an encyclopedia of finite element definitions

Morley

 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

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.
$$\displaystyle l_{3}:v\mapsto\nabla{v}(\tfrac{1}{2},\tfrac{1}{2})\cdot\hat{\boldsymbol{n}}_{0}$$
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.
$$\displaystyle l_{5}:v\mapsto\nabla{v}(\tfrac{1}{2},0)\cdot\hat{\boldsymbol{n}}_{2}$$
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.

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]

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 Element added 09 January 2021 Element last updated 05 June 2021