Page 71 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

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THE FINITE ELEMENT METHOD
There are two sets of relations that must be defined when using the finite element
method. One set determines the shape of the element and the other set defines the order
of the interpolation function for the field variable. It is not necessary to use the same
shape functions for the coordinate transformation and the interpolation equation. Thus, two
different sets of global nodes can exist. Both sets of global nodes are identical in the case
of isoparametric elements.
The natural coordinate system for the one-dimensional element is the length ratio defined
such that −1 ≤ ζ ≤ 1, where ζ is the natural coordinate. The origin of the coordinate is
at the mid-point of the line segment. For a one-dimensional linear element (substituting
x = ζ, x 1 =−1and x 2 = 1 into Equation 3.31), we obtain
ζ − 1 1
N i = = (1 − ζ)
−1 − 1 2
ζ − (−1) 1
N j = = (1 + ζ) (3.103)
1 − (−1) 2
where i and j are the two nodes of a one-dimensional element. For a one-dimensional
quadratic element, we have (Equation 3.33)

(ζ − 0)(ζ − 1) ζ
N i = =− (1 − ζ)
(−1 − 0)(−1 − 1) 2
(ζ − (−1))(ζ − 1)
2
N j = = (1 − ζ )
(0 − (−1))(0 − 1)
(ζ − (−1)) (ζ − 0) ζ
N k = = (1 + ζ) (3.104)
(1 − (−1)) (1 − 0) 2
where i, j and k represent the three nodes of the quadratic element. In order to calculate
the stiffness matrix, we need the derivative of the shape functions with respect to the global
coordinate, that is, with regard to x in this case. Therefore, a coordinate transformation of
the type shown in Figure 3.17 should be determined. In either case, the functions g(ζ) and
g(x) are assumed to be one-to-one mappings.
The coordinate transformation can be written using the same functions as given in
Equation 3.104, but substituting the coordinate value for the nodal parameter. Thus, the
coordinate transformation becomes

x = N i x i + N j x j + N k x k (3.105)

where N i ,N j and N k are given by Equation 3.104. The ζ derivative is

dN i dN i dx dN i
= = J i (3.106)
dζ dx dζ dx
which gives
dN i −1 dN i
= J i (3.107)
dx dζ