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Degree 0 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: \(\left(\begin{array}{ccc}\displaystyle 1&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle 1&\displaystyle 0\\\displaystyle 1&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle 0&\displaystyle 1\\\displaystyle 0&\displaystyle 0&\displaystyle 0\\\displaystyle 1&\displaystyle 0&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 1&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle 1\\\displaystyle 0&\displaystyle 1&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle 1\end{array}\right)\)
- \(\mathcal{L}=\{l_0,...,l_{5}\}\)
- Functionals and basis functions:
\(\displaystyle l_{0}:\mathbf{V}\mapsto\displaystyle\int_{f_{0}}|{f_{0}}|\hat{\boldsymbol{n}}^{\text{t}}_{0}\mathbf{V}\hat{\boldsymbol{n}}_{0}\)
where \(f_{0}\) is the 0th face;
and \(\hat{\boldsymbol{n}}_{0}\) is the normal to facet 0.
\(\displaystyle \mathbf{\Phi}_{0} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle \frac{1}{3}&\displaystyle \frac{1}{3}\\\displaystyle \frac{1}{3}&\displaystyle 0&\displaystyle \frac{1}{3}\\\displaystyle \frac{1}{3}&\displaystyle \frac{1}{3}&\displaystyle 0\end{array}\right)\)
This DOF is associated with face 0 of the reference cell.
where \(f_{0}\) is the 0th face;
and \(\hat{\boldsymbol{n}}_{0}\) is the normal to facet 0.
\(\displaystyle \mathbf{\Phi}_{0} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle \frac{1}{3}&\displaystyle \frac{1}{3}\\\displaystyle \frac{1}{3}&\displaystyle 0&\displaystyle \frac{1}{3}\\\displaystyle \frac{1}{3}&\displaystyle \frac{1}{3}&\displaystyle 0\end{array}\right)\)
This DOF is associated with face 0 of the reference cell.
\(\displaystyle l_{1}:\mathbf{V}\mapsto\displaystyle\int_{f_{1}}|{f_{1}}|\hat{\boldsymbol{n}}^{\text{t}}_{1}\mathbf{V}\hat{\boldsymbol{n}}_{1}\)
where \(f_{1}\) is the 1st face;
and \(\hat{\boldsymbol{n}}_{1}\) is the normal to facet 1.
\(\displaystyle \mathbf{\Phi}_{1} = \left(\begin{array}{ccc}\displaystyle 2&\displaystyle - \frac{1}{3}&\displaystyle - \frac{1}{3}\\\displaystyle - \frac{1}{3}&\displaystyle 0&\displaystyle - \frac{1}{3}\\\displaystyle - \frac{1}{3}&\displaystyle - \frac{1}{3}&\displaystyle 0\end{array}\right)\)
This DOF is associated with face 1 of the reference cell.
where \(f_{1}\) is the 1st face;
and \(\hat{\boldsymbol{n}}_{1}\) is the normal to facet 1.
\(\displaystyle \mathbf{\Phi}_{1} = \left(\begin{array}{ccc}\displaystyle 2&\displaystyle - \frac{1}{3}&\displaystyle - \frac{1}{3}\\\displaystyle - \frac{1}{3}&\displaystyle 0&\displaystyle - \frac{1}{3}\\\displaystyle - \frac{1}{3}&\displaystyle - \frac{1}{3}&\displaystyle 0\end{array}\right)\)
This DOF is associated with face 1 of the reference cell.
\(\displaystyle l_{2}:\mathbf{V}\mapsto\displaystyle\int_{f_{2}}|{f_{2}}|\hat{\boldsymbol{n}}^{\text{t}}_{2}\mathbf{V}\hat{\boldsymbol{n}}_{2}\)
where \(f_{2}\) is the 2nd face;
and \(\hat{\boldsymbol{n}}_{2}\) is the normal to facet 2.
\(\displaystyle \mathbf{\Phi}_{2} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle - \frac{1}{3}&\displaystyle - \frac{1}{3}\\\displaystyle - \frac{1}{3}&\displaystyle 2&\displaystyle - \frac{1}{3}\\\displaystyle - \frac{1}{3}&\displaystyle - \frac{1}{3}&\displaystyle 0\end{array}\right)\)
This DOF is associated with face 2 of the reference cell.
where \(f_{2}\) is the 2nd face;
and \(\hat{\boldsymbol{n}}_{2}\) is the normal to facet 2.
