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Degree 1 Bernardi–Raugel on a tetrahedron
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- \(R\) is the reference tetrahedron. The following numbering of the subentities of the reference is used:
- \(\mathcal{V}\) is spanned by: \(\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 1\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle y\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle y\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle y\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle z\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle z\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle z\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle \frac{\sqrt{3} x y z}{3}\\\displaystyle \frac{\sqrt{3} x y z}{3}\\\displaystyle \frac{\sqrt{3} x y z}{3}\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle y z \left(- x - y - z + 1\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x z \left(x + y + z - 1\right)\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x y \left(- x - y - z + 1\right)\end{array}\right)\)
- \(\mathcal{L}=\{l_0,...,l_{15}\}\)
- Functionals and basis functions:
\(\displaystyle l_{0}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,0,0)\cdot\left(\begin{array}{c}1\\0\\0\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{0} = \left(\begin{array}{c}\displaystyle 20 x y z - x + 20 y^{2} z + 20 y z^{2} - 20 y z - y - z + 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{0} = \left(\begin{array}{c}\displaystyle 20 x y z - x + 20 y^{2} z + 20 y z^{2} - 20 y z - y - z + 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle l_{1}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,0,0)\cdot\left(\begin{array}{c}0\\1\\0\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{1} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 20 x^{2} z + 20 x y z + 20 x z^{2} - 20 x z - x - y - z + 1\\\displaystyle 0\end{array}\right)\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{1} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 20 x^{2} z + 20 x y z + 20 x z^{2} - 20 x z - x - y - z + 1\\\displaystyle 0\end{array}\right)\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle l_{2}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,0,0)\cdot\left(\begin{array}{c}0\\0\\1\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{2} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 20 x^{2} y + 20 x y^{2} + 20 x y z - 20 x y - x - y - z + 1\end{array}\right)\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{2} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 20 x^{2} y + 20 x y^{2} + 20 x y z - 20 x y - x - y - z + 1\end{array}\right)\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle l_{3}:\boldsymbol{v}\mapsto\boldsymbol{v}(1,0,0)\cdot\left(\begin{array}{c}1\\0\\0\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{3} = \left(\begin{array}{c}\displaystyle \frac{x \left(- 20 y z + 3\right)}{3}\\\displaystyle - \frac{20 x y z}{3}\\\displaystyle - \frac{20 x y z}{3}\end{array}\right)\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{3} = \left(\begin{array}{c}\displaystyle \frac{x \left(- 20 y z + 3\right)}{3}\\\displaystyle - \frac{20 x y z}{3}\\\displaystyle - \frac{20 x y z}{3}\end{array}\right)\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle l_{4}:\boldsymbol{v}\mapsto\boldsymbol{v}(1,0,0)\cdot\left(\begin{array}{c}0\\1\\0\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{4} = \left(\begin{array}{c}\displaystyle - \frac{20 x y z}{3}\\\displaystyle \frac{x \left(60 x z + 40 y z + 60 z^{2} - 60 z + 3\right)}{3}\\\displaystyle - \frac{20 x y z}{3}\end{array}\right)\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{4} = \left(\begin{array}{c}\displaystyle - \frac{20 x y z}{3}\\\displaystyle \frac{x \left(60 x z + 40 y z + 60 z^{2} - 60 z + 3\right)}{3}\\\displaystyle - \frac{20 x y z}{3}\end{array}\right)\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle l_{5}:\boldsymbol{v}\mapsto\boldsymbol{v}(1,0,0)\cdot\left(\begin{array}{c}0\\0\\1\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{5} = \left(\begin{array}{c}\displaystyle - \frac{20 x y z}{3}\\\displaystyle - \frac{20 x y z}{3}\\\displaystyle \frac{x \left(60 x y + 60 y^{2} + 40 y z - 60 y + 3\right)}{3}\end{array}\right)\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{5} = \left(\begin{array}{c}\displaystyle - \frac{20 x y z}{3}\\\displaystyle - \frac{20 x y z}{3}\\\displaystyle \frac{x \left(60 x y + 60 y^{2} + 40 y z - 60 y + 3\right)}{3}\end{array}\right)\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle l_{6}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,1,0)\cdot\left(\begin{array}{c}1\\0\\0\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{6} = \left(\begin{array}{c}\displaystyle \frac{y \left(40 x z + 60 y z + 60 z^{2} - 60 z + 3\right)}{3}\\\displaystyle - \frac{20 x y z}{3}\\\displaystyle - \frac{20 x y z}{3}\end{array}\right)\)
This DOF is associated with vertex 2 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{6} = \left(\begin{array}{c}\displaystyle \frac{y \left(40 x z + 60 y z + 60 z^{2} - 60 z + 3\right)}{3}\\\displaystyle - \frac{20 x y z}{3}\\\displaystyle - \frac{20 x y z}{3}\end{array}\right)\)
This DOF is associated with vertex 2 of the reference element.
