an encyclopedia of finite element definitions

# Argyris

 Orders $$k=5$$ Reference elements triangle Polynomial set $$\mathcal{P}_{k}$$↓ Show polynomial set definitions ↓ DOFs On each vertex: point evaluations,point evaluations of derivatives in coordinate directions, and point evaluations of components of Jacobian On each edge: point evaluations of normal derivatives at midpoints Number of DOFs triangle: $$21$$ Categories Scalar-valued elements

## Implementations

 Symfem string "Argyris"↓ Show Symfem examples ↓ UFL string "Argyris"↓ Show UFL examples ↓

## Examples

triangle
order 5
• $$R$$ is the reference triangle. The following numbering of the subentities of the reference is used:
• $$\mathcal{V}$$ is spanned by: $$1$$, $$x$$, $$x^{2}$$, $$x^{3}$$, $$x^{4}$$, $$x^{5}$$, $$y$$, $$x y$$, $$x^{2} y$$, $$x^{3} y$$, $$x^{4} y$$, $$y^{2}$$, $$x y^{2}$$, $$x^{2} y^{2}$$, $$x^{3} y^{2}$$, $$y^{3}$$, $$x y^{3}$$, $$x^{2} y^{3}$$, $$y^{4}$$, $$x y^{4}$$, $$y^{5}$$
• $$\mathcal{L}=\{l_0,...,l_{20}\}$$
• Functionals and basis functions:
$$\displaystyle l_{0}:v\mapsto v(0,0)$$

$$\displaystyle \phi_{0} = - 6 x^{5} + 15 x^{4} + 30 x^{3} y^{2} - 10 x^{3} + 30 x^{2} y^{3} - 30 x^{2} y^{2} - 6 y^{5} + 15 y^{4} - 10 y^{3} + 1$$

This DOF is associated with vertex 0 of the reference element.
$$\displaystyle l_{1}:v\mapsto\nabla{v}(0,0)\cdot\left(\begin{array}{c}1\\0\end{array}\right)$$

$$\displaystyle \phi_{1} = x \left(- 3 x^{4} + 8 x^{3} + x^{2} y^{2} - 6 x^{2} - 10 x y^{3} + 10 x y^{2} - 8 y^{4} + 18 y^{3} - 11 y^{2} + 1\right)$$

This DOF is associated with vertex 0 of the reference element.
$$\displaystyle l_{2}:v\mapsto\nabla{v}(0,0)\cdot\left(\begin{array}{c}0\\1\end{array}\right)$$

$$\displaystyle \phi_{2} = y \left(- 8 x^{4} - 10 x^{3} y + 18 x^{3} + x^{2} y^{2} + 10 x^{2} y - 11 x^{2} - 3 y^{4} + 8 y^{3} - 6 y^{2} + 1\right)$$

This DOF is associated with vertex 0 of the reference element.
$$\displaystyle l_{3}:v\mapsto\frac{\partial^2v}{\partial x\partial x}(0,0)$$

$$\displaystyle \phi_{3} = \frac{x^{2} \left(- x^{3} + 3 x^{2} + 3 x y^{2} - 3 x + 2 y^{3} - 3 y^{2} + 1\right)}{2}$$

This DOF is associated with vertex 0 of the reference element.
$$\displaystyle l_{4}:v\mapsto\frac{\partial^2v}{\partial y\partial x}(0,0)$$

$$\displaystyle \phi_{4} = x y \left(- 2 x^{3} - 6 x^{2} y + 5 x^{2} - 6 x y^{2} + 10 x y - 4 x - 2 y^{3} + 5 y^{2} - 4 y + 1\right)$$

This DOF is associated with vertex 0 of the reference element.
$$\displaystyle l_{5}:v\mapsto\frac{\partial^2v}{\partial y\partial y}(0,0)$$

$$\displaystyle \phi_{5} = \frac{y^{2} \left(2 x^{3} + 3 x^{2} y - 3 x^{2} - y^{3} + 3 y^{2} - 3 y + 1\right)}{2}$$

This DOF is associated with vertex 0 of the reference element.
$$\displaystyle l_{6}:v\mapsto v(1,0)$$

$$\displaystyle \phi_{6} = x^{2} \left(6 x^{3} - 15 x^{2} - 15 x y^{2} + 10 x - 15 y^{3} + 15 y^{2}\right)$$

This DOF is associated with vertex 1 of the reference element.
$$\displaystyle l_{7}:v\mapsto\nabla{v}(1,0)\cdot\left(\begin{array}{c}1\\0\end{array}\right)$$

$$\displaystyle \phi_{7} = \frac{x^{2} \left(- 6 x^{3} + 14 x^{2} + 7 x y^{2} - 8 x + 7 y^{3} - 7 y^{2}\right)}{2}$$

