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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: \(\left(\begin{array}{cc}\displaystyle 1&\displaystyle 0\\\displaystyle 0&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{cc}\displaystyle 0&\displaystyle 1\\\displaystyle 1&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{cc}\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 1\end{array}\right)\), \(\left(\begin{array}{cc}\displaystyle x&\displaystyle 0\\\displaystyle 0&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{cc}\displaystyle 0&\displaystyle x\\\displaystyle x&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{cc}\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle x\end{array}\right)\), \(\left(\begin{array}{cc}\displaystyle y&\displaystyle 0\\\displaystyle 0&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{cc}\displaystyle 0&\displaystyle y\\\displaystyle y&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{cc}\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle y\end{array}\right)\)
- \(\mathcal{L}=\{l_0,...,l_{8}\}\)
- 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;
\(\hat{\boldsymbol{n}}_{0}\) is the normal to facet 0;
and \(s_{0},s_{1}\) is a parametrisation of \(e_{0}\).
\(\displaystyle \mathbf{\Phi}_{0} = \left(\begin{array}{cc}\displaystyle 0&\displaystyle 3 x - 1\\\displaystyle 3 x - 1&\displaystyle 0\end{array}\right)\)
This DOF is associated with edge 0 of the reference element.
where \(e_{0}\) is the 0th edge;
\(\hat{\boldsymbol{n}}_{0}\) is the normal to facet 0;
and \(s_{0},s_{1}\) is a parametrisation of \(e_{0}\).
\(\displaystyle \mathbf{\Phi}_{0} = \left(\begin{array}{cc}\displaystyle 0&\displaystyle 3 x - 1\\\displaystyle 3 x - 1&\displaystyle 0\end{array}\right)\)
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;
\(\hat{\boldsymbol{n}}_{0}\) is the normal to facet 0;
and \(s_{0},s_{1}\) is a parametrisation of \(e_{0}\).
\(\displaystyle \mathbf{\Phi}_{1} = \left(\begin{array}{cc}\displaystyle 0&\displaystyle 3 y - 1\\\displaystyle 3 y - 1&\displaystyle 0\end{array}\right)\)
This DOF is associated with edge 0 of the reference element.
where \(e_{0}\) is the 0th edge;
\(\hat{\boldsymbol{n}}_{0}\) is the normal to facet 0;
and \(s_{0},s_{1}\) is a parametrisation of \(e_{0}\).
\(\displaystyle \mathbf{\Phi}_{1} = \left(\begin{array}{cc}\displaystyle 0&\displaystyle 3 y - 1\\\displaystyle 3 y - 1&\displaystyle 0\end{array}\right)\)
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;
\(\hat{\boldsymbol{n}}_{1}\) is the normal to facet 1;
and \(s_{0},s_{1}\) is a parametrisation of \(e_{1}\).
\(\displaystyle \mathbf{\Phi}_{2} = \left(\begin{array}{cc}\displaystyle - 6 x - 6 y + 4&\displaystyle 3 x + 3 y - 2\\\displaystyle 3 x + 3 y - 2&\displaystyle 0\end{array}\right)\)
This DOF is associated with edge 1 of the reference element.
where \(e_{1}\) is the 1st edge;
\(\hat{\boldsymbol{n}}_{1}\) is the normal to facet 1;
and \(s_{0},s_{1}\) is a parametrisation of \(e_{1}\).
\(\displaystyle \mathbf{\Phi}_{2} = \left(\begin{array}{cc}\displaystyle - 6 x - 6 y + 4&\displaystyle 3 x + 3 y - 2\\\displaystyle 3 x + 3 y - 2&\displaystyle 0\end{array}\right)\)
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;
\(\hat{\boldsymbol{n}}_{1}\) is the normal to facet 1;
and \(s_{0},s_{1}\) is a parametrisation of \(e_{1}\).
\(\displaystyle \mathbf{\Phi}_{3} = \left(\begin{array}{cc}\displaystyle 6 y - 2&\displaystyle 1 - 3 y\\\displaystyle 1 - 3 y&\displaystyle 0\end{array}\right)\)
This DOF is associated with edge 1 of the reference element.
where \(e_{1}\) is the 1st edge;
\(\hat{\boldsymbol{n}}_{1}\) is the normal to facet 1;
and \(s_{0},s_{1}\) is a parametrisation of \(e_{1}\).
\(\displaystyle \mathbf{\Phi}_{3} = \left(\begin{array}{cc}\displaystyle 6 y - 2&\displaystyle 1 - 3 y\\\displaystyle 1 - 3 y&\displaystyle 0\end{array}\right)\)
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;
\(\hat{\boldsymbol{n}}_{2}\) is the normal to facet 2;
and \(s_{0},s_{1}\) is a parametrisation of \(e_{2}\).
