HIGHER-ORDER ELEMENTS 75
Figure 3.15 Quadratic element
The functions can also be expressed as complete cubic polynomials.
Each component now involves ten parameters. In order to satisfy
interelement compatibility the function for a side (which is cubic)
must be represented with four nodal quantities. One possibility is to
work with cornerpoint nodes and two interior nodes per side (Figure
3.16(a)), taking as nodal quantities, the u or q components. An
additional interior node is needed to maintain completeness of the
polynomial, since if the polynomials is not complete, its properties
will have preferred directions which is not desirable. It is convenient
to take the interior node at the centroid. We can also attempt to work
only with corner nodes. Since four quantities are necessary for
interelement compatibility, one must include derivatives of the
function as nodal quantities, i.e.
Mnodei = W
du du
2 /node i
(3.50)
3 (u,
a) (b)
Figure 3.16 Cubic element
du
du_
<
As before one additional quantity not associated with the boundaries
is required, i.e. the value of u at the centroid, u
c
.
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