4. Material nonlinearity

This part treats the most common problems encountered when using nonlinear behavioral laws.

4.1. Parameters related to behavior resolution

Most of Code_Aster’s laws of behavior are integrated analytically. In the event of an error in the integration of the law of behavior, there is therefore no numerical parameter on which to play. The usual solution consists in re-dividing the global time step (or in choosing a finer list of moments).

Some laws of behavior are resolved locally via an iterative numerical method, for which the user can choose the maximum allowed residue and the maximum number of iterations locally. Most of the time, the default settings (RESI_INTE_RELA = \({10}^{-6}\) and ITER_INTE_MAXI = 20) are sufficient. In the event of an error in integrating the law, as before, you can subdivide the time step, but also allow additional local iterations (ITER_INTE_MAXI). For example, for MONOCRISTAL, it’s common to allow 100 or 200 local iterations. The choice of the residue is more delicate. Indeed, it is important that the behavior is properly integrated. This means that it is imperative not to increase the residue beyond the default value (\({10}^{-4}\) for example), except for behaviors integrated by an explicit algorithm. On the other hand, it is sometimes advisable to tighten the residue (for example RESI_INTE_RELA = \({10}^{-10}\)). In fact, depending on how the behavior is implemented, the residue does not have the same physical meaning (stress, deformation…). It is then necessary to refer to the reference documentation for the law of conduct in question.

4.2. Management of plastic incompressibility

If the material is incompressible (\(\nu >0.45\)) or in case of strong plastic deformations, oscillations on the stresses or on the stress trace may occur. Some special finite elements, such as sub-integrated elements or mixed elements, make it possible to deal with these problems [U2.01.10]. If the use of sub-integrated elements does not solve the problem, then we recommend using the mixed formulation with 2 fields (displacement and pressure) in small deformations and the mixed formulation with 3 fields in large deformations. Note that the latter only works on quadratic meshes.

4.3. Landfill

In the event of a discharge, problems of convergence of the Newton algorithm may be encountered. We recommend using elastic prediction (in STAT_NON_LINE under the NEWTON keyword). It is also useful to activate the sub-division of the time step in case of a singular matrix (this is automatic in version 11, but to be activated by hand in previous versions via STOP_SINGULIER = “DECOUPE” = “” in STAT_NON_LINE under the keyword SOLVEUR).

4.4. Plane stresses

The treatment of the planar stress condition is carried out in the general framework by the Deborst method. It is advisable to update the tangent matrix often (every one to three Newton iterations). In some cases, convergence is achieved for Newton’s algorithm, but not for the verification of the state of plane constraints, which leads to additional iterations, or even an excessive re-division of the time step. It is then advisable to activate an additional loop to better satisfy the plane constraints during Newton iterations: ITER_CPLAN_MAXI must be chosen at least equal to 5 [U4.51.11].

4.5. Damage (softening problems)

To deal with damage issues, a lot of advice is given in [U2.05.06]. In general, it is recommended to use piloting, possibly with mixed linear research.

For certain laws of behavior (ENDO_FRAGILE and ENDO_ISOT_BETON), the IMPLEX method is proposed as an alternative to Newton’s method. We activate this method in STAT_NON_LINE under the NEWTON keyword. This method is based on an explicit extrapolation of internal variables to determine the displacements from which the behavior is implicitly integrated. The nullity of the balance is not verified. As a result, it introduces an approximation of the resolution but makes it possible to guarantee the robustness of the calculation. An error control and an optimization of the time steps are possible via the operator DEFI_LIST_INST by choosing an automatic management of the time step and a method of calculating the time steps specific to IMPLEX. From a practical point of view, the method requires the matrix to be updated at each increment and only one iteration. Before using method IMPLEX, document [R5.03.81] must be consulted.

4.6. Thermo-Hydro-Mechanical (THM)

The problems of THM involve very specific concepts. It is recommended to consult the documentation [U2.04.05]. One gives here some general tips for resolution. The initialization of fields is a delicate step that must be taken care of (ETAT_INIT). It is necessary to use the re-updated tangent matrix. In case of convergence problems, it can be very useful to activate linear search (preferably mixed). However, linear search does not systematically improve convergence, so it should be handled with care as it can increase the CPU cost.