an encyclopedia of finite element definitions
Degree 2 Lagrange on a pyramid
◀ Back to Lagrange definition page
- \(R\) is the reference pyramid. The following numbering of the subentities of the reference is used:
- \(\mathcal{V}\) is spanned by: \(1\), \(x\), \(x^{2}\), \(y\), \(x y\), \(y^{2}\), \(z\), \(x z\), \(y z\), \(z^{2}\), \(\frac{x^{2} y^{2}}{z^{2} - 2 z + 1}\), \(- \frac{x y z}{z - 1}\), \(- \frac{x y^{2}}{z - 1}\), \(- \frac{x^{2} y}{z - 1}\)
- \(\mathcal{L}=\{l_0,...,l_{13}\}\)
- Functionals and basis functions:
\(\displaystyle l_{0}:v\mapsto v(0,0,0)\)
\(\displaystyle \phi_{0} = \frac{4 x^{2} y^{2} \left(z - 1\right) + x y \left(6 x + 6 y + z\right) \left(z^{2} - 2 z + 1\right) + \left(z - 1\right) \left(z^{2} - 2 z + 1\right) \left(2 x^{2} + 9 x y + 4 x z - 3 x + 2 y^{2} + 4 y z - 3 y + 2 z^{2} - 3 z + 1\right)}{\left(z - 1\right) \left(z^{2} - 2 z + 1\right)}\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle \phi_{0} = \frac{4 x^{2} y^{2} \left(z - 1\right) + x y \left(6 x + 6 y + z\right) \left(z^{2} - 2 z + 1\right) + \left(z - 1\right) \left(z^{2} - 2 z + 1\right) \left(2 x^{2} + 9 x y + 4 x z - 3 x + 2 y^{2} + 4 y z - 3 y + 2 z^{2} - 3 z + 1\right)}{\left(z - 1\right) \left(z^{2} - 2 z + 1\right)}\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle l_{1}:v\mapsto v(1,0,0)\)
\(\displaystyle \phi_{1} = \frac{x \left(4 x y^{2} \left(z - 1\right) + y \left(6 x + 2 y - z\right) \left(z^{2} - 2 z + 1\right) + \left(z - 1\right) \left(2 x + 3 y - 1\right) \left(z^{2} - 2 z + 1\right)\right)}{\left(z - 1\right) \left(z^{2} - 2 z + 1\right)}\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle \phi_{1} = \frac{x \left(4 x y^{2} \left(z - 1\right) + y \left(6 x + 2 y - z\right) \left(z^{2} - 2 z + 1\right) + \left(z - 1\right) \left(2 x + 3 y - 1\right) \left(z^{2} - 2 z + 1\right)\right)}{\left(z - 1\right) \left(z^{2} - 2 z + 1\right)}\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle l_{2}:v\mapsto v(0,1,0)\)
\(\displaystyle \phi_{2} = \frac{y \left(4 x^{2} y \left(z - 1\right) + x \left(2 x + 6 y - z\right) \left(z^{2} - 2 z + 1\right) + \left(z - 1\right) \left(3 x + 2 y - 1\right) \left(z^{2} - 2 z + 1\right)\right)}{\left(z - 1\right) \left(z^{2} - 2 z + 1\right)}\)
This DOF is associated with vertex 2 of the reference element.
\(\displaystyle \phi_{2} = \frac{y \left(4 x^{2} y \left(z - 1\right) + x \left(2 x + 6 y - z\right) \left(z^{2} - 2 z + 1\right) + \left(z - 1\right) \left(3 x + 2 y - 1\right) \left(z^{2} - 2 z + 1\right)\right)}{\left(z - 1\right) \left(z^{2} - 2 z + 1\right)}\)
This DOF is associated with vertex 2 of the reference element.
\(\displaystyle l_{3}:v\mapsto v(1,1,0)\)
\(\displaystyle \phi_{3} = \frac{x y \left(4 x y + 2 x z - 2 x + 2 y z - 2 y + 2 z^{2} - 3 z + 1\right)}{z^{2} - 2 z + 1}\)
This DOF is associated with vertex 3 of the reference element.
\(\displaystyle \phi_{3} = \frac{x y \left(4 x y + 2 x z - 2 x + 2 y z - 2 y + 2 z^{2} - 3 z + 1\right)}{z^{2} - 2 z + 1}\)
This DOF is associated with vertex 3 of the reference element.
\(\displaystyle l_{4}:v\mapsto v(0,0,1)\)
\(\displaystyle \phi_{4} = z \left(2 z - 1\right)\)
This DOF is associated with vertex 4 of the reference element.
\(\displaystyle \phi_{4} = z \left(2 z - 1\right)\)
This DOF is associated with vertex 4 of the reference element.
\(\displaystyle l_{5}:v\mapsto v(\tfrac{1}{2},0,0)\)
\(\displaystyle \phi_{5} = \frac{4 x \left(2 x y^{2} \cdot \left(1 - z\right) - y \left(3 x + 2 y\right) \left(z^{2} - 2 z + 1\right) + \left(z - 1\right) \left(z^{2} - 2 z + 1\right) \left(- x - 3 y - z + 1\right)\right)}{\left(z - 1\right) \left(z^{2} - 2 z + 1\right)}\)
This DOF is associated with edge 0 of the reference element.
