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.