\(\displaystyle l_{7}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,1,0)\cdot\left(\begin{array}{c}0\\1\\0\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{7} = \left(\begin{array}{c}\displaystyle - \frac{20 x y z}{3}\\\displaystyle \frac{y \left(- 20 x z + 3\right)}{3}\\\displaystyle - \frac{20 x y z}{3}\end{array}\right)\)
This DOF is associated with vertex 2 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{7} = \left(\begin{array}{c}\displaystyle - \frac{20 x y z}{3}\\\displaystyle \frac{y \left(- 20 x z + 3\right)}{3}\\\displaystyle - \frac{20 x y z}{3}\end{array}\right)\)
This DOF is associated with vertex 2 of the reference element.
\(\displaystyle l_{8}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,1,0)\cdot\left(\begin{array}{c}0\\0\\1\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{8} = \left(\begin{array}{c}\displaystyle - \frac{20 x y z}{3}\\\displaystyle - \frac{20 x y z}{3}\\\displaystyle \frac{y \left(60 x^{2} + 60 x y + 40 x z - 60 x + 3\right)}{3}\end{array}\right)\)
This DOF is associated with vertex 2 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{8} = \left(\begin{array}{c}\displaystyle - \frac{20 x y z}{3}\\\displaystyle - \frac{20 x y z}{3}\\\displaystyle \frac{y \left(60 x^{2} + 60 x y + 40 x z - 60 x + 3\right)}{3}\end{array}\right)\)
This DOF is associated with vertex 2 of the reference element.
\(\displaystyle l_{9}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,0,1)\cdot\left(\begin{array}{c}1\\0\\0\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{9} = \left(\begin{array}{c}\displaystyle \frac{z \left(40 x y + 60 y^{2} + 60 y z - 60 y + 3\right)}{3}\\\displaystyle - \frac{20 x y z}{3}\\\displaystyle - \frac{20 x y z}{3}\end{array}\right)\)
This DOF is associated with vertex 3 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{9} = \left(\begin{array}{c}\displaystyle \frac{z \left(40 x y + 60 y^{2} + 60 y z - 60 y + 3\right)}{3}\\\displaystyle - \frac{20 x y z}{3}\\\displaystyle - \frac{20 x y z}{3}\end{array}\right)\)
This DOF is associated with vertex 3 of the reference element.
\(\displaystyle l_{10}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,0,1)\cdot\left(\begin{array}{c}0\\1\\0\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{10} = \left(\begin{array}{c}\displaystyle - \frac{20 x y z}{3}\\\displaystyle \frac{z \left(60 x^{2} + 40 x y + 60 x z - 60 x + 3\right)}{3}\\\displaystyle - \frac{20 x y z}{3}\end{array}\right)\)
This DOF is associated with vertex 3 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{10} = \left(\begin{array}{c}\displaystyle - \frac{20 x y z}{3}\\\displaystyle \frac{z \left(60 x^{2} + 40 x y + 60 x z - 60 x + 3\right)}{3}\\\displaystyle - \frac{20 x y z}{3}\end{array}\right)\)
This DOF is associated with vertex 3 of the reference element.
