D modeling
==============


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

Modeling of a quarter cylinder.

3D shell elements: TRIA7


.. image:: images/Object_14.svg
    :width: 288
    :height: 192


.. _RefImage_Object_14.svg:

Discretized geometry is shown above.


.. csv-table::

    ":math:`\mathrm{point}` "," :math:`\mathrm{nœud}`"
    ":math:`K` "," :math:`\mathrm{NO72}`"
    ":math:`L` "," :math:`\mathrm{NO1}`"
    ":math:`M` "," :math:`\mathrm{NO33}`"
    ":math:`N` "," :math:`\mathrm{NO39}`"
    ":math:`P` "," :math:`\mathrm{NO186}`"
    ":math:`Q` "," :math:`\mathrm{NO190}`"


The shear correction factor :math:`{A}_{\mathrm{CIS}}` is :math:`5/6` (Reissner shell theory).


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

Number of external nodes: 153

Number of meshes and types: 64 TRIA7 + 8 SEG3


Boundary conditions in movement and rotation
-------------------------------------------------


Gravity
~~~~~~~~

Move :math:`\mathrm{DZ}` is stuck on the group of nodes :math:`\mathrm{LM}`.

The movement :math:`\mathrm{DY}` as well as the rotations around the axes :math:`X` and :math:`Z` are blocked on the group of nodes :math:`\mathrm{KL}`.

The movement :math:`\mathrm{DX}` as well as the rotations around the axes :math:`Y` and :math:`Z` are blocked on the group of nodes :math:`\mathrm{MN}`.


Rotation
~~~~~~~

The :math:`\mathrm{DZ}` movement as well as the rotations around the :math:`X` and :math:`Y` axes are blocked on the node groups :math:`\mathrm{KNSANSKN}` and :math:`\mathrm{LMSANSLM}`.

The movement :math:`\mathrm{DY}` as well as the rotations around the axes :math:`X` and :math:`Z` are blocked on the group of nodes :math:`\mathrm{KL}`.

The movement :math:`\mathrm{DX}` as well as the rotations around the axes :math:`Y` and :math:`Z` are blocked on the group of nodes :math:`\mathrm{MN}`.

The :math:`\mathrm{DZ}` movement and the rotation around the :math:`Y` axis are blocked on the group of nodes :math:`\mathrm{KETL}`.

The :math:`\mathrm{DZ}` movement and the rotation around the :math:`X` axis are blocked on the group of nodes :math:`\mathrm{METN}`.


Thermal expansion case no. 1
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The :math:`\mathrm{DZ}` movement as well as the rotations around the :math:`X` and :math:`Y` axes are blocked on the node groups :math:`\mathrm{KNSANSKN}` and :math:`\mathrm{LMSANSLM}`.

The movement :math:`\mathrm{DY}` as well as the rotations around the axes :math:`X` and :math:`Z` are blocked on the group of nodes :math:`\mathrm{KL}`.

The movement :math:`\mathrm{DX}` as well as the rotations around the axes :math:`Y` and :math:`Z` are blocked on the group of nodes :math:`\mathrm{MN}`.

The :math:`\mathrm{DZ}` movement and the rotation around the :math:`Y` axis are blocked on the group of nodes :math:`\mathrm{KETL}`.

The :math:`\mathrm{DZ}` movement and the rotation around the :math:`X` axis are blocked on the group of nodes :math:`\mathrm{METN}`.


Thermal expansion case no. 2
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The :math:`\mathrm{DZ}` movement as well as the rotations around the :math:`X` and :math:`Y` axes are blocked on the node groups :math:`\mathrm{KNSANSKN}` and :math:`\mathrm{LMSANSLM}`.

The movement :math:`\mathrm{DY}` as well as the rotations around the axes :math:`X` and :math:`Z` are blocked on the group of nodes :math:`\mathrm{KL}`.

The movement :math:`\mathrm{DX}` as well as the rotations around the axes :math:`Y` and :math:`Z` are blocked on the group of nodes :math:`\mathrm{MN}`.

The :math:`\mathrm{DZ}` movement and the rotation around the :math:`Y` axis are blocked on the group of nodes :math:`\mathrm{KETL}`.

The :math:`\mathrm{DZ}` movement and the rotation around the :math:`X` axis are blocked on the group of nodes :math:`\mathrm{METN}`.


