Linear Brick Shape Functions

Shape functions define the geometry of an element in its computational (intrinsic) domain. As was seen in Finite Element Formulation, physical coordinates are transformed into simpler computational intrinsic coordinates so that integration of values is numerically more efficient.

Where, $r \equiv ξ$ , $s \equiv η$ , and $t \equiv ζ$ .

The shape functions of an 8 node brick element, shown in Figure 1, are given by:(1)

Φ_{1} = \frac{1}{8} (1 - ξ) (1 - η) (1 - ζ)

(2)

Φ_{2} = \frac{1}{8} (1 - ξ) (1 - η) (1 + ζ)

(3)

Φ_{3} = \frac{1}{8} (1 + ξ) (1 - η) (1 + ζ)

(4)

Φ_{4} = \frac{1}{8} (1 + ξ) (1 - η) (1 - ζ)

(5)

Φ_{5} = \frac{1}{8} (1 - ξ) (1 + η) (1 - ζ)

(6)

Φ_{6} = \frac{1}{8} (1 - ξ) (1 + η) (1 + ζ)

(7)

Φ_{7} = \frac{1}{8} (1 + ξ) (1 + η) (1 + ζ)

(8)

Φ_{8} = \frac{1}{8} (1 + ξ) (1 + η) (1 - ζ)

The element velocity field is related by:(9)

v_{i} = \sum_{I = 1}^{8} Φ_{I} . v_{i I}

Where, $v_{i I}$ are the nodal velocities.