A new radiality criterion for the evaluation of the time discretization error

17 January 2012

by J.-M. Proix, EDF R&D / AMA

For elasto-(visco)-plastic models, non-linear material behavior requires the use of a numerical integration scheme. The integration scheme is intended to transcribe the continuous formulation of the model to a discrete-time expression. The majority of models available in Code_Aster are integrated according to an implicit scheme with radial return. These schemes are very robust, unconditionally stable, and flexible enough to be adapted to most of behaviors. These schemes are also known for their accuracy when they are applied to the assessment of the internal variables of the material point solution.

However, for non-radial loading paths, ie with a change in the loading direction, the radial-return schemes (like other schemes) significantly lose their accuracy for large time steps (but converge to the solution if the time step tends to zero). It is therefore necessary to take into account these errors and alert the users or help them in the continuation of their calculation, for example by applying a finer discretization of the load, or automatically subdivide the time step when the radiality criterion loss reaches a critical value.

A new criterion is available via the DERA_ELGA option, operator CALC_ELEM. Its value is an approximation of the error of stress or strain during the integration of the constitutive law, according to the time discretization.

This criterion is available for the constitutive laws VMIS_ISOT_*, VMIS_CINE_LINE, VMIS_ECMI_*, and all the Chaboche behaviours : VMIS_CIN1_CHAB, VMIS_CIN2_CHAB, VMIS_CIN2_MEMO. During the integration, the code rises a return code if the criterion is not verified. This makes it possible to reduce automatically the time step. This criterion is activated by the keyword RESI_RADI_RELA in COMP_INCR, see for example the test cases SSNP14, SSNP15.

Test case of the perforated plate: it is found that the radiality criterion actually decreases with the time step, and the adaptive time step provides the most accurate result.
SSNP15 test case (non proportional traction and shear on a volume element) - evolution of plastic deformation versus time for a calculation with coarse discretization, a calculation with time step adaptation on a Delta p criterion (DEFI_LIT_INST / DELTA_GRANDEUR), and a calculation with time step adaptation on a radiality criterion (RESI_RADI_RELA=0.02, with a maximum error of 2%). We note that the solution with coarse time steps is not accurate. The radiality criterion do not lead to a subdivision at the beginning of the load (because the traction and shear are then proportional, there is no integration error). So we save computation time.