an encyclopedia of finite element definitions
Hellan–Herrmann–Johnson
Click here to read what the information on this page means.
| Orders | \(0\leqslant k\) |
| Reference elements | triangle |
| Polynomial set | \(\mathcal{Z}^{(4)}_{k}\) ↓ Show polynomial set definitions ↓ |
| DOFs | On each edge: integral moments of inner products of normal to edge with an order \(k\) Lagrange space On each face: point evaluations of tensor products with symmetric matrices whose entries are in (lagrange,k-1) |
| Number of DOFs | triangle: \(3(k+1)(k+2)/2\) (A045943) |
| Categories | Matrix-valued elements |
Implementations
| Symfem | "HHJ"↓ Show Symfem examples ↓ |
| UFL | "HHJ"↓ Show UFL examples ↓ |
Examples
triangleorder 1triangle
order 2
- \(R\) is the reference triangle. The following numbering of the subentities of the reference is used:
- \(\mathcal{V}\) is spanned by: \(\left(\begin{matrix}1 & 0\\0 & 0\end{matrix}\right)\), \(\left(\begin{matrix}0 & 1\\1 & 0\end{matrix}\right)\), \(\left(\begin{matrix}0 & 0\\0 & 1\end{matrix}\right)\), \(\left(\begin{matrix}x & 0\\0 & 0\end{matrix}\right)\), \(\left(\begin{matrix}0 & x\\x & 0\end{matrix}\right)\), \(\left(\begin{matrix}0 & 0\\0 & x\end{matrix}\right)\), \(\left(\begin{matrix}y & 0\\0 & 0\end{matrix}\right)\), \(\left(\begin{matrix}0 & y\\y & 0\end{matrix}\right)\), \(\left(\begin{matrix}0 & 0\\0 & y\end{matrix}\right)\)
- \(\mathcal{L}=\{l_0,...,l_{8}\}\)
- 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 \(s_{0}\) is a parametrisation of \(e_{0}\).
\(\displaystyle \mathbf{\Phi}_{0} = \left(\begin{matrix}0 & 3 x - 1\\3 x - 1 & 0\end{matrix}\right)\)
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 \(s_{0}\) is a parametrisation of \(e_{0}\).
\(\displaystyle \mathbf{\Phi}_{0} = \left(\begin{matrix}0 & 3 x - 1\\3 x - 1 & 0\end{matrix}\right)\)
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 \(s_{0}\) is a parametrisation of \(e_{0}\).
\(\displaystyle \mathbf{\Phi}_{1} = \left(\begin{matrix}0 & 3 y - 1\\3 y - 1 & 0\end{matrix}\right)\)
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 \(s_{0}\) is a parametrisation of \(e_{0}\).
\(\displaystyle \mathbf{\Phi}_{1} = \left(\begin{matrix}0 & 3 y - 1\\3 y - 1 & 0\end{matrix}\right)\)
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 \(s_{0}\) is a parametrisation of \(e_{1}\).
\(\displaystyle \mathbf{\Phi}_{2} = \left(\begin{matrix}- 6 x - 6 y + 4 & 3 x + 3 y - 2\\3 x + 3 y - 2 & 0\end{matrix}\right)\)
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 \(s_{0}\) is a parametrisation of \(e_{1}\).
\(\displaystyle \mathbf{\Phi}_{2} = \left(\begin{matrix}- 6 x - 6 y + 4 & 3 x + 3 y - 2\\3 x + 3 y - 2 & 0\end{matrix}\right)\)
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 \(s_{0}\) is a parametrisation of \(e_{1}\).
\(\displaystyle \mathbf{\Phi}_{3} = \left(\begin{matrix}6 y - 2 & 1 - 3 y\\1 - 3 y & 0\end{matrix}\right)\)
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 \(s_{0}\) is a parametrisation of \(e_{1}\).
\(\displaystyle \mathbf{\Phi}_{3} = \left(\begin{matrix}6 y - 2 & 1 - 3 y\\1 - 3 y & 0\end{matrix}\right)\)
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 \(s_{0}\) is a parametrisation of \(e_{2}\).
