Civil Engineering Reference
In-Depth Information
which is the same as
y
=−
2
x
+
7
as desired.
For
(
r
,
s
)
=
(1, 0
.
5)
, we obtain
5
.
5
2
0
.
5
2
x
=
−
(0
.
5)
=
2
.
625
3
2
+
1
2
(0
.
5)
=
1
.
75
y
=
In formulating element characteristic matrices, various derivatives of the in-
terpolation functions with respect to the global coordinates are required, as pre-
viously demonstrated. In isoparametric elements, both element geometry and
variation of the interpolation functions are expressed in terms of the natural
coordinates of the parent element, so some additional mathematical complica-
tion arises. Specifically, we must compute
∂
N
i
/∂
x
and
∂
N
i
/∂
y
(and, possibly,
higher-order derivatives). Since the interpolation functions are expressed in
(
r
,
s
)
coordinates, we can formally write these derivatives as
∂
∂
∂
x
+
∂
∂
N
i
∂
N
i
∂
r
N
i
∂
s
x
=
r
∂
s
∂
x
(6.82)
∂
N
i
∂
∂
N
i
∂
∂
r
y
+
∂
N
i
∂
∂
s
y
=
r
∂
s
∂
y
However, unless we invert the relations in Equation 6.80, the partial derivatives
of the natural coordinates with respect to the global coordinates are not known.
As it is virtually impossible to invert Equation 6.80 to explicit algebraic expres-
sions, a different approach must be taken.
We take an indirect approach, by first examining the partial derivatives of the
field variable with respect to the natural coordinates. From Equation 6.81, the
partial derivatives of the field variable with respect to the natural coordinates can
be expressed formally as
∂
∂
r
=
∂
x
∂
x
r
+
∂
y
∂
y
∂
∂
∂
∂
r
(6.83)
∂
∂
s
=
∂
x
∂
x
s
+
∂
y
∂
y
∂
∂
∂
∂
s
In light of Equation 6.81, computation of the partial derivatives of the field vari-
able requires the partial derivatives of each interpolation function as
∂
N
i
∂
r
=
∂
N
i
∂
∂
x
r
+
∂
N
i
∂
∂
y
∂
∂
x
y
r
=
(6.84)
i
1, 4
∂
N
i
∂
s
=
∂
N
i
∂
∂
x
s
+
∂
N
i
∂
∂
y
x
∂
y
∂
s