\(\displaystyle \mathbf{\Phi}_{2} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle - \frac{1}{3}&\displaystyle - \frac{1}{3}\\\displaystyle - \frac{1}{3}&\displaystyle 2&\displaystyle - \frac{1}{3}\\\displaystyle - \frac{1}{3}&\displaystyle - \frac{1}{3}&\displaystyle 0\end{array}\right)\)
This DOF is associated with face 2 of the reference cell.
\(\displaystyle l_{3}:\mathbf{V}\mapsto\displaystyle\int_{f_{3}}|{f_{3}}|\hat{\boldsymbol{n}}^{\text{t}}_{3}\mathbf{V}\hat{\boldsymbol{n}}_{3}\)
where \(f_{3}\) is the 3rd face;
and \(\hat{\boldsymbol{n}}_{3}\) is the normal to facet 3.
\(\displaystyle \mathbf{\Phi}_{3} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle - \frac{1}{3}&\displaystyle - \frac{1}{3}\\\displaystyle - \frac{1}{3}&\displaystyle 0&\displaystyle - \frac{1}{3}\\\displaystyle - \frac{1}{3}&\displaystyle - \frac{1}{3}&\displaystyle 2\end{array}\right)\)
This DOF is associated with face 3 of the reference cell.
where \(f_{3}\) is the 3rd face;
and \(\hat{\boldsymbol{n}}_{3}\) is the normal to facet 3.
\(\displaystyle \mathbf{\Phi}_{3} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle - \frac{1}{3}&\displaystyle - \frac{1}{3}\\\displaystyle - \frac{1}{3}&\displaystyle 0&\displaystyle - \frac{1}{3}\\\displaystyle - \frac{1}{3}&\displaystyle - \frac{1}{3}&\displaystyle 2\end{array}\right)\)
This DOF is associated with face 3 of the reference cell.
\(\displaystyle l_{4}:v\mapsto\displaystyle\int_{R}(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle 0&\displaystyle -1\\\displaystyle 0&\displaystyle 0&\displaystyle 1\\\displaystyle -1&\displaystyle 1&\displaystyle 0\end{array}\right))v\)
where \(R\) is the reference element.
\(\displaystyle \mathbf{\Phi}_{4} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle 1&\displaystyle -2\\\displaystyle 1&\displaystyle 0&\displaystyle 1\\\displaystyle -2&\displaystyle 1&\displaystyle 0\end{array}\right)\)
This DOF is associated with volume 0 of the reference cell.
where \(R\) is the reference element.
\(\displaystyle \mathbf{\Phi}_{4} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle 1&\displaystyle -2\\\displaystyle 1&\displaystyle 0&\displaystyle 1\\\displaystyle -2&\displaystyle 1&\displaystyle 0\end{array}\right)\)
This DOF is associated with volume 0 of the reference cell.
\(\displaystyle l_{5}:v\mapsto\displaystyle\int_{R}(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle -1&\displaystyle 0\\\displaystyle -1&\displaystyle 0&\displaystyle 1\\\displaystyle 0&\displaystyle 1&\displaystyle 0\end{array}\right))v\)
where \(R\) is the reference element.
\(\displaystyle \mathbf{\Phi}_{5} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle -2&\displaystyle 1\\\displaystyle -2&\displaystyle 0&\displaystyle 1\\\displaystyle 1&\displaystyle 1&\displaystyle 0\end{array}\right)\)
This DOF is associated with volume 0 of the reference cell.
where \(R\) is the reference element.
\(\displaystyle \mathbf{\Phi}_{5} = \left(\begin{array}{ccc}\displaystyle 0&\displaystyle -2&\displaystyle 1\\\displaystyle -2&\displaystyle 0&\displaystyle 1\\\displaystyle 1&\displaystyle 1&\displaystyle 0\end{array}\right)\)
This DOF is associated with volume 0 of the reference cell.