This DOF is associated with vertex 1 of the reference element.
$$\displaystyle l_{8}:v\mapsto\nabla{v}(1,0)\cdot\left(\begin{array}{c}0\\1\end{array}\right)$$

$$\displaystyle \phi_{8} = \frac{x^{2} y \left(- 16 x^{2} - 37 x y + 28 x - 27 y^{2} + 37 y - 10\right)}{2}$$

This DOF is associated with vertex 1 of the reference element.
$$\displaystyle l_{9}:v\mapsto\frac{\partial^2v}{\partial x\partial x}(1,0)$$

$$\displaystyle \phi_{9} = \frac{x^{2} \left(2 x^{3} - 4 x^{2} - x y^{2} + 2 x - y^{3} + y^{2}\right)}{4}$$

This DOF is associated with vertex 1 of the reference element.
$$\displaystyle l_{10}:v\mapsto\frac{\partial^2v}{\partial y\partial x}(1,0)$$

$$\displaystyle \phi_{10} = \frac{x^{2} y \left(4 x^{2} + 7 x y - 6 x + 5 y^{2} - 7 y + 2\right)}{2}$$

This DOF is associated with vertex 1 of the reference element.
$$\displaystyle l_{11}:v\mapsto\frac{\partial^2v}{\partial y\partial y}(1,0)$$

$$\displaystyle \phi_{11} = \frac{x^{2} y^{2} \left(- 3 x - 5 y + 5\right)}{4}$$

This DOF is associated with vertex 1 of the reference element.
$$\displaystyle l_{12}:v\mapsto v(0,1)$$

$$\displaystyle \phi_{12} = y^{2} \left(- 15 x^{3} - 15 x^{2} y + 15 x^{2} + 6 y^{3} - 15 y^{2} + 10 y\right)$$

This DOF is associated with vertex 2 of the reference element.
$$\displaystyle l_{13}:v\mapsto\nabla{v}(0,1)\cdot\left(\begin{array}{c}1\\0\end{array}\right)$$

$$\displaystyle \phi_{13} = \frac{x y^{2} \left(- 27 x^{2} - 37 x y + 37 x - 16 y^{2} + 28 y - 10\right)}{2}$$

This DOF is associated with vertex 2 of the reference element.
$$\displaystyle l_{14}:v\mapsto\nabla{v}(0,1)\cdot\left(\begin{array}{c}0\\1\end{array}\right)$$

$$\displaystyle \phi_{14} = \frac{y^{2} \left(7 x^{3} + 7 x^{2} y - 7 x^{2} - 6 y^{3} + 14 y^{2} - 8 y\right)}{2}$$

This DOF is associated with vertex 2 of the reference element.
$$\displaystyle l_{15}:v\mapsto\frac{\partial^2v}{\partial x\partial x}(0,1)$$

$$\displaystyle \phi_{15} = \frac{x^{2} y^{2} \left(- 5 x - 3 y + 5\right)}{4}$$

This DOF is associated with vertex 2 of the reference element.
$$\displaystyle l_{16}:v\mapsto\frac{\partial^2v}{\partial y\partial x}(0,1)$$

$$\displaystyle \phi_{16} = \frac{x y^{2} \left(5 x^{2} + 7 x y - 7 x + 4 y^{2} - 6 y + 2\right)}{2}$$

This DOF is associated with vertex 2 of the reference element.
$$\displaystyle l_{17}:v\mapsto\frac{\partial^2v}{\partial y\partial y}(0,1)$$

$$\displaystyle \phi_{17} = \frac{y^{2} \left(- x^{3} - x^{2} y + x^{2} + 2 y^{3} - 4 y^{2} + 2 y\right)}{4}$$

This DOF is associated with vertex 2 of the reference element.
$$\displaystyle l_{18}:v\mapsto\nabla{v}(\tfrac{1}{2},\tfrac{1}{2})\cdot\hat{\boldsymbol{n}}_{0}$$

$$\displaystyle \phi_{18} = 8 \sqrt{2} x^{2} y^{2} \left(- x - y + 1\right)$$

This DOF is associated with edge 0 of the reference element.
$$\displaystyle l_{19}:v\mapsto\nabla{v}(0,\tfrac{1}{2})\cdot\hat{\boldsymbol{n}}_{1}$$

$$\displaystyle \phi_{19} = 16 x y^{2} \left(- x^{2} - 2 x y + 2 x - y^{2} + 2 y - 1\right)$$

This DOF is associated with edge 1 of the reference element.
$$\displaystyle l_{20}:v\mapsto\nabla{v}(\tfrac{1}{2},0)\cdot\hat{\boldsymbol{n}}_{2}$$

$$\displaystyle \phi_{20} = 16 x^{2} y \left(x^{2} + 2 x y - 2 x + y^{2} - 2 y + 1\right)$$

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

## References

• Argyris, J. H., Fried, I., and Scharpf, D. W. The TUBA Family of Plate Elements for the Matrix Displacement Method, The Aeronautical Journal 72(692), 701–709, 1968. [DOI: 10.1017/S000192400008489X] [BibTeX]