Features and verification
===============================

To perform a calculation in *Code_Aster* in limit analysis with the Norton-Hoff-Friaâ regularization method with the VonMises resistance criterion, you must:


* define the 2D (plane or axis) or 3D model with the almost incompressible finite elements, 3D models_ INCO_UPG, D_ PLAN_INCO_UPG, or AXIS_INCO_UPG;


* ensure the incompressibility condition: GONF =0 in AFFE_CHAR_MECA;


* define only the material characteristic :math:`{s}_{y}`, the limit load being independent of

    .. image:: images/Object_209.svg
        :width: 15
        :height: 17


    .. _RefImage_Object_209.svg:

and :math:`\nu`,


* define the permanent load and the one set by :math:`\lambda`;


* define discretization in time, (in practice between :math:`{t}_{\mathit{min}}\mathrm{=}1` and :math:`{t}_{\mathit{max}}\mathrm{=}2` to :math:`5`);


* perform a non-linear calculation with the behavior relationship NORTON_HOFF with the command STAT_NON_LINE [:external:ref:`U4.51.03 <U4.51.03>`], and the control ANA_LIM. In practice, linear search can be used to improve convergence, and the subdivision of the time step,


* post-process the calculation to get the load limit with the POST_ELEM [:external:ref:`U4.81.22 <U4.81.22>`] command.

The use of these commands is detailed in document [:external:ref:`U2.05.04 <U2.05.04>`].

As far as post-processing is concerned, the operator POST_ELEM then produces a table which gives 2 parameters for each moment of the calculation, i.e. for increasingly weaker regularizations:


* the parameter 'CHAR_LIMI_SUP' contains an upper bound of the limit load, by integration on each finite element and a sum over all the elements of the model: :math:`{\stackrel{ˆ}{\lambda }}_{m}={\int }_{\Omega }{\sigma }_{y}\sqrt{\frac{2}{3}\varepsilon ({u}_{m})\text{.}\varepsilon ({u}_{m})}d\Omega \text{}-{L}_{0}({u}_{m})`


* and, in the absence of constant loading, (CHAR_CSTE =' NON '), the parameter' CHAR_LIMI_ESTIMEE 'contains an estimate of a lower bound :math:`{\underline{\lambda }}_{m}` corresponding to: :math:`{\underline{\lambda }}_{m}={\int }_{\Omega }\frac{A(m)}{m}\text{.}{(\sqrt{\varepsilon ({u}_{m})\text{.}\varepsilon ({u}_{m})})}^{m}d\Omega \text{.}{(\underset{x\in \Omega }{\text{Sup}}(\frac{\sqrt{\frac{3}{2}{\sigma }^{D}({u}_{m})\text{.}{\sigma }^{D}({u}_{m})}}{{\sigma }_{y}}))}^{\text{-}1}\le {\stackrel{ˆ}{\lambda }}_{m}`

If a constant loading is present, (then it is imperative to enter CHAR_CSTE = 'OUI'), the parameter PUIS_CHAR_CSTE represents the power of the constant loading in the speed field solution of the problem.

Several verification tests are available, in particular test SSNV124 [:external:ref:`V6.04.124 <V6.04.124>`]. On this very simple problem, an analytical calculation makes it possible to obtain the exact limit load in the direction of loading, as well as the estimates produced by the regularization method. For more details, refer to [:ref:`bib4 <bib4>`] and [:ref:`bib5 <bib5>`].

On the other hand, additional validations have been carried out as part of comparative studies, such as the European benchmark LISA [:ref:`bib8, bib10 <bib8, bib10>`]: on limit load calculations in 2D, 2D axis and 3D, the regularized kinematic method presented here makes it possible to gain a factor of 6 to 10 in the calculation time compared to an incremental elastoplastic calculation, and makes it possible to obtain a framework for the limit load, unlike the methods of the other participants.