an encyclopedia of finite element definitions
Degree 1 Tiniest tensor on a hexahedron
◀ Back to Tiniest tensor definition page
- \(R\) is the reference hexahedron. The following numbering of the subentities of the reference is used:
- \(\mathcal{V}\) is spanned by: \(1\), \(z\), \(y\), \(y z\), \(x\), \(x z\), \(x y\), \(x y z\), \(\frac{3 x \left(x y z - x y - x z + x - y z + y + z - 1\right)}{2}\), \(\frac{3 x z \left(- x y + x + y - 1\right)}{2}\), \(\frac{3 x y \left(- x z + x + z - 1\right)}{2}\), \(\frac{3 x y z \left(x - 1\right)}{2}\), \(\frac{3 y \left(x y z - x y - x z + x - y z + y + z - 1\right)}{2}\), \(\frac{3 y z \left(- x y + x + y - 1\right)}{2}\), \(\frac{3 x y \left(- y z + y + z - 1\right)}{2}\), \(\frac{3 x y z \left(y - 1\right)}{2}\), \(\frac{3 z \left(x y z - x y - x z + x - y z + y + z - 1\right)}{2}\), \(\frac{3 y z \left(- x z + x + z - 1\right)}{2}\), \(\frac{3 x z \left(- y z + y + z - 1\right)}{2}\), \(\frac{3 x y z \left(z - 1\right)}{2}\)
- \(\mathcal{L}=\{l_0,...,l_{19}\}\)
- Functionals and basis functions:
\(\displaystyle l_{0}:v\mapsto v(0,0,0)\)
\(\displaystyle \phi_{0} = 3 x^{2} y z - 3 x^{2} y - 3 x^{2} z + 3 x^{2} + 3 x y^{2} z - 3 x y^{2} + 3 x y z^{2} - 10 x y z + 7 x y - 3 x z^{2} + 7 x z - 4 x - 3 y^{2} z + 3 y^{2} - 3 y z^{2} + 7 y z - 4 y + 3 z^{2} - 4 z + 1\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle \phi_{0} = 3 x^{2} y z - 3 x^{2} y - 3 x^{2} z + 3 x^{2} + 3 x y^{2} z - 3 x y^{2} + 3 x y z^{2} - 10 x y z + 7 x y - 3 x z^{2} + 7 x z - 4 x - 3 y^{2} z + 3 y^{2} - 3 y z^{2} + 7 y z - 4 y + 3 z^{2} - 4 z + 1\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle l_{1}:v\mapsto v(1,0,0)\)
\(\displaystyle \phi_{1} = x \left(3 x y z - 3 x y - 3 x z + 3 x - 3 y^{2} z + 3 y^{2} - 3 y z^{2} + 4 y z - y + 3 z^{2} - z - 2\right)\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle \phi_{1} = x \left(3 x y z - 3 x y - 3 x z + 3 x - 3 y^{2} z + 3 y^{2} - 3 y z^{2} + 4 y z - y + 3 z^{2} - z - 2\right)\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle l_{2}:v\mapsto v(0,1,0)\)
\(\displaystyle \phi_{2} = y \left(- 3 x^{2} z + 3 x^{2} + 3 x y z - 3 x y - 3 x z^{2} + 4 x z - x - 3 y z + 3 y + 3 z^{2} - z - 2\right)\)
This DOF is associated with vertex 2 of the reference element.
\(\displaystyle \phi_{2} = y \left(- 3 x^{2} z + 3 x^{2} + 3 x y z - 3 x y - 3 x z^{2} + 4 x z - x - 3 y z + 3 y + 3 z^{2} - z - 2\right)\)
This DOF is associated with vertex 2 of the reference element.
\(\displaystyle l_{3}:v\mapsto v(1,1,0)\)
\(\displaystyle \phi_{3} = x y \left(- 3 x z + 3 x - 3 y z + 3 y + 3 z^{2} + 2 z - 5\right)\)
This DOF is associated with vertex 3 of the reference element.
\(\displaystyle \phi_{3} = x y \left(- 3 x z + 3 x - 3 y z + 3 y + 3 z^{2} + 2 z - 5\right)\)
This DOF is associated with vertex 3 of the reference element.