\(\displaystyle \mathbf{\Phi}_{4} = \left(\begin{matrix}0 & 3 x + 3 y - 2\\3 x + 3 y - 2 & - 6 x - 6 y + 4\end{matrix}\right)\)
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 \(s_{0}\) is a parametrisation of \(e_{2}\).
\(\displaystyle \mathbf{\Phi}_{4} = \left(\begin{matrix}0 & 3 x + 3 y - 2\\3 x + 3 y - 2 & - 6 x - 6 y + 4\end{matrix}\right)\)
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 \(s_{0}\) is a parametrisation of \(e_{2}\).
\(\displaystyle \mathbf{\Phi}_{5} = \left(\begin{matrix}0 & 1 - 3 x\\1 - 3 x & 6 x - 2\end{matrix}\right)\)
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 \(s_{0}\) is a parametrisation of \(e_{2}\).
\(\displaystyle \mathbf{\Phi}_{5} = \left(\begin{matrix}0 & 1 - 3 x\\1 - 3 x & 6 x - 2\end{matrix}\right)\)
This DOF is associated with edge 2 of the reference element.
\(\displaystyle l_{6}:\mathbf{V}\mapsto\displaystyle\int_{R}\mathbf{V}:\left(\begin{array}{cc}1&0\\0&0\end{array}\right)\)
where \(R\) is the reference element.
\(\displaystyle \mathbf{\Phi}_{6} = \left(\begin{matrix}6 x & - 6 x - 3 y + 3\\- 6 x - 3 y + 3 & 0\end{matrix}\right)\)
This DOF is associated with face 0 of the reference element.
where \(R\) is the reference element.
\(\displaystyle \mathbf{\Phi}_{6} = \left(\begin{matrix}6 x & - 6 x - 3 y + 3\\- 6 x - 3 y + 3 & 0\end{matrix}\right)\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{7}:\mathbf{V}\mapsto\displaystyle\int_{R}\mathbf{V}:\left(\begin{array}{cc}0&1\\1&0\end{array}\right)\)
where \(R\) is the reference element.
\(\displaystyle \mathbf{\Phi}_{7} = \left(\begin{matrix}0 & - 3 x - 3 y + 3\\- 3 x - 3 y + 3 & 0\end{matrix}\right)\)
This DOF is associated with face 0 of the reference element.
where \(R\) is the reference element.
\(\displaystyle \mathbf{\Phi}_{7} = \left(\begin{matrix}0 & - 3 x - 3 y + 3\\- 3 x - 3 y + 3 & 0\end{matrix}\right)\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{8}:\mathbf{V}\mapsto\displaystyle\int_{R}\mathbf{V}:\left(\begin{array}{cc}0&0\\0&1\end{array}\right)\)
where \(R\) is the reference element.
\(\displaystyle \mathbf{\Phi}_{8} = \left(\begin{matrix}0 & - 3 x - 6 y + 3\\- 3 x - 6 y + 3 & 6 y\end{matrix}\right)\)
This DOF is associated with face 0 of the reference element.
where \(R\) is the reference element.
\(\displaystyle \mathbf{\Phi}_{8} = \left(\begin{matrix}0 & - 3 x - 6 y + 3\\- 3 x - 6 y + 3 & 6 y\end{matrix}\right)\)
This DOF is associated with face 0 of the reference element.
