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Morley
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| Orders | \(k=2\) |
| Reference elements | triangle |
| Polynomial set | \(\mathcal{P}_{k}\) ↓ Show polynomial set definitions ↓ |
| DOFs | On each vertex: point evaluations On each edge: point evaluations of normal derivatives at midpoints |
| Number of DOFs | triangle: \(6\) |
| Categories | Scalar-valued elements |
Implementations
| Symfem | "Morley"↓ Show Symfem examples ↓ |
| UFL | "Morley"↓ Show UFL examples ↓ |
Examples
triangleorder 2
- \(R\) is the reference triangle. The following numbering of the subentities of the reference is used:
- \(\mathcal{V}\) is spanned by: \(1\), \(x\), \(x^{2}\), \(y\), \(x y\), \(y^{2}\)
- \(\mathcal{L}=\{l_0,...,l_{5}\}\)
- Functionals and basis functions:
\(\displaystyle l_{0}:v\mapsto v(0,0)\)
\(\displaystyle \phi_{0} = 2 x y - x - y + 1\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle \phi_{0} = 2 x y - x - y + 1\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle l_{1}:v\mapsto v(1,0)\)
\(\displaystyle \phi_{1} = \frac{x^{2}}{2} - x y + \frac{x}{2} - \frac{y^{2}}{2} + \frac{y}{2}\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle \phi_{1} = \frac{x^{2}}{2} - x y + \frac{x}{2} - \frac{y^{2}}{2} + \frac{y}{2}\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle l_{2}:v\mapsto v(0,1)\)
\(\displaystyle \phi_{2} = - \frac{x^{2}}{2} - x y + \frac{x}{2} + \frac{y^{2}}{2} + \frac{y}{2}\)
This DOF is associated with vertex 2 of the reference element.
\(\displaystyle \phi_{2} = - \frac{x^{2}}{2} - x y + \frac{x}{2} + \frac{y^{2}}{2} + \frac{y}{2}\)
This DOF is associated with vertex 2 of the reference element.
\(\displaystyle l_{3}:v\mapsto\nabla{v}(\tfrac{1}{2},\tfrac{1}{2})\cdot\hat{\boldsymbol{n}}_{0}\)
where \(\hat{\boldsymbol{n}}_{0}\) is the normal to facet 0.
\(\displaystyle \phi_{3} = \frac{\sqrt{2} \left(- x^{2} - 2 x y + x - y^{2} + y\right)}{2}\)
This DOF is associated with edge 0 of the reference element.
where \(\hat{\boldsymbol{n}}_{0}\) is the normal to facet 0.
\(\displaystyle \phi_{3} = \frac{\sqrt{2} \left(- x^{2} - 2 x y + x - y^{2} + y\right)}{2}\)
This DOF is associated with edge 0 of the reference element.
\(\displaystyle l_{4}:v\mapsto\nabla{v}(0,\tfrac{1}{2})\cdot\hat{\boldsymbol{n}}_{1}\)
where \(\hat{\boldsymbol{n}}_{1}\) is the normal to facet 1.
\(\displaystyle \phi_{4} = x \left(x - 1\right)\)
This DOF is associated with edge 1 of the reference element.
where \(\hat{\boldsymbol{n}}_{1}\) is the normal to facet 1.
\(\displaystyle \phi_{4} = x \left(x - 1\right)\)
This DOF is associated with edge 1 of the reference element.
\(\displaystyle l_{5}:v\mapsto\nabla{v}(\tfrac{1}{2},0)\cdot\hat{\boldsymbol{n}}_{2}\)
where \(\hat{\boldsymbol{n}}_{2}\) is the normal to facet 2.
\(\displaystyle \phi_{5} = y \left(1 - y\right)\)
This DOF is associated with edge 2 of the reference element.
where \(\hat{\boldsymbol{n}}_{2}\) is the normal to facet 2.
\(\displaystyle \phi_{5} = y \left(1 - y\right)\)
This DOF is associated with edge 2 of the reference element.
References
- Morley, L. S. The triangular equilibrium element in the solution of plate bending problems, The Aeronautical Quarterly 19(2), 149–169, 1968. [DOI: 10.1017/S0001925900004546] [BibTeX]
DefElement stats
| Element added | 09 January 2021 |
| Element last updated | 05 June 2021 |