5. C modeling

5.1. Characteristics of modeling

Modeling DKT (TRIA3)

5.2. Characteristics of the mesh

Number of knots: 6030

Number of meshes and types: 12000 TRIA3

(The circle is cut into 30 segments)

5.3. 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 |||| +——-+———+——-+———-+———+——-+———-++++

5.4. notes

We couldn’t find the clean mode for \(n=1\) and \(m=1\).

Thanks to the symmetry of the pipe structure, we observe that the natural modes of \(n=2-6\) and \(m=1\) 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).

Fig. 5. Examples of triangle meshes