\displaystyle z&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle z\end{array}\right)\)
- \(\mathcal{L}=\{l_0,...,l_{23}\}\)
- Functionals and basis functions:
\(\displaystyle l_{0}:\mathbf{V}\mapsto\displaystyle\int_{f_{0}}(- s_{0} - s_{1} + 1)|{f_{0}}|\hat{\boldsymbol{n}}^{\text{t}}_{0}\mathbf{V}\hat{\boldsymbol{n}}_{0}\)
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 \(f_{0}\).
\(\displaystyle \mathbf{\Phi}_{0} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle 4 x - 1&\displaystyle 4 x - 1\\\displaystyle 4 x - 1&\displaystyle 0&\displaystyle 4 x - 1\\\displaystyle 4 x - 1&\displaystyle 4 x - 1&\displaystyle 0\end{array}\right)\)
This DOF is associated with face 0 of the reference cell.
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 \(f_{0}\).
\(\displaystyle \mathbf{\Phi}_{0} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle 4 x - 1&\displaystyle 4 x - 1\\\displaystyle 4 x - 1&\displaystyle 0&\displaystyle 4 x - 1\\\displaystyle 4 x - 1&\displaystyle 4 x - 1&\displaystyle 0\end{array}\right)\)
This DOF is associated with face 0 of the reference cell.
\(\displaystyle l_{1}:\mathbf{V}\mapsto\displaystyle\int_{f_{0}}(s_{0})|{f_{0}}|\hat{\boldsymbol{n}}^{\text{t}}_{0}\mathbf{V}\hat{\boldsymbol{n}}_{0}\)
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 \(f_{0}\).
\(\displaystyle \mathbf{\Phi}_{1} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle 4 y - 1&\displaystyle 4 y - 1\\\displaystyle 4 y - 1&\displaystyle 0&\displaystyle 4 y - 1\\\displaystyle 4 y - 1&\displaystyle 4 y - 1&\displaystyle 0\end{array}\right)\)
This DOF is associated with face 0 of the reference cell.
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 \(f_{0}\).
\(\displaystyle \mathbf{\Phi}_{1} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle 4 y - 1&\displaystyle 4 y - 1\\\displaystyle 4 y - 1&\displaystyle 0&\displaystyle 4 y - 1\\\displaystyle 4 y - 1&\displaystyle 4 y - 1&\displaystyle 0\end{array}\right)\)
This DOF is associated with face 0 of the reference cell.
\(\displaystyle l_{2}:\mathbf{V}\mapsto\displaystyle\int_{f_{0}}(s_{1})|{f_{0}}|\hat{\boldsymbol{n}}^{\text{t}}_{0}\mathbf{V}\hat{\boldsymbol{n}}_{0}\)
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 \(f_{0}\).
\(\displaystyle \mathbf{\Phi}_{2} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle 4 z - 1&\displaystyle 4 z - 1\\\displaystyle 4 z - 1&\displaystyle 0&\displaystyle 4 z - 1\\\displaystyle 4 z - 1&\displaystyle 4 z - 1&\displaystyle 0\end{array}\right)\)
This DOF is associated with face 0 of the reference cell.
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 \(f_{0}\).
\(\displaystyle \mathbf{\Phi}_{2} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle 4 z - 1&\displaystyle 4 z - 1\\\displaystyle 4 z - 1&\displaystyle 0&\displaystyle 4 z - 1\\\displaystyle 4 z - 1&\displaystyle 4 z - 1&\displaystyle 0\end{array}\right)\)
This DOF is associated with face 0 of the reference cell.
\(\displaystyle l_{3}:\mathbf{V}\mapsto\displaystyle\int_{f_{1}}(- s_{0} - s_{1} + 1)|{f_{1}}|\hat{\boldsymbol{n}}^{\text{t}}_{1}\mathbf{V}\hat{\boldsymbol{n}}_{1}\)
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 \(f_{1}\).
\(\displaystyle \mathbf{\Phi}_{3} = \left(\begin{array}{ccc}\displaystyle - 24 x - 24 y - 24 z + 18&\displaystyle 4 x + 4 y + 4 z - 3&\displaystyle 4 x + 4 y + 4 z - 3\\\displaystyle 4 x + 4 y + 4 z - 3&\displaystyle 0&\displaystyle 4 x + 4 y + 4 z - 3\\\displaystyle 4 x + 4 y + 4 z - 3&\displaystyle 4 x + 4 y + 4 z - 3&\displaystyle 0\end{array}\right)\)
This DOF is associated with face 1 of the reference cell.
