Benchmark solution
=====================


Calculation method used for the reference solution
--------------------------------------------------------

Comparison with the numerical results of a volume modeling with Samcef V7.0 with 50 time steps (module THERNL for calculating the transient temperature field and repeated with module MECANL for non-linear mechanical calculation).

An implicit schema is used to integrate the heat equation.

The non-linear mechanical calculation was performed with SAMCEF using axisymmetric modeling with 5 time steps (geometric nonlinear modeling) and volume modeling (a quarter cylinder) with 5 time steps (geometric nonlinear modeling).


Benchmark results
----------------------

**Evolution of the temperature field according to the thickness on the lower side of the cylinder**


+-------------------------------------------+-------------+
|Axisymmetric calculation with 50 time steps              |
+-------------------------------------------+-------------+
|Time = 0.5 sec                             |Time = 25 sec|
+-------------------------------------------+-------------+
|1.25355E+01                                |8.64267E+01  |
+-------------------------------------------+-------------+
|5.71233E+00                                |7.65695E+01  |
+-------------------------------------------+-------------+
|2.44526E+00                                |6.77355E+01  |
+-------------------------------------------+-------------+
|1.12189E+00                                |5.98610E+01  |
+-------------------------------------------+-------------+
|4.83644E-01                                |5.28476E+01  |
+-------------------------------------------+-------------+
|2.22443E-01                                |4.66462E+01  |
+-------------------------------------------+-------------+
|9.62036E-02                                |4.11801E+01  |
+-------------------------------------------+-------------+
|4.43444E-02                                |3.64106E+01  |
+-------------------------------------------+-------------+
|1.92310E-02                                |3.22765E+01  |
+-------------------------------------------+-------------+
|8.87973E-03                                |2.87468E+01  |
+-------------------------------------------+-------------+
|3.85960E-03                                |2.57723E+01  |
+-------------------------------------------+-------------+
|1.78524E-03                                |2.33283E+01  |
+-------------------------------------------+-------------+
|7.78650E-04                                |2.13761E+01  |
+-------------------------------------------+-------------+
|3.63114E-04                                |1.98963E+01  |
+-------------------------------------------+-------------+
|1.64448E-04                                |1.88596E+01  |
+-------------------------------------------+-------------+
|8.80577E-05                                |1.82514E+01  |
+-------------------------------------------+-------------+
|6.54904E-05                                |1.80507E+01  |
+-------------------------------------------+-------------+


+-------------------------------------+-------------+
|Volume calculation with 50 time steps              |
+-------------------------------------+-------------+
|Time = 0.5 sec                       |Time = 25 sec|
+-------------------------------------+-------------+
|1.26658E+01                          |8.64175E+01  |
+-------------------------------------+-------------+
|5.64556E+00                          |7.65753E+01  |
+-------------------------------------+-------------+
|2.50690E+00                          |6.77332E+01  |
+-------------------------------------+-------------+
|1.10448E+00                          |5.98571E+01  |
+-------------------------------------+-------------+
|5.06396E-01                          |5.28510E+01  |
+-------------------------------------+-------------+
|2.23330E-01                          |4.66453E+01  |
+-------------------------------------+-------------+
|9.98984E-02                          |4.11819E+01  |
+-------------------------------------+-------------+
|4.41255E-02                          |3.64095E+01  |
+-------------------------------------+-------------+
|2.02976E-02                          |3.22781E+01  |
+-------------------------------------+-------------+
|8.97177E-03                          |2.87460E+01  |
+-------------------------------------+-------------+
|4.05352E-03                          |2.57736E+01  |
+-------------------------------------+-------------+
|1.79338E-03                          |2.33275E+01  |
+-------------------------------------+-------------+
|8.25938E-04                          |2.13766E+01  |
+-------------------------------------+-------------+
|3.68531E-04                          |1.98979E+01  |
+-------------------------------------+-------------+
|1.74367E-04                          |1.88647E+01  |
+-------------------------------------+-------------+
|8.91805E-05                          |1.82450E+01  |
+-------------------------------------+-------------+
|6.98548E-05                          |1.80607E+01  |
+-------------------------------------+-------------+

