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


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

the reference solution proposed by NAFEMS [:ref:`bib1 <bib1>`] was obtained from a finite element calculation :math:`\mathrm{3D}` with brick elements at 20 nodes and with a mesh :math:`4\times 4` (plane :math:`\mathrm{xy}`) and 1 element depending on the thickness.


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

The first 7 non-zero frequencies and the associated natural modes, the first three modes are those of rigid bodies:


.. csv-table::

    "* Frequency (mode 4 off plane)", ":", ":math:`44.762\mathrm{hz}`"
    "* Frequency (modes 5 & 6 off plane)", ":", ":math:`110.52\mathrm{hz}`"
    "* Frequency (mode 7 out of plane)", ":", ":math:`169.08\mathrm{hz}`"
    "* Frequency (Mode 8 in the plan)", ":", ":math:`193.93\mathrm{hz}`"
    "* Frequency (mode 9 & 10 in the plan)", ":", ":math:`206.64\mathrm{hz}`"



.. image:: images/1000000000000275000000CB04B52C92D4573BCC.png
    :width: 3.0398in
    :height: 0.9752in


.. _RefImage_1000000000000275000000CB04B52C92D4573BCC.png:


+-----------------------------------------------------------------------------------------------------------------------------+-----------------------------------------------------------------------------------------------------------------------------+
|                                                                                                                             |                                                                                                                             |
+                                                                                                                             + .. image:: images/1000000000000269000000BAD13E83C29D00E004.png                                                              +
|                                                                                                                             |    :width: 2.9209in                                                                                                         |
+                                                                                                                             +    :height: 0.8744in                                                                                                        +
|                                                                                                                             |                                                                                                                             |
+                                                                                                                             +                                                                                                                             +
|                                                                                                                             |                                                                                                                             |
+-----------------------------------------------------------------------------------------------------------------------------+-----------------------------------------------------------------------------------------------------------------------------+
|Mode 4 off plane                                                                                                             |5&6 mode off plan                                                                                                            |
+-----------------------------------------------------------------------------------------------------------------------------+-----------------------------------------------------------------------------------------------------------------------------+
|                                                                                                                             |                                                                                                                             |
+ .. image:: images/1000000000000260000000B7845CDD7072BF6385.png                                                              + .. image:: images/1000000000000281000001C1D746A6F2798DC9AD.png                                                              +
|    :width: 3.1417in                                                                                                         |    :width: 1.911in                                                                                                          |
+    :height: 0.9445in                                                                                                        +    :height: 1.2417in                                                                                                        +
|                                                                                                                             |                                                                                                                             |
+                                                                                                                             +                                                                                                                             +
|                                                                                                                             |                                                                                                                             |
+-----------------------------------------------------------------------------------------------------------------------------+-----------------------------------------------------------------------------------------------------------------------------+
|Mode 7 off plan                                                                                                              |Mode 8 in the plan                                                                                                           |
+-----------------------------------------------------------------------------------------------------------------------------+-----------------------------------------------------------------------------------------------------------------------------+
|                                                                                                                             |                                                                                                                             |
+ .. image:: images/100000000000020A000001DDBDD065004DECCAA5.png                                                              +                                                                                                                             +
|    :width: 1.6047in                                                                                                         |                                                                                                                             |
+    :height: 1.3618in                                                                                                        +                                                                                                                             +
|                                                                                                                             |                                                                                                                             |
+                                                                                                                             +                                                                                                                             +
|                                                                                                                             |                                                                                                                             |
+-----------------------------------------------------------------------------------------------------------------------------+-----------------------------------------------------------------------------------------------------------------------------+
|9&10 mode in the map                                                                                                         |                                                                                                                             |
+-----------------------------------------------------------------------------------------------------------------------------+-----------------------------------------------------------------------------------------------------------------------------+



Uncertainties about the solution
----------------------------

:math:`\text{< 2\%}` for a mesh identical to that of [:ref:`§2.1 <§2.1>`], i.e. with few elements.


Bibliographical references
---------------------------


1. NAFEMS: The standard NAFEMS Benchmarks, TNSB, Rev. 3, October 5, 1990.