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 \(f_{1}\).
\(\displaystyle \mathbf{\Phi}_{3} = \left(\begin{array}{ccc}\displaystyle - 24 x - 24 y - 24 z + 18&\displaystyle 4 x + 4 y + 4 z - 3&\displaystyle 4 x + 4 y + 4 z - 3\\\displaystyle 4 x + 4 y + 4 z - 3&\displaystyle 0&\displaystyle 4 x + 4 y + 4 z - 3\\\displaystyle 4 x + 4 y + 4 z - 3&\displaystyle 4 x + 4 y + 4 z - 3&\displaystyle 0\end{array}\right)\)
This DOF is associated with face 1 of the reference cell.
\(\displaystyle l_{4}:\mathbf{V}\mapsto\displaystyle\int_{f_{1}}(s_{0})|{f_{1}}|\hat{\boldsymbol{n}}^{\text{t}}_{1}\mathbf{V}\hat{\boldsymbol{n}}_{1}\)
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 \(f_{1}\).
\(\displaystyle \mathbf{\Phi}_{4} = \left(\begin{array}{ccc}\displaystyle 24 y - 6&\displaystyle 1 - 4 y&\displaystyle 1 - 4 y\\\displaystyle 1 - 4 y&\displaystyle 0&\displaystyle 1 - 4 y\\\displaystyle 1 - 4 y&\displaystyle 1 - 4 y&\displaystyle 0\end{array}\right)\)
This DOF is associated with face 1 of the reference cell.
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 \(f_{1}\).
\(\displaystyle \mathbf{\Phi}_{4} = \left(\begin{array}{ccc}\displaystyle 24 y - 6&\displaystyle 1 - 4 y&\displaystyle 1 - 4 y\\\displaystyle 1 - 4 y&\displaystyle 0&\displaystyle 1 - 4 y\\\displaystyle 1 - 4 y&\displaystyle 1 - 4 y&\displaystyle 0\end{array}\right)\)
This DOF is associated with face 1 of the reference cell.
\(\displaystyle l_{5}:\mathbf{V}\mapsto\displaystyle\int_{f_{1}}(s_{1})|{f_{1}}|\hat{\boldsymbol{n}}^{\text{t}}_{1}\mathbf{V}\hat{\boldsymbol{n}}_{1}\)
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 \(f_{1}\).
\(\displaystyle \mathbf{\Phi}_{5} = \left(\begin{array}{ccc}\displaystyle 24 z - 6&\displaystyle 1 - 4 z&\displaystyle 1 - 4 z\\\displaystyle 1 - 4 z&\displaystyle 0&\displaystyle 1 - 4 z\\\displaystyle 1 - 4 z&\displaystyle 1 - 4 z&\displaystyle 0\end{array}\right)\)
This DOF is associated with face 1 of the reference cell.
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 \(f_{1}\).
\(\displaystyle \mathbf{\Phi}_{5} = \left(\begin{array}{ccc}\displaystyle 24 z - 6&\displaystyle 1 - 4 z&\displaystyle 1 - 4 z\\\displaystyle 1 - 4 z&\displaystyle 0&\displaystyle 1 - 4 z\\\displaystyle 1 - 4 z&\displaystyle 1 - 4 z&\displaystyle 0\end{array}\right)\)
This DOF is associated with face 1 of the reference cell.
\(\displaystyle l_{6}:\mathbf{V}\mapsto\displaystyle\int_{f_{2}}(- s_{0} - s_{1} + 1)|{f_{2}}|\hat{\boldsymbol{n}}^{\text{t}}_{2}\mathbf{V}\hat{\boldsymbol{n}}_{2}\)
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 \(f_{2}\).
\(\displaystyle \mathbf{\Phi}_{6} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle 4 x + 4 y + 4 z - 3&\displaystyle 4 x + 4 y + 4 z - 3\\\displaystyle 4 x + 4 y + 4 z - 3&\displaystyle - 24 x - 24 y - 24 z + 18&\displaystyle 4 x + 4 y + 4 z - 3\\\displaystyle 4 x + 4 y + 4 z - 3&\displaystyle 4 x + 4 y + 4 z - 3&\displaystyle 0\end{array}\right)\)
This DOF is associated with face 2 of the reference cell.