**Evolution of the longitudinal stress** :math:`{\sigma }_{T}`


.. image:: images/10000FC2000069BB00004F7AA120A248C7C5FF1D.svg
    :width: 316
    :height: 237


.. _RefImage_10000FC2000069BB00004F7AA120A248C7C5FF1D.svg:

**Cases where the pressure is not following**


+------------+---------------------+--------------------------------------------------------+---------------+
|            |                     |Geometric nonlinear volume calculation with 5 time steps                |
+------------+---------------------+--------------------------------------------------------+---------------+
|:math:`t(s)`|:math:`\sigma`       |Point :math:`A`                                         |Point :math:`B`|
+------------+---------------------+--------------------------------------------------------+---------------+
|15.0        |:math:`{\sigma }_{T}`|1.21692E8                                               |1.67285E8      |
+------------+---------------------+--------------------------------------------------------+---------------+
|20.0        |:math:`{\sigma }_{T}`|1.39804E8                                               |1.65266E8      |
+------------+---------------------+--------------------------------------------------------+---------------+
|25.0        |:math:`{\sigma }_{T}`|1.28967E8                                               |1.57980E8      |
+------------+---------------------+--------------------------------------------------------+---------------+


+------------+---------------------+--------------------------------------------------------------+---------------+
|            |                     |Geometric nonlinear axisymmetric calculation with 5 time steps                |
+------------+---------------------+--------------------------------------------------------------+---------------+
|:math:`t(s)`|:math:`\sigma`       |Point :math:`A`                                               |Point :math:`B`|
+------------+---------------------+--------------------------------------------------------------+---------------+
|15.0        |:math:`{\sigma }_{T}`|1.20472E8                                                     |1.66917E8      |
+------------+---------------------+--------------------------------------------------------------+---------------+
|20.0        |:math:`{\sigma }_{T}`|1.39164E8                                                     |1.70412E8      |
+------------+---------------------+--------------------------------------------------------------+---------------+
|25.0        |:math:`{\sigma }_{T}`|1.28421E8                                                     |1.63168E8      |
+------------+---------------------+--------------------------------------------------------------+---------------+

**Cases where the pressure is follow**


+------------+---------------------+--------------------------------------------------------+---------------+
|            |                     |Geometric nonlinear volume calculation with 5 time steps                |
+------------+---------------------+--------------------------------------------------------+---------------+
|:math:`t(s)`|:math:`\sigma`       |Point :math:`A`                                         |Point :math:`B`|
+------------+---------------------+--------------------------------------------------------+---------------+
|15.0        |:math:`{\sigma }_{T}`|1.21692E8                                               |1.67285E8      |
+------------+---------------------+--------------------------------------------------------+---------------+
|20.0        |:math:`{\sigma }_{T}`|1.39894E8                                               |1.65313E8      |
+------------+---------------------+--------------------------------------------------------+---------------+
|25.0        |:math:`{\sigma }_{T}`|1.29032E8                                               |1.58007E8      |
+------------+---------------------+--------------------------------------------------------+---------------+


+------------+---------------------+--------------------------------------------------------------+---------------+
|            |                     |Geometric nonlinear axisymmetric calculation with 5 time steps                |
+------------+---------------------+--------------------------------------------------------------+---------------+
|:math:`t(s)`|:math:`\sigma`       |Point :math:`A`                                               |Point :math:`B`|
+------------+---------------------+--------------------------------------------------------------+---------------+
|15.0        |:math:`{\sigma }_{T}`|1.20472E8                                                     |1.65934E8      |
+------------+---------------------+--------------------------------------------------------------+---------------+
|20.0        |:math:`{\sigma }_{T}`|1.39252E8                                                     |1.64075E8      |
+------------+---------------------+--------------------------------------------------------------+---------------+
|25.0        |:math:`{\sigma }_{T}`|1.28485E8                                                     |1.56886E8      |
+------------+---------------------+--------------------------------------------------------------+---------------+


Uncertainty about the solution
---------------------------

Uncertainty less than :math:`\text{1 \%}` for thermal calculation, uncertainty less than :math:`\text{0.5 \%}` for mechanical calculation.