\(\displaystyle l_{4}:v\mapsto v(0,0,1)\)
\(\displaystyle \phi_{4} = z \left(- 3 x^{2} y + 3 x^{2} - 3 x y^{2} + 3 x y z + 4 x y - 3 x z - x + 3 y^{2} - 3 y z - y + 3 z - 2\right)\)
This DOF is associated with vertex 4 of the reference element.
\(\displaystyle \phi_{4} = z \left(- 3 x^{2} y + 3 x^{2} - 3 x y^{2} + 3 x y z + 4 x y - 3 x z - x + 3 y^{2} - 3 y z - y + 3 z - 2\right)\)
This DOF is associated with vertex 4 of the reference element.
\(\displaystyle l_{5}:v\mapsto v(1,0,1)\)
\(\displaystyle \phi_{5} = x z \left(- 3 x y + 3 x + 3 y^{2} - 3 y z + 2 y + 3 z - 5\right)\)
This DOF is associated with vertex 5 of the reference element.
\(\displaystyle \phi_{5} = x z \left(- 3 x y + 3 x + 3 y^{2} - 3 y z + 2 y + 3 z - 5\right)\)
This DOF is associated with vertex 5 of the reference element.
\(\displaystyle l_{6}:v\mapsto v(0,1,1)\)
\(\displaystyle \phi_{6} = y z \left(3 x^{2} - 3 x y - 3 x z + 2 x + 3 y + 3 z - 5\right)\)
This DOF is associated with vertex 6 of the reference element.
\(\displaystyle \phi_{6} = y z \left(3 x^{2} - 3 x y - 3 x z + 2 x + 3 y + 3 z - 5\right)\)
This DOF is associated with vertex 6 of the reference element.
\(\displaystyle l_{7}:v\mapsto v(1,1,1)\)
\(\displaystyle \phi_{7} = x y z \left(3 x + 3 y + 3 z - 8\right)\)
This DOF is associated with vertex 7 of the reference element.
\(\displaystyle \phi_{7} = x y z \left(3 x + 3 y + 3 z - 8\right)\)
This DOF is associated with vertex 7 of the reference element.
\(\displaystyle l_{8}:\mathbf{v}\mapsto\displaystyle\int_{e_{0}}v\)
where \(e_{0}\) is the 0th edge.
\(\displaystyle \phi_{8} = 6 x \left(- x y z + x y + x z - x + y z - y - z + 1\right)\)
This DOF is associated with edge 0 of the reference element.
where \(e_{0}\) is the 0th edge.
\(\displaystyle \phi_{8} = 6 x \left(- x y z + x y + x z - x + y z - y - z + 1\right)\)
This DOF is associated with edge 0 of the reference element.
\(\displaystyle l_{9}:\mathbf{v}\mapsto\displaystyle\int_{e_{1}}v\)
where \(e_{1}\) is the 1st edge.
\(\displaystyle \phi_{9} = 6 y \left(- x y z + x y + x z - x + y z - y - z + 1\right)\)
This DOF is associated with edge 1 of the reference element.
where \(e_{1}\) is the 1st edge.
\(\displaystyle \phi_{9} = 6 y \left(- x y z + x y + x z - x + y z - y - z + 1\right)\)
This DOF is associated with edge 1 of the reference element.
\(\displaystyle l_{10}:\mathbf{v}\mapsto\displaystyle\int_{e_{2}}v\)
where \(e_{2}\) is the 2nd edge.
\(\displaystyle \phi_{10} = 6 z \left(- x y z + x y + x z - x + y z - y - z + 1\right)\)
This DOF is associated with edge 2 of the reference element.
where \(e_{2}\) is the 2nd edge.
\(\displaystyle \phi_{10} = 6 z \left(- x y z + x y + x z - x + y z - y - z + 1\right)\)
This DOF is associated with edge 2 of the reference element.
\(\displaystyle l_{11}:\mathbf{v}\mapsto\displaystyle\int_{e_{3}}v\)
where \(e_{3}\) is the 3rd edge.
\(\displaystyle \phi_{11} = 6 x y \left(y z - y - z + 1\right)\)
This DOF is associated with edge 3 of the reference element.
where \(e_{3}\) is the 3rd edge.