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 \(f_{2}\).
\(\displaystyle \mathbf{\Phi}_{6} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle 4 x + 4 y + 4 z - 3&\displaystyle 4 x + 4 y + 4 z - 3\\\displaystyle 4 x + 4 y + 4 z - 3&\displaystyle - 24 x - 24 y - 24 z + 18&\displaystyle 4 x + 4 y + 4 z - 3\\\displaystyle 4 x + 4 y + 4 z - 3&\displaystyle 4 x + 4 y + 4 z - 3&\displaystyle 0\end{array}\right)\)
This DOF is associated with face 2 of the reference cell.
\(\displaystyle l_{7}:\mathbf{V}\mapsto\displaystyle\int_{f_{2}}(s_{0})|{f_{2}}|\hat{\boldsymbol{n}}^{\text{t}}_{2}\mathbf{V}\hat{\boldsymbol{n}}_{2}\)
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 \(f_{2}\).
\(\displaystyle \mathbf{\Phi}_{7} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle 1 - 4 x&\displaystyle 1 - 4 x\\\displaystyle 1 - 4 x&\displaystyle 24 x - 6&\displaystyle 1 - 4 x\\\displaystyle 1 - 4 x&\displaystyle 1 - 4 x&\displaystyle 0\end{array}\right)\)
This DOF is associated with face 2 of the reference cell.
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 \(f_{2}\).
\(\displaystyle \mathbf{\Phi}_{7} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle 1 - 4 x&\displaystyle 1 - 4 x\\\displaystyle 1 - 4 x&\displaystyle 24 x - 6&\displaystyle 1 - 4 x\\\displaystyle 1 - 4 x&\displaystyle 1 - 4 x&\displaystyle 0\end{array}\right)\)
This DOF is associated with face 2 of the reference cell.
\(\displaystyle l_{8}:\mathbf{V}\mapsto\displaystyle\int_{f_{2}}(s_{1})|{f_{2}}|\hat{\boldsymbol{n}}^{\text{t}}_{2}\mathbf{V}\hat{\boldsymbol{n}}_{2}\)
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 \(f_{2}\).
\(\displaystyle \mathbf{\Phi}_{8} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle 1 - 4 z&\displaystyle 1 - 4 z\\\displaystyle 1 - 4 z&\displaystyle 24 z - 6&\displaystyle 1 - 4 z\\\displaystyle 1 - 4 z&\displaystyle 1 - 4 z&\displaystyle 0\end{array}\right)\)
This DOF is associated with face 2 of the reference cell.
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 \(f_{2}\).
\(\displaystyle \mathbf{\Phi}_{8} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle 1 - 4 z&\displaystyle 1 - 4 z\\\displaystyle 1 - 4 z&\displaystyle 24 z - 6&\displaystyle 1 - 4 z\\\displaystyle 1 - 4 z&\displaystyle 1 - 4 z&\displaystyle 0\end{array}\right)\)
This DOF is associated with face 2 of the reference cell.
\(\displaystyle l_{9}:\mathbf{V}\mapsto\displaystyle\int_{f_{3}}(- s_{0} - s_{1} + 1)|{f_{3}}|\hat{\boldsymbol{n}}^{\text{t}}_{3}\mathbf{V}\hat{\boldsymbol{n}}_{3}\)
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 \(f_{3}\).
\(\displaystyle \mathbf{\Phi}_{9} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle 4 x + 4 y + 4 z - 3&\displaystyle 4 x + 4 y + 4 z - 3\\\displaystyle 4 x + 4 y + 4 z - 3&\displaystyle 0&\displaystyle 4 x + 4 y + 4 z - 3\\\displaystyle 4 x + 4 y + 4 z - 3&\displaystyle 4 x + 4 y + 4 z - 3&\displaystyle - 24 x - 24 y - 24 z + 18\end{array}\right)\)
This DOF is associated with face 3 of the reference cell.
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 \(f_{3}\).
