an encyclopedia of finite element definitions

# Bell

 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: integral moments of normal derivatives with an order $$0$$ Lagrange space Number of DOFs triangle: $$21$$ Categories Scalar-valued elements

## Implementations

 Symfem "Bell"↓ Show Symfem examples ↓ UFL "Bell"↓ 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.
$$\displaystyle l_{1}:v\mapsto\frac{\partial}{\partial x}v(0,0)$$

$$\displaystyle \phi_{1} = x \left(- 3 x^{4} + 8 x^{3} - 6 x^{2} y^{2} - 6 x^{2} - 24 x y^{3} + 24 x y^{2} - 15 y^{4} + 32 y^{3} - 18 y^{2} + 1\right)$$

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

$$\displaystyle \phi_{2} = y \left(- 15 x^{4} - 24 x^{3} y + 32 x^{3} - 6 x^{2} y^{2} + 24 x^{2} y - 18 x^{2} - 3 y^{4} + 8 y^{3} - 6 y^{2} + 1\right)$$

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

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

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

$$\displaystyle \phi_{4} = \frac{x y \left(- 5 x^{3} - 15 x^{2} y + 12 x^{2} - 15 x y^{2} + 24 x y - 9 x - 5 y^{3} + 12 y^{2} - 9 y + 2\right)}{2}$$

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

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

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

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

This DOF is associated with vertex 1 of the reference element.
$$\displaystyle l_{7}:v\mapsto\frac{\partial}{\partial x}v(1,0)$$

$$\displaystyle \phi_{7} = x^{3} \left(- 3 x^{2} + 7 x - 4\right)$$

This DOF is associated with vertex 1 of the reference element.
$$\displaystyle l_{8}:v\mapsto\frac{\partial}{\partial y}v(1,0)$$

$$\displaystyle \phi_{8} = x^{2} y \left(- 15 x^{2} - 36 x y + 28 x - 24 y^{2} + 36 y - 12\right)$$

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

$$\displaystyle \phi_{9} = \frac{x^{3} \left(x^{2} - 2 x + 1\right)}{2}$$

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

$$\displaystyle \phi_{10} = \frac{x^{2} y \left(5 x^{2} + 9 x y - 8 x + 6 y^{2} - 9 y + 3\right)}{2}$$

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

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

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.
$$\displaystyle l_{13}:v\mapsto\frac{\partial}{\partial x}v(0,1)$$

$$\displaystyle \phi_{13} = x y^{2} \left(- 24 x^{2} - 36 x y + 36 x - 15 y^{2} + 28 y - 12\right)$$

This DOF is associated with vertex 2 of the reference element.
$$\displaystyle l_{14}:v\mapsto\frac{\partial}{\partial y}v(0,1)$$

$$\displaystyle \phi_{14} = y^{3} \left(- 3 y^{2} + 7 y - 4\right)$$

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

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

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

$$\displaystyle \phi_{16} = \frac{x y^{2} \cdot \left(6 x^{2} + 9 x y - 9 x + 5 y^{2} - 8 y + 3\right)}{2}$$

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

$$\displaystyle \phi_{17} = \frac{y^{3} \left(y^{2} - 2 y + 1\right)}{2}$$

This DOF is associated with vertex 2 of the reference element.
$$\displaystyle l_{18}:v\mapsto\displaystyle\int_{e_{0}}\frac{\partial v}{\partial\hat{\boldsymbol{n}}_{0}}$$
where $$e_{0}$$ is the 0th edge;
and $$\hat{\boldsymbol{n}}_{0}$$ is the normal to facet 0.

$$\displaystyle \phi_{18} = 15 x^{2} y^{2} \left(- x - y + 1\right)$$

This DOF is associated with edge 0 of the reference element.
$$\displaystyle l_{19}:v\mapsto\displaystyle\int_{e_{1}}\frac{\partial v}{\partial\hat{\boldsymbol{n}}_{1}}$$
where $$e_{1}$$ is the 1st edge;
and $$\hat{\boldsymbol{n}}_{1}$$ is the normal to facet 1.

$$\displaystyle \phi_{19} = 30 x y^{2} \left(- x^{2} - 2 x y + 2 x - y^{2} + 2 y - 1\right)$$

This DOF is associated with edge 1 of the reference element.
$$\displaystyle l_{20}:v\mapsto\displaystyle\int_{e_{2}}\frac{\partial v}{\partial\hat{\boldsymbol{n}}_{2}}$$
where $$e_{2}$$ is the 2nd edge;
and $$\hat{\boldsymbol{n}}_{2}$$ is the normal to facet 2.

$$\displaystyle \phi_{20} = 30 x^{2} y \left(x^{2} + 2 x y - 2 x + y^{2} - 2 y + 1\right)$$

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

## References

• Bell, Kolbein. A refined triangular plate bending finite element, International Journal for Numerical Methods in Engineering 1(1), 101–122, 1969. [DOI: 10.1002/nme.1620010108] [BibTeX]

## DefElement stats

 Element added 20 February 2021 Element last updated 10 February 2022