Reference modeling
=========================


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

3D solid elements


.. image:: images/1000000000000193000001FA5CF5D97A6F260EEF.png
    :width: 3.4165in
    :height: 4.289in


.. _RefImage_1000000000000193000001FA5CF5D97A6F260EEF.png:

Discretized geometry is shown above. The 3D elements are solid with 8 nodes of the HEXA8 type. The number of knots on the circumference is 600, on the thickness 9 and on the length 9.


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

Number of knots: 48600

Number of meshes and type: 38400 HEXA8


Reference values tested
----------------------------

Frequencies of natural modes of ovalization, trifoliate and out-of-plane.


.. csv-table::

    "**Mode**", "**Natural Frequencies** :math:`(\mathrm{Hz})`"
    "ovalization", "210.55
    210.55"
    "trifoliate", "587.92
    587.92"
    "off plan", "205.89
    205.89
    588.88
    588.88"



notes
---------

The axisymmetric problem has dual in-plane and out-of-plane modes.

Off-plane modes have the following deformations:


+--------------------------+------------------------------------------------------------------------------------------------------------------+--------------------------+-----------------------------------------------------------------------------------------------------------------------------+
|:math:`205.89\mathrm{Hz}`:|                                                                                                                  |:math:`588.88\mathrm{Hz}`:|                                                                                                                             |
+                          + .. image:: images/1000079C0000068C000018FF5B49767D09204555.svg                                                   +                          + .. image:: images/10000A5800000C0400001D93FE963813BA96FB92.svg                                                              +
|                          |    :width: 35                                                                                                    |                          |    :width: 0.6008in                                                                                                         |
+                          +    :height: 144                                                                                                  +                          +    :height: 1.5465in                                                                                                        +
|                          |                                                                                                                  |                          |                                                                                                                             |
+                          +                                                                                                                  +                          +                                                                                                                             +
|                          |                                                                                                                  |                          |                                                                                                                             |
+--------------------------+------------------------------------------------------------------------------------------------------------------+--------------------------+-----------------------------------------------------------------------------------------------------------------------------+



Uncertainties
------------

The uncertainty comes from the convergence analysis of the mesh where the natural frequencies with the reference mesh :math:`600\times 8\times 8` are compared to those of the mesh :math:`500\times 7\times 9`.


+----------------------+-------------+----------+---------------------------------------------------+----------------------------------+
|**Number of elements**                         |**Natural frequencies (** :math:`\mathrm{Hz}` **)**                                   |
+----------------------+-------------+----------+---------------------------------------------------+----------------------------------+
|**Circumference**     |**Thickness**|**Length**|**Ovalization**                                    |**Trifoliate**                    |
+----------------------+-------------+----------+---------------------------------------------------+----------------------------------+
|100 200 400 500 600   |3 3 5 7 8    |3 3 8 9 8 |232.03 225.20 221.02 210.64 210.55                 |648.62 628.74 616.34 588.18 587.92|
+----------------------+-------------+----------+---------------------------------------------------+----------------------------------+

The uncertainty about the frequency is 0.05%.