Civil Engineering Reference
In-Depth Information
The triangularized system then gives the nodal temperatures in succession as
T
4
=
82
.
57
◦
C
T
3
=
85
.
15
◦
C
T
2
=
90
.
14
◦
C
T
1
=
95
.
15
◦
C
The fifth equation of the system is
−
4
.
40
T
4
+
4
.
40(80)
=−
0
.
0028
q
5
which, on substitution of the computed value of
T
4
, results in
6 W/m
2
q
5
=
4038
.
As this is assumed to be a steady-state situation, the heat flow from the right-hand end of
the cylinder, node 5, should be exactly equal to the inflow at the left end. The discrepancy
in this case is due simply to round-off error in the computations, which were accom-
plished via a hand calculator for this example. If the values are computed to “machine
accuracy” and no intermediate rounding is used, the value of the heat flow at node 5 is
found to be exactly 4000 W/m
2
. In fact, it can be shown that, for this example, the finite
element solution is exact.
5.6 CLOSING REMARKS
The method of weighted residuals, especially the embodiment of the Galerkin
finite element method, is a powerful mathematical tool that provides a technique
for formulating a finite element solution approach to practically any problem for
which the governing differential equation and boundary conditions can be writ-
ten. For situations in which a principle such as the first theorem of Castigliano
or the principle of minimum potential energy is applicable, the Galerkin method
produces exactly the same formulation. In subsequent chapters, the Galerkin
method is extended to two- and three-dimensional cases of structural analysis,
heat transfer, and fluid flow. Prior to examining specific applications, we exam-
ine, in the next chapter, the general requirements of interpolation functions for
the formulation of a finite element approach to any type of problem.
REFERENCES
1.
Stasa, F. L.
Applied Element Analysis for Engineers.
New York: Holt, Rinehart, and
Winston, 1985.
2.
Burnett, D. S.
Finite Element Analysis.
Reading, MA: Addison-Wesley, 1987.
