Reference problem
=====================

Geometry
---------


.. image:: images/100000000000024200000104D5C0F4BEE56929B4.png
    :width: 6.0102in
    :height: 2.6984in


.. _RefImage_100000000000024200000104D5C0F4BEE56929B4.png:


.. csv-table::

    "Ray", ":math:`R\mathrm{=}10.m`"
    "Thickness", ":math:`t\mathrm{=}0.04`"


Coordinates of the points:

.. csv-table::

    "", ":math:`A` "," :math:`B` "," :math:`C`"
    ":math:`x` ", "10. ", "0. ", "0."
    ":math:`y` ", "0. ", "10. ", "0."
    ":math:`z` ", "0. ", "0. ", "10."



Material properties
-----------------------

The hemisphere is composed of concrete reinforced by grids. As the objective is only to test controls, the same properties are given to concrete and to grids.

:math:`E\mathrm{=}200000\mathit{Pa}`, :math:`\nu \mathrm{=}0.3`


Boundary conditions and loads
-------------------------------------

Over a quarter of the hemisphere:

.. csv-table::

    "Point :math:`C` ", "no move in :math:`z`"
    "Side :math:`\mathrm{AC}` ", "symmetry with respect to plane :math:`\mathit{xz}`"
    "Side :math:`\mathrm{BC}` ", "symmetry with respect to plane :math:`\mathit{yz}`"
    "Side :math:`\mathrm{AB}` ", "free"


Internal pressure: :math:`P\mathrm{=}10.\mathit{Pa}`


Orientation problem
---------------------------

It is desired that the reinforcing grids be oriented circumferentially with respect to the given axis. However, given the hemispherical shape considered, a Python function is proposed to achieve it.

The principle is as follows:


1. the axis of the hemisphere is noted :math:`Z`


2. for each cell, we calculate the vector indicating the ciconferential direction by :math:`V\mathrm{=}Z\mathrm{\wedge }N` where :math:`N` is the normal to the mesh. Note :math:`V\mathrm{=}\mathrm{[}\mathit{Vx},\mathit{Vy},\mathit{Vz}\mathrm{]}` and define its projection on the :math:`(\mathit{XOY})` plane by :math:`W\mathrm{=}\mathrm{[}\mathit{Vx},\mathit{Vy}\mathrm{,0}\mathrm{]}`


3. we calculate the values of ANGL_REP to be assigned to the current mesh by :math:`\alpha =\mathrm{arctan}(\mathrm{Vy}/\mathrm{Vx})` and :math:`\beta \mathrm{=}\mathrm{-}\mathrm{arctan}(\mathit{Vz}\mathrm{/}\mathit{norm}(W))`

The function called LIST_CARA_CIRCONF loops through all grid cells, computes :math:`\alpha` and :math:`\beta`, and creates the keywords:

_F (SECTION =20.0,

 MAILLE =Current_Mesh-Name,

 EXCENTREMENT =0.0, ANGL_REP =(,)),

 COEF_RIGI_DRZ =1.E-10

),

and he adds them to a list. This list is then provided to AFFE_CARA_ELEM. So we get the commands:

LIST_GRI = LIST_CARA_CIRCONF (

 GROUP_MA =' GRILLE ',

 AXE =( 0.,0.,1. ),

 MODELE = MODEL,

 GRILLE =_F (SECTION =20. , EXCENTREMENT =0.,)

)

CARA_COQ = AFFE_CARA_ELEM (

 MODELE = MODEL,

 COEF_RIGI_DRZ =0.,),

 GRILLE = LIST_GRI,

)

Finally, note that, formally, LIST_CARA_CIRCONF obeys the following catalog:

LIST_CARA_CIRCONF (

 GROUP_MA = SIMP (status='o', typ=grma, max='**'),

 AXE = SIMP (status='o', typ='R', max=3, min=3),

 MODE = SIMP (status='o', typ=modele_sdaster),

 GRILLE = FACT (status='f', max=1,

     SECTION = SIMP (status='o', typ='R'),

     EXCENTREMENT = SIMP (status='f', typ='R'),

     COEF_RIGI_DRZ = SIMP (status='f', typ='R'),

 ),

)