\(\displaystyle \mathbf{\Phi}_{9} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle 4 x + 4 y + 4 z - 3&\displaystyle 4 x + 4 y + 4 z - 3\\\displaystyle 4 x + 4 y + 4 z - 3&\displaystyle 0&\displaystyle 4 x + 4 y + 4 z - 3\\\displaystyle 4 x + 4 y + 4 z - 3&\displaystyle 4 x + 4 y + 4 z - 3&\displaystyle - 24 x - 24 y - 24 z + 18\end{array}\right)\)
This DOF is associated with face 3 of the reference cell.
\(\displaystyle l_{10}:\mathbf{V}\mapsto\displaystyle\int_{f_{3}}(s_{0})|{f_{3}}|\hat{\boldsymbol{n}}^{\text{t}}_{3}\mathbf{V}\hat{\boldsymbol{n}}_{3}\)
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 \(f_{3}\).
\(\displaystyle \mathbf{\Phi}_{10} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle 1 - 4 x&\displaystyle 1 - 4 x\\\displaystyle 1 - 4 x&\displaystyle 0&\displaystyle 1 - 4 x\\\displaystyle 1 - 4 x&\displaystyle 1 - 4 x&\displaystyle 24 x - 6\end{array}\right)\)
This DOF is associated with face 3 of the reference cell.
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 \(f_{3}\).
\(\displaystyle \mathbf{\Phi}_{10} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle 1 - 4 x&\displaystyle 1 - 4 x\\\displaystyle 1 - 4 x&\displaystyle 0&\displaystyle 1 - 4 x\\\displaystyle 1 - 4 x&\displaystyle 1 - 4 x&\displaystyle 24 x - 6\end{array}\right)\)
This DOF is associated with face 3 of the reference cell.
\(\displaystyle l_{11}:\mathbf{V}\mapsto\displaystyle\int_{f_{3}}(s_{1})|{f_{3}}|\hat{\boldsymbol{n}}^{\text{t}}_{3}\mathbf{V}\hat{\boldsymbol{n}}_{3}\)
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 \(f_{3}\).
\(\displaystyle \mathbf{\Phi}_{11} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle 1 - 4 y&\displaystyle 1 - 4 y\\\displaystyle 1 - 4 y&\displaystyle 0&\displaystyle 1 - 4 y\\\displaystyle 1 - 4 y&\displaystyle 1 - 4 y&\displaystyle 24 y - 6\end{array}\right)\)
This DOF is associated with face 3 of the reference cell.
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 \(f_{3}\).
\(\displaystyle \mathbf{\Phi}_{11} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle 1 - 4 y&\displaystyle 1 - 4 y\\\displaystyle 1 - 4 y&\displaystyle 0&\displaystyle 1 - 4 y\\\displaystyle 1 - 4 y&\displaystyle 1 - 4 y&\displaystyle 24 y - 6\end{array}\right)\)
This DOF is associated with face 3 of the reference cell.
\(\displaystyle l_{12}:v\mapsto\displaystyle\int_{R}(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle 1&\displaystyle 1\\\displaystyle 1&\displaystyle 0&\displaystyle 1\\\displaystyle 1&\displaystyle 1&\displaystyle 0\end{array}\right))v\)
where \(R\) is the reference element.
\(\displaystyle \mathbf{\Phi}_{12} = \left(\begin{array}{ccc}\displaystyle 4 x&\displaystyle - \frac{20 x}{3} - \frac{20 y}{3} - \frac{20 z}{3} + 6&\displaystyle - \frac{20 x}{3} - \frac{20 y}{3} - \frac{20 z}{3} + 6\\\displaystyle - \frac{20 x}{3} - \frac{20 y}{3} - \frac{20 z}{3} + 6&\displaystyle 4 y&\displaystyle - \frac{20 x}{3} - \frac{20 y}{3} - \frac{20 z}{3} + 6\\\displaystyle - \frac{20 x}{3} - \frac{20 y}{3} - \frac{20 z}{3} + 6&\displaystyle - \frac{20 x}{3} - \frac{20 y}{3} - \frac{20 z}{3} + 6&\displaystyle 4 z\end{array}\right)\)
This DOF is associated with volume 0 of the reference cell.
where \(R\) is the reference element.
