Computation of contour integrals of the deformation (Hencky model)

As a first tests, we consider the computation of the integral J_u(u) and, for simplicity, take $\psi$ $\equiv$ n, the outer normal unit vector along S,

$$\int_{S}^{} \int_{\Omega_S}^{}$$ (51)

where  $\Omega_{S}^{}$  is the domain with boundary  $\partial$$\Omega_{S}^{}$ = S .

Our weighted error estimator turns ourt to be rather sharp even on relatively coarse meshes. This indicates that the strategy of evaluating the weights  $\omega_{K}^{}$  computationally works also for the present nonlinear problem. However, we also see that in this case the meshes generated on the basis of this estimator are not more economical than those obtained by the other estimators. This is obviously due to the dominant effect of the corner singularity which is seen by all three estimators. The same effect has already been observed in the linear elastic case (see [31]).

Tabelle: Results for  J_u(u_h)  with adaptivity based on the weighted error estimator

N	Ju(uh)	Erelweight	Ratioweight
1,000	1.6760e-04	1.5291e-02	2.8855e+00
2,000	1.6817e-04	1.1934e-02	2.0956e+00
4,000	1.6875e-04	8.5311e-03	1.6575e+00
8,000	1.6926e-04	5.5153e-03	1.4065e+00
16,000	1.6963e-04	3.3684e-03	1.2725e+00
32,000	1.6986e-04	1.9982e-03	1.1725e+00
64,000	1.7000e-04	1.1569e-03	1.0546e+00
128,000	1.7010e-04	5.9107e-04	1.0955e+00
$\infty$	1.7020e-04

Tabelle: Comparison of results for  J_u(u_h)  for the different error estimators

N	Erelweight	ErelE	ErelZZ
1,000	1.5291e-02	1.4659e-02	1.5863e-02
2,000	1.1934e-02	1.0196e-02	1.0583e-02
4,000	8.5311e-03	7.8349e-03	7.6686e-03
6,000	5.7685e-03	4.5658e-03	5.2068e-03
8,000	3.4042e-03	2.5006e-03	3.3872e-03
10,000	2.0141e-03	1.1310e-03	2.3120e-03
12,000	1.1733e-03	2.1034e-04	1.7368e-03
14,000	7.7850e-04	4.0717e-04	1.4336e-03
16,000	5.4994e-04	5.4407e-04	1.0811e-03
18,000	4.4653e-04	7.0153e-04	8.3255e-04

Abbildung: Relative error for  J_u(u_h)  on grids based on the different estimators.

Abbildung: Structure of grids for  J_u(u_h)  with  N $\approx$  8, 400