\(\displaystyle \mathbf{\Phi}_{4} = \left(\begin{array}{cc}\displaystyle 0&\displaystyle 3 x + 3 y - 2\\\displaystyle 3 x + 3 y - 2&\displaystyle - 6 x - 6 y + 4\end{array}\right)\)
This DOF is associated with edge 2 of the reference element.
where \(e_{2}\) is the 2nd edge;
\(\hat{\boldsymbol{n}}_{2}\) is the normal to facet 2;
and \(s_{0},s_{1}\) is a parametrisation of \(e_{2}\).
\(\displaystyle \mathbf{\Phi}_{4} = \left(\begin{array}{cc}\displaystyle 0&\displaystyle 3 x + 3 y - 2\\\displaystyle 3 x + 3 y - 2&\displaystyle - 6 x - 6 y + 4\end{array}\right)\)
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;
\(\hat{\boldsymbol{n}}_{2}\) is the normal to facet 2;
and \(s_{0},s_{1}\) is a parametrisation of \(e_{2}\).
\(\displaystyle \mathbf{\Phi}_{5} = \left(\begin{array}{cc}\displaystyle 0&\displaystyle 1 - 3 x\\\displaystyle 1 - 3 x&\displaystyle 6 x - 2\end{array}\right)\)
This DOF is associated with edge 2 of the reference element.
where \(e_{2}\) is the 2nd edge;
\(\hat{\boldsymbol{n}}_{2}\) is the normal to facet 2;
and \(s_{0},s_{1}\) is a parametrisation of \(e_{2}\).
\(\displaystyle \mathbf{\Phi}_{5} = \left(\begin{array}{cc}\displaystyle 0&\displaystyle 1 - 3 x\\\displaystyle 1 - 3 x&\displaystyle 6 x - 2\end{array}\right)\)
This DOF is associated with edge 2 of the reference element.
\(\displaystyle l_{6}:v\mapsto\displaystyle\int_{R}(\left(\begin{array}{cc}\displaystyle 0&\displaystyle 1\\\displaystyle 1&\displaystyle 0\end{array}\right))v\)
where \(R\) is the reference element.
\(\displaystyle \mathbf{\Phi}_{6} = \left(\begin{array}{cc}\displaystyle 3 x&\displaystyle - \frac{15 x}{2} - \frac{15 y}{2} + 6\\\displaystyle - \frac{15 x}{2} - \frac{15 y}{2} + 6&\displaystyle 3 y\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
where \(R\) is the reference element.
\(\displaystyle \mathbf{\Phi}_{6} = \left(\begin{array}{cc}\displaystyle 3 x&\displaystyle - \frac{15 x}{2} - \frac{15 y}{2} + 6\\\displaystyle - \frac{15 x}{2} - \frac{15 y}{2} + 6&\displaystyle 3 y\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{7}:v\mapsto\displaystyle\int_{R}(\left(\begin{array}{cc}\displaystyle -2&\displaystyle 1\\\displaystyle 1&\displaystyle 0\end{array}\right))v\)
where \(R\) is the reference element.
\(\displaystyle \mathbf{\Phi}_{7} = \left(\begin{array}{cc}\displaystyle - 3 x&\displaystyle 3 x + \frac{3 y}{2} - \frac{3}{2}\\\displaystyle 3 x + \frac{3 y}{2} - \frac{3}{2}&\displaystyle 0\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
where \(R\) is the reference element.
\(\displaystyle \mathbf{\Phi}_{7} = \left(\begin{array}{cc}\displaystyle - 3 x&\displaystyle 3 x + \frac{3 y}{2} - \frac{3}{2}\\\displaystyle 3 x + \frac{3 y}{2} - \frac{3}{2}&\displaystyle 0\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{8}:v\mapsto\displaystyle\int_{R}(\left(\begin{array}{cc}\displaystyle 0&\displaystyle -1\\\displaystyle -1&\displaystyle 2\end{array}\right))v\)
where \(R\) is the reference element.
\(\displaystyle \mathbf{\Phi}_{8} = \left(\begin{array}{cc}\displaystyle 0&\displaystyle - \frac{3 x}{2} - 3 y + \frac{3}{2}\\\displaystyle - \frac{3 x}{2} - 3 y + \frac{3}{2}&\displaystyle 3 y\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
where \(R\) is the reference element.
\(\displaystyle \mathbf{\Phi}_{8} = \left(\begin{array}{cc}\displaystyle 0&\displaystyle - \frac{3 x}{2} - 3 y + \frac{3}{2}\\\displaystyle - \frac{3 x}{2} - 3 y + \frac{3}{2}&\displaystyle 3 y\end{array}\right)\)
This DOF is associated with face 0 of the reference element.