Civil Engineering Reference
In-Depth Information
element. Given the geometry and the form of Equations 6.77 and 6.78, each
function
G
i
(
x
,
y
)
must evaluate to unity at its associated node and to zero at each
of the other three nodes.
These conditions are exactly the same as those imposed on the interpolation
functions of the parent element. Consequently, the interpolation functions for the
parent element can be used for the geometric functions, if we map the coordi-
nates so that
(
r
,
s
)
=
(
−
1,
−
1)
⇒
(
x
1
,
y
1
)
(
r
,
s
)
=
(1,
−
1)
⇒
(
x
2
,
y
2
)
(6.79)
(
r
,
s
)
=
(1, 1)
⇒
(
x
3
,
y
3
)
(
r
,
s
)
=
(
−
1, 1)
⇒
(
x
4
,
y
4
)
where the symbol
⇒
is read as “maps to” or “corresponds to.” Note that the
(
r
,
s
)
coordinates used here are
not
the same as those defined by Equation 6.54. In-
stead, these are the actual rectangular coordinates of the 2 unit by 2 unit parent
element.
Consequently, the geometric expressions become
4
x
=
N
i
(
r
,
s
)
x
i
i
=
1
(6.80)
4
=
y
N
i
(
r
,
s
)
y
i
i
=
1
Clearly, we can also express the field variable variation in the quadrilateral ele-
ment as
4
(
x
,
y
)
=
(
r
,
s
)
=
N
i
(
r
,
s
)
i
(6.81)
i
=
1
if the mapping of Equation 6.79 is used, since all required nodal conditions are
satisfied. Since the same interpolation functions are used for both the field vari-
able and description of element geometry, the procedure is known as
isopara-
metric
(constant parameter)
mapping.
The element defined by such a procedure
is known as an
isoparametric element.
The mapping of element boundaries is
illustrated in the following example.
EXAMPLE 6.3
Figure 6.22 shows a quadrilateral element in global coordinates. Show that the mapping
described by Equation 6.80 correctly describes the line connecting nodes 2 and 3 and
determine the
(
x
,
y
)
coordinates corresponding to
(
r
,
s
)
=
(1, 0
.
5)
■
Solution
First, we determine the equation of the line passing through nodes 2 and 3 strictly by
geometry, using the equation of a two-dimensional straight line
y
=
mx
+
b
. Using the
