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Degree 1 Hellan–Herrmann–Johnson on a triangle
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- \(R\) is the reference triangle. The following numbering of the subentities of the reference is used:
- \(\mathcal{V}\) is spanned by: \(\left(\begin{array}{cc}\displaystyle1&\displaystyle0\\\displaystyle0&\displaystyle0\end{array}\right)\), \(\left(\begin{array}{cc}\displaystyle0&\displaystyle1\\\displaystyle1&\displaystyle0\end{array}\right)\), \(\left(\begin{array}{cc}\displaystyle0&\displaystyle0\\\displaystyle0&\displaystyle1\end{array}\right)\), \(\left(\begin{array}{cc}\displaystylex&\displaystyle0\\\displaystyle0&\displaystyle0\end{array}\right)\), \(\left(\begin{array}{cc}\displaystyle0&\displaystylex\\\displaystylex&\displaystyle0\end{array}\right)\), \(\left(\begin{array}{cc}\displaystyle0&\displaystyle0\\\displaystyle0&\displaystylex\end{array}\right)\), \(\left(\begin{array}{cc}\displaystyley&\displaystyle0\\\displaystyle0&\displaystyle0\end{array}\right)\), \(\left(\begin{array}{cc}\displaystyle0&\displaystyley\\\displaystyley&\displaystyle0\end{array}\right)\), \(\left(\begin{array}{cc}\displaystyle0&\displaystyle0\\\displaystyle0&\displaystyley\end{array}\right)\)
- \(\mathcal{L}=\{l_0,...,l_{8}\}\)
- Functionals and basis functions:
\(\displaystyle l_{0}:\mathbf{V}\mapsto\displaystyle\int_{e_{0}}(1 - s_{0})|{e_{0}}|\hat{\boldsymbol{n}}^{\text{t}}_{0}\mathbf{V}\hat{\boldsymbol{n}}_{0}\)
where \(e_{0}\) is the 0th edge;
and \(\hat{\boldsymbol{n}}_{0}\) is the normal to facet 0.
\(\displaystyle \mathbf{\Phi}_{0} = \left(\begin{array}{cc}\displaystyle0&\displaystyle3 x - 1\\\displaystyle3 x - 1&\displaystyle0\end{array}\right)\)
This DOF is associated with edge 0 of the reference element.
where \(e_{0}\) is the 0th edge;
and \(\hat{\boldsymbol{n}}_{0}\) is the normal to facet 0.
\(\displaystyle \mathbf{\Phi}_{0} = \left(\begin{array}{cc}\displaystyle0&\displaystyle3 x - 1\\\displaystyle3 x - 1&\displaystyle0\end{array}\right)\)
This DOF is associated with edge 0 of the reference element.
\(\displaystyle l_{1}:\mathbf{V}\mapsto\displaystyle\int_{e_{0}}(s_{0})|{e_{0}}|\hat{\boldsymbol{n}}^{\text{t}}_{0}\mathbf{V}\hat{\boldsymbol{n}}_{0}\)
where \(e_{0}\) is the 0th edge;
and \(\hat{\boldsymbol{n}}_{0}\) is the normal to facet 0.
\(\displaystyle \mathbf{\Phi}_{1} = \left(\begin{array}{cc}\displaystyle0&\displaystyle3 y - 1\\\displaystyle3 y - 1&\displaystyle0\end{array}\right)\)
This DOF is associated with edge 0 of the reference element.
where \(e_{0}\) is the 0th edge;
and \(\hat{\boldsymbol{n}}_{0}\) is the normal to facet 0.
\(\displaystyle \mathbf{\Phi}_{1} = \left(\begin{array}{cc}\displaystyle0&\displaystyle3 y - 1\\\displaystyle3 y - 1&\displaystyle0\end{array}\right)\)
This DOF is associated with edge 0 of the reference element.
\(\displaystyle l_{2}:\mathbf{V}\mapsto\displaystyle\int_{e_{1}}(1 - s_{0})|{e_{1}}|\hat{\boldsymbol{n}}^{\text{t}}_{1}\mathbf{V}\hat{\boldsymbol{n}}_{1}\)
where \(e_{1}\) is the 1st edge;
and \(\hat{\boldsymbol{n}}_{1}\) is the normal to facet 1.
\(\displaystyle \mathbf{\Phi}_{2} = \left(\begin{array}{cc}\displaystyle- 6 x - 6 y + 4&\displaystyle3 x + 3 y - 2\\\displaystyle3 x + 3 y - 2&\displaystyle0\end{array}\right)\)
This DOF is associated with edge 1 of the reference element.
where \(e_{1}\) is the 1st edge;
and \(\hat{\boldsymbol{n}}_{1}\) is the normal to facet 1.
\(\displaystyle \mathbf{\Phi}_{2} = \left(\begin{array}{cc}\displaystyle- 6 x - 6 y + 4&\displaystyle3 x + 3 y - 2\\\displaystyle3 x + 3 y - 2&\displaystyle0\end{array}\right)\)
This DOF is associated with edge 1 of the reference element.
\(\displaystyle l_{3}:\mathbf{V}\mapsto\displaystyle\int_{e_{1}}(s_{0})|{e_{1}}|\hat{\boldsymbol{n}}^{\text{t}}_{1}\mathbf{V}\hat{\boldsymbol{n}}_{1}\)
where \(e_{1}\) is the 1st edge;
and \(\hat{\boldsymbol{n}}_{1}\) is the normal to facet 1.
