Graphics Reference
In-Depth Information
CHAPTER 19
The Finite Element Method
19.1
Introduction
This is another very brief chapter, which is only included because the finite element
method (FEM), also referred to as finite element analysis (FEA), is very important for
some tasks for which geometric modelers are used and most modelers support it in
one form or another. Even though it is a huge topic, it is rather specialized and tan-
gential to the main thrust of this topic. Nevertheless, it seems appropriate to say a few
things about it because the term “finite element method” is often mentioned in con-
nection with geometric modelers and so a person wanting to be knowledgeable about
geometric modeling should know something about it. Two general-purpose references
are [Buch95] and [John87].
Section 19.2 gives a brief overview of the topic. The mathematics behind it are
described in general terms in Section 19.3 and we work through one example in
Section 19.4. Finally, Section 19.5 summarizes some main points.
19.2
What Is It All About?
The explanation for most physical phenomena in science reduces to solving differen-
tial (or integral) equations. An exact solution for these equations is usually very diffi-
cult if not impossible to obtain and so what is wanted is a computable approximation
to the solution. (The need for computable approximations would exist even if one
could solve the equation.) The classical numerical approach to solving differential
equations is using the method of finite differences. With this method, derivatives are
replaced with quotients of differences and solved using the values at some finite mesh
of points. See [ConD72] or [Horn75] for a general introduction to the subject. The
finite element method is also a method to solve differential and integral equations.
Introduced by engineers in the early 1950s and 1960s to obtain solutions to partial
differential equations in structural engineering, such as elasticity and plate equations,
it soon became clear that it had much wider applications. In subsequent years it came
to be used in a wide variety of engineering applications. The method also subdivides
