Civil Engineering Reference
In-Depth Information
opposite pairs,” ensuring that each independent variable plays an equal role in
the polynomial. For example, the four-term incomplete quadratic polynomial
P
(
x
,
y
)
a
3
x
2
(6.27)
is not symmetric, as there is a quadratic term in
x
but the corresponding quadratic
term in
y
does not appear. On the other hand, the incomplete quadratic polynomial
P
(
x
,
y
)
=
a
0
+
a
1
x
+
a
2
y
+
a
3
xy
(6.28)
is symmetric, as the quadratic term gives equal “weight” to both variables.
A very convenient way of visualizing some of the commonly used incom-
plete but symmetric polynomials of a given order is also afforded by the Pascal
triangle. Again referring to Figure 6.5, the dashed lines show the terms that must
be included in an incomplete yet symmetric polynomial of a given order. (These
are, of course, not the only incomplete, symmetric polynomials that can be
constructed.) All terms above the dashed lines must be included in a polynomial
representation if the function is to exhibit geometric isotropy. This feature of
polynomials is utilized to a significant extent in following the development of
various element interpolation functions.
As in the two-dimensional case, to satisfy the geometric isotropy require-
ments, the polynomial expression of the field variable in three dimensions must
be complete or incomplete but symmetric. Completeness and symmetry can also
be depicted graphically by the “Pascal pyramid” shown in Figure 6.6. While the
three-dimensional case is a bit more difficult to visualize, the basic premise
=
a
0
+
a
1
x
+
a
2
y
+
1
x
z
y
xz
x
2
z
2
xy
yz
y
2
x
2
z
x
3
xz
2
z
3
xyz
x
2
y
yz
2
xy
2
y
2
z
y
3
Figure 6.6
Pascal “pyramid” for
polynomials in three dimensions.
