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


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

**z**

3D, :math:`\mathrm{H20}` and :math:`\mathrm{P15}` meshes

**y, v**

**F**

**B**

**y**

**x**

**A, E**

**B, F**

.. image:: images/1000000000000E6500000C960E3F99FDBFE68C10.png
    :width: 6.1417in
    :height: 5.3701in


.. _RefImage_1000000000000E6500000C960E3F99FDBFE68C10.png:

**x, u**


.. csv-table::

    "Point position:", ":math:`A,B` in section :math:`z=0`"
    "", ":math:`E,F` in the middle section :math:`z=L/2`"
    "", ""
    "Breakdown:", "20 elements depending on the length"
    "", "2 elements according to the radius, 8 elements according to the circumference."


Since the load is symmetric, only half of the cylinder is modelled.

Boundary conditions:


1. recessed ends section (:math:`u=v=w=0`)


2. symmetry conditions in plane :math:`\mathrm{xz}`: :math:`v=0`


.. csv-table::

    "1)", "Circumference pressure (field :math:`\mathrm{Up}`)
    The surface of the cylinder is divided into 8 rows of elements according to the circumference (1 row of elements represents a sector of :math:`\pi /8` radians.
    Since the pressure is in :math:`\mathrm{cos}\theta`, it is assumed to be uniform on each row. For any point on the corner surface :math:`\theta`, (between :math:`{\theta }_{1}` and and between and :math:`{\theta }_{2}`, :math:`{\theta }_{1}=(n–1)\frac{\pi }{8}`,, :math:`{\theta }_{2}=n\frac{\pi }{8}`, :math:`1\le n\le 8`), the pressure value assigned to the row of elements containing this point is taken to be equal to:
    :math:`\frac{\mathrm{p0}}{2}(\mathrm{cos}{\theta }_{1}+\mathrm{cos}{\theta }_{2})`."
    "", ""
    "2)", "Vertical gravity next :math:`x` (field :math:`\mathrm{Ug}`)"


Node names:


.. csv-table::

    ":math:`A=\mathrm{N845}` "," :math:`B=\mathrm{N965}` "," :math:`E=\mathrm{N865}` "," :math:`F=\mathrm{N995}`"



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

Number of knots: 1285

Number of meshes and types: 160 HEXA20, 80 PENTA15


Tested values
---------------


.. csv-table::

    "**Location**", "**Value type**", "**Reference**", "**Aster**", "**% difference**"
    "Field :math:`\mathrm{Up}` ", "", "", "", ""
    "Point E", ":math:`u(m)`
    :math:`v(m)` ", "0. ", "—7.82 x 10—6
    10—21", ""
    "Point :math:`F` "," :math:`u(m)`
    :math:`v(m)` ", "0. ", "—7.816 x 10—6
    10—21", ""
    "Point :math:`B` "," :math:`{\sigma }_{\mathrm{xx}}(\mathrm{Pa})`
    :math:`{\sigma }_{\mathrm{yy}}(\mathrm{Pa})`
    :math:`{\sigma }_{\mathrm{zz}}(\mathrm{Pa})` ", "", "1.63 x 106
    1.65 x 106
    5.51 x 106", ""
    "Field :math:`\mathrm{Up}+\mathrm{Ug}` ", "", "", "", ""
    "Point :math:`E` "," :math:`u(m)`
    :math:`v(m)` ", "0. ", "—7.46 x 10—6
    10—21", ""
    "Point :math:`F` "," :math:`u(m)`
    :math:`v(m)` ", "0. ", "—7.44 x 10—6
    10—21", ""
    "Point :math:`B` "," :math:`{\sigma }_{\mathrm{xx}}(\mathrm{Pa})`
    :math:`{\sigma }_{\mathrm{yy}}(\mathrm{Pa})`
    :math:`{\sigma }_{\mathrm{zz}}(\mathrm{Pa})` ", "", "1.56 x 106
    1.57 x 106
    5.25 x 106", ""



notes
---------


* There are no reference values for this modeling. The results are to be compared with those of the AXIS_FOURIER :math:`(A,B,D)` models.


* At point :math:`B` (located in the plane of symmetry), we have: :math:`{\sigma }_{\mathrm{rr}}={\sigma }_{\mathrm{xx}}`, :math:`{\sigma }_{\theta \theta }={\sigma }_{\mathrm{yy}}`