4. Use in Code_Aster#

4.1. 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 \(\alpha\) otherwise);

  • “RAPPORT”: the ratio between the current size \({h}_{E}\) and the new size \({h}_{E}^{\ast }\) of the finite element (\({h}_{E}/{h}_{E}^{\ast }=1/{r}_{E}\));

  • “TAILLE”: the new \({h}_{E}^{\ast }\) size of the finite element (\({h}_{E}^{\ast }={r}_{E}{h}_{E}\)).

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 \({\varepsilon }_{0}\) precision of equation [éq 2.2-1] to be calculated as follows: \({\varepsilon }_{0}=\text{PREC\_ERR}\ast {\text{Erreur}}_{\text{totale}}\). 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.

4.2. 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. [§2]), the calculation of the order of the singularity is obtained from the theoretical energy at the crack point [éq 2.3.1-2], an equation valid only in elasticity. The use of this option in elastoplasticity should therefore be handled with caution.