- \(R\) is the reference triangle. The following numbering of the subentities of the reference is used:
- \(\mathcal{V}\) is spanned by: \(\left(\begin{matrix}1 & 0\\0 & 0\end{matrix}\right)\), \(\left(\begin{matrix}0 & 1\\1 & 0\end{matrix}\right)\), \(\left(\begin{matrix}0 & 0\\0 & 1\end{matrix}\right)\), \(\left(\begin{matrix}x & 0\\0 & 0\end{matrix}\right)\), \(\left(\begin{matrix}0 & x\\x & 0\end{matrix}\right)\), \(\left(\begin{matrix}0 & 0\\0 & x\end{matrix}\right)\), \(\left(\begin{matrix}x^{2} & 0\\0 & 0\end{matrix}\right)\), \(\left(\begin{matrix}0 & x^{2}\\x^{2} & 0\end{matrix}\right)\), \(\left(\begin{matrix}0 & 0\\0 & x^{2}\end{matrix}\right)\), \(\left(\begin{matrix}y & 0\\0 & 0\end{matrix}\right)\), \(\left(\begin{matrix}0 & y\\y & 0\end{matrix}\right)\), \(\left(\begin{matrix}0 & 0\\0 & y\end{matrix}\right)\), \(\left(\begin{matrix}x y & 0\\0 & 0\end{matrix}\right)\), \(\left(\begin{matrix}0 & x y\\x y & 0\end{matrix}\right)\), \(\left(\begin{matrix}0 & 0\\0 & x y\end{matrix}\right)\), \(\left(\begin{matrix}y^{2} & 0\\0 & 0\end{matrix}\right)\), \(\left(\begin{matrix}0 & y^{2}\\y^{2} & 0\end{matrix}\right)\), \(\left(\begin{matrix}0 & 0\\0 & y^{2}\end{matrix}\right)\)
- \(\mathcal{L}=\{l_0,...,l_{17}\}\)
- Functionals and basis functions:
\(\displaystyle l_{0}:\mathbf{V}\mapsto\displaystyle\int_{e_{0}}(2 s_{0}^{2} - 3 s_{0} + 1)|{e_{0}}|\hat{\boldsymbol{n}}^{\text{t}}_{0}\mathbf{V}\hat{\boldsymbol{n}}_{0}\)
where \(e_{0}\) is the 0th edge;
\(\hat{\boldsymbol{n}}_{0}\) is the normal to facet 0;
and \(s_{0}\) is a parametrisation of \(e_{0}\).
\(\displaystyle \mathbf{\Phi}_{0} = \left(\begin{matrix}0 & 15 x^{2} - 12 x + \frac{3}{2}\\15 x^{2} - 12 x + \frac{3}{2} & 0\end{matrix}\right)\)
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 \(s_{0}\) is a parametrisation of \(e_{0}\).
\(\displaystyle \mathbf{\Phi}_{0} = \left(\begin{matrix}0 & 15 x^{2} - 12 x + \frac{3}{2}\\15 x^{2} - 12 x + \frac{3}{2} & 0\end{matrix}\right)\)
This DOF is associated with edge 0 of the reference element.
\(\displaystyle l_{1}:\mathbf{V}\mapsto\displaystyle\int_{e_{0}}(s_{0} \left(2 s_{0} - 1\right))|{e_{0}}|\hat{\boldsymbol{n}}^{\text{t}}_{0}\mathbf{V}\hat{\boldsymbol{n}}_{0}\)
where \(e_{0}\) is the 0th edge;
\(\hat{\boldsymbol{n}}_{0}\) is the normal to facet 0;
and \(s_{0}\) is a parametrisation of \(e_{0}\).
\(\displaystyle \mathbf{\Phi}_{1} = \left(\begin{matrix}0 & 15 y^{2} - 12 y + \frac{3}{2}\\15 y^{2} - 12 y + \frac{3}{2} & 0\end{matrix}\right)\)
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 \(s_{0}\) is a parametrisation of \(e_{0}\).
\(\displaystyle \mathbf{\Phi}_{1} = \left(\begin{matrix}0 & 15 y^{2} - 12 y + \frac{3}{2}\\15 y^{2} - 12 y + \frac{3}{2} & 0\end{matrix}\right)\)
This DOF is associated with edge 0 of the reference element.
\(\displaystyle l_{2}:\mathbf{V}\mapsto\displaystyle\int_{e_{0}}(4 s_{0} \left(1 - s_{0}\right))|{e_{0}}|\hat{\boldsymbol{n}}^{\text{t}}_{0}\mathbf{V}\hat{\boldsymbol{n}}_{0}\)
where \(e_{0}\) is the 0th edge;
\(\hat{\boldsymbol{n}}_{0}\) is the normal to facet 0;
and \(s_{0}\) is a parametrisation of \(e_{0}\).
