Discretization
==============



Degrees of freedom
------------------

The degrees of freedom are, for each node of the support mesh:


* the six displacement components of the average fiber (three translations and three rotations),


* three degrees of freedom corresponding to modes 0 and 1,


* for each Fourier mode 6 degrees of freedom (:math:`U` corresponds to warping, :math:`V` to orthoradial displacement, :math:`W` to radial displacement).


+--------------------+--------------------------------------------+---+---+---+---+---+-----------+
|**Finished element**|**Degrees of freedom (at each vertex node)**                    |**Remarks**|
+--------------------+--------------------------------------------+---+---+---+---+---+-----------+
|TUYAU_3M            |DX                                          |DY |DZ |DRX|DRY|DRZ|           |
+--------------------+--------------------------------------------+---+---+---+---+---+-----------+
|                    |W0                                          |WI1|WO1|   |   |   |           |
+--------------------+--------------------------------------------+---+---+---+---+---+-----------+
|                    |UI2                                         |VI2|WI2|UO2|VO2|WO2|Fashion 2  |
+--------------------+--------------------------------------------+---+---+---+---+---+-----------+
|                    |UI3                                         |VI3|WI3|UO3|VO3|WO3|Fashion 3  |
+--------------------+--------------------------------------------+---+---+---+---+---+-----------+
|TUYAU_6M            |DX                                          |DY |DZ |DRX|DRY|DRZ|           |
+--------------------+--------------------------------------------+---+---+---+---+---+-----------+
|                    |W0                                          |WI1|WO1|   |   |   |           |
+--------------------+--------------------------------------------+---+---+---+---+---+-----------+
|                    |UI2                                         |VI2|WI2|UO2|VO2|WO2|Fashion 2  |
+--------------------+--------------------------------------------+---+---+---+---+---+-----------+
|                    |UI3                                         |VI3|WI3|UO3|VO3|WO3|Fashion 3  |
+--------------------+--------------------------------------------+---+---+---+---+---+-----------+
|                    |UI4                                         |VI4|WI4|UO4|VO4|WO4|Mode 4     |
+--------------------+--------------------------------------------+---+---+---+---+---+-----------+
|                    |UI5                                         |VI5|WI5|UO5|VO5|WO5|Fashion 5  |
+--------------------+--------------------------------------------+---+---+---+---+---+-----------+
|                    |UI6                                         |VI6|WI6|UO6|VO6|WO6|Fashion 6  |
+--------------------+--------------------------------------------+---+---+---+---+---+-----------+




Support mesh for stiffness matrices
------------------------

Finite element support meshes, in displacement formulation, are segments with 3 or 4 knots.


.. csv-table::

    "**Modeling**", "**Mesh**", "**Finished Element**", "**Remarks**"
    "TUYAU_3M "," SEG3 "," MET3SEG3 ", ""
    "", "SEG4 "," MET3SEG4 ", ""
    "TUYAU_6M "," SEG3 "," MET6SEG3 ", ""