Civil Engineering Reference
In-Depth Information
4.5
Exact
Two elements
Four elements
4.0
3.5
3.0
2.5
2.0
1.5
1.0
0.25
0.5
0.75
1.0
x
L
Figure 1.5
Comparison of the computed axial stress value in a
tapered cylinder:
0
F
A
0
.
the derived variables are not necessarily continuous. In the limiting process of
mesh refinement, the derived variables become closer and closer to continuity.
Our example shows how the finite element solution converges to a
known
exact solution (the exactness of the solution in this case is that of strength of
materials theory). If we know the exact solution, we would not be applying the
finite element method! So how do we assess the accuracy of a finite element solu-
tion for a problem with an unknown solution? The answer to this question is not
simple. If we did not have the dashed line in Figure 1.3 representing the exact
solution, we could still discern convergence to
a
solution. Convergence of a
numerical method (such as the finite element method) is by no means assurance
that the convergence is to the correct solution. A person using the finite element
analysis technique must examine the solution analytically in terms of (1) numeri-
cal convergence, (2) reasonableness (does the result make sense?), (3) whether the
physical laws of the problem are satisfied (is the structure in equilibrium? Does the
heat output balance with the heat input?), and (4) whether the discontinuities in
value of derived variables across element boundaries are reasonable. Many
such questions must be posed and examined prior to accepting the results of a finite
element analysis as representative of a correct solution useful for design purposes.
1.2.2 Comparison of Finite Element and Finite
Difference Methods
The
finite difference
method is another numerical technique frequently used to
obtain approximate solutions of problems governed by differential equations.
Details of the technique are discussed in Chapter 7 in the context of transient heat
