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

Geometry and modeling
-------------------------

The mesh is composed of:

• 9 SEG2 cells on which the 6 types of beams and the 2 types of discrete elements with two nodes are modelled (there are two elements in POU_D_T).

• 2 SEG3 meshes on which the 2 types of 3-knot pipes are modelled.

• 1 SEG4 mesh on which a 4-knot pipe is modelled.

• 2 POI1sur meshes which model the 2 types of discrete elements at a node.

All cells that have a length are oriented according to the vector :math:`(\mathrm{1,}\mathrm{1,}0)`.


Orientation of the local coordinate system
---------------------------

In order to define the local coordinate system for these elements, we use the keywords ANGL_VRIL for beams and discrete elements with two nodes, GENE_TUYAU for pipes and ANGL_NAUT for discrete elements at one node of the keyword factor ORIENTATION of the AFFE_CARA_ELEM operator (see U4.42.01).

The table above gives the orientations chosen for each element:

.. csv-table::

    "Beams", "ANGL_VRIL "," :math:`90`"
    "Discreet with two knots", "ANGL_VRIL "," :math:`\mathrm{-}90`"
    "Discreet to a knot", "ANGL_NAUT "," :math:`(\mathrm{90,}\mathrm{-}90.0\mathrm{,90}.0)`"
    "Pipes", "GENE_TUYAU "," :math:`(0.\mathrm{,0}\mathrm{.}\mathrm{,1}\mathrm{.})`"



Calculation of local landmarks
-------------------------

The local landmarks are formed by the vectors :math:`x`, :math:`y`, and :math:`z`.


Beams
~~~~~~

The vector :math:`x` is defined by geometry and is therefore equal to :math:`\left(\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2},0\right)`. The :math:`90` value of ANGL_VRIL rotates the default coordinate system of :math:`90°`, which results in :math:`y\mathrm{=}(\mathrm{0,0}\mathrm{,1})` and :math:`z=\left(\frac{\sqrt{2}}{2},-\frac{\sqrt{2}}{2},0\right)`.


Discreet with two knots
~~~~~~~~~~~~~~~~~~~~~~~

As for the :math:`x=\left(\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2},0\right)` beams, but this time we pivot in the other direction which gives :math:`y\mathrm{=}(\mathrm{0,0},\mathrm{-}1)` and :math:`z=\left(\frac{-\sqrt{2}}{2},\frac{\sqrt{2}}{2},0\right)`.


pipes
~~~~~

No change for :math:`x`. We gave the value :math:`(0.\mathrm{,0}\mathrm{.}\mathrm{,1}\mathrm{.})` to GENE_TUYAU, the vector :math:`y` is then the projection of :math:`(0.\mathrm{,0}\mathrm{.}\mathrm{,1}\mathrm{.})` on the plane orthogonal to :math:`x`, that is to say :math:`(0.\mathrm{,0}\mathrm{.}\mathrm{,1}\mathrm{.})` itself.

So we have :math:`z=\left(\frac{\sqrt{2}}{2},-\frac{\sqrt{2}}{2},0\right)`.

But a different treatment of angle GAMMA1 in Code_Aster induces an additional rotation of 90° around :math:`x` which finally gives:

:math:`y=\left(\frac{\sqrt{2}}{2},-\frac{\sqrt{2}}{2},0\right)` and :math:`z\mathrm{=}(0.\mathrm{,0}\mathrm{.},\mathrm{-}1.)`

**Note:**

Pipes carried with SEG4 links are not compatible with those carried by SEG3 links. They are therefore treated separately.


Discreet at a knot
~~~~~~~~~~~~~~~~~~~

In this case the local coordinate system is only defined by the values of ANGL_NAUT. The second component of the given vector gives :math:`x\mathrm{=}(0.\mathrm{,0}\mathrm{.}\mathrm{,1}\mathrm{.})`. From the three components we determine that :math:`y\mathrm{=}(0.,\mathrm{-}1.\mathrm{,0}\mathrm{.})` and :math:`z\mathrm{=}(1.\mathrm{,0}\mathrm{.}\mathrm{,0}\mathrm{.})`.


Tested sizes
------------------

The tested results are shown in the following table:

.. csv-table::

    "MAILLE ", "Vector", "Component", "**Reference Value**", "**Tolerance**"
    "POU1 "," :math:`x` ", "X", ":math:`0.707106781186E0` "," :math:`1.E\mathrm{-}8`"
    "POU3 "," :math:`x` ", "Y", ":math:`0.707106781186E0` "," :math:`1.E\mathrm{-}8`"
    "POU5 "," :math:`x` ", "X", ":math:`0.707106781186E0` "," :math:`1.E\mathrm{-}8`"
    "POU7 "," :math:`x` ", "Y", ":math:`0.707106781186E0` "," :math:`1.E\mathrm{-}8`"
    "DISL1 "," :math:`x` ", "X", ":math:`0.707106781186E0` "," :math:`1.E\mathrm{-}8`"
    "TUY32 "," :math:`x` ", "Y", ":math:`0.707106781186E0` "," :math:`1.E\mathrm{-}8`"
    "DISN2 "," :math:`x` ", "Z", ":math:`1.0` "," :math:`1.E\mathrm{-}8`"
    "POU2 "," :math:`y` ", "Z", ":math:`1.0` "," :math:`1.E\mathrm{-}8`"
    "POU4 "," :math:`y` ", "Z", ":math:`1.0` "," :math:`1.E\mathrm{-}8`"
    "POU6 "," :math:`y` ", "Z", ":math:`1.0` "," :math:`1.E\mathrm{-}8`"
    "DISL2 "," :math:`y` ", "Z", ":math:`-1.0` "," :math:`1.E\mathrm{-}8`"
    "DISN1 "," :math:`y` ", "Y", ":math:`-1.0` "," :math:`1.E\mathrm{-}8`"
    "TUY31 "," :math:`y` ", "Y", ":math:`-0.707106781186E0` "," :math:`1.E\mathrm{-}8`"
    "TUY41 "," :math:`x` ", "X", ":math:`0.707106781186E0` "," :math:`1.E\mathrm{-}8`"
    "TUY41 "," :math:`x` ", "Y", ":math:`0.707106781186E0` "," :math:`1.E\mathrm{-}8`"
    "TUY41 "," :math:`y` ", "X", ":math:`0.707106781186E0` "," :math:`1.E\mathrm{-}8`"
    "TUY41 "," :math:`y` ", "Y", ":math:`-0.707106781186E0` "," :math:`1.E\mathrm{-}8`"
    "TUY41 "," :math:`z` ", "Z", ":math:`-1.0` "," :math:`1.E\mathrm{-}8`"