Tested values
---------------


+--------------------------------+----------------------------------+-----------------------------------+-------------+
|**Identification**              |**Knot (mesh)**                   |**Value tested**                   |**Reference**|
+--------------------------------+----------------------------------+-----------------------------------+-------------+
|                                |                                  |                                   |             |
+--------------------------------+----------------------------------+-----------------------------------+-------------+
|**Gravity**                     |:math:`K`                         |:math:`\mathrm{DX}(\mathrm{mm})`   |—2.40000 10—8|
+                                +----------------------------------+-----------------------------------+-------------+
|                                |:math:`N`                         |:math:`\mathrm{DY}(\mathrm{mm})`   |—2.40000 10—8|
+                                +----------------------------------+-----------------------------------+-------------+
|                                |:math:`P`                         |:math:`\mathrm{DZ}(\mathrm{mm})`   |5.00000 10—9 |
+                                +----------------------------------+-----------------------------------+-------------+
|                                |:math:`Q`                         |:math:`\mathrm{DZ}(\mathrm{mm})`   |5.00000 10—9 |
+                                +----------------------------------+-----------------------------------+-------------+
|                                |:math:`P`                         |:math:`-\mathrm{DRY}`              |2.40000 10—9 |
+                                +----------------------------------+-----------------------------------+-------------+
|                                |:math:`Q`                         |:math:`\mathrm{DRX}`               |2.40000 10—9 |
+                                +----------------------------------+-----------------------------------+-------------+
|                                |:math:`K(\mathrm{M60})`           |:math:`\mathrm{NYY}(N)`            |8.00000 10—4 |
+                                +----------------------------------+-----------------------------------+-------------+
|                                |:math:`N(\mathrm{M56})`           |:math:`\mathrm{NYY}(N)`            |8.00000 10—4 |
+                                +----------------------------------+-----------------------------------+-------------+
|                                |:math:`K(\mathrm{M60})` Inner skin|:math:`\mathrm{SIYY}(\mathrm{MPa})`|8.00000 10—4 |
+                                +----------------------------------+-----------------------------------+-------------+
|                                |:math:`N(\mathrm{M56})` Inner skin|:math:`\mathrm{SIYY}(\mathrm{MPa})`|8.00000 10—4 |
+--------------------------------+----------------------------------+-----------------------------------+-------------+
|                                |                                  |                                   |             |
+--------------------------------+----------------------------------+-----------------------------------+-------------+
|**Rotation — centrifugal force**|:math:`L`                         |:math:`\mathrm{DX}(\mathrm{mm})`   |2.91200 10—7 |
+                                +----------------------------------+-----------------------------------+-------------+
|                                |:math:`M`                         |:math:`\mathrm{DY}(\mathrm{mm})`   |2.91200 10—7 |
+                                +----------------------------------+-----------------------------------+-------------+
|                                |:math:`L(\mathrm{M25})`           |:math:`\mathrm{NYY}(N)`            |9.60000 10—4 |
+                                +----------------------------------+-----------------------------------+-------------+
|                                |:math:`M(\mathrm{M53})`           |:math:`\mathrm{NYY}(N)`            |9.60000 10—4 |
+                                +----------------------------------+-----------------------------------+-------------+
|                                |:math:`L(\mathrm{M25})` Inner skin|:math:`\mathrm{SIYY}(\mathrm{MPa})`|9.84600 10—4 |
+                                +----------------------------------+-----------------------------------+-------------+
|                                |:math:`M(\mathrm{M53})` Inner skin|:math:`\mathrm{SIYY}(\mathrm{MPa})`|9.84600 10—4 |
+--------------------------------+----------------------------------+-----------------------------------+-------------+
|                                |                                  |                                   |             |
+--------------------------------+----------------------------------+-----------------------------------+-------------+
|**Dilation** **case 1**         |:math:`L(\mathrm{M25})`           |:math:`\mathrm{MYY}(\mathrm{N.mm})`|—2.38095 10—1|
+                                +----------------------------------+-----------------------------------+-------------+
|                                |:math:`M(\mathrm{M53})`           |:math:`\mathrm{MYY}(\mathrm{N.mm})`|—2.38095 10—1|
+                                +----------------------------------+-----------------------------------+-------------+
|                                |:math:`L(\mathrm{M25})` Inner skin|:math:`\mathrm{SIYY}(\mathrm{MPa})`|1.428571     |
+                                +----------------------------------+-----------------------------------+-------------+
|                                |:math:`M(\mathrm{M53})` Inner skin|:math:`\mathrm{SIYY}(\mathrm{MPa})`|1.428571     |
+--------------------------------+----------------------------------+-----------------------------------+-------------+
|                                |                                  |                                   |             |
+--------------------------------+----------------------------------+-----------------------------------+-------------+
|**Dilation** **case 2**         |:math:`L`                         |:math:`\mathrm{DX}(\mathrm{mm})`   |26.0000 10—6 |
+                                +----------------------------------+-----------------------------------+-------------+
|                                |:math:`M`                         |:math:`\mathrm{DY}(\mathrm{mm})`   |26.0000 10—6 |
+                                +----------------------------------+-----------------------------------+-------------+
|                                |:math:`L(\mathrm{M25})`           |:math:`\mathrm{NYY}(N)`            |—2.00000 10—1|
+                                +----------------------------------+-----------------------------------+-------------+
|                                |:math:`M(\mathrm{M53})`           |:math:`\mathrm{NYY}(N)`            |—2.00000 10—1|
+                                +----------------------------------+-----------------------------------+-------------+
|                                |:math:`L(\mathrm{M25})` Inner skin|:math:`\mathrm{SIYY}(\mathrm{MPa})`|—1.97800 10—1|
+                                +----------------------------------+-----------------------------------+-------------+
|                                |:math:`M(\mathrm{M53})` Inner skin|:math:`\mathrm{SIYY}(\mathrm{MPa})`|—1.97800 10—1|
+--------------------------------+----------------------------------+-----------------------------------+-------------+


notes
---------

Satisfactory and identical results for calculations with MECA_STATIQUE and STAT_NON_LINE.