Graphics Reference
In-Depth Information
Figure 19.2.
Quadratic one-dimensional elements and nodes.
expensive. Suppose that one wanted to use quadratic polynomials. The values at the
endpoints of intervals would be insufficient to completely specify the polynomial since
it has three degrees of freedom. What one would do is introduce an extra node on the
interior of intervals. Figure 19.2 shows how a function is approximated by two quad-
ratic elements, each of which is defined by three nodes. For a cubic one would use
two extra interior nodes. As an aside, recall that a cubic is completely specified on an
interval if one knows the values and derivatives at the endpoints. It might therefore
occur to the reader that this would avoid the introduction of interior nodes. However,
having to find the value a derivative would be expensive computationally. In any case
what we have is a global solution (actually an approximation), which, being a collec-
tion of local solutions - the
elements
, depends on a finite set of unknowns, namely,
some to be specified data at the
nodes
. As we shall see, the local solutions will actu-
ally be indexed by the nodes, not the elements. To summarize, the key element to
getting solutions using the approach just described is having a ready collection of
interpolating splines on hand to serve as basis functions for each element. These basis
functions are called the
local shape functions
.
Higher-dimensional problems are handled in a similar fashion. For example, in
the two-dimensional case one is trying to approximate a function defined by a differ-
ential equation over a region
A
in the plane. This region could be arbitrary and does
not need to be rectangular. One has to subdivide the region by means of a mesh of
nodes. Typical shapes for the elements are quadrilaterals or triangles. Over each of
these regions with their associated nodes one defines approximations, which are
typically low-degree polynomials (splines). One ends up with a global approximation
that depends on a finite number of unknowns coming from values at the nodes. Again,
the choice of basis function for each element is up to the user. Figure 19.3(a) shows
a linear triangular element. Figure 19.3(b) shows a four-point interpolating surface
for a quadrilateral element (see Section 12.6, equation (12.17)).
19.3
The Mathematics Behind FEM
This section gives a brief overview of the mathematics behind the FEM. As we men-
tioned earlier, there are basically two approaches.
