Computation of mean normal stresses (Hencky model)

As a second test, we compute the mean normal stress over that portion of the boundary along which the body is fixed,

$$\sigma \int_{\Gamma_{u}}^{} \sigma_{nn}^{}$$ (52)

In this example, our weighted error estimator is again resonably accurate, but at the same time yields more economical meshes than the other error estimators. This is also in agreement with our observations in the linear elastic case.

Tabelle: Results for  J_$\sigma$(u_h)  with adaptivity based on the weighted approach

N	J$\sigma$(uh)	Erelweight	Ratioweight
1,000	2.2224e+02	1.5757e-02	1.6650e+00
2,000	2.2344e+02	1.0430e-02	2.1377e+00
4,000	2.2405e+02	7.7387e-03	1.7360e+00
6,000	2.2431e+02	6.6019e-03	1.7071e+00
8,000	2.2475e+02	4.6647e-03	1.6205e+00
10,000	2.2487e+02	4.1337e-03	1.7561e+00
12,000	2.2509e+02	3.1559e-03	1.5455e+00
14,000	2.2533e+02	2.0855e-03	2.4058e+00
16,000	2.2532e+02	2.1360e-03	1.7726e+00
18,000	2.2534e+02	2.0496e-03	1.7669e+00
$\infty$	2.2580e+02

Tabelle: Comparison of results for  J_$\sigma$(u_h)  for the different error estimators

N	Erelweight	ErelE	ErelZZ
1,000	1.5757e-02	1.7511e-02	1.4933e-02
2,000	1.0430e-02	1.6593e-02	1.4773e-02
4,000	7.7387e-03	9.6399e-03	8.5943e-03
6,000	6.6019e-03	9.2378e-03	8.2834e-03
8,000	4.6647e-03	8.1205e-03	8.4455e-03
10,000	4.1337e-03	8.4973e-03	7.5992e-03
12,000	3.1559e-03	7.8574e-03	7.1838e-03
14,000	2.0855e-03	8.1488e-03	5.9035e-03
16,000	2.1360e-03	8.3264e-03	5.9473e-03

Abbildung: Relative error for  J_$\sigma$(u_h)  on grids based on the different estimators.

Abbildung: Stucture of grids for  J_$\sigma$(u_h)  with  N $\approx$ 8, 100