\(\displaystyle \mathbf{\Phi}_{2} = \left(\begin{matrix}0 & \frac{15 x^{2}}{4} + 15 x y - 6 x + \frac{15 y^{2}}{4} - 6 y + \frac{3}{2}\\\frac{15 x^{2}}{4} + 15 x y - 6 x + \frac{15 y^{2}}{4} - 6 y + \frac{3}{2} & 0\end{matrix}\right)\)
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 \(s_{0}\) is a parametrisation of \(e_{0}\).
\(\displaystyle \mathbf{\Phi}_{2} = \left(\begin{matrix}0 & \frac{15 x^{2}}{4} + 15 x y - 6 x + \frac{15 y^{2}}{4} - 6 y + \frac{3}{2}\\\frac{15 x^{2}}{4} + 15 x y - 6 x + \frac{15 y^{2}}{4} - 6 y + \frac{3}{2} & 0\end{matrix}\right)\)
This DOF is associated with edge 0 of the reference element.
\(\displaystyle l_{3}:\mathbf{V}\mapsto\displaystyle\int_{e_{1}}(2 s_{0}^{2} - 3 s_{0} + 1)|{e_{1}}|\hat{\boldsymbol{n}}^{\text{t}}_{1}\mathbf{V}\hat{\boldsymbol{n}}_{1}\)
where \(e_{1}\) is the 1st edge;
\(\hat{\boldsymbol{n}}_{1}\) is the normal to facet 1;
and \(s_{0}\) is a parametrisation of \(e_{1}\).
\(\displaystyle \mathbf{\Phi}_{3} = \left(\begin{matrix}30 x^{2} + 60 x y - 36 x + 30 y^{2} - 36 y + 9 & - 15 x^{2} - 30 x y + 18 x - 15 y^{2} + 18 y - \frac{9}{2}\\- 15 x^{2} - 30 x y + 18 x - 15 y^{2} + 18 y - \frac{9}{2} & 0\end{matrix}\right)\)
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 \(s_{0}\) is a parametrisation of \(e_{1}\).
\(\displaystyle \mathbf{\Phi}_{3} = \left(\begin{matrix}30 x^{2} + 60 x y - 36 x + 30 y^{2} - 36 y + 9 & - 15 x^{2} - 30 x y + 18 x - 15 y^{2} + 18 y - \frac{9}{2}\\- 15 x^{2} - 30 x y + 18 x - 15 y^{2} + 18 y - \frac{9}{2} & 0\end{matrix}\right)\)
This DOF is associated with edge 1 of the reference element.
\(\displaystyle l_{4}:\mathbf{V}\mapsto\displaystyle\int_{e_{1}}(s_{0} \left(2 s_{0} - 1\right))|{e_{1}}|\hat{\boldsymbol{n}}^{\text{t}}_{1}\mathbf{V}\hat{\boldsymbol{n}}_{1}\)
where \(e_{1}\) is the 1st edge;
\(\hat{\boldsymbol{n}}_{1}\) is the normal to facet 1;
and \(s_{0}\) is a parametrisation of \(e_{1}\).
\(\displaystyle \mathbf{\Phi}_{4} = \left(\begin{matrix}30 y^{2} - 24 y + 3 & - 15 y^{2} + 12 y - \frac{3}{2}\\- 15 y^{2} + 12 y - \frac{3}{2} & 0\end{matrix}\right)\)
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 \(s_{0}\) is a parametrisation of \(e_{1}\).
\(\displaystyle \mathbf{\Phi}_{4} = \left(\begin{matrix}30 y^{2} - 24 y + 3 & - 15 y^{2} + 12 y - \frac{3}{2}\\- 15 y^{2} + 12 y - \frac{3}{2} & 0\end{matrix}\right)\)
This DOF is associated with edge 1 of the reference element.
\(\displaystyle l_{5}:\mathbf{V}\mapsto\displaystyle\int_{e_{1}}(4 s_{0} \left(1 - s_{0}\right))|{e_{1}}|\hat{\boldsymbol{n}}^{\text{t}}_{1}\mathbf{V}\hat{\boldsymbol{n}}_{1}\)
where \(e_{1}\) is the 1st edge;
\(\hat{\boldsymbol{n}}_{1}\) is the normal to facet 1;
and \(s_{0}\) is a parametrisation of \(e_{1}\).
