Use in Code_Aster
===========================




The orders
-------------

The order of the singularity and the size change map are calculated by the CALC_ERREUR command by activating the options' SING_ELEM '(constant field per element) or' SING_ELNO '(field at the nodes per element).

The 'SING_ELEM' option calculates two components for each element:


*'DEGRE': the order of the singularity, i.e. the value of the coefficient :math:`{q}_{E}` (which is worth*p* if the element is not connected to a singular node and which is equal to :math:`\alpha` otherwise);


* 'RAPPORT': the ratio between the current size :math:`{h}_{E}` and the new size :math:`{h}_{E}^{\ast }` of the finite element (:math:`{h}_{E}/{h}_{E}^{\ast }=1/{r}_{E}`);


* 'TAILLE': the new :math:`{h}_{E}^{\ast }` size of the finite element (:math:`{h}_{E}^{\ast }={r}_{E}{h}_{E}`).

The calculation of this option requires, first, the calculation of an error estimator (the absolute component is used and is hard-coded in*Code_Aster*) and the total deformation energy. If one of these options is not calculated, an alarm message is sent and the option 'SING_ELEM' is not calculated.

The user can enter the optional keyword 'TYPE_ESTI' by specifying one of the following options:


* 'ERME_ELEM' for the residual-based estimator;


* 'ERZ (1 or 2) _ ELEM_SIGM' for the estimator based on smoothed constraints (Zhu‑Zienkiewicz version 1 or 2);


* 'QIRE_ELEM' for the estimator in quantity of interest based on residuals;


* 'QIZ (1 or 2) _ ELEM_SIGM' for the quantity of interest estimator based on the smoothed constraints (Zhu‑Zienkiewicz version 1 or 2).

If this keyword is not entered then the estimator based on the residuals' ERME_ELEM 'is chosen by default (alarm message sent). If both Zhu-Zienkiewicz estimators are present, we choose 'ERZ1_ELEM'.

For the total deformation energy, the following are used:


* With STAT_NON_LINE: 'ETOT_ELEM' which is the total deformation energy on a finite element (valid for elastic behavior and for elastoplastic behavior 'VMIS_ISOT_XXX').


* With MECA_STATIQUE: 'EPOT_ELEM' which is the potential energy of elastic deformation on a finite and integrated element based on displacements and temperature (valid only for elastic behavior).

The user must also enter the keyword 'PREC_ERR' which allows the :math:`{\varepsilon }_{0}` precision of equation [:ref:`éq 2.2-1 <éq 2.2-1>`] to be calculated as follows: :math:`{\varepsilon }_{0}=\text{PREC\_ERR}\ast {\text{Erreur}}_{\text{totale}}`. The value of 'PREC_ERR' is strictly between 0 and 1 (a fatal message is sent if this condition is not met).

For the 'SING_ELNO' option, it is a copy of the values of 'SING_ELEM' to the nodes of the element. The prior calculation of 'SING_ELEM' is therefore necessary. If 'SING_ELEM' is not present, an alarm message is sent and the 'SING_ELNO' option is not calculated.




Scope of use
-----------------------

The scope of use is the same (but smaller) than that of the error estimator chosen, namely:


* For the residual estimator: finite elements of continuous media in 2D (triangles and quadrangles) or 3D (only tetrahedra) for elastoplastic behavior,


* For the Zhu-Zienkiewicz estimator: finite elements of 2D continuous media (triangles and quadrangles) for elastic behavior.

Strictly (cf. [:ref:`§2 <§2>`]), the calculation of the order of the singularity is obtained from the theoretical energy at the crack point [:ref:`éq 2.3.1-2 <éq 2.3.1-2>`], an equation valid only in elasticity. The use of this option in elastoplasticity should therefore be handled with caution.


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