Modeling A
==============


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

The modeling of elasto-acoustic beams is in formulation :math:`u,\theta ,p,\phi`.

It is achieved by assigning elements PHENOMENE = 'MECANIQUE', MODELISATION = 'FLUI_STRU' to SEG2 meshes (segments with 2 nodes).

The circular cross-section characteristics are assigned to the elements:

.. csv-table::

    "outer radius", ":math:`{R}_{\mathit{ext}}=\mathrm{0,100}m` ", ""
    "thickness", ":math:`\mathit{ep}=\mathrm{0,010}m` ", "cf. [:ref:`§1.1 <§1.1>`]"


We also assign to these elements a mixed behavior material at the same time ELAS:

.. csv-table::

    "Young's module", ":math:`E=\mathrm{1,0}.1010\mathit{Pa}` ", ""
    "Poisson's ratio", ":math:`\nu =\mathrm{0,3}` ", ""
    "density", ":math:`{\rho }_{s}=1000\mathit{kg}/{m}^{3}` ", ""


and FLUIDE:

.. csv-table::

    "speed", ":math:`c=1000m/s` ", ""
    "density", ":math:`{\rho }_{f}=1000\mathit{kg}/{m}^{3}` ", "cf. [:ref:`§1.2 <§1.2>`]"


The degrees of freedom (DDL) of translation in :math:`y` and :math:`z` (DY and DZ) and all the degrees of freedom of rotation (DRX, DRY and DRZ) of all the nodes are blocked.

In order to embed the :math:`A` end of the pipe, the degree of freedom of translation is also blocked in :math:`x` (DX) of the node :math:`\mathit{NO1}`.

For the fluid the reservoir condition at the :math:`A` end is imposed by PRES = 0. and PHI = 0. at the node :math:`\mathit{NO1}`.


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


.. image:: images/1000000000000C90000001B5F514AD72F187E3BC.png
    :width: 5.3598in
    :height: 0.7283in


.. _RefImage_1000000000000C90000001B5F514AD72F187E3BC.png:

The total number of nodes used for this mesh is 26.

The stitches are 25 in number and of type SEG2.

The mesh file is in ASTER format.


Calculus
------

We want to validate the elasto-acoustic beam elements.

The frequency of the first axially coupled mode is calculated with operator CALC_MODES.


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

The test focuses on the frequency of the first coupled axial mode of the fluid-containing pipe.

The tolerance for relative deviation from the analytical value is 0.1%.


.. csv-table::

    "**Mode Number**", "**Analytical Value**"
    "1"," :math:`\mathrm{157,93981}\mathit{Hz}`
    Non-regression value"



note
--------

The reference values are both the analytical values and also those obtained by *Code_Aster* during the retrieval of the test case, which will therefore make it possible to verify the subsequent non-regression of the code during its evolution.