\(\displaystyle \phi_{11} = 6 x y \left(y z - y - z + 1\right)\)
This DOF is associated with edge 3 of the reference element.
\(\displaystyle l_{12}:\mathbf{v}\mapsto\displaystyle\int_{e_{4}}v\)
where \(e_{4}\) is the 4th edge.
\(\displaystyle \phi_{12} = 6 x z \left(y z - y - z + 1\right)\)
This DOF is associated with edge 4 of the reference element.
where \(e_{4}\) is the 4th edge.
\(\displaystyle \phi_{12} = 6 x z \left(y z - y - z + 1\right)\)
This DOF is associated with edge 4 of the reference element.
\(\displaystyle l_{13}:\mathbf{v}\mapsto\displaystyle\int_{e_{5}}v\)
where \(e_{5}\) is the 5th edge.
\(\displaystyle \phi_{13} = 6 x y \left(x z - x - z + 1\right)\)
This DOF is associated with edge 5 of the reference element.
where \(e_{5}\) is the 5th edge.
\(\displaystyle \phi_{13} = 6 x y \left(x z - x - z + 1\right)\)
This DOF is associated with edge 5 of the reference element.
\(\displaystyle l_{14}:\mathbf{v}\mapsto\displaystyle\int_{e_{6}}v\)
where \(e_{6}\) is the 6th edge.
\(\displaystyle \phi_{14} = 6 y z \left(x z - x - z + 1\right)\)
This DOF is associated with edge 6 of the reference element.
where \(e_{6}\) is the 6th edge.
\(\displaystyle \phi_{14} = 6 y z \left(x z - x - z + 1\right)\)
This DOF is associated with edge 6 of the reference element.
\(\displaystyle l_{15}:\mathbf{v}\mapsto\displaystyle\int_{e_{7}}v\)
where \(e_{7}\) is the 7th edge.
\(\displaystyle \phi_{15} = 6 x y z \left(1 - z\right)\)
This DOF is associated with edge 7 of the reference element.
where \(e_{7}\) is the 7th edge.
\(\displaystyle \phi_{15} = 6 x y z \left(1 - z\right)\)
This DOF is associated with edge 7 of the reference element.
\(\displaystyle l_{16}:\mathbf{v}\mapsto\displaystyle\int_{e_{8}}v\)
where \(e_{8}\) is the 8th edge.
\(\displaystyle \phi_{16} = 6 x z \left(x y - x - y + 1\right)\)
This DOF is associated with edge 8 of the reference element.
where \(e_{8}\) is the 8th edge.
\(\displaystyle \phi_{16} = 6 x z \left(x y - x - y + 1\right)\)
This DOF is associated with edge 8 of the reference element.
\(\displaystyle l_{17}:\mathbf{v}\mapsto\displaystyle\int_{e_{9}}v\)
where \(e_{9}\) is the 9th edge.
\(\displaystyle \phi_{17} = 6 y z \left(x y - x - y + 1\right)\)
This DOF is associated with edge 9 of the reference element.
where \(e_{9}\) is the 9th edge.
\(\displaystyle \phi_{17} = 6 y z \left(x y - x - y + 1\right)\)
This DOF is associated with edge 9 of the reference element.
\(\displaystyle l_{18}:\mathbf{v}\mapsto\displaystyle\int_{e_{10}}v\)
where \(e_{10}\) is the 10th edge.
\(\displaystyle \phi_{18} = 6 x y z \left(1 - y\right)\)
This DOF is associated with edge 10 of the reference element.
where \(e_{10}\) is the 10th edge.
\(\displaystyle \phi_{18} = 6 x y z \left(1 - y\right)\)
This DOF is associated with edge 10 of the reference element.
\(\displaystyle l_{19}:\mathbf{v}\mapsto\displaystyle\int_{e_{11}}v\)
where \(e_{11}\) is the 11th edge.
\(\displaystyle \phi_{19} = 6 x y z \left(1 - x\right)\)
This DOF is associated with edge 11 of the reference element.
where \(e_{11}\) is the 11th edge.
\(\displaystyle \phi_{19} = 6 x y z \left(1 - x\right)\)
This DOF is associated with edge 11 of the reference element.