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


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


.. image:: images/100000000000060200000295E490978F1801EEDF.png
    :width: 4.9217in
    :height: 2.1138in


.. _RefImage_100000000000060200000295E490978F1801EEDF.png:

Figure 3.1-1: Problem geometry mesh

In this modeling, the complete structure is meshed using 3D solid elements.

The flatness constraint is expressed as follows. Be it,


* point A :math:`X=0`, :math:`Y=h/2`, :math:`Z=0`: :math:`\mathit{DX}=\mathit{DY}=0`, :math:`\mathit{DZ}\ne 0`;


* point B :math:`X=0`, :math:`Y=h/2`, :math:`Z=b`: :math:`\mathit{DX}=\mathit{DY}=0`, :math:`\mathit{DZ}\ne 0`;


* the C point :math:`X=0`, :math:`Y=h`, :math:`Z=b/2`: :math:`\mathit{DX}\ne \mathit{DY}\ne 0`, :math:`\mathit{DZ}=0`

By neglecting second-order terms in displacements, the flatness constraint results in a linear relationship between the :math:`\mathit{DX}=0` displacements of the points :math:`A`, :math:`B`, :math:`C` and :math:`P`, any point on the face:

:math:`∣\begin{array}{cccc}{\mathit{DX}}_{P}& {Y}_{P}& {Z}_{P}& 1\\ {\mathit{DX}}_{A}& h/2& 0& 1\\ {\mathit{DX}}_{B}& h/2& b& 1\\ {\mathit{DX}}_{C}& h& b/2& 1\end{array}∣=0`

The condition is then written for :math:`{\mathit{DX}}_{A}=0`:

:math:`{\mathit{DX}}_{P}={\mathit{DX}}_{C}\times (\frac{2{Y}_{P}}{h-1})`

In :math:`X=L`, the flatness condition is obtained in the same way but for any :math:`{\mathit{DX}}_{A}`. The relationship is then written:

:math:`{\mathit{DX}}_{P}={\mathit{DX}}_{C}\times (\frac{2{Y}_{P}}{h-1})+2\times (1-\frac{{Y}_{P}}{h})\times {\mathit{DX}}_{A}`


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

Number of knots: 1077

Number of meshes and types: 160 HEXA20


Tested sizes and results
------------------------------


.. csv-table::

    "Mode", "Reference Value", "Reference Type", "Tolerance (%)"
    "1"," :math:`115.7` ", "'ANALYTIQUE'", ":math:`1.0`"
    "2"," :math:`442.2` ", "'ANALYTIQUE'", ":math:`1.0`"
    "3"," :math:`648.6` ", "'ANALYTIQUE'", ":math:`1.0`"
    "4"," :math:`931.6` ", "'ANALYTIQUE'", ":math:`1.0`"
    "5"," :math:`1534.0` ", "'ANALYTIQUE'", ":math:`1.0`"