Barna Szabó - Finite Element Analysis

Exercise 1.12Let

where fk is a constant. Compute numerically in terms of fk and ℓk using 3, 4 and 5 Gauss points. See Appendix E. Use the Legendre basis functions.

Exercise 1.13Assume that is a linear function on I . Using the Lagrange shape functions for , compute .

Having computed the coefficient matrices and right hand side vectors for each element, it is necessary to form the coefficient matrix and right hand side vector for the entire mesh. This process, called assembly, executes the summation in equations (1.62), (1.67)and (1.73). The local and global numbering of variables is reconciled in the assembly process. The algorithm is illustrated by the following example.

Example 1.6Consider the three‐element mesh shown in Fig. 1.5. The polynomial degrees , , are assigned to elements 1, 2, 3 respectively. The basis functions shown in Fig. 1.5are composed of the mapped Legendre shape functions. For instance, the basis function is composed of the mapped shape function N 2from element 1 and the mapped shape function N 1from element 2. This basis function is zero over element 3. Basis function is the mapped shape function N 3from element 3. This basis function is zero over elements 1 and 2.

Figure 1.5 Typical finite element basis functions in one dimension.

Table 1.1 Local and global numbering in Example 1.6.

Each basis function is assigned a unique number, called a global number, and this number is associated with those element numbers and the shape function numbers from which the basis function is composed. The global and local numbers in this example are indicated in Table 1.1.

We denote and, using equations (1.62)and (1.67), write in the following form:

where the elements within the brackets are in the local numbering system whereas the coefficients aj and bi outside of the brackets are in the global system. The superscripts indicate the element numbers. The terms multiplied by are summed to obtain the elements of the assembled coefficient matrix which will be denoted by . For example,

Assuming that the boundary conditions do not include Dirichlet conditions, the bilinear form can be written in terms of the coefficient matrix as:

(1.76)

The treatment of Dirichlet conditions will be discussed separately in the next section.

The assembly of the right hand side vector from the element‐level right hand side vectors is analogous to the procedure just described. Referring to eq. (1.73)we write in the following form:

where , , , etc.

Each element has internal basis functions. Those elements of the coefficient matrix which are associated with the internal basis functions can be eliminated at the element level. This process is called condensation.

Let us partition the coefficient matrix and right hand side vector of a finite element with such that

where the and . The coefficient matrix is symmetric therefore . Using