Page 82 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

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8
7
14
16 15 THE FINITE ELEMENT METHOD
20 13
6
5 19
17
4
3 18
11
12 10
1 9 2
Figure 3.23 20-node hexahedral element

Mid-side nodes
1 2
N i = (1 − ζ )(1 + ηη i )(1 + ρρ i ) with i = 9, 13, 15, 11
4
1 2
N i = (1 − η )(1 + ζζ i )(1 + ρρ i ) with i = 10, 14, 16, 12
4
1 2
N i = (1 − ρ )(1 + ζζ i )(1 + ηη i ) with i = 18, 19, 20, 17 (3.167)
4
The shape functions for a linear pentahedran element (which is used in cylindrical
geometries) can be generated from the product of triangular and one-dimensional interpo-
lation functions (Refer to Figure 3.21(c)).
1
N 1 = L 1 (1 − w)
2
1
N 2 = L 2 (1 − w)
2
1
N 3 = L 3 (1 − w)
2
1
N 4 = L 1 (1 + w)
2
1
N 5 = L 2 (1 + w)
2
1
N 6 = L 3 (1 + w) (3.168)
2
where w =−1 at the bottom surface and 1 at the top surface. In conclusion, isoparametric
elements are very useful as they can be used for modelling irregular solids and the element
can be mapped onto a unit cube.