Civil Engineering Reference
In-Depth Information
6.4 POLYNOMIAL FORMS:
GEOMETRIC ISOTROPY
The previous discussion of one-dimensional (line) elements revealed that the
polynomial representation of the field variable must contain the same number of
terms as the number of nodal degrees of freedom. In addition, to satisfy the com-
pleteness requirement, the polynomial representation for an
M
-degree of free-
dom element should contain all powers of the independent variable up to and
including
M
1
. Another way of stating the latter requirement is that the poly-
nomial is
complete
. In two and three dimensions, polynomial representations of
the field variable, in general, satisfy the compatibility and completeness require-
ments if the polynomial exhibits the property known as
geometric isotropy
[1]. A
mathematical function satisfies geometric isotropy if the functional form does
not change under a translation or rotation of coordinates. In two dimensions, a
complete polynomial of order
M
can be expressed as
−
N
(2)
t
a
k
x
i
y
j
P
M
(
x
,
y
)
=
i
+
j
≤
M
(6.26)
k
=
0
where
N
(2)
t
2
is the total number of terms. A complete
polynomial as expressed by Equation 6.26 satisfies the condition of geometric
isotropy, since the two variables,
x
and
y
, are included in each term in similar
powers. Therefore, a translation or rotation of coordinates is not prejudicial to
either independent variable.
A graphical method of depicting complete two-dimensional polynomials is
the so-called Pascal triangle shown in Figure 6.5. Each horizontal line represents
a polynomial of order
M
. A complete polynomial of order
M
must contain all
terms shown above the horizontal line. For example, a complete quadratic poly-
nomial in two dimensions must contain six terms. Hence, for use in a finite ele-
ment representation of a field variable, a complete quadratic expression requires
six nodal degrees of freedom in the element. We examine this particular case in
the context of triangular elements in the next section.
In addition to the complete polynomials, incomplete polynomials also
exhibit geometric isotropy if the incomplete polynomial is
symmetric
. In this
context, symmetry implies that the independent variables appear as “equal and
=
[(
M
+
1)(
M
+
2)]
/
1
x
y
Linear
Quadratic
Cubic
Quartic
x
2
y
2
xy
x
3
x
2
y
xy
2
y
3
x
4
x
3
y
x
2
y
2
xy
3
y
4
Figure 6.5
Pascal triangle for polynomials
in two dimensions.
