Finite Element Analysis. Barna Szabó

      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.

      1.3.6 Condensation

      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

(1.77)

      we get

      (1.78)

The condensed stiffness matrices and load vectors are assembled and the Dirichlet boundary conditions are enforced as described in the following section. Upon solving the assembled system of equations the coefficients of the internal basis functions are computed from eq. (1.77) for each element.

      1.3.7 Enforcement of Dirichlet boundary conditions

      When Dirichlet conditions are specified on either or both boundary points then is split into two functions; a function and an arbitrary specific function from