Finite Element Analysis. Barna Szabó

      Using five elements of equal length on the interval and assigned to each element, find the finite element solution for this problem.

Referring to equations (1.66) and (1.70), the element‐level coefficient matrix for each element is

      where we used , , . The assembled unconstrained coefficient matrix is:

      Upon enforcement of the Dirichlet conditions the system of equations is

alternatively:

      where the first and sixth equations are placeholders for the boundary conditions , . The solution is:

      Exercise 1.14 Solve the problem in Example 1.7 with the boundary conditions , .

      Exercise 1.15 Solve the problem in Example 1.7 with the boundary conditions , .

1.4 Post‐solution operations

      Following assembly of the coefficient matrix and enforcement of the essential boundary conditions (when applicable) the resulting system of simultaneous equations is solved by one of several methods designed to exploit the symmetry and sparsity of the coefficient matrix. The solvers are classified into two broad categories; direct and iterative solvers. Optimal choice of a solver in a particular application is based on consideration of the size of the problem and the available computational resources.

      At the end of the solution process the finite element solution is available in the form

      (1.81)

      where the indices reference the global numbering and Nu is the number of degrees of freedom plus the number of Dirichlet conditions.

      The basis functions are decomposed into their constituent shape functions and the element‐level solution records are created in the local numbering convention. Therefore the finite element solution on the kth element is available in the following form:

      (1.82)

      1.4.1 Computation of the quantities of interest

      The computation of typical engineering quantities of interest (QoI) by direct and indirect methods is outlined in this section.

      Computation of uFE(x₀)

Direct computation of in the point involves a search to identify the element Ik in which point x₀ lies and, using the inverse of the mapping function defined by