an encyclopedia of finite element definitions

an encyclopedia of finite element definitions

- \(R\) is the reference hexahedron. The following numbering of the subentities of the reference is used:
- \(\mathcal{V}\) is spanned by: \(\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 1\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x\\\displaystyle y\\\displaystyle z\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 2 x\\\displaystyle - y\\\displaystyle - z\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x\\\displaystyle - 2 y\\\displaystyle z\end{array}\right)\)
- \(\mathcal{L}=\{l_0,...,l_{5}\}\)
- Functionals and basis functions:

\(\displaystyle l_{0}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{0}}\boldsymbol{v}\cdot\hat{\boldsymbol{n}}_{0}\)

where \(f_{0}\) is the 0th face;

and \(\hat{\boldsymbol{n}}_{0}\) is the normal to facet 0.

\(\displaystyle \boldsymbol{\phi}_{0} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 1 - z\end{array}\right)\)

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.

\(\displaystyle \boldsymbol{\phi}_{0} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 1 - z\end{array}\right)\)

This DOF is associated with face 0 of the reference element.

\(\displaystyle l_{1}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{1}}\boldsymbol{v}\cdot\hat{\boldsymbol{n}}_{1}\)

where \(f_{1}\) is the 1st face;

and \(\hat{\boldsymbol{n}}_{1}\) is the normal to facet 1.

\(\displaystyle \boldsymbol{\phi}_{1} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle y - 1\\\displaystyle 0\end{array}\right)\)

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.

\(\displaystyle \boldsymbol{\phi}_{1} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle y - 1\\\displaystyle 0\end{array}\right)\)

This DOF is associated with face 1 of the reference element.

\(\displaystyle l_{2}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{2}}\boldsymbol{v}\cdot\hat{\boldsymbol{n}}_{2}\)

where \(f_{2}\) is the 2nd face;

and \(\hat{\boldsymbol{n}}_{2}\) is the normal to facet 2.

\(\displaystyle \boldsymbol{\phi}_{2} = \left(\begin{array}{c}\displaystyle 1 - x\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)

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.

\(\displaystyle \boldsymbol{\phi}_{2} = \left(\begin{array}{c}\displaystyle 1 - x\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)

This DOF is associated with face 2 of the reference element.

\(\displaystyle l_{3}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{3}}\boldsymbol{v}\cdot\hat{\boldsymbol{n}}_{3}\)

where \(f_{3}\) is the 3th face;

and \(\hat{\boldsymbol{n}}_{3}\) is the normal to facet 3.

\(\displaystyle \boldsymbol{\phi}_{3} = \left(\begin{array}{c}\displaystyle x\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)

This DOF is associated with face 3 of the reference element.

where \(f_{3}\) is the 3th face;

and \(\hat{\boldsymbol{n}}_{3}\) is the normal to facet 3.

\(\displaystyle \boldsymbol{\phi}_{3} = \left(\begin{array}{c}\displaystyle x\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)

This DOF is associated with face 3 of the reference element.

\(\displaystyle l_{4}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{4}}\boldsymbol{v}\cdot\hat{\boldsymbol{n}}_{4}\)

where \(f_{4}\) is the 4th face;

and \(\hat{\boldsymbol{n}}_{4}\) is the normal to facet 4.

\(\displaystyle \boldsymbol{\phi}_{4} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle - y\\\displaystyle 0\end{array}\right)\)

This DOF is associated with face 4 of the reference element.

where \(f_{4}\) is the 4th face;

and \(\hat{\boldsymbol{n}}_{4}\) is the normal to facet 4.

\(\displaystyle \boldsymbol{\phi}_{4} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle - y\\\displaystyle 0\end{array}\right)\)

This DOF is associated with face 4 of the reference element.

\(\displaystyle l_{5}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{5}}\boldsymbol{v}\cdot\hat{\boldsymbol{n}}_{5}\)

where \(f_{5}\) is the 5th face;

and \(\hat{\boldsymbol{n}}_{5}\) is the normal to facet 5.

\(\displaystyle \boldsymbol{\phi}_{5} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle z\end{array}\right)\)

This DOF is associated with face 5 of the reference element.

where \(f_{5}\) is the 5th face;

and \(\hat{\boldsymbol{n}}_{5}\) is the normal to facet 5.

\(\displaystyle \boldsymbol{\phi}_{5} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle z\end{array}\right)\)

This DOF is associated with face 5 of the reference element.