Graphics Reference
In-Depth Information
tion N
i
(x) to the node x
i
. These functions are defined on
all
of
R
. Their graphs
are shown in Figures 19.5(c)-(e). The functions N
i
(x) should have the following
properties:
(1) They are defined in terms of basis functions for the elements.
(2) N
i
(x) is nonzero at the ith node but vanishes at all the other nodes. More pre-
cisely, we want
()
=d .
Nx
i
j
ij
(3) N
i
(x) vanishes on all elements other than the ones adjacent to the ith node.
If we had decided to use a higher degree approximation, then the only thing that
would change is the local shape function basis.
Our approximation to T(x) now has the form
()
=
()
+
()
+
()
Hx
TN x
TN x
TN x
3 3
,
(19.12)
11
2 2
and we solve for the T
i
by substituting H(x) for T(x) and N
i
(x) for w(x) into equation
(19.11). One will get three equations. Each integration will have to be broken up into
two parts, one for each element, since there is no one formula for the N
i
(x) over whole
interval [0, L]. We get
L
2
3
Ê
Á
Ê
Á
dN
dx
ˆ
˜
-
ˆ
˜
K
dN
dx
TNQ xNK
dT
dx
L
2
È
Í
Ê
Ë
ˆ
¯
˘
˙
j
i
Â
Ú
+
-
+
j
i
i
0
0
j
=
1
L
3
Ê
Á
Ê
Á
dN
dx
ˆ
˜
-
ˆ
˜
K
dN
dx
TNQ xNK
dT
dx
È
Í
Ê
Ë
ˆ
¯
˘
˙
L
j
i
Â
Ú
+
-
=
0,
i
=
1123
,,.
(19.13)
j
i
i
L
2
L
2
j
=
1
Knowing the functions N
i
(x), and hence also their derivatives, we can perform the
integration in (19.13). Finally, using facts about where the N
i
(x) vanish and initial
conditions (19.7) and (19.8), the equations (19.13) simplify to
K
L
14
-
16
T
T
QL
1
40
6
q
K
L
T
L
2
16
-
Ê
Ë
ˆ
¯
Ê
Ë
ˆ
¯
Ê
Ë
ˆ
¯
+
Ê
ˆ
¯
+
Ê
Ë
ˆ
¯
.
1
0
=
(19.14)
Ë
6
6
-
16
32
2
Note that the variable T
3
does not appear since we know its value, which is T
L
. The
value of the heat flux at x = L is
K
dT
dx
K
L
TT T
QL
()
=
(
)
+
-
L
16
-
2
-
14
.
(19.15)
2
1
L
6
6
Although we have omitted some details, which can be found in [PepH92], the steps
outlined above show the general thrust of the FEM. To summarize, our approximate
solution to (19.11) is the function H(x) in (19.12). The only unknowns in the formula
