Base Elements' Geometry

Elements in 2D

For 2D, we have in total 4 elements:

T3 Element	T6 Element

Q4 Element	Q8 Element

And so, we have that for

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And we can have all the interpolation functions, using the point as the form (s, t).

Element T3

i	Pi	Ni	dNi/ds	dNi/dt
1	(0, 0)	1-s-t	-1	-1
2	(1, 0)	s	1	0
3	(0, 1)	t	0	1

Element T6

i	Pi	Ni	dNi/ds	dNi/dt
1	(0, 0)	(s+t-1)(2s+2t-1)	4(s+t)-3	4(s+t)-3
2	(0.5, 0)	-4s(s+t-1)	4(1-2s-t)	-4s
3	(1, 0)	s(2s - 1)	4s - 1	0
4	(0.5, 0.5)	4st	4t	4s
5	(0, 1)	t(2t - 1)	0	4t - 1
6	(0, 0.5)	-4t(s+t-1)	-4t	4(1-s-2t)

Element Q4

i	Pi	Ni	dNi/ds	dNi/dt
1	(-1, -1)	(s-1)(t-1)/4	(t-1)/4	(s-1)
2	(1, -1)	(s+1)(1-t)/4	(1-t)/4	-(s+1)
3	(1, 1)	(s+1)(t+1)/4	(t+1)/4	(s+1)
4	(-1, 1)	(1-s)(t+1)/4	-(t+1)/4	(1-s)

Element Q8

i	Pi	Ni	dNi/ds	dNi/dt
1	(-1, -1)	(1-s)(t-1)(s+t+1)/4	(2s+t)(1-t)/4	-(s-1)(s+2t)/4
2	(0, -1)	(s^2-1)(t-1)/2	s(t-1)	(s^2-1)/2
3	(1, -1)	(s+1)(1-t)(s-t-1)/4	(t-2s)(t-1)/4	(-s+2t)(s+1)/4
4	(-1, 0)	(s+1)(1-t^2)/2	(1-t^2)/2	-t(s+1)
5	(1, 0)	(s+1)(t+1)(s+t-1)/4	(2s+t)(t+1)/4	(s+1)(s+2t)/4
6	(-1, 1)	(1-s^2)(t+1)/2	-s(t+1)	(1-s^2)/2
7	(0, 1)	(s-1)(t+1)(s-t+1)/4	(2s-t)(t+1)/4	(s-1)(s-2t)/4
8	(1, 1)	(s-1)(t^2-1)/2	(t^2-1)/2	t(s-1)