Page 54 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

THE FINITE ELEMENT METHOD
46
or
(3.22)
T = N i T i + N j T j + N k T k
Hence the shape functions for a one-dimensional quadratic element are obtained from
Equation 3.21 as follows:
3x 2x
2
N i = 1 − + 2
l l
x x
2
N j = 4 − 4
l l 2
2
x x
N k = 2 − (3.23)
l 2 l
The variation of temperature and shape functions of a typical quadratic element is
shown in Figure 3.5. The first derivative of temperature can now be written as
dT dN i dN j dN k
= T i + T j + T k (3.24)
dx dx dx dx
or
dT 4x 3 4 8x 4x 1
= − T i + − T j + − T k (3.25)
dx l 2 l l l 2 l 2 l

N j
N N
i k

1 1

i j k

T k
T j
T i l/2 l/2
i j k

l

Figure 3.5 Variation of shape functions and their derivatives over a one-dimensional
quadratic element