Finite Element Analysis. Barna Szabó

Finite Element Analysis

or 5.22%. When using the exact value of the potential energy for reference then the relative error is the same as the estimated relative error to within three digits of accuracy:

left-parenthesis e Subscript r Baseline right-parenthesis Subscript upper E Baseline equals StartRoot StartFraction pi left-parenthesis u Subscript upper F upper E Baseline right-parenthesis minus pi left-parenthesis u Subscript upper E upper X Baseline right-parenthesis Over double-vertical-bar u Subscript upper E upper X Baseline double-vertical-bar Subscript upper E left-parenthesis upper I right-parenthesis Superscript 2 Baseline EndFraction EndRoot equals StartRoot StartFraction negative 1.41382648 plus 1.41768859 Over 1.41768859 EndFraction EndRoot equals 0.0522 period

Exercise 1.20 Compare the estimated and exact values of the relative error in energy norm for the problem in Example 1.10 for upper M left-parenthesis normal upper Delta right-parenthesis equals 100 , alpha equals 0.7 .

Example 1.11 Let us consider once again model problems in the form of eq. (1.103) with the data: script l equals 1 , kappa equals 1 , c equals 50 and exact solutions corresponding to alpha equals 0.6 comma 0.7 comma 0.8 comma 0.9 , see eq. (1.104). Using a sequence of uniform finite element meshes with upper M left-parenthesis normal upper Delta right-parenthesis equals 10 comma 100 comma 1000 comma 10 comma 000 and p equals 2 assigned to each element, the results shown in Fig. 1.9 are obtained. The values of β were computed by linear regression using eq. (1.101). We observe that beta equals alpha minus 1 slash 2 . This is consistent with the asymptotic estimate given by eq. (1.91).

Example 1.12 Let us consider model problems in the form of eq. (1.103) with the data: script l equals 1 , kappa equals 1 , c equals 50 and exact solutions corresponding to alpha equals 0.6 comma 0.7 comma 0.8 comma 0.9 , see eq. (1.104). Using a uniform finite element mesh with upper M left-parenthesis normal upper Delta right-parenthesis equals 10 and p equals 2 comma 3 comma 4 comma 5 assigned to each element, the results shown in Fig. 1.10 are obtained. The values of β were computed by linear regression using eq. (1.101). We observe that beta equals 2 left-parenthesis alpha minus 1 slash 2 right-parenthesis , that is, the rate of convergence is twice that in Example 1.11. This is consistent with the theoretical results in [22, 84]: The rate of p‐convergence is at least twice the rate of h‐convergence when the singular point is a nodal point.

1.5.4 Error in the extracted QoI

In Example 1.9 it was demonstrated that the QoI can be extracted from the finite element solution efficiently and accurately even when the discretization was very poorly chosen. Let us consider a quantity of interest normal upper Phi left-parenthesis u right-parenthesis and the corresponding extraction function w element-of upper E left-parenthesis upper I right-parenthesis . The extracted value of the QoI is

(1.109) normal upper Phi left-parenthesis u Subscript upper F upper E Baseline right-parenthesis equals upper F left-parenthesis w right-parenthesis minus upper B left-parenthesis u Subscript upper F upper E Baseline comma w right-parenthesis

Graph depicts relative error in energy norm. M(Δ)=10,100,1000,10000, p=2.

Figure 1.9 Relative error in energy norm.

Graph depicts relative error in energy norm. M(Δ)=10, p=2,3,4,5.

Figure 1.10 Relative error in energy norm.

and the exact value of the QoI is

(1.110) normal upper Phi left-parenthesis u Subscript upper E upper X Baseline right-parenthesis equals upper F left-parenthesis w right-parenthesis minus upper B left-parenthesis u Subscript upper E upper X Baseline comma w right-parenthesis period

Subtracting eq. (1.109) from eq. (1.110) we get

(1.111)Скачать книгу