Shape Functions¶
About shape function for pyramid:¶
In order to be coherent with the facial elements:
Base: $\( \xi\in[-1,1],\eta\in[-1,1],\zeta=0 \)\( Axis: \)\( \xi=0,\eta=0,\zeta=[0,1] \)$
Linear Pyramid5: $\( \left(\begin{array}{c} -\frac{\left(\eta +\zeta -1\right)\,\left(\xi +\zeta -1\right)}{4\,\left(\zeta -1\right)}\\ \frac{\left(\eta +\zeta -1\right)\,\left(\xi -\zeta +1\right)}{4\,\left(\zeta -1\right)}\\ -\frac{\left(\eta -\zeta +1\right)\,\left(\xi -\zeta +1\right)}{4\,\left(\zeta -1\right)}\\ \frac{\left(\xi +\zeta -1\right)\,\left(\eta -\zeta +1\right)}{4\,\left(\zeta -1\right)}\\ \zeta \end{array}\right) \)$
Quad Pyramid14: $\( \left(\begin{array}{c} \frac{\eta \,\xi \,\left(2\,\zeta -1\right)\,\left(2\,\zeta -2\right)\,\left(\eta +\zeta -1\right)\,\left(\xi +\zeta -1\right)}{8\,{\left(\zeta -1\right)}^4}\\ \frac{\eta \,\xi \,\left(2\,\zeta -1\right)\,\left(2\,\zeta -2\right)\,\left(\eta +\zeta -1\right)\,\left(\xi -\zeta +1\right)}{8\,{\left(\zeta -1\right)}^4}\\ \frac{\eta \,\xi \,\left(2\,\zeta -1\right)\,\left(2\,\zeta -2\right)\,\left(\eta -\zeta +1\right)\,\left(\xi -\zeta +1\right)}{8\,{\left(\zeta -1\right)}^4}\\ \frac{\eta \,\xi \,\left(2\,\zeta -1\right)\,\left(2\,\zeta -2\right)\,\left(\xi +\zeta -1\right)\,\left(\eta -\zeta +1\right)}{8\,{\left(\zeta -1\right)}^4}\\ \zeta \,\left(2\,\zeta -1\right)\\ -\frac{\eta \,\left(2\,\zeta -1\right)\,\left(2\,\zeta -2\right)\,\left(\eta +\zeta -1\right)\,\left(\xi +\zeta -1\right)\,\left(\xi -\zeta +1\right)}{4\,{\left(\zeta -1\right)}^4}\\ -\frac{\xi \,\left(2\,\zeta -1\right)\,\left(2\,\zeta -2\right)\,\left(\eta +\zeta -1\right)\,\left(\eta -\zeta +1\right)\,\left(\xi -\zeta +1\right)}{4\,{\left(\zeta -1\right)}^4}\\ -\frac{\eta \,\left(2\,\zeta -1\right)\,\left(2\,\zeta -2\right)\,\left(\xi +\zeta -1\right)\,\left(\eta -\zeta +1\right)\,\left(\xi -\zeta +1\right)}{4\,{\left(\zeta -1\right)}^4}\\ -\frac{\xi \,\left(2\,\zeta -1\right)\,\left(2\,\zeta -2\right)\,\left(\eta +\zeta -1\right)\,\left(\xi +\zeta -1\right)\,\left(\eta -\zeta +1\right)}{4\,{\left(\zeta -1\right)}^4}\\ -\frac{\zeta \,\left(2\,\zeta -2\right)\,\left(\eta +\zeta -1\right)\,\left(\xi +\zeta -1\right)}{2\,{\left(\zeta -1\right)}^2}\\ \frac{\zeta \,\left(2\,\zeta -2\right)\,\left(\eta +\zeta -1\right)\,\left(\xi -\zeta +1\right)}{2\,{\left(\zeta -1\right)}^2}\\ -\frac{\zeta \,\left(2\,\zeta -2\right)\,\left(\eta -\zeta +1\right)\,\left(\xi -\zeta +1\right)}{2\,{\left(\zeta -1\right)}^2}\\ \frac{\zeta \,\left(2\,\zeta -2\right)\,\left(\xi +\zeta -1\right)\,\left(\eta -\zeta +1\right)}{2\,{\left(\zeta -1\right)}^2}\\ \frac{\left(2\,\zeta -1\right)\,\left(2\,\zeta -2\right)\,\left(\eta +\zeta -1\right)\,\left(\xi +\zeta -1\right)\,\left(\eta -\zeta +1\right)\,\left(\xi -\zeta +1\right)}{2\,{\left(\zeta -1\right)}^4} \end{array}\right) \)$
Here we slice N3 from Pyramid5.
At \(\xi = 0\):
At \(\xi = 0.01\):
At \(\xi = 0.1\):
At \(\xi = 0.5\):
\(\xi=0\) leads to a purely linear function in slice,while \(\xi \neq 0\) leads to more complex distribution. Note that although singularity exists, none of them exist inside the pyramid.
The shape functions and first derivatives have singularities and discontinuities along \(\zeta =0\), but for the cone inside the Pyramid, the functions seem to be regular. ?