Overscript p minus 1 Endscripts upper N Subscript i plus 2 Baseline left-parenthesis xi overbar right-parenthesis StartRoot StartFraction 2 i plus 1 Over 2 EndFraction EndRoot integral Subscript negative 1 Superscript 1 Baseline upper P Subscript i Baseline left-parenthesis xi right-parenthesis left-parenthesis upper P 0 left-parenthesis xi right-parenthesis plus upper P 1 left-parenthesis xi right-parenthesis right-parenthesis d xi equals negative three thirty-seconds dot EndLayout"/>
Taking the orthogonality of the Legendre polynomials (see eq. (D.13)) into account, the sum has to be evaluated only for
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
Exercise 1.17 For the problem in Example 1.9 let
Nodal forces
The vector of nodal forces associated with element k, denoted by
(1.88)
where
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
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 rate of convergence of a discretization scheme, given information about the regularity (smoothness) of the exact solution and (b) a posteriori estimators that provide estimates of the error in energy norm for the finite element solution of a particular problem.
There is a very substantial body of work in the mathematical literature on the a priori estimation of the rate of convergence, given a quantitative measure of the regularity of the exact solution and a sequence of discretizations. The underlying theory is outside of the scope of this book; however, understanding the main results is important for practitioners of finite element analysis. For details we refer to [28, 45, 70, 84].
1.5.1 Regularity
Let us consider problems the exact solution of which has the functional form
where
