C modeling
==============


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

Modeling DKT (TRIA3)


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

Number of knots: 6030

Number of meshes and types: 12000 TRIA3

(The circle is cut into 30 segments)


Tested sizes and results
------------------------------

Tab. 4. C modeling results


+-------+---------+-------+----------+---------+--------+----------+---------+-------+----------+
|**m=1**|**FF**                      |**SF**                       |**SS**                      |
+       +---------+-------+----------+---------+--------+----------+---------+-------+----------+
|       |**Aster**|**Ref**|**%** diff|**Aster**|**Ref** |**%** diff|**Aster**|**Ref**|**%** diff|
+-------+---------+-------+----------+---------+--------+----------+---------+-------+----------+
|**n=2**|656,432  |654,216|0.34      |657,048  |657,308 |0.04      |666,260  |663,543|0.41      |
+-------+---------+-------+----------+---------+--------+----------+---------+-------+----------+
|**n=3**|1853,03  |1850,29|0.15      |1853,95  |1851,81 |0.12      |1860,07  |1855,44|0.25      |
+-------+---------+-------+----------+---------+--------+----------+---------+-------+----------+
|**n=4**|3544,79  |3547.66|0.08      |3545.74  |3549,113|0.10      |3553,92  |3552.95|0.03      |
+-------+---------+-------+----------+---------+--------+----------+---------+-------+----------+
|**n=5**|5717.42  |5737.23|0.35      |5718.28  |5738.70 |0.36      |5728.62  |5742,74|0.25      |
+-------+---------+-------+----------+---------+--------+----------+---------+-------+----------+
|**n=6**|8362,13  |8416.32|0.64      |8362.94  |8417.79 |0.65      |8374,69  |8421.97|0.56      |
+-------+---------+-------+----------+---------+--------+----------+---------+-------+----------+


+-------+---------+-------+----------+---------+-------+----------++++
|**m=1**|**SC**                      |**CC**                      |  |
+       +---------+-------+----------+---------+-------+----------++++
|       |**Aster**|**Ref**|**%** diff|**Aster**|**Ref**|**%** diff||||
+-------+---------+-------+----------+---------+-------+----------++++
|**n=2**|674,87   |671,113|0.56      |689,418  |687,249|0.32      ||||
+-------+---------+-------+----------+---------+-------+----------++++
|**n=3**|1861.27  |1856,45|0.26      |1862.87  |1858,30|0.25      ||||
+-------+---------+-------+----------+---------+-------+----------++++
|**n=4**|3554,36  |3553,29|0.03      |3554,85  |3553,91|0.03      ||||
+-------+---------+-------+----------+---------+-------+----------++++
|**n=5**|5728.83  |5742,87|0.24      |5729.07  |5743,21|0.25      ||||
+-------+---------+-------+----------+---------+-------+----------++++
|**n=6**|8374,82  |8422.00|0.56      |8374,96  |8422,24|0.56      ||||
+-------+---------+-------+----------+---------+-------+----------++++


notes
---------

We couldn't find the clean mode for :math:`n=1` and :math:`m=1`.

Thanks to the symmetry of the pipe structure, we observe that the natural modes of :math:`n=2-6` and :math:`m=1` always appear in pairs.

**Warning**: when using triangle meshes, the results are very sensitive to mesh quality. For example, when using the mesh in Fig. 5 (b), we always observe a twist in the natural modes and also a frequency difference between the pairs of the symmetric modes.

Although this error decreases with a finer mesh, meshes with well-distributed elements in the structure are recommended, as in Fig. 5 (a).


.. image:: images/10000201000004300000023826E366EA3CB9AB6D.png
    :width: 3.1492in
    :height: 1.6417in


.. _RefImage_10000201000004300000023826E366EA3CB9AB6D.png:


.. image:: images/100002010000043E000002361B89A174B1AC67FA.png
    :width: 3.1492in
    :height: 1.5043in


.. _RefImage_100002010000043E000002361B89A174B1AC67FA.png:

(a) (b)

Fig. 5. Examples of triangle meshes