\(\displaystyle \mathbf{\Phi}_{12} = \left(\begin{array}{ccc}\displaystyle 4 x&\displaystyle - \frac{20 x}{3} - \frac{20 y}{3} - \frac{20 z}{3} + 6&\displaystyle - \frac{20 x}{3} - \frac{20 y}{3} - \frac{20 z}{3} + 6\\\displaystyle - \frac{20 x}{3} - \frac{20 y}{3} - \frac{20 z}{3} + 6&\displaystyle 4 y&\displaystyle - \frac{20 x}{3} - \frac{20 y}{3} - \frac{20 z}{3} + 6\\\displaystyle - \frac{20 x}{3} - \frac{20 y}{3} - \frac{20 z}{3} + 6&\displaystyle - \frac{20 x}{3} - \frac{20 y}{3} - \frac{20 z}{3} + 6&\displaystyle 4 z\end{array}\right)\)
This DOF is associated with volume 0 of the reference cell.
\(\displaystyle l_{13}:v\mapsto\displaystyle\int_{R}(\left(\begin{array}{ccc}\displaystyle -6&\displaystyle 1&\displaystyle 1\\\displaystyle 1&\displaystyle 0&\displaystyle 1\\\displaystyle 1&\displaystyle 1&\displaystyle 0\end{array}\right))v\)
where \(R\) is the reference element.
\(\displaystyle \mathbf{\Phi}_{13} = \left(\begin{array}{ccc}\displaystyle - 4 x&\displaystyle \frac{4 x}{3} + \frac{2 y}{3} + \frac{2 z}{3} - \frac{2}{3}&\displaystyle \frac{4 x}{3} + \frac{2 y}{3} + \frac{2 z}{3} - \frac{2}{3}\\\displaystyle \frac{4 x}{3} + \frac{2 y}{3} + \frac{2 z}{3} - \frac{2}{3}&\displaystyle 0&\displaystyle \frac{4 x}{3} + \frac{2 y}{3} + \frac{2 z}{3} - \frac{2}{3}\\\displaystyle \frac{4 x}{3} + \frac{2 y}{3} + \frac{2 z}{3} - \frac{2}{3}&\displaystyle \frac{4 x}{3} + \frac{2 y}{3} + \frac{2 z}{3} - \frac{2}{3}&\displaystyle 0\end{array}\right)\)
This DOF is associated with volume 0 of the reference cell.
where \(R\) is the reference element.
\(\displaystyle \mathbf{\Phi}_{13} = \left(\begin{array}{ccc}\displaystyle - 4 x&\displaystyle \frac{4 x}{3} + \frac{2 y}{3} + \frac{2 z}{3} - \frac{2}{3}&\displaystyle \frac{4 x}{3} + \frac{2 y}{3} + \frac{2 z}{3} - \frac{2}{3}\\\displaystyle \frac{4 x}{3} + \frac{2 y}{3} + \frac{2 z}{3} - \frac{2}{3}&\displaystyle 0&\displaystyle \frac{4 x}{3} + \frac{2 y}{3} + \frac{2 z}{3} - \frac{2}{3}\\\displaystyle \frac{4 x}{3} + \frac{2 y}{3} + \frac{2 z}{3} - \frac{2}{3}&\displaystyle \frac{4 x}{3} + \frac{2 y}{3} + \frac{2 z}{3} - \frac{2}{3}&\displaystyle 0\end{array}\right)\)
This DOF is associated with volume 0 of the reference cell.
\(\displaystyle l_{14}:v\mapsto\displaystyle\int_{R}(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle 1&\displaystyle 1\\\displaystyle 1&\displaystyle -6&\displaystyle 1\\\displaystyle 1&\displaystyle 1&\displaystyle 0\end{array}\right))v\)
where \(R\) is the reference element.
\(\displaystyle \mathbf{\Phi}_{14} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle \frac{2 x}{3} + \frac{4 y}{3} + \frac{2 z}{3} - \frac{2}{3}&\displaystyle \frac{2 x}{3} + \frac{4 y}{3} + \frac{2 z}{3} - \frac{2}{3}\\\displaystyle \frac{2 x}{3} + \frac{4 y}{3} + \frac{2 z}{3} - \frac{2}{3}&\displaystyle - 4 y&\displaystyle \frac{2 x}{3} + \frac{4 y}{3} + \frac{2 z}{3} - \frac{2}{3}\\\displaystyle \frac{2 x}{3} + \frac{4 y}{3} + \frac{2 z}{3} - \frac{2}{3}&\displaystyle \frac{2 x}{3} + \frac{4 y}{3} + \frac{2 z}{3} - \frac{2}{3}&\displaystyle 0\end{array}\right)\)
This DOF is associated with volume 0 of the reference cell.
where \(R\) is the reference element.
