B modeling
==============

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

The axisymmetric modeling is carried out on a slice of the tube.


Thermal analysis
~~~~~~~~~~~~~~~~~~~~

We test the unit heat shock :math:`(\Delta {T}_{U}=1°C)` and the thermal shock :math:`\mathrm{\Delta }T=50°C` using the proposed methodology, with the "AXIS_DIAG" modeling.

**Unit case** :math:`(\Delta {T}_{U}=1°C)`

 Boundary conditions:


* Extracts: :math:`{T}_{\mathit{imposée}}={T}_{\mathit{Initiale}}`


* Intrados: Convection :math:`h=20000W/{m}^{2}/°C` :math:`{T}_{\mathit{EXT}}={T}_{\mathit{Fluide}}={T}_{\mathit{Initiale}}+\Delta {T}_{U}`



 Initial conditions: :math:`{T}_{\mathit{Initiale}}=0.°C`

**Case** :math:`(\Delta T=50°C;\Delta T=100°C;\Delta T=200°C)`

 Boundary conditions:


* Extracts: :math:`{T}_{\mathit{imposée}}={T}_{\mathit{Initiale}}`


* Intrados: Convection :math:`h=20000W/{m}^{2}/°C` :math:`{T}_{\mathit{EXT}}={T}_{\mathit{Fluide}}={T}_{\mathit{Initiale}}+\Delta T`

 Initial conditions: :math:`{T}_{\mathit{Initiale}}=20.°C`


Mechanical analysis
~~~~~~~~~~~~~~~~~~

We test the mechanical response for unit shock :math:`(\Delta {T}_{U}=1°C)` and thermal shock :math:`\mathrm{\Delta }T=50°C` using the proposed methodology, with the "AXIS" modeling.


* Boundary conditions: EF rating: :math:`\mathit{DY}=0.`


* Reference temperature: :math:`{T}_{\mathit{Référence}}=0.°C`


* Thermal loads: :math:`\mathrm{\Delta }T=50°C`


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

To properly take into account the thermal shock on the inner wall of the tube, refinement has been imposed.

For mechanical modeling, the same mesh is used but with quadratic cells.


* Number of nodes: 504 linear


* Number of meshes and types: 460 QUAD4 (thermal analysis)


* Number of meshes and types: 460 QUAD8 (mechanical analysis)


B modeling results
------------------------------

Unitary case
~~~~~~~~~~~~

Only a non-regression test is used on this calculation, which does not use the CALC_THERMECA_MULT macro command.


Heat shock :math:`\mathrm{\Delta }T=50°C`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

These results were obtained with CALC_THERMECA_MULT and the following coefficients:

:math:`{T(x,t)}_{\Delta T}=\beta \mathrm{.}{\stackrel{̃}{T}}_{\Delta {T}_{U}}(x,t)+{T}_{\mathit{initiale}}`


.. csv-table::

    "Thermal shock", ":math:`\beta` "," :math:`{T}_{\mathit{initiale}}(°C)`"
    ":math:`\Delta T=50°C` ", "50. ", "20."



+--------------------------------------+--------+--------------+------------------+-------------+
|Temperature (°C) :math:`\Delta T=50°C`                                                         |
+--------------------------------------+--------+--------------+------------------+-------------+
|Time                                  |Location|Reference type|Reference         |Tolerance (%)|
+--------------------------------------+--------+--------------+------------------+-------------+
|0.1 sec                               |Intrados|'AUTRE_ASTER' |39.60439148764347 |             |
+                                      +--------+--------------+------------------+-------------+
|                                      |A       |'AUTRE_ASTER' |26.057424514530002|             |
+                                      +--------+--------------+------------------+-------------+
|                                      |B       |'AUTRE_ASTER' |22.127310880200298|             |
+                                      +--------+--------------+------------------+-------------+
|                                      |C       |'AUTRE_ASTER' |20.072931376414274|             |
+                                      +--------+--------------+------------------+-------------+
|                                      |D       |'AUTRE_ASTER' |20.000074233637914|             |
+--------------------------------------+--------+--------------+------------------+-------------+
|3.0 s                                 |Intrados|'AUTRE_ASTER' |62.791991296545994|             |
+                                      +--------+--------------+------------------+-------------+
|                                      |A       |'AUTRE_ASTER' |57.91447605545711 |             |
+                                      +--------+--------------+------------------+-------------+
|                                      |B       |'AUTRE_ASTER' |54.26645281787711 |             |
+                                      +--------+--------------+------------------+-------------+
|                                      |C       |'AUTRE_ASTER' |45.020909667984306|             |
+                                      +--------+--------------+------------------+-------------+
|                                      |D       |'AUTRE_ASTER' |31.02483220030017 |             |
+--------------------------------------+--------+--------------+------------------+-------------+


Mechanical analyses
~~~~~~~~~~~~~~~~~~~~~