\(\displaystyle \mathbf{\Phi}_{3} = \left(\begin{array}{cc}\displaystyle6 y - 2&\displaystyle1 - 3 y\\\displaystyle1 - 3 y&\displaystyle0\end{array}\right)\)
This DOF is associated with edge 1 of the reference element.
where \(e_{1}\) is the 1st edge;
and \(\hat{\boldsymbol{n}}_{1}\) is the normal to facet 1.
\(\displaystyle \mathbf{\Phi}_{3} = \left(\begin{array}{cc}\displaystyle6 y - 2&\displaystyle1 - 3 y\\\displaystyle1 - 3 y&\displaystyle0\end{array}\right)\)
This DOF is associated with edge 1 of the reference element.
\(\displaystyle l_{4}:\mathbf{V}\mapsto\displaystyle\int_{e_{2}}(1 - s_{0})|{e_{2}}|\hat{\boldsymbol{n}}^{\text{t}}_{2}\mathbf{V}\hat{\boldsymbol{n}}_{2}\)
where \(e_{2}\) is the 2nd edge;
and \(\hat{\boldsymbol{n}}_{2}\) is the normal to facet 2.
\(\displaystyle \mathbf{\Phi}_{4} = \left(\begin{array}{cc}\displaystyle0&\displaystyle3 x + 3 y - 2\\\displaystyle3 x + 3 y - 2&\displaystyle- 6 x - 6 y + 4\end{array}\right)\)
This DOF is associated with edge 2 of the reference element.
where \(e_{2}\) is the 2nd edge;
and \(\hat{\boldsymbol{n}}_{2}\) is the normal to facet 2.
\(\displaystyle \mathbf{\Phi}_{4} = \left(\begin{array}{cc}\displaystyle0&\displaystyle3 x + 3 y - 2\\\displaystyle3 x + 3 y - 2&\displaystyle- 6 x - 6 y + 4\end{array}\right)\)
This DOF is associated with edge 2 of the reference element.
\(\displaystyle l_{5}:\mathbf{V}\mapsto\displaystyle\int_{e_{2}}(s_{0})|{e_{2}}|\hat{\boldsymbol{n}}^{\text{t}}_{2}\mathbf{V}\hat{\boldsymbol{n}}_{2}\)
where \(e_{2}\) is the 2nd edge;
and \(\hat{\boldsymbol{n}}_{2}\) is the normal to facet 2.
\(\displaystyle \mathbf{\Phi}_{5} = \left(\begin{array}{cc}\displaystyle0&\displaystyle1 - 3 x\\\displaystyle1 - 3 x&\displaystyle6 x - 2\end{array}\right)\)
This DOF is associated with edge 2 of the reference element.
where \(e_{2}\) is the 2nd edge;
and \(\hat{\boldsymbol{n}}_{2}\) is the normal to facet 2.
\(\displaystyle \mathbf{\Phi}_{5} = \left(\begin{array}{cc}\displaystyle0&\displaystyle1 - 3 x\\\displaystyle1 - 3 x&\displaystyle6 x - 2\end{array}\right)\)
This DOF is associated with edge 2 of the reference element.
\(\displaystyle l_{6}:v\mapsto\displaystyle\int_{R}(\left(\begin{array}{cc}\displaystyle1&\displaystyle0\\\displaystyle0&\displaystyle0\end{array}\right))v\)
where \(R\) is the reference element.
\(\displaystyle \mathbf{\Phi}_{6} = \left(\begin{array}{cc}\displaystyle6 x&\displaystyle- 6 x - 3 y + 3\\\displaystyle- 6 x - 3 y + 3&\displaystyle0\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
where \(R\) is the reference element.
\(\displaystyle \mathbf{\Phi}_{6} = \left(\begin{array}{cc}\displaystyle6 x&\displaystyle- 6 x - 3 y + 3\\\displaystyle- 6 x - 3 y + 3&\displaystyle0\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{7}:v\mapsto\displaystyle\int_{R}(\left(\begin{array}{cc}\displaystyle0&\displaystyle1\\\displaystyle1&\displaystyle0\end{array}\right))v\)
where \(R\) is the reference element.
\(\displaystyle \mathbf{\Phi}_{7} = \left(\begin{array}{cc}\displaystyle0&\displaystyle- 3 x - 3 y + 3\\\displaystyle- 3 x - 3 y + 3&\displaystyle0\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
where \(R\) is the reference element.
\(\displaystyle \mathbf{\Phi}_{7} = \left(\begin{array}{cc}\displaystyle0&\displaystyle- 3 x - 3 y + 3\\\displaystyle- 3 x - 3 y + 3&\displaystyle0\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{8}:v\mapsto\displaystyle\int_{R}(\left(\begin{array}{cc}\displaystyle0&\displaystyle0\\\displaystyle0&\displaystyle1\end{array}\right))v\)
where \(R\) is the reference element.
\(\displaystyle \mathbf{\Phi}_{8} = \left(\begin{array}{cc}\displaystyle0&\displaystyle- 3 x - 6 y + 3\\\displaystyle- 3 x - 6 y + 3&\displaystyle6 y\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
where \(R\) is the reference element.
\(\displaystyle \mathbf{\Phi}_{8} = \left(\begin{array}{cc}\displaystyle0&\displaystyle- 3 x - 6 y + 3\\\displaystyle- 3 x - 6 y + 3&\displaystyle6 y\end{array}\right)\)
This DOF is associated with face 0 of the reference element.