\(\displaystyle \mathbf{\Phi}_{14} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle \frac{2 x}{3} + \frac{4 y}{3} + \frac{2 z}{3} - \frac{2}{3}&\displaystyle \frac{2 x}{3} + \frac{4 y}{3} + \frac{2 z}{3} - \frac{2}{3}\\\displaystyle \frac{2 x}{3} + \frac{4 y}{3} + \frac{2 z}{3} - \frac{2}{3}&\displaystyle - 4 y&\displaystyle \frac{2 x}{3} + \frac{4 y}{3} + \frac{2 z}{3} - \frac{2}{3}\\\displaystyle \frac{2 x}{3} + \frac{4 y}{3} + \frac{2 z}{3} - \frac{2}{3}&\displaystyle \frac{2 x}{3} + \frac{4 y}{3} + \frac{2 z}{3} - \frac{2}{3}&\displaystyle 0\end{array}\right)\)
This DOF is associated with volume 0 of the reference cell.
\(\displaystyle l_{15}:v\mapsto\displaystyle\int_{R}(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle 1&\displaystyle 1\\\displaystyle 1&\displaystyle 0&\displaystyle 1\\\displaystyle 1&\displaystyle 1&\displaystyle -6\end{array}\right))v\)
where \(R\) is the reference element.
\(\displaystyle \mathbf{\Phi}_{15} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle \frac{2 x}{3} + \frac{2 y}{3} + \frac{4 z}{3} - \frac{2}{3}&\displaystyle \frac{2 x}{3} + \frac{2 y}{3} + \frac{4 z}{3} - \frac{2}{3}\\\displaystyle \frac{2 x}{3} + \frac{2 y}{3} + \frac{4 z}{3} - \frac{2}{3}&\displaystyle 0&\displaystyle \frac{2 x}{3} + \frac{2 y}{3} + \frac{4 z}{3} - \frac{2}{3}\\\displaystyle \frac{2 x}{3} + \frac{2 y}{3} + \frac{4 z}{3} - \frac{2}{3}&\displaystyle \frac{2 x}{3} + \frac{2 y}{3} + \frac{4 z}{3} - \frac{2}{3}&\displaystyle - 4 z\end{array}\right)\)
This DOF is associated with volume 0 of the reference cell.
where \(R\) is the reference element.
\(\displaystyle \mathbf{\Phi}_{15} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle \frac{2 x}{3} + \frac{2 y}{3} + \frac{4 z}{3} - \frac{2}{3}&\displaystyle \frac{2 x}{3} + \frac{2 y}{3} + \frac{4 z}{3} - \frac{2}{3}\\\displaystyle \frac{2 x}{3} + \frac{2 y}{3} + \frac{4 z}{3} - \frac{2}{3}&\displaystyle 0&\displaystyle \frac{2 x}{3} + \frac{2 y}{3} + \frac{4 z}{3} - \frac{2}{3}\\\displaystyle \frac{2 x}{3} + \frac{2 y}{3} + \frac{4 z}{3} - \frac{2}{3}&\displaystyle \frac{2 x}{3} + \frac{2 y}{3} + \frac{4 z}{3} - \frac{2}{3}&\displaystyle - 4 z\end{array}\right)\)
This DOF is associated with volume 0 of the reference cell.
\(\displaystyle l_{16}:v\mapsto\displaystyle\int_{R}(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle 0&\displaystyle s_{0} + s_{1} + s_{2} - 1\\\displaystyle 0&\displaystyle 0&\displaystyle - s_{0} - s_{1} - s_{2} + 1\\\displaystyle s_{0} + s_{1} + s_{2} - 1&\displaystyle - s_{0} - s_{1} - s_{2} + 1&\displaystyle 0\end{array}\right))v\)
where \(R\) is the reference element;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{16} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle - 20 x - 20 y - 20 z + 16&\displaystyle 40 x + 40 y + 40 z - 32\\\displaystyle - 20 x - 20 y - 20 z + 16&\displaystyle 0&\displaystyle - 20 x - 20 y - 20 z + 16\\\displaystyle 40 x + 40 y + 40 z - 32&\displaystyle - 20 x - 20 y - 20 z + 16&\displaystyle 0\end{array}\right)\)
This DOF is associated with volume 0 of the reference cell.
