It is possible to assume that for each of the 
-nodes in a two- or
three-dimensional element, the number of associated shape functions is
different. For a central node, it is also possible to use different orders
of approximation in both directions. For example, the irregular
-element shown in Figure 9 has a maximum order of
approximation of
, total number of degrees of freedom of 11 (4 linear, 5 on edges, and 2 in
the center), a linear approximation along the lower edge (missing -node),
and maximum directional orders of approximation 
.
As we can see, the central node can have two different orders in each
direction, so in the element database this node can be referred to as
two -nodes.
For the purpose of numerical integration, we need to estimate the
number and location of integration points. Usually, tensor products of
one-dimensional Gauss-Legendre quadratures are used, with the number of
Gauss points in each direction estimated by the maximum order of
-nodes in each direction. Thus, the element in
Figure 9 will require a
12 point integration formula (
). For simple low order
elements, it is possible to derive special (non-tensor product)
integration formulas with fewer integration points. Such formulas may
improve the speed of computations. As for elements of higher order, a
significant portion of the CPU time is used in element assembly over
integration points.
As a special, interesting case of irregular -elements, we consider
``serendipity'' -elements, which have a lower number of degrees of
freedom than regular elements of order , but they still contain all of the
terms of order 
. For example, a regular element of order 
has 9 degrees of freedom, but the central degree of freedom is
associated with the shape function containing (among others)
expression 
, obviously not required here. When using
serendipity elements, we can save 1 dof for quadratic elements (8 dofs
out of 9), 4 dofs for a cubic element (12 out of 16) and 8 for a quartic
element (17 out of 25, with only one dof in the centerpoint). See
Figure 10 for examples of serendipity elements. The
savings are more
dramatic for three-dimensional elements, especially when combined
with optimized integration rules (using less Gauss points than tensor
product formulas).
Figure 10: Serendipity elements