\(\displaystyle l_{11}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,0,1)\cdot\left(\begin{array}{c}0\\0\\1\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{11} = \left(\begin{array}{c}\displaystyle - \frac{20 x y z}{3}\\\displaystyle - \frac{20 x y z}{3}\\\displaystyle \frac{z \left(- 20 x y + 3\right)}{3}\end{array}\right)\)
This DOF is associated with vertex 3 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{11} = \left(\begin{array}{c}\displaystyle - \frac{20 x y z}{3}\\\displaystyle - \frac{20 x y z}{3}\\\displaystyle \frac{z \left(- 20 x y + 3\right)}{3}\end{array}\right)\)
This DOF is associated with vertex 3 of the reference element.
\(\displaystyle l_{12}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{0}}\boldsymbol{v}\cdot(1)\hat{\boldsymbol{n}}_{0}\)
where \(f_{0}\) is the 0th face;
and \(\hat{\boldsymbol{n}}_{0}\) is the normal to facet 0.
\(\displaystyle \boldsymbol{\phi}_{12} = \left(\begin{array}{c}\displaystyle 40 x y z\\\displaystyle 40 x y z\\\displaystyle 40 x y z\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
where \(f_{0}\) is the 0th face;
and \(\hat{\boldsymbol{n}}_{0}\) is the normal to facet 0.
\(\displaystyle \boldsymbol{\phi}_{12} = \left(\begin{array}{c}\displaystyle 40 x y z\\\displaystyle 40 x y z\\\displaystyle 40 x y z\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{13}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{1}}\boldsymbol{v}\cdot(1)\hat{\boldsymbol{n}}_{1}\)
where \(f_{1}\) is the 1st face;
and \(\hat{\boldsymbol{n}}_{1}\) is the normal to facet 1.
\(\displaystyle \boldsymbol{\phi}_{13} = \left(\begin{array}{c}\displaystyle 120 y z \left(- x - y - z + 1\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)
This DOF is associated with face 1 of the reference element.
where \(f_{1}\) is the 1st face;
and \(\hat{\boldsymbol{n}}_{1}\) is the normal to facet 1.
\(\displaystyle \boldsymbol{\phi}_{13} = \left(\begin{array}{c}\displaystyle 120 y z \left(- x - y - z + 1\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)
This DOF is associated with face 1 of the reference element.
\(\displaystyle l_{14}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{2}}\boldsymbol{v}\cdot(1)\hat{\boldsymbol{n}}_{2}\)
where \(f_{2}\) is the 2nd face;
and \(\hat{\boldsymbol{n}}_{2}\) is the normal to facet 2.
\(\displaystyle \boldsymbol{\phi}_{14} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 120 x z \left(x + y + z - 1\right)\\\displaystyle 0\end{array}\right)\)
This DOF is associated with face 2 of the reference element.
where \(f_{2}\) is the 2nd face;
and \(\hat{\boldsymbol{n}}_{2}\) is the normal to facet 2.
\(\displaystyle \boldsymbol{\phi}_{14} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 120 x z \left(x + y + z - 1\right)\\\displaystyle 0\end{array}\right)\)
This DOF is associated with face 2 of the reference element.
\(\displaystyle l_{15}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{3}}\boldsymbol{v}\cdot(1)\hat{\boldsymbol{n}}_{3}\)
where \(f_{3}\) is the 3rd face;
and \(\hat{\boldsymbol{n}}_{3}\) is the normal to facet 3.
\(\displaystyle \boldsymbol{\phi}_{15} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 120 x y \left(- x - y - z + 1\right)\end{array}\right)\)
This DOF is associated with face 3 of the reference element.
where \(f_{3}\) is the 3rd face;
and \(\hat{\boldsymbol{n}}_{3}\) is the normal to facet 3.
\(\displaystyle \boldsymbol{\phi}_{15} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 120 x y \left(- x - y - z + 1\right)\end{array}\right)\)
This DOF is associated with face 3 of the reference element.