\(\displaystyle \phi_{5} = \frac{4 x \left(2 x y^{2} \cdot \left(1 - z\right) - y \left(3 x + 2 y\right) \left(z^{2} - 2 z + 1\right) + \left(z - 1\right) \left(z^{2} - 2 z + 1\right) \left(- x - 3 y - z + 1\right)\right)}{\left(z - 1\right) \left(z^{2} - 2 z + 1\right)}\)
This DOF is associated with edge 0 of the reference element.
\(\displaystyle l_{6}:v\mapsto v(0,\tfrac{1}{2},0)\)
\(\displaystyle \phi_{6} = \frac{4 y \left(2 x^{2} y \left(1 - z\right) - x \left(2 x + 3 y\right) \left(z^{2} - 2 z + 1\right) + \left(z - 1\right) \left(z^{2} - 2 z + 1\right) \left(- 3 x - y - z + 1\right)\right)}{\left(z - 1\right) \left(z^{2} - 2 z + 1\right)}\)
This DOF is associated with edge 1 of the reference element.
\(\displaystyle \phi_{6} = \frac{4 y \left(2 x^{2} y \left(1 - z\right) - x \left(2 x + 3 y\right) \left(z^{2} - 2 z + 1\right) + \left(z - 1\right) \left(z^{2} - 2 z + 1\right) \left(- 3 x - y - z + 1\right)\right)}{\left(z - 1\right) \left(z^{2} - 2 z + 1\right)}\)
This DOF is associated with edge 1 of the reference element.
\(\displaystyle l_{7}:v\mapsto v(0,0,\tfrac{1}{2})\)
\(\displaystyle \phi_{7} = \frac{4 z \left(- x y + \left(z - 1\right) \left(- x - y - z + 1\right)\right)}{z - 1}\)
This DOF is associated with edge 2 of the reference element.
\(\displaystyle \phi_{7} = \frac{4 z \left(- x y + \left(z - 1\right) \left(- x - y - z + 1\right)\right)}{z - 1}\)
This DOF is associated with edge 2 of the reference element.
\(\displaystyle l_{8}:v\mapsto v(1,\tfrac{1}{2},0)\)
\(\displaystyle \phi_{8} = \frac{4 x y \left(- 2 x y - 2 x z + 2 x - y z + y - z^{2} + 2 z - 1\right)}{z^{2} - 2 z + 1}\)
This DOF is associated with edge 3 of the reference element.
\(\displaystyle \phi_{8} = \frac{4 x y \left(- 2 x y - 2 x z + 2 x - y z + y - z^{2} + 2 z - 1\right)}{z^{2} - 2 z + 1}\)
This DOF is associated with edge 3 of the reference element.
\(\displaystyle l_{9}:v\mapsto v(\tfrac{1}{2},0,\tfrac{1}{2})\)
\(\displaystyle \phi_{9} = \frac{4 x z \left(y + z - 1\right)}{z - 1}\)
This DOF is associated with edge 4 of the reference element.
\(\displaystyle \phi_{9} = \frac{4 x z \left(y + z - 1\right)}{z - 1}\)
This DOF is associated with edge 4 of the reference element.
\(\displaystyle l_{10}:v\mapsto v(\tfrac{1}{2},1,0)\)
\(\displaystyle \phi_{10} = \frac{4 x y \left(- 2 x y - x z + x - 2 y z + 2 y - z^{2} + 2 z - 1\right)}{z^{2} - 2 z + 1}\)
This DOF is associated with edge 5 of the reference element.
\(\displaystyle \phi_{10} = \frac{4 x y \left(- 2 x y - x z + x - 2 y z + 2 y - z^{2} + 2 z - 1\right)}{z^{2} - 2 z + 1}\)
This DOF is associated with edge 5 of the reference element.
\(\displaystyle l_{11}:v\mapsto v(0,\tfrac{1}{2},\tfrac{1}{2})\)
\(\displaystyle \phi_{11} = \frac{4 y z \left(x + z - 1\right)}{z - 1}\)
This DOF is associated with edge 6 of the reference element.
\(\displaystyle \phi_{11} = \frac{4 y z \left(x + z - 1\right)}{z - 1}\)
This DOF is associated with edge 6 of the reference element.
\(\displaystyle l_{12}:v\mapsto v(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})\)
\(\displaystyle \phi_{12} = - \frac{4 x y z}{z - 1}\)
This DOF is associated with edge 7 of the reference element.
\(\displaystyle \phi_{12} = - \frac{4 x y z}{z - 1}\)
This DOF is associated with edge 7 of the reference element.
\(\displaystyle l_{13}:v\mapsto v(\tfrac{1}{2},\tfrac{1}{2},0)\)
\(\displaystyle \phi_{13} = \frac{16 x y \left(x y + x z - x + y z - y + z^{2} - 2 z + 1\right)}{z^{2} - 2 z + 1}\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle \phi_{13} = \frac{16 x y \left(x y + x z - x + y z - y + z^{2} - 2 z + 1\right)}{z^{2} - 2 z + 1}\)
This DOF is associated with face 0 of the reference element.