Civil Engineering Reference
In-Depth Information
compute additional, derived variables, such as reaction forces, element stresses,
and heat flow.
As it is not uncommon for a finite element model to be represented by tens
of thousands of equations, special solution techniques are used to reduce data
storage requirements and computation time. For static, linear problems, a
wave
front solver,
based on Gauss elimination (Appendix C), is commonly used. While
a complete discussion of the various algorithms is beyond the scope of this text,
the interested reader will find a thorough discussion in the Bathe topic [1].
1.3.3 Postprocessing
Analysis and evaluation of the solution results is referred to as
postprocessing
.
Postprocessor software contains sophisticated routines used for sorting, printing,
and plotting selected results from a finite element solution. Examples of opera-
tions that can be accomplished include
Sort element stresses in order of magnitude.
Check equilibrium.
Calculate factors of safety.
Plot deformed structural shape.
Animate dynamic model behavior.
Produce color-coded temperature plots.
While solution data can be manipulated many ways in postprocessing, the most
important objective is to apply sound engineering judgment in determining
whether the solution results are physically reasonable.
1.4 BRIEF HISTORY OF THE FINITE
ELEMENT METHOD
The mathematical roots of the finite element method dates back at least a half
century. Approximate methods for solving differential equations using trial solu-
tions are even older in origin. Lord Rayleigh [2] and Ritz [3] used trial functions
(in our context, interpolation functions) to approximate solutions of differential
equations. Galerkin [4] used the same concept for solution. The drawback in the
earlier approaches, compared to the modern finite element method, is that the
trial functions must apply over the
entire
domain of the problem of concern.
While the Galerkin method provides a very strong basis for the finite element
method (Chapter 5), not until the 1940s, when Courant [5] introduced the con-
cept of piecewise-continuous functions in a subdomain, did the finite element
method have its real start.
In the late 1940s, aircraft engineers were dealing with the invention of the jet
engine and the needs for more sophisticated analysis of airframe structures to
withstand larger loads associated with higher speeds. These engineers, without
the benefit of modern computers, developed matrix methods of force analysis,
