Shape functions¶
This chapter describes the shape functions for the finite elements.
Understanding shape function notations¶
The notations used in shape functions are listed below:
- u: displacement in x (or s) direction.
- v: displacement in y (or t) direction.
- w: displacement in z (or s) direction.
- \(\theta_{x}\): Rotation about x direction.
- \(\theta_{y}\): Rotation about y direction.
- \(\theta_{z}\): Rotation about z direction.
- \(A_{x}\): x-component of vector magnetic potential.
- \(A_{y}\): y-component of vector magnetic potential.
- \(A_{z}\): z-component of vector magnetic potential.
- C: Concentration.
- P: Pressure.
- T: Temperature.
- V: Electric potential or source current.
- For the shell element, the u and v represent in-plane motions, and w denotes the out-of-plane motion.
3D shell elements¶
This section describes the shape functions for 3D shell elements that are applied in the WELSIM application.
3-Node triangle¶
The shape functions for the 3-node triangular shell elements are:
\[ u=u_{0}L_{0}+u_{1}L_{1}+u_{2}L_{2} \]
\[ v=v_{0}L_{0}+v_{1}L_{1}+v_{2}L_{2} \]
\[ w=w_{0}L_{0}+w_{1}L_{1}+w_{2}L_{2} \]
\[ A_{x}=A_{x0}L_{0}+A_{x1}L_{1}+A_{x2}L_{2} \]
\[ A_{y}=A_{y0}L_{0}+A_{y1}L_{1}+A_{y2}L_{2} \]
\[ A_{z}=A_{z0}L_{0}+A_{z1}L_{1}+A_{z2}L_{2} \]
\[ T=T_{0}L_{0}+T_{1}L_{1}+T_{2}L_{2} \]
\[ V=V_{0}L_{0}+V_{1}L_{1}+V_{2}L_{2} \]
6-Node triangle¶
The shape functions for the 6-node triangular shell elements are:
\[ u=u_{0}(2L_{0}-1)L_{0}+u_{1}(2L_{1}-1)L_{1}+u_{2}(2L_{2}-1)L_{2}+u_{3}(4L_{0}L_{1})+u_{4}(4L_{1}L_{2})+u_{5}(4L_{2}L_{0}) \]
\[ v=v_{0}(2L_{0}-1)L_{0}+v_{1}(2L_{1}-1)L_{1}+v_{2}(2L_{2}-1)L_{2}+v_{3}(4L_{0}L_{1})+v_{4}(4L_{1}L_{2})+v_{5}(4L_{2}L_{0}) \]
\[ w=w_{0}(2L_{0}-1)L_{0}+w_{1}(2L_{1}-1)L_{1}+w_{2}(2L_{2}-1)L_{2}+w_{3}(4L_{0}L_{1})+w_{4}(4L_{1}L_{2})+w_{5}(4L_{2}L_{0}) \]
3D solid elements¶
This section describes the shape functions for the 3D solid elements that are applied in the WELSIM application.
4-Node tetrahedra¶
The 4-node tetrahedra is also called liner tetrahedra element. The shape functions are:
\[ u=u_{0}L_{0}+u_{1}L_{1}+u_{2}L_{2}+u_{3}L_{3} \]
\[ v=v_{0}L_{0}+v_{1}L_{1}+v_{2}L_{2}+v_{3}L_{3} \]
\[ w=w_{0}L_{0}+w_{1}L_{1}+w_{2}L_{2}+w_{3}L_{3} \]
10-Node tetrahedra¶
The 10-node tetrahedra is also called bilinear tetrahedra element. The shape functions are:
\[ u=u_{0}(2L_{0}-1)L_{0}+u_{1}(2L_{1}-1)L_{1}+u_{2}(2L_{2}-1)L_{2}+u_{3}(2L_{3}-1)L_{3}+4u_{4}L_{0}L_{1}+u_{5}L_{1}L_{2}+u_{6}L_{0}L_{2}+u_{7}L_{0}L_{3}+u_{8}L_{1}L_{3}+u_{9}L_{2}L_{3} \]
\[ v=...\text{(analogous to u)} \]
\[ w=...\text{(analogous to u)} \]