+---------------------------------+--------+----------+--------------+-------------------+---------+
|Stress (Pa) :math:`\Delta T=50°C`                                                                 |
+---------------------------------+--------+----------+--------------+-------------------+---------+
|Time                             |Location|Constraint|Reference type|Reference          |Tolerance|
+---------------------------------+--------+----------+--------------+-------------------+---------+
|0.1 sec                          |Intrados|SIXX      |'AUTRE_ASTER' |-2223.323126038972 |         |
+                                 +        +----------+--------------+-------------------+---------+
|                                 |        |SIYY      |'AUTRE_ASTER' |-78755606.54685567 |1. %     |
+                                 +        +----------+--------------+-------------------+---------+
|                                 |        |SIZZ      |'AUTRE_ASTER' |-78753557.89057745 |1. %     |
+                                 +        +----------+--------------+-------------------+---------+
|                                 |        |VMIS      |'AUTRE_ASTER' |78752358.91557704  |1. %     |
+                                 +--------+----------+--------------+-------------------+---------+
|                                 |A       |SIXX      |'AUTRE_ASTER' |-1225712.1785479188|1. %     |
+                                 +        +----------+--------------+-------------------+---------+
|                                 |        |SIYY      |'AUTRE_ASTER' |-22196222.627284177|1. %     |
+                                 +        +----------+--------------+-------------------+---------+
|                                 |        |SIZZ      |'AUTRE_ASTER' |-20970978.954915553|1. %     |
+                                 +        +----------+--------------+-------------------+---------+
|                                 |        |VMIS      |'AUTRE_ASTER' |20385606.92198515  |1. %     |
+                                 +--------+----------+--------------+-------------------+---------+
|                                 |B       |SIXX      |'AUTRE_ASTER' |-1428428.090007163 |1. %     |
+                                 +        +----------+--------------+-------------------+---------+
|                                 |        |SIYY      |'AUTRE_ASTER' |-2329006.724762771 |1. %     |
+                                 +        +----------+--------------+-------------------+---------+
|                                 |        |SIZZ      |'AUTRE_ASTER' |-900846.4000951699 |1. %     |
+                                 +        +----------+--------------+-------------------+---------+
|                                 |        |VMIS      |'AUTRE_ASTER' |1247569.5007786928 |1. %     |
+                                 +--------+----------+--------------+-------------------+---------+
|                                 |C       |SIXX      |'AUTRE_ASTER' |-1152140.4888492671|1. %     |
+                                 +        +----------+--------------+-------------------+---------+
|                                 |        |SIYY      |'AUTRE_ASTER' |4900600.802635529  |1. %     |
+                                 +        +----------+--------------+-------------------+---------+
|                                 |        |SIZZ      |'AUTRE_ASTER' |6052925.081161121  |1. %     |
+                                 +        +----------+--------------+-------------------+---------+
|                                 |        |VMIS      |'AUTRE_ASTER' |6703597.4346061945 |1. %     |
+                                 +--------+----------+--------------+-------------------+---------+
|                                 |D       |SIXX      |'AUTRE_ASTER' |-499617.21189450816|1. %     |
+                                 +        +----------+--------------+-------------------+---------+
|                                 |        |SIYY      |'AUTRE_ASTER' |5263164.8315241365 |1. %     |
+                                 +        +----------+--------------+-------------------+---------+
|                                 |        |SIZZ      |'AUTRE_ASTER' |5763009.107557128  |1. %     |
+                                 +        +----------+--------------+-------------------+---------+
|                                 |        |VMIS      |'AUTRE_ASTER' |6028267.390181367  |1. %     |
+                                 +--------+----------+--------------+-------------------+---------+
|                                 |Extrados|SIXX      |'AUTRE_ASTER' |1437.4754262985766 |         |
+                                 +        +----------+--------------+-------------------+---------+
|                                 |        |SIYY      |'AUTRE_ASTER' |5263716.868909629  |1. %     |
+                                 +        +----------+--------------+-------------------+---------+
|                                 |        |SIZZ      |'AUTRE_ASTER' |5262478.893439884  |1. %     |
+                                 +        +----------+--------------+-------------------+---------+
|                                 |        |VMIS      |'AUTRE_ASTER' |5261660.514976556  |1. %     |
+---------------------------------+--------+----------+--------------+-------------------+---------+


notes
---------


* For modeling B, if we take a zero temperature for the unit case, we notice larger relative differences for the unit case (Max 10%). This difference is much smaller for 50° C. The origin of these values is mainly due to the presence of the non-zero initial temperature for 50° C. or for the current version of the unit calculation.

Example: Given two values T1= 1 and T2=1.1, the relative error is:


*
    *
        *
            * by 10% with an initial temperature of 0°C


            * by 0.47% with an initial temperature of 20°C