Modeling, loading
=========================

To assign the Fourier axisymmetric finite elements to the mesh, the AFFE_MODELE operator is used in the following way:


.. code-block:: text

    mo = AFFE_MODELE (MAILLAGE = my,
AFFE = _F (TOUT = 'OUI',
PHENOMENE = 'MECANIQUE' or 'THERMIQUE',
MODELISATION = 'AXIS_FOURIER')
);

The decomposition of the load into Fourier series must have been done beforehand by the user

Be :math:`f=\sum _{l=0}^{N}\left[(\begin{array}{ccc}\mathrm{cos}l\theta & & 0\\ & \mathrm{cos}l\theta & \\ 0& & -\mathrm{sin}l\theta \end{array}){F}_{l}^{s}(r,z)+(\begin{array}{ccc}\mathrm{sin}l\theta & & 0\\ & \mathrm{sin}l\theta & \\ 0& & \mathrm{cos}l\theta \end{array}){F}_{l}^{a}(r,z)\right]`

with :math:`{F}_{l}^{s(a)}=({f}_{rl}^{\mathrm{ss}(a)},{f}_{zl}^{s(a)},{f}_{\theta l}^{s(a)})`

The charges :math:`{F}_{l}^{s}` and :math:`{F}_{l}^{a}` are introduced harmonic by harmonic and type by type by the operator AFFE_CHAR_MECA. The mode or type is not specified at this level.

Example: a loading with distributed pressure in symmetric mode 1 and in pure torsional mode (antisymmetric mode 0) is assumed.

We will write:


.. code-block:: text

    ch1sym = AFFE_CHAR_MECA (Template= mo,
PRES_REP = _F (GROUP_MA = 'grma',
PRES = p));

    ch0anti = AFFE_CHAR_MECA (Template = mo,
FORCE_NODALE = _F (FZ = f,
NOEUD = 'N1')));

Dirichlet-type boundary conditions will be introduced in a separate load:


.. code-block:: text

    chdir = AFFE_CHAR_MECA (Template= mo,
DDL_IMPO = _F (GROUP_NO = 'big',
DX=0., DY=0., DZ=0.,));

The loads allowed by Fourier elements are:

**in elasticity:**


.. csv-table::

    "**Elements**", "**Nature of load**", "**Keyword** **AFFE_CHAR_MECA**"
    "TRIA3 - TRIA6
    QUAD4 - QUAD8 - QUAD9 ", "Volume forces
    Rotation
    Gravity
    Point forces", "FORCE_INTERNE
    ROTATION
    PESANTEUR
    FORCE_NODALE"
    "SEG2 - SEG3 ", "Pressure
    Surface forces", "PRES_REP
    FORCE_CONTOUR"


**in thermal:**


.. csv-table::

    "**Elements**", "**Nature of load**", "**Keyword** **AFFE_CHAR_THER**"
    "Surface", "Heat Source", "SOURCE"
    "Edge", "Normal flow imposed
    Exchange", "FLUX_REP
    ECHANGE"



Special case of gravity
-------------------------------

We consider the cylinder below whose shaded part corresponds to the meshed part on which an axisymmetric Fourier calculation is performed (plane :math:`\mathrm{rZ}`).

This cylinder is subject to its own weight :math:`P` directed along the :math:`-\text{OX}` axis.

R

:math:`\theta`

The load to be applied here is :math:`\overrightarrow{P}=({P}_{r},{P}_{z},{P}_{\theta })`

:math:`\theta =0`

:math:`\theta`

With :math:`{P}_{r}=-g\mathrm{cos}\theta` :math:`{P}_{z}=0` :math:`{P}_{\theta }=g\mathrm{sin}\theta`

:math:`\theta =\frac{3}{2}\pi`

We check that for any value of :math:`\theta` we have a loading
R
with amplitude :math:`g>0` directed downwards.

The desired load is therefore :math:`g(-\mathrm{1,0},-1)` according to the Fourier mode decomposition (symmetric harmonic and cosine series development for the radial component).
:math:`\theta =\pi`

:math:`\theta`

Attention, the direction is under the assumption that :math:`P` is directed along the axis

:math:`-\text{OX}`. For another orientation of :math:`P`, the calculation must be adapted.
R

:math:`X(r)`

:math:`Y(Z)`

:math:`Z(\theta )`

:math:`\overrightarrow{P}`

However, since the loading direction vector is normalized in AFFE_CHAR_MECA, it is necessary to multiply by this norm to obtain the desired load.

The load to be introduced in AFFE_CHAR_MECA behind the PESANTEURest keyword is therefore :math:`\sqrt{2}g(-\mathrm{1,0},-1)`.

In MACRO_ELAS_MULT, this load will be assigned to symmetric harmonic 1 (cosine series expansion for the radial component).