Upper Bound and Mesh Convergence

For testing the upper bound and mesh convergence of FE solution, we select the following 2 problems shown in Figures 17.2(a) and 17.2(b).

Figure 17.2 Problems for Upper Bound and Mesh Convergence

The exact solutions of the above problems are given by :

,   for problem (a),
,  for problem (b)	(17.4)

Note that these solutions are not polynomials in x . We calculate the strain energy of the exact solution using eq. (9.25) :

(17.5)

Now we choose a uniform mesh of N elements with some value for N . Further, we choose the order of approximation p as one. Then, we obtain the FE solution by using the FE code. We calculate strain energy of the FE solution using eq. (10.10) :

(17.6)

where { F } is the global force vector and { U } is the vector containing nodal values of the displacement. We compare with to test whether the inequality

(17.7)

is satisfied or not. Next, we obtain by keeping the value of p fixed but by refining the mesh, i.e., by increasing N . Then also the inequality (17.7) should be satisfied. Further, as N is increased, should approach monotonically. If we choose some other value of p instead of one, then also the inequality (17.7) and the monotonic convergence of to from below should be observed.