\(\displaystyle \mathbf{\Phi}_{5} = \left(\begin{matrix}\frac{15 x^{2}}{2} - 15 x y - 3 x - 15 y^{2} + 15 y - \frac{3}{2} & - \frac{15 x^{2}}{4} + \frac{15 x y}{2} + \frac{3 x}{2} + \frac{15 y^{2}}{2} - \frac{15 y}{2} + \frac{3}{4}\\- \frac{15 x^{2}}{4} + \frac{15 x y}{2} + \frac{3 x}{2} + \frac{15 y^{2}}{2} - \frac{15 y}{2} + \frac{3}{4} & 0\end{matrix}\right)\)
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 \(s_{0}\) is a parametrisation of \(e_{1}\).
\(\displaystyle \mathbf{\Phi}_{5} = \left(\begin{matrix}\frac{15 x^{2}}{2} - 15 x y - 3 x - 15 y^{2} + 15 y - \frac{3}{2} & - \frac{15 x^{2}}{4} + \frac{15 x y}{2} + \frac{3 x}{2} + \frac{15 y^{2}}{2} - \frac{15 y}{2} + \frac{3}{4}\\- \frac{15 x^{2}}{4} + \frac{15 x y}{2} + \frac{3 x}{2} + \frac{15 y^{2}}{2} - \frac{15 y}{2} + \frac{3}{4} & 0\end{matrix}\right)\)
This DOF is associated with edge 1 of the reference element.
\(\displaystyle l_{6}:\mathbf{V}\mapsto\displaystyle\int_{e_{2}}(2 s_{0}^{2} - 3 s_{0} + 1)|{e_{2}}|\hat{\boldsymbol{n}}^{\text{t}}_{2}\mathbf{V}\hat{\boldsymbol{n}}_{2}\)
where \(e_{2}\) is the 2nd edge;
\(\hat{\boldsymbol{n}}_{2}\) is the normal to facet 2;
and \(s_{0}\) is a parametrisation of \(e_{2}\).
\(\displaystyle \mathbf{\Phi}_{6} = \left(\begin{matrix}0 & - 15 x^{2} - 30 x y + 18 x - 15 y^{2} + 18 y - \frac{9}{2}\\- 15 x^{2} - 30 x y + 18 x - 15 y^{2} + 18 y - \frac{9}{2} & 30 x^{2} + 60 x y - 36 x + 30 y^{2} - 36 y + 9\end{matrix}\right)\)
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 \(s_{0}\) is a parametrisation of \(e_{2}\).
\(\displaystyle \mathbf{\Phi}_{6} = \left(\begin{matrix}0 & - 15 x^{2} - 30 x y + 18 x - 15 y^{2} + 18 y - \frac{9}{2}\\- 15 x^{2} - 30 x y + 18 x - 15 y^{2} + 18 y - \frac{9}{2} & 30 x^{2} + 60 x y - 36 x + 30 y^{2} - 36 y + 9\end{matrix}\right)\)
This DOF is associated with edge 2 of the reference element.
\(\displaystyle l_{7}:\mathbf{V}\mapsto\displaystyle\int_{e_{2}}(s_{0} \left(2 s_{0} - 1\right))|{e_{2}}|\hat{\boldsymbol{n}}^{\text{t}}_{2}\mathbf{V}\hat{\boldsymbol{n}}_{2}\)
where \(e_{2}\) is the 2nd edge;
\(\hat{\boldsymbol{n}}_{2}\) is the normal to facet 2;
and \(s_{0}\) is a parametrisation of \(e_{2}\).
\(\displaystyle \mathbf{\Phi}_{7} = \left(\begin{matrix}0 & - 15 x^{2} + 12 x - \frac{3}{2}\\- 15 x^{2} + 12 x - \frac{3}{2} & 30 x^{2} - 24 x + 3\end{matrix}\right)\)
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 \(s_{0}\) is a parametrisation of \(e_{2}\).
\(\displaystyle \mathbf{\Phi}_{7} = \left(\begin{matrix}0 & - 15 x^{2} + 12 x - \frac{3}{2}\\- 15 x^{2} + 12 x - \frac{3}{2} & 30 x^{2} - 24 x + 3\end{matrix}\right)\)
This DOF is associated with edge 2 of the reference element.