where \(R\) is the reference element;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{16} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle - 20 x - 20 y - 20 z + 16&\displaystyle 40 x + 40 y + 40 z - 32\\\displaystyle - 20 x - 20 y - 20 z + 16&\displaystyle 0&\displaystyle - 20 x - 20 y - 20 z + 16\\\displaystyle 40 x + 40 y + 40 z - 32&\displaystyle - 20 x - 20 y - 20 z + 16&\displaystyle 0\end{array}\right)\)
This DOF is associated with volume 0 of the reference cell.
\(\displaystyle l_{17}:v\mapsto\displaystyle\int_{R}(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle s_{0} + s_{1} + s_{2} - 1&\displaystyle 0\\\displaystyle s_{0} + s_{1} + s_{2} - 1&\displaystyle 0&\displaystyle - s_{0} - s_{1} - s_{2} + 1\\\displaystyle 0&\displaystyle - s_{0} - s_{1} - s_{2} + 1&\displaystyle 0\end{array}\right))v\)
where \(R\) is the reference element;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{17} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle 40 x + 40 y + 40 z - 32&\displaystyle - 20 x - 20 y - 20 z + 16\\\displaystyle 40 x + 40 y + 40 z - 32&\displaystyle 0&\displaystyle - 20 x - 20 y - 20 z + 16\\\displaystyle - 20 x - 20 y - 20 z + 16&\displaystyle - 20 x - 20 y - 20 z + 16&\displaystyle 0\end{array}\right)\)
This DOF is associated with volume 0 of the reference cell.
where \(R\) is the reference element;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{17} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle 40 x + 40 y + 40 z - 32&\displaystyle - 20 x - 20 y - 20 z + 16\\\displaystyle 40 x + 40 y + 40 z - 32&\displaystyle 0&\displaystyle - 20 x - 20 y - 20 z + 16\\\displaystyle - 20 x - 20 y - 20 z + 16&\displaystyle - 20 x - 20 y - 20 z + 16&\displaystyle 0\end{array}\right)\)
This DOF is associated with volume 0 of the reference cell.
\(\displaystyle l_{18}:v\mapsto\displaystyle\int_{R}(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle 0&\displaystyle - s_{0}\\\displaystyle 0&\displaystyle 0&\displaystyle s_{0}\\\displaystyle - s_{0}&\displaystyle s_{0}&\displaystyle 0\end{array}\right))v\)
where \(R\) is the reference element;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{18} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle 20 x - 4&\displaystyle 8 - 40 x\\\displaystyle 20 x - 4&\displaystyle 0&\displaystyle 20 x - 4\\\displaystyle 8 - 40 x&\displaystyle 20 x - 4&\displaystyle 0\end{array}\right)\)
This DOF is associated with volume 0 of the reference cell.
where \(R\) is the reference element;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{18} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle 20 x - 4&\displaystyle 8 - 40 x\\\displaystyle 20 x - 4&\displaystyle 0&\displaystyle 20 x - 4\\\displaystyle 8 - 40 x&\displaystyle 20 x - 4&\displaystyle 0\end{array}\right)\)
This DOF is associated with volume 0 of the reference cell.
\(\displaystyle l_{19}:v\mapsto\displaystyle\int_{R}(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle - s_{0}&\displaystyle 0\\\displaystyle - s_{0}&\displaystyle 0&\displaystyle s_{0}\\\displaystyle 0&\displaystyle s_{0}&\displaystyle 0\end{array}\right))v\)
where \(R\) is the reference element;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{19} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle 8 - 40 x&\displaystyle 20 x - 4\\\displaystyle 8 - 40 x&\displaystyle 0&\displaystyle 20 x - 4\\\displaystyle 20 x - 4&\displaystyle 20 x - 4&\displaystyle 0\end{array}\right)\)
This DOF is associated with volume 0 of the reference cell.
where \(R\) is the reference element;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{19} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle 8 - 40 x&\displaystyle 20 x - 4\\\displaystyle 8 - 40 x&\displaystyle 0&\displaystyle 20 x - 4\\\displaystyle 20 x - 4&\displaystyle 20 x - 4&\displaystyle 0\end{array}\right)\)
This DOF is associated with volume 0 of the reference cell.
