Civil Engineering Reference
In-Depth Information
remains that each independent variable must be of equal “strength” in the poly-
nomial. For example, the 3-D quadratic polynomial
a
4
x
2
a
5
y
2
a
6
z
2
P
(
x
,
y
,
z
)
=
a
0
+
a
1
x
+
a
2
y
+
a
3
z
+
+
+
+
a
7
xy
+
a
8
xz
+
a
9
yz
(6.29)
is complete and could be applied to an element having 10 nodes. Similarly, an
incomplete, symmetric form such as
a
4
x
2
a
5
y
2
a
6
z
2
P
(
x
,
y
,
z
)
=
a
0
+
a
1
x
+
a
2
y
+
a
3
z
+
+
+
(6.30)
or
P
(
x
,
y
,
z
)
=
a
0
+
a
1
x
+
a
2
y
+
a
3
z
+
a
4
xy
+
a
5
xz
+
a
6
yz
(6.31)
could be used for elements having seven nodal degrees of freedom (an unlikely
case, however).
Geometric isotropy is not an
absolute
requirement for field variable
repesentation [1], hence, interpolation functions. As demonstrated by many
researchers, incomplete representations are quite often used and solution conver-
gence attained. However, in terms of
h-
refinement, use of geometrically isotropic
representations guarantees satisfaction of the compatibility and completeness
requirements. For the
p-
refinement method, the reader is reminded that the inter-
polation functions in any finite element analysis solution are approximations to
the power series expansion of the problem solution. As we increase the number
of element nodes, the order of the interpolation functions increases and, in the
limit, as the number of nodes approaches infinity, the polynomial expression of
the field variable approaches the power series expansion of the solution.
6.5 TRIANGULAR ELEMENTS
The interpolation functions for triangular elements are inherently formulated in
two dimensions and a family of such elements exists. Figure 6.7 depicts the first
three elements (linear, quadratic, and cubic) of the family. Note that, in the case
of the cubic element, an internal node exists. The internal node is required to
(a)
(b)
(c)
Figure 6.7
Triangular elements:
(a) 3-node linear, (b) 6-node quadratic,
(c) 10-node cubic.
