Modeling I
==============


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

Modeling of a quarter cylinder.

Plate elements DKQ: QUAD4. Assignment of characteristics of homogenized plates corresponding to the material of :math:`\S 1.2`.


.. image:: images/Object_19.svg
    :width: 288
    :height: 192


.. _RefImage_Object_19.svg:

Discretized geometry is shown above.


.. csv-table::

    "point", "knot"
    ":math:`K` "," :math:`\mathrm{NO160}`"
    ":math:`L` "," :math:`\mathrm{NO203}`"
    ":math:`M` "," :math:`\mathrm{NO11}`"
    ":math:`N` "," :math:`\mathrm{NO1}`"
    ":math:`P` "," :math:`\mathrm{NO226}`"
    ":math:`Q` "," :math:`\mathrm{NO6}`"



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

Number of external nodes: 231

Number of meshes and types: 200 QUAD4 + 80 SEG2


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}`.


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}`.


Modelling results I
------------------------------


+-----------------------+-----------------------------------+-----------------------------------+-------------+
|**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{DRX}`              |2.40000 10—9 |
+                       +-----------------------------------+-----------------------------------+-------------+
|                       |:math:`Q`                          |:math:`\mathrm{DRX}`               |2.40000 10—9 |
+                       +-----------------------------------+-----------------------------------+-------------+
|                       |:math:`K(\mathrm{M181})`           |:math:`\mathrm{NYY}(N)`            |8.00000 10—4 |
+                       +-----------------------------------+-----------------------------------+-------------+
|                       |:math:`N(\mathrm{M200})`           |:math:`\mathrm{NYY}(N)`            |8.00000 10—4 |
+-----------------------+-----------------------------------+-----------------------------------+-------------+
|                       |                                   |                                   |             |
+-----------------------+-----------------------------------+-----------------------------------+-------------+
|**Dilation** **case 1**|:math:`L(\mathrm{M1})`             |:math:`\mathrm{MYY}(\mathrm{N.mm})`|—2.38095 10—1|
+                       +-----------------------------------+-----------------------------------+-------------+
|                       |:math:`M(M20)`                     |:math:`\mathrm{MYY}(\mathrm{N.mm})`|—2.38095 10—1|
+                       +-----------------------------------+-----------------------------------+-------------+
|                       |:math:`L(\mathrm{M1})`, inner skin |:math:`\mathrm{SIYY}(\mathrm{Pa})` |1.428571     |
+                       +-----------------------------------+-----------------------------------+-------------+
|                       |:math:`M(\mathrm{M20})`, inner skin|:math:`\mathrm{SIYY}(\mathrm{Pa})` |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(M1)`                      |:math:`\mathrm{NYY}(N)`            |—2.00000 10—1|
+                       +-----------------------------------+-----------------------------------+-------------+
|                       |:math:`M(\mathrm{M20})`            |:math:`\mathrm{NYY}(N)`            |—2.00000 10—1|
+                       +-----------------------------------+-----------------------------------+-------------+
|                       |:math:`L(\mathrm{M1})`             |:math:`\mathrm{SIYY}(\mathrm{Pa})` |—2.00000 10—1|
+                       +-----------------------------------+-----------------------------------+-------------+
|                       |:math:`M(\mathrm{M20})`            |:math:`\mathrm{SIYY}(\mathrm{Pa})` |—2.00000 10—1|
+-----------------------+-----------------------------------+-----------------------------------+-------------+