\(\displaystyle l_{20}:v\mapsto\displaystyle\int_{R}(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle 0&\displaystyle - s_{1}\\\displaystyle 0&\displaystyle 0&\displaystyle s_{1}\\\displaystyle - s_{1}&\displaystyle s_{1}&\displaystyle 0\end{array}\right))v\)
where \(R\) is the reference element;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{20} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle 20 y - 4&\displaystyle 8 - 40 y\\\displaystyle 20 y - 4&\displaystyle 0&\displaystyle 20 y - 4\\\displaystyle 8 - 40 y&\displaystyle 20 y - 4&\displaystyle 0\end{array}\right)\)
This DOF is associated with volume 0 of the reference cell.
where \(R\) is the reference element;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{20} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle 20 y - 4&\displaystyle 8 - 40 y\\\displaystyle 20 y - 4&\displaystyle 0&\displaystyle 20 y - 4\\\displaystyle 8 - 40 y&\displaystyle 20 y - 4&\displaystyle 0\end{array}\right)\)
This DOF is associated with volume 0 of the reference cell.
\(\displaystyle l_{21}:v\mapsto\displaystyle\int_{R}(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle - s_{1}&\displaystyle 0\\\displaystyle - s_{1}&\displaystyle 0&\displaystyle s_{1}\\\displaystyle 0&\displaystyle s_{1}&\displaystyle 0\end{array}\right))v\)
where \(R\) is the reference element;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{21} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle 8 - 40 y&\displaystyle 20 y - 4\\\displaystyle 8 - 40 y&\displaystyle 0&\displaystyle 20 y - 4\\\displaystyle 20 y - 4&\displaystyle 20 y - 4&\displaystyle 0\end{array}\right)\)
This DOF is associated with volume 0 of the reference cell.
where \(R\) is the reference element;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{21} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle 8 - 40 y&\displaystyle 20 y - 4\\\displaystyle 8 - 40 y&\displaystyle 0&\displaystyle 20 y - 4\\\displaystyle 20 y - 4&\displaystyle 20 y - 4&\displaystyle 0\end{array}\right)\)
This DOF is associated with volume 0 of the reference cell.
\(\displaystyle l_{22}:v\mapsto\displaystyle\int_{R}(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle 0&\displaystyle - s_{2}\\\displaystyle 0&\displaystyle 0&\displaystyle s_{2}\\\displaystyle - s_{2}&\displaystyle s_{2}&\displaystyle 0\end{array}\right))v\)
where \(R\) is the reference element;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{22} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle 20 z - 4&\displaystyle 8 - 40 z\\\displaystyle 20 z - 4&\displaystyle 0&\displaystyle 20 z - 4\\\displaystyle 8 - 40 z&\displaystyle 20 z - 4&\displaystyle 0\end{array}\right)\)
This DOF is associated with volume 0 of the reference cell.
where \(R\) is the reference element;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{22} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle 20 z - 4&\displaystyle 8 - 40 z\\\displaystyle 20 z - 4&\displaystyle 0&\displaystyle 20 z - 4\\\displaystyle 8 - 40 z&\displaystyle 20 z - 4&\displaystyle 0\end{array}\right)\)
This DOF is associated with volume 0 of the reference cell.
\(\displaystyle l_{23}:v\mapsto\displaystyle\int_{R}(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle - s_{2}&\displaystyle 0\\\displaystyle - s_{2}&\displaystyle 0&\displaystyle s_{2}\\\displaystyle 0&\displaystyle s_{2}&\displaystyle 0\end{array}\right))v\)
where \(R\) is the reference element;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{23} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle 8 - 40 z&\displaystyle 20 z - 4\\\displaystyle 8 - 40 z&\displaystyle 0&\displaystyle 20 z - 4\\\displaystyle 20 z - 4&\displaystyle 20 z - 4&\displaystyle 0\end{array}\right)\)
This DOF is associated with volume 0 of the reference cell.
where \(R\) is the reference element;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{23} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle 8 - 40 z&\displaystyle 20 z - 4\\\displaystyle 8 - 40 z&\displaystyle 0&\displaystyle 20 z - 4\\\displaystyle 20 z - 4&\displaystyle 20 z - 4&\displaystyle 0\end{array}\right)\)
This DOF is associated with volume 0 of the reference cell.