\(\displaystyle l_{8}:\mathbf{V}\mapsto\displaystyle\int_{e_{2}}(4 s_{0} \left(1 - s_{0}\right))|{e_{2}}|\hat{\boldsymbol{n}}^{\text{t}}_{2}\mathbf{V}\hat{\boldsymbol{n}}_{2}\)
where \(e_{2}\) is the 2nd edge;
\(\hat{\boldsymbol{n}}_{2}\) is the normal to facet 2;
and \(s_{0}\) is a parametrisation of \(e_{2}\).
\(\displaystyle \mathbf{\Phi}_{8} = \left(\begin{matrix}0 & \frac{15 x^{2}}{2} + \frac{15 x y}{2} - \frac{15 x}{2} - \frac{15 y^{2}}{4} + \frac{3 y}{2} + \frac{3}{4}\\\frac{15 x^{2}}{2} + \frac{15 x y}{2} - \frac{15 x}{2} - \frac{15 y^{2}}{4} + \frac{3 y}{2} + \frac{3}{4} & - 15 x^{2} - 15 x y + 15 x + \frac{15 y^{2}}{2} - 3 y - \frac{3}{2}\end{matrix}\right)\)
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 \(s_{0}\) is a parametrisation of \(e_{2}\).
\(\displaystyle \mathbf{\Phi}_{8} = \left(\begin{matrix}0 & \frac{15 x^{2}}{2} + \frac{15 x y}{2} - \frac{15 x}{2} - \frac{15 y^{2}}{4} + \frac{3 y}{2} + \frac{3}{4}\\\frac{15 x^{2}}{2} + \frac{15 x y}{2} - \frac{15 x}{2} - \frac{15 y^{2}}{4} + \frac{3 y}{2} + \frac{3}{4} & - 15 x^{2} - 15 x y + 15 x + \frac{15 y^{2}}{2} - 3 y - \frac{3}{2}\end{matrix}\right)\)
This DOF is associated with edge 2 of the reference element.
\(\displaystyle l_{9}:\mathbf{V}\mapsto\displaystyle\int_{R}\mathbf{V}:\left(\begin{array}{cc}- s_{0} - s_{1} + 1&0\\0&0\end{array}\right)\)
where \(R\) is the reference element;
and \((s_{0},s_{1})\) is a parametrisation of \(R(2)\).
\(\displaystyle \mathbf{\Phi}_{9} = \left(\begin{matrix}24 x \left(- 5 x - 5 y + 4\right) & 90 x^{2} + 120 x y - 96 x + 30 y^{2} - 48 y + 18\\90 x^{2} + 120 x y - 96 x + 30 y^{2} - 48 y + 18 & 0\end{matrix}\right)\)
This DOF is associated with face 0 of the reference element.
where \(R\) is the reference element;
and \((s_{0},s_{1})\) is a parametrisation of \(R(2)\).
\(\displaystyle \mathbf{\Phi}_{9} = \left(\begin{matrix}24 x \left(- 5 x - 5 y + 4\right) & 90 x^{2} + 120 x y - 96 x + 30 y^{2} - 48 y + 18\\90 x^{2} + 120 x y - 96 x + 30 y^{2} - 48 y + 18 & 0\end{matrix}\right)\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{10}:\mathbf{V}\mapsto\displaystyle\int_{R}\mathbf{V}:\left(\begin{array}{cc}0&- s_{0} - s_{1} + 1\\- s_{0} - s_{1} + 1&0\end{array}\right)\)
where \(R\) is the reference element;
and \((s_{0},s_{1})\) is a parametrisation of \(R(2)\).
\(\displaystyle \mathbf{\Phi}_{10} = \left(\begin{matrix}0 & 30 x^{2} + 60 x y - 48 x + 30 y^{2} - 48 y + 18\\30 x^{2} + 60 x y - 48 x + 30 y^{2} - 48 y + 18 & 0\end{matrix}\right)\)
This DOF is associated with face 0 of the reference element.
where \(R\) is the reference element;
and \((s_{0},s_{1})\) is a parametrisation of \(R(2)\).
