Page 81 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

THE FINITE ELEMENT METHOD
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Figure 3.22 Quadratic tetrahedral element

N 5 = 4L 4 L 1

N 6 = 4L 3 L 4
N 7 = 4L 3 L 2
N 8 = 4L 1 L 2

N 9 = 4L 2 L 4
N 10 = 4L 1 L 3 (3.162)
The brick, or hexahedron element shown in Figure 3.21(b), is a simple element, which
is easy to visualize when the domain is discretized. The bilinear interpolation function is
T = α 1 + α 2 x + α 3 y + α 4 z + α 5 xy + α 6 yz + α 7 zx + α 8 xyz (3.163)

which can be written as
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T = N i T i (3.164)
i=1

where
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N i = (1 + ζζ i )(1 + ηη i )(1 + ρρ i ) (3.165)
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where ζ i ,η i and ρ i are the local coordinates.
For a quadratic 20-node hexahedron, which can represent arbitrary solids with curved
surfaces as shown in Figure 3.23, the shape functions can be written as follows.
Corner nodes
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N i = (1 + ζζ i )(1 + ηη i )(1 + ρρ i )(ζζ i + ηη i + ρρ i − 1) with i = 1, 2,... , 8.
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(3.166)