Page 55 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

47
THE FINITE ELEMENT METHOD
In matrix form,
The [B] matrix is given as g = [B]{T} (3.26)

4x 3 4 8x 4x 1
[B] = − − − (3.27)
l 2 l l l 2 l 2 l
Equation 3.23 shows that N i =1at i and0at j and k, N j =1at j and0at i and k
and N k =1at k and 0 at i and j.
It can be verified easily that within an element the summation over the shape functions
is equal to unity, that is,
3

N i = 1 (3.28)
i=1

For example at the point x = l/4, the shape function values are
3 2 6
N i = 1 − + =
4 16 16
4 12
N j = 1 − =
16 16
2 1 2
N k = − =− (3.29)
16 4 16
and it can be easily seen that the sum of the above three shape functions is equal to 1.
It can also be observed that even though the derivatives of the quadratic element are
functions of the independent variable x, they will not be continuous at the inter-element
nodes. The type of interpolation used here is known as Lagrangian (as they can be generated
by Lagrangian interpolation formulae) and it only guarantees the continuity of the function
0
across the inter-element boundaries. These types of elements are known as C elements,in
which the superscript indicates that only derivatives of zero order are continuous, that is,
only the function is continuous. The elements that also assure the continuity of derivatives
across inter-element boundaries, in addition to the continuity of functions, are known as
1
C elements and such functions are known as Hermite polynomials.
0
The C shape functions can be determined in a general way by using Lagrangian poly-
nomial formulae. The one-dimensional (n − 1) th order Lagrange interpolation polynomial
is the ratio of two products. For an element with n nodes, (n − 1) order polynomial, the
interpolation function is
e
N (x) = n i=1 x − x i (3.30)
k
x k − x i
Note that in the above equation k = i. For a one-dimensional linear element, the shape
functions can be written using Equation 3.30, as (n = 2)

x − x 2 x − x 1
N 1 = and N 2 = (3.31)
x 1 − x 2 x 2 − x 1