\(\displaystyle \mathbf{\Phi}_{10} = \left(\begin{matrix}0 & 30 x^{2} + 60 x y - 48 x + 30 y^{2} - 48 y + 18\\30 x^{2} + 60 x y - 48 x + 30 y^{2} - 48 y + 18 & 0\end{matrix}\right)\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{11}:\mathbf{V}\mapsto\displaystyle\int_{R}\mathbf{V}:\left(\begin{array}{cc}0&0\\0&- s_{0} - s_{1} + 1\end{array}\right)\)
where \(R\) is the reference element;
and \((s_{0},s_{1})\) is a parametrisation of \(R(2)\).
\(\displaystyle \mathbf{\Phi}_{11} = \left(\begin{matrix}0 & 30 x^{2} + 120 x y - 48 x + 90 y^{2} - 96 y + 18\\30 x^{2} + 120 x y - 48 x + 90 y^{2} - 96 y + 18 & 24 y \left(- 5 x - 5 y + 4\right)\end{matrix}\right)\)
This DOF is associated with face 0 of the reference element.
where \(R\) is the reference element;
and \((s_{0},s_{1})\) is a parametrisation of \(R(2)\).
\(\displaystyle \mathbf{\Phi}_{11} = \left(\begin{matrix}0 & 30 x^{2} + 120 x y - 48 x + 90 y^{2} - 96 y + 18\\30 x^{2} + 120 x y - 48 x + 90 y^{2} - 96 y + 18 & 24 y \left(- 5 x - 5 y + 4\right)\end{matrix}\right)\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{12}:\mathbf{V}\mapsto\displaystyle\int_{R}\mathbf{V}:\left(\begin{array}{cc}s_{0}&0\\0&0\end{array}\right)\)
where \(R\) is the reference element;
and \((s_{0},s_{1})\) is a parametrisation of \(R(2)\).
\(\displaystyle \mathbf{\Phi}_{12} = \left(\begin{matrix}12 x \left(5 x - 2\right) & - 90 x^{2} - 60 x y + 84 x + 12 y - 12\\- 90 x^{2} - 60 x y + 84 x + 12 y - 12 & 0\end{matrix}\right)\)
This DOF is associated with face 0 of the reference element.
where \(R\) is the reference element;
and \((s_{0},s_{1})\) is a parametrisation of \(R(2)\).
\(\displaystyle \mathbf{\Phi}_{12} = \left(\begin{matrix}12 x \left(5 x - 2\right) & - 90 x^{2} - 60 x y + 84 x + 12 y - 12\\- 90 x^{2} - 60 x y + 84 x + 12 y - 12 & 0\end{matrix}\right)\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{13}:\mathbf{V}\mapsto\displaystyle\int_{R}\mathbf{V}:\left(\begin{array}{cc}0&s_{0}\\s_{0}&0\end{array}\right)\)
where \(R\) is the reference element;
and \((s_{0},s_{1})\) is a parametrisation of \(R(2)\).
\(\displaystyle \mathbf{\Phi}_{13} = \left(\begin{matrix}0 & - 60 x^{2} - 60 x y + 72 x + 12 y - 12\\- 60 x^{2} - 60 x y + 72 x + 12 y - 12 & 0\end{matrix}\right)\)
This DOF is associated with face 0 of the reference element.
where \(R\) is the reference element;
and \((s_{0},s_{1})\) is a parametrisation of \(R(2)\).
\(\displaystyle \mathbf{\Phi}_{13} = \left(\begin{matrix}0 & - 60 x^{2} - 60 x y + 72 x + 12 y - 12\\- 60 x^{2} - 60 x y + 72 x + 12 y - 12 & 0\end{matrix}\right)\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{14}:\mathbf{V}\mapsto\displaystyle\int_{R}\mathbf{V}:\left(\begin{array}{cc}0&0\\0&s_{0}\end{array}\right)\)
where \(R\) is the reference element;
and \((s_{0},s_{1})\) is a parametrisation of \(R(2)\).
