Modeling A
==============


Characteristics of modeling
-----------------------------------

Modeling in shell elements (DKT).


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    :width: 3.1709in
    :height: 3.298in


.. _RefImage_100000000000049C00000423B5D5829E0095EB29.png:


.. image:: images/10000000000003EC0000044F660BC956C9F3E610.png
    :width: 2.8965in
    :height: 3.4465in


.. _RefImage_10000000000003EC0000044F660BC956C9F3E610.png:

**Figure 3.1-a: geometry and mesh of the half cylinder (initial RC)**


Characteristics of the mesh
----------------------------

The mesh is regenerated at each iteration due to the change in the radius of curvature of the geometry. However, the topological characteristics of the mesh are unchanged:


* 1013 meshes (900 TRIA3, 110 SEG2, 3 POI3)


* 507 knots.

The mesh groups correspond to:


* :math:`\mathrm{GM30}` surface of :math:`\mathrm{TUYAU}`


* :math:`\mathrm{GM28}\iff` section B (effort)


* :math:`\mathrm{GM31}\iff` point :math:`\mathrm{A1}` :math:`(-R,\mathrm{0,}0)`


* :math:`\mathrm{GM27}\iff` section :math:`A` (embedding)


* :math:`\mathrm{GM29}\iff \mathrm{SYMETRIE}`


Aster commands
---------------

This paragraph describes the algorithm used in the command file and introduces the *Code_Aster* commands used.


* Initialization of some variables:


* Radius of curvature: :math:`\mathrm{Rc}=0.3`


* Convergence criterion: :math:`\mathrm{crit}=2.0E+09` (target Von Mises constraint)


* We enter the python loop whose stopping criterion relates to the Von Mises max constraint. As long as the criterion is not met, the following instructions are carried out:


* Reconstruction of the mesh:


* we redefine :math:`\mathrm{Rc}` in the GMSH geometry.geo file


* we launch gmsh via python to generate the.msh mesh file


* Reading the mesh (PRE_GMSH) and generating the mesh (LIRE_MAILLAGE). We use DEFI_GROUP to rename mesh groups according to correspondence:


.. code-block:: text
 # GM30 <=> TUYAU # GM28 <=> EFOND
# GM31 <=> A1 # GM27 <=> ENCAST
# GM29 <=> SYMETRIE


* Definition of the finite elements used (AFFE_MODELE). The straight pipes and the elbow are modeled by shell elements (DKT).


* Reorienting the normals to the elements: we use MODI_MAILLAGE to orient all the elements in the same way, with a normal facing inwards.


* Definition and assignment of the material (DEFI_MATERIAU and AFFE_MATERIAU). The mechanical characteristics are identical throughout the structure.

Python loop for optimizing the radius of curvature


* Assigning the characteristics of shell elements (AFFE_CARA_ELEM): thickness, vector V defining the stripping coordinate system (keyword ANGL_REP)


* Definition of boundary conditions and loading (AFFE_CHAR_MECA).


* The pipe is embedded at its base, on all the nodes located in the Y=0 plane. The pipe has a :math:`Z=0` plane of symmetry.


* We calculate a distributed force :math:`\mathrm{FY}` directed along the :math:`Y` axis and applied to the section :math:`B`, (the distributed force is such that the resultant :math:`2\pi \mathrm{.}\mathrm{RMOY.}\mathrm{FY}=\mathrm{FTOT}`, :math:`\mathrm{FTOT}` being the total force that one wishes to apply). To apply the effort to section B, we'll use FORCE_ARETE.


* Solving the linear elastic problem (MECA_STATIQUE).


* Calculation of the stress field by elements at the nodes for each load case (option 'SIGM_ELNO'). The constraints are calculated in the local coordinate system defined for each element using the vector :math:`V` (previous ANG_REP keyword). Use NIVE_COUCHE to define the level of calculation in the thickness.


* Calculation of the field of equivalent constraints by elements at the nodes calculated from the constraint field (option 'SIEQ_ELNO').


* Calculation of the fields preceding the nodes (options' SIGM_NOEU ',' SIEQ_NOEU ')


* Printing results (IMPR_RESU).


* Determination of a table containing the calculations of averages of the field of stresses equivalent to the nodes. ('POST_RELEVE_T')


* Extracting component VMIS from the previous table via python.


* Stop test:


* If VMIS is greater than CRIT, then repeat with :math:`\mathrm{Rc}=\mathrm{Rc}+0.2`


* Otherwise we leave the python loop.