Page 56 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

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1
j
i 0 x THE FINITE ELEMENT METHOD
x/l (1 − x/l)
Figure 3.6 A one-dimensional linear element represented by local coordinates

Note that N 1 and N 2 are the shape functions corresponding to the two nodes of a
one-dimensional linear element (N i and N j ).
If we use local coordinates, as shown in Figure 3.6, with x 1 =0 and x 2 = 1, then the
shape functions (Equation 3.31) become
x
x
N i = 1 − = L i and N j = = L j (3.32)
l l
where L i and L j are the shape functions defined by the local coordinate system. For a
one-dimensional quadratic element, the shape functions using Lagrangian multipliers are
given as follows:
x − x 2 x − x 3
N 1 =
x 1 − x 2 x 1 − x 3
x − x 1 x − x 3
N 2 =
x 2 − x 1 x 2 − x 3
x − x 1 x − x 2
N 3 = (3.33)
x 3 − x 1 x 3 − x 2

If we substitute x 1 = 0,x 2 = l/2and x 3 = l, in the above equation, we can immediately
verify that the resulting equations are identical to the one derived from Equation 3.23.
Similarly, cubic elements, or any other one-dimensional higher-order element shape
functions, can easily be derived using the Lagrangian interpolation formula.
For the case of quadratic and cubic elements, a better approximation of curved shapes
is possible as we have more than two points placed along the boundaries of an element.

3.2.3 Two-dimensional linear triangular elements

When one-dimensional approximations are insufficient, multi-dimensional solution proce-
dures need to be employed. In this section, we introduce for the first time a two-dimensional
element. The simplest geometric shape that can be employed to approximate irregular sur-
faces is the triangle and it is one of the popular elements currently used in finite element
calculations. This is partly due to the advances made on unstructured and adaptive mesh
generation techniques in recent times (Thompson et al. 1999).
The two-dimensional linear triangular element, also known as a simplex element,is
represented by
T (x, y) = α 1 + α 2 x + α 3 y (3.34)