\(\displaystyle \mathbf{\Phi}_{14} = \left(\begin{matrix}0 & - 60 x^{2} - 120 x y + 72 x + 24 y - 12\\- 60 x^{2} - 120 x y + 72 x + 24 y - 12 & 24 y \left(5 x - 1\right)\end{matrix}\right)\)
This DOF is associated with face 0 of the reference element.
where \(R\) is the reference element;
and \((s_{0},s_{1})\) is a parametrisation of \(R(2)\).
\(\displaystyle \mathbf{\Phi}_{14} = \left(\begin{matrix}0 & - 60 x^{2} - 120 x y + 72 x + 24 y - 12\\- 60 x^{2} - 120 x y + 72 x + 24 y - 12 & 24 y \left(5 x - 1\right)\end{matrix}\right)\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{15}:\mathbf{V}\mapsto\displaystyle\int_{R}\mathbf{V}:\left(\begin{array}{cc}s_{1}&0\\0&0\end{array}\right)\)
where \(R\) is the reference element;
and \((s_{0},s_{1})\) is a parametrisation of \(R(2)\).
\(\displaystyle \mathbf{\Phi}_{15} = \left(\begin{matrix}24 x \left(5 y - 1\right) & - 120 x y + 24 x - 60 y^{2} + 72 y - 12\\- 120 x y + 24 x - 60 y^{2} + 72 y - 12 & 0\end{matrix}\right)\)
This DOF is associated with face 0 of the reference element.
where \(R\) is the reference element;
and \((s_{0},s_{1})\) is a parametrisation of \(R(2)\).
\(\displaystyle \mathbf{\Phi}_{15} = \left(\begin{matrix}24 x \left(5 y - 1\right) & - 120 x y + 24 x - 60 y^{2} + 72 y - 12\\- 120 x y + 24 x - 60 y^{2} + 72 y - 12 & 0\end{matrix}\right)\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{16}:\mathbf{V}\mapsto\displaystyle\int_{R}\mathbf{V}:\left(\begin{array}{cc}0&s_{1}\\s_{1}&0\end{array}\right)\)
where \(R\) is the reference element;
and \((s_{0},s_{1})\) is a parametrisation of \(R(2)\).
\(\displaystyle \mathbf{\Phi}_{16} = \left(\begin{matrix}0 & - 60 x y + 12 x - 60 y^{2} + 72 y - 12\\- 60 x y + 12 x - 60 y^{2} + 72 y - 12 & 0\end{matrix}\right)\)
This DOF is associated with face 0 of the reference element.
where \(R\) is the reference element;
and \((s_{0},s_{1})\) is a parametrisation of \(R(2)\).
\(\displaystyle \mathbf{\Phi}_{16} = \left(\begin{matrix}0 & - 60 x y + 12 x - 60 y^{2} + 72 y - 12\\- 60 x y + 12 x - 60 y^{2} + 72 y - 12 & 0\end{matrix}\right)\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{17}:\mathbf{V}\mapsto\displaystyle\int_{R}\mathbf{V}:\left(\begin{array}{cc}0&0\\0&s_{1}\end{array}\right)\)
where \(R\) is the reference element;
and \((s_{0},s_{1})\) is a parametrisation of \(R(2)\).
\(\displaystyle \mathbf{\Phi}_{17} = \left(\begin{matrix}0 & - 60 x y + 12 x - 90 y^{2} + 84 y - 12\\- 60 x y + 12 x - 90 y^{2} + 84 y - 12 & 12 y \left(5 y - 2\right)\end{matrix}\right)\)
This DOF is associated with face 0 of the reference element.
where \(R\) is the reference element;
and \((s_{0},s_{1})\) is a parametrisation of \(R(2)\).
\(\displaystyle \mathbf{\Phi}_{17} = \left(\begin{matrix}0 & - 60 x y + 12 x - 90 y^{2} + 84 y - 12\\- 60 x y + 12 x - 90 y^{2} + 84 y - 12 & 12 y \left(5 y - 2\right)\end{matrix}\right)\)
This DOF is associated with face 0 of the reference element.
References
- Arnold, D. N. and Walker, S. W. The Hellan–Herrmann–Johnson method with curved elements, arΧiv: 1909.09687, 2020. [BibTeX]
DefElement stats
| Element added | 06 February 2021 |
| Element last updated | 13 June 2021 |