an encyclopedia of finite element definitions

# Bogner–Fox–Schmitt

 Orders $$k=3$$ Reference elements quadrilateral Polynomial set $$\mathcal{Q}_{k}$$↓ Show polynomial set definitions ↓ DOFs On each vertex: point evaluations, and point evaluations of derivatives in coordinate directions Number of DOFs quadrilateral: $$16$$ Categories Scalar-valued elements

## Implementations

 Symfem "BFS"↓ Show Symfem examples ↓

## Examples

order 3
• $$R$$ is the reference quadrilateral. The following numbering of the subentities of the reference is used:
• $$\mathcal{V}$$ is spanned by: $$1$$, $$y$$, $$y^{2}$$, $$y^{3}$$, $$x$$, $$x y$$, $$x y^{2}$$, $$x y^{3}$$, $$x^{2}$$, $$x^{2} y$$, $$x^{2} y^{2}$$, $$x^{2} y^{3}$$, $$x^{3}$$, $$x^{3} y$$, $$x^{3} y^{2}$$, $$x^{3} y^{3}$$
• $$\mathcal{L}=\{l_0,...,l_{15}\}$$
• Functionals and basis functions:
$$\displaystyle l_{0}:v\mapsto v(0,0)$$

$$\displaystyle \phi_{0} = 4 x^{3} y^{3} - 6 x^{3} y^{2} + 2 x^{3} - 6 x^{2} y^{3} + 9 x^{2} y^{2} - 3 x^{2} + 2 y^{3} - 3 y^{2} + 1$$

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

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

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

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

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

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

This DOF is associated with vertex 0 of the reference element.
$$\displaystyle l_{4}:v\mapsto v(1,0)$$

$$\displaystyle \phi_{4} = x^{2} \left(- 4 x y^{3} + 6 x y^{2} - 2 x + 6 y^{3} - 9 y^{2} + 3\right)$$

This DOF is associated with vertex 1 of the reference element.
$$\displaystyle l_{5}:v\mapsto\frac{\partial}{\partial x}v(1,0)$$

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

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

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

This DOF is associated with vertex 1 of the reference element.
$$\displaystyle l_{7}:v\mapsto\frac{\partial^{2}}{\partial x\partial y}v(1,0)$$

$$\displaystyle \phi_{7} = x^{2} y \left(x y^{2} - 2 x y + x - y^{2} + 2 y - 1\right)$$

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

$$\displaystyle \phi_{8} = y^{2} \left(- 4 x^{3} y + 6 x^{3} + 6 x^{2} y - 9 x^{2} - 2 y + 3\right)$$

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

$$\displaystyle \phi_{9} = x y^{2} \left(- 2 x^{2} y + 3 x^{2} + 4 x y - 6 x - 2 y + 3\right)$$

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

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

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

$$\displaystyle \phi_{11} = x y^{2} \left(x^{2} y - x^{2} - 2 x y + 2 x + y - 1\right)$$

This DOF is associated with vertex 2 of the reference element.
$$\displaystyle l_{12}:v\mapsto v(1,1)$$

$$\displaystyle \phi_{12} = x^{2} y^{2} \left(4 x y - 6 x - 6 y + 9\right)$$

This DOF is associated with vertex 3 of the reference element.
$$\displaystyle l_{13}:v\mapsto\frac{\partial}{\partial x}v(1,1)$$

$$\displaystyle \phi_{13} = x^{2} y^{2} \left(- 2 x y + 3 x + 2 y - 3\right)$$

This DOF is associated with vertex 3 of the reference element.
$$\displaystyle l_{14}:v\mapsto\frac{\partial}{\partial y}v(1,1)$$

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

This DOF is associated with vertex 3 of the reference element.
$$\displaystyle l_{15}:v\mapsto\frac{\partial^{2}}{\partial x\partial y}v(1,1)$$

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

This DOF is associated with vertex 3 of the reference element.

## References

• Bogner, F. K., Fox, R. L., and Schmit, L. A. The generation of interelement compatible stiffness and mass matrices by the use of interpolation formulae, Proceedings of the Conference on Matrix Methods in Structural Mechanics, 397–444, 1965. [BibTeX]

## DefElement stats

 Element added 06 March 2021 Element last updated 21 April 2021