  Robust global/goal-oriented error estimation is used nowadays to control the
approximate finite element solutions obtained from simulation. In the context
of Computational Mechanics, the construction of admissible stress fields (\ie
stress tensors which verify the equilibrium equations) is required to set up
strict and guaranteed error bounds (using residual based error estimators) and
plays an important role in the quality of the error estimates. This work
focuses on the different procedures used in the calculation of admissible
stress fields, which is a crucial and technically complicated point. The three
main techniques that currently exist, called the element equilibration
technique (EET), the star-patch equilibration technique (SPET), and the element
equilibration + star-patch technique (EESPT), are investigated and compared
with respect to three different criteria, namely the quality of associated
error estimators, computational cost and easiness of practical implementation
into commercial finite element codes. The numerical results which are presented
focus on industrial problems; they highlight the main advantages and drawbacks
of the different methods and show that the behavior of the three estimators,
which have the same convergence rate as the exact global error, is consistent.
Two- and three-dimensional experiments have been carried out in order to
compare the performance and the computational cost of the three different
approaches. The analysis of the results reveals that the SPET is more accurate
than EET and EESPT methods, but the corresponding computational cost is higher.
Overall, the numerical tests prove the interest of the hybrid method EESPT and
show that it is a correct compromise between quality of the error estimate,
practical implementation and computational cost.
