Morley

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Orders	\(k=2\)
Reference elements	triangle
Polynomial set	\(\mathcal{P}_{k}\) ↓ Show polynomial set definitions ↓↑ Hide polynomial set definitions ↑ \(\mathcal{P}_k=\operatorname{span}\left\{\prod_{i=1}^dx_i^{p_i}\middle\|\sum_{i=1}^dp_i\leqslant k\right\}\)
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 string	`"Morley"` ↓ Show Symfem examples ↓↑ Hide Symfem examples ↑ Before trying this example, you must install Symfem: pip3 install symfem This element can then be created with the following lines of Python: import symfem # Create Morley order 2 on a triangle element = symfem.create_element("triangle", "Morley", 2)
UFL string	`"Morley"` ↓ Show UFL examples ↓↑ Hide UFL examples ↑ Before trying this example, you must install UFL: pip3 install UFL This element can then be created with the following lines of Python: import ufl # Create Morley order 2 on a triangle element = ufl.FiniteElement("Morley", "triangle", 2)

Examples

triangle
order 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.

\(\displaystyle l_{1}:v\mapsto v(1,0)\)

\(\displaystyle \phi_{1} = \frac{x^{2}}{2} - x y + \frac{x}{2} - \frac{y^{2}}{2} + \frac{y}{2}\)

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

\(\displaystyle l_{2}:v\mapsto v(0,1)\)

\(\displaystyle \phi_{2} = - \frac{x^{2}}{2} - x y + \frac{x}{2} + \frac{y^{2}}{2} + \frac{y}{2}\)

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

\(\displaystyle l_{3}:v\mapsto\nabla{v}(\tfrac{1}{2},\tfrac{1}{2})\cdot\hat{\boldsymbol{n}}_{0}\)

\(\displaystyle \phi_{3} = \frac{\sqrt{2} \left(- x^{2} - 2 x y + x - y^{2} + y\right)}{2}\)

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

\(\displaystyle l_{4}:v\mapsto\nabla{v}(0,\tfrac{1}{2})\cdot\hat{\boldsymbol{n}}_{1}\)

\(\displaystyle \phi_{4} = x \left(x - 1\right)\)

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

\(\displaystyle l_{5}:v\mapsto\nabla{v}(\tfrac{1}{2},0)\cdot\hat{\boldsymbol{n}}_{2}\)

\(\displaystyle \phi_{5} = y \left(1 - y\right)\)

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]