Finite Element Analysis. Barna Szabó

computed by the direct method.

      which is the exact solution. The choice was exceptionally fortuitous because it happens to be the Green's function (also known as the influence function) for . Therefore the extracted value is independent of the solution .

      Let us choose for the extraction function. In this case

      Substituting for :

      Taking the orthogonality of the Legendre polynomials (see eq. (D.13)) into account, the sum has to be evaluated only for . The extracted value of for is (31.25% error).

      An explanation of why the extraction method is much more efficient than direct computation is given in Section 1.5.4.

      Exercise 1.16 Find for the problem in Example 1.7 by the direct and indirect methods. Compute the relative errors.

      Exercise 1.17 For the problem in Example 1.9 let be the extraction function. Calculate the extracted value of for .

      Nodal forces

      The vector of nodal forces associated with element k, denoted by , is defined as follows:

      (1.88)

      where is the stiffness matrix, is the solution vector and is the load vector corresponding to traction forces, concentrated forces and thermal loads acting on element k.

      The sign convention for nodal forces is different from the sign convention for the bar force: Whereas the bar force is positive when tensile, a nodal force is positive when acting in the direction of the positive coordinate axis.

Exercise 1.18 Assume that hierarchic basis functions based on Legendre polynomials are used. Show that when κ is constant and on Ik then

Figure 1.8 Exercise 1.18. Notation.

independently of the polynomial degree pk. For sign convention refer to Fig. 1.8. Consider both thermal and traction loads. This exercise demonstrates that nodal forces are in equilibrium independently of the finite element solution. Therefore equilibrium of nodal forces is not an indicator of the quality of finite element solutions.

1.5 Estimation of error in energy norm

We have seen that the finite element solution minimizes the error in energy norm in the sense of eq. (1.48). It is natural therefore to use the energy norm as a measure of the error of approximation. There are two types of error estimators: (a) A priori estimators that establish the asymptotic