Thus far we have made no specifications on the form of the basis
functions, 
which will be associated with the unknown
solution coefficient, 
. It is apparent that once we have
specified these basis functions, we can perform the various
integrations defining the matrices in equation (
), and solve
the matrix equation for the unknown coefficients, 
, from which
the shape of the solution can be constructed using equation
(). We do require that they be square-integrable over the
domain. We will impose one further condition that will make the
identification of the 
's more transparent. If we specify that
each basis function be equal to one at precisely one element boundary
or node, and precisely zero at all other nodes, then the solution for
the 
's yields us precisely the solution for the u's at each
of these nodes. With the condition
equation () becomes
where 
replaces 
and can be identified as the value
of the solution at the node 
.
We can build such basis functions in a piece-wise manner over the
smooth subdomains or elements by combining element shape functions
which are only defined within a particular element. For example, in
Figure 1 we show a 6-node, 5-element 1-dimensional domain. Also
shown by the solid line on the figure is the basis function
, corresponding to the fourth node. Note that this basis
function is precisely 1 at node 4, and precisely zero at all others. It
is apparent that this basis function satisfies the requirements of
being square-integrable over the domain, and that it has the required
differentiability; i.e., that first derivatives exist.
Focusing on the region of the basis function where it is nonzero (from
node 3 to node 5), we see that it is composed of two pieces, one that is
zero at node 3 and 1 at node 4 (element 3) and one that is 1 at node 4
and zero at node 5 (element 4). These two pieces represent individual
shape functions from these two elements. Also note that each element
contains pieces that will contribute to precisely two basis functions.
We define the shape functions, 
, on a particular element to have
the properties that 
is precisely one at the 
node of
the element, and that it is precisely zero at all the other nodes in
the element. Thus basis function 
is composed of shape
function 
from element 3 and shape function 
from
element 4. By the same reasoning we could assemble in a piece-wise
manner basis functions for each of the nodes in the domain from
precisely two of the shape functions from the two adjacent elements.
Now we look back at the integrals from 0 to L of equations
() and () that were used to define the various
entries in the stiffness matrix and the load vector. First we notice
that integrals of a single basis function, or of a basis function times
itself, need only be integrated over the nonzero region of the basis
function. Thus 
and 
only need only be integrated from node 3
to node 5 as opposed to over the entire domain. Terms involving
dissimilar basis functions, such as 
or 
, need only be
integrated over the region where the two basis functions overlap,
namely element 3. If we break up our global integration from 0 to L
into separate integrations over each element, we find that there are
only four contributions to the global stiffness matrix, and that there
are only two contributions to the global load vector. Thus for each
element we will have expressions analogous to equations ()
and (). These are the element stiffness matrix and load
vector, and the indices i and j range from 1 to the number of nodes
in an element, 
.
Performing the global integration of the N basis functions over the
entire domain collapses to performing local integration of the 
shape functions over the individual elements and judiciously combining
these element matrix entries together to form the global matrices.
Traditionally one defines the shape functions in terms of local element
coordinates. Integration is then performed in terms of these local
coordinates and the results scaled by the Jacobian of the
transformation to the global coordinate system. This allows the
automation of the matrix formation to be performed quite efficiently
for irregular and complex arrangement of the elements.
