Modeling A ============== The list of moments is used to control the Norton-Hoff regularization method by means of a coefficient :math:`t`, (:math:`m=1+{\text{10}}^{1-t}`), and not the evolution of the load as during an ordinary calculation. To perform a calculation in *code_aster* in limit analysis with the Norton-Hoff-Friaâ regularization method with the vonMises resistance criterion, it is necessary: * 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 characteristic of the material :math:`{s}_{y}`, the limit load being independent of .. image:: images/Object_43.svg :width: 15 :height: 17 .. _RefImage_Object_43.svg: and :math:`\mathrm{\nu }`; * define the permanent load and the one set by :math:`\lambda`; * define discretization in time, (in practice between :math:`{t}_{\mathit{min}}=1` and :math:`{t}_{\mathit{max}}=2` to :math:`5`); * perform a non-linear calculation with the behavior relationship NORTON_HOFF with the command STAT_NON_LINE [:ref:`U4.51.03 `], and the control ANA_LIM. In practice, linear research 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 [:ref:`U4.81.22 `] command. Characteristics of modeling ----------------------------------- We consider a cylinder modelled by incompressible axisymmetric elements of type QUAD8. Characteristics of the mesh ---------------------------- +---------------------------------------------------------------------------------------------------------------------------------------------------------------------+-----------------------------------------------------------------------------------------------------------------------------+ |Number of knots: 141 | | + + .. image:: images/100002010000032000000243A3A4CAE37FDC21AA.png + | Number of meshes: 51 | :width: 2.9846in | + + :height: 2.1598in + | Type of mesh: 34 meshes of type QUAD8incompressible; 17 meshes of type SEG3pour when applying pressure. | | + + + | | | +---------------------------------------------------------------------------------------------------------------------------------------------------------------------+-----------------------------------------------------------------------------------------------------------------------------+ Tested sizes and results ------------------------------ With the mesh in question, the calculation of the test case is stopped at :math:`t=\mathrm{2,450}s`. Additional details on the calculation are provided in the user support document [:ref:`U2.05.04 `]. The Norton-Hoff coefficient corresponding to this moment is :math:`m=\mathrm{1,0355}`. The table below shows the extraction of the values of :math:`\mathrm{\lambda }` at certain times when the calculation is stopped, as well as the comparison with the reference values. +--------------------+------------------------+-------------------------+------------------------+------------------------+ |**Modeling** |Estimated upper value |Estimated lower value | +--------------------+------------------------+-------------------------+------------------------+------------------------+ |**EDF** |:math:`t=2` |:math:`m=\mathrm{1,1}` |:math:`\mathrm{3,94863}`|:math:`\mathrm{3,27091}`| + +------------------------+-------------------------+------------------------+------------------------+ | |:math:`t=\mathrm{2,125}`|:math:`m=\mathrm{1,0750}`|:math:`\mathrm{3,93504}`|:math:`\mathrm{3,40992}`| + +------------------------+-------------------------+------------------------+------------------------+ | |:math:`t=\mathrm{2,25}` |:math:`m=\mathrm{1,0562}`|:math:`\mathrm{3,92505}`|:math:`\mathrm{3,52215}`| + +------------------------+-------------------------+------------------------+------------------------+ | |:math:`t=\mathrm{2,450}`|:math:`m=\mathrm{1,0355}`|:math:`\mathrm{3,91429}`|:math:`\mathrm{3,65061}`| +--------------------+------------------------+-------------------------+------------------------+------------------------+ |Univ. of Liège/ LTAS |:math:`\mathrm{3,931}` |nought | +--------------------+------------------------+-------------------------+------------------------+------------------------+ |Research Center FZJ |nought |:math:`\mathrm{3,997}` | +--------------------+------------------------+-------------------------+------------------------+------------------------+ .. csv-table:: "**Value tested**", "**Reference -** SOURCE_EXTERNE ", "**Accuracy**" "CHAR_LIMI_SUPà :math:`t=2` "," :math:`\mathrm{3,931}` "," :math:`\mathrm{0,01}`"