Reference problem
=====================


Geometry
---------

We consider a periodic network of :math:`4\mathrm{\times }4` beams [:ref:`fig 1.1-a <fig 1.1-a>`]. The domain period is :math:`\varepsilon Y`. Figure [:ref:`fig 1.1-b <fig 1.1-b>`] represents an enlargement of :math:`1/\varepsilon` of the period. Each beam is straight and has a square cross section.


.. image:: images/100025600000153000001FBBAE7DD2C4810357C0.svg
    :width: 273
    :height: 409


.. _RefImage_100025600000153000001FBBAE7DD2C4810357C0.svg:

**Figure 1.1-a: Geometry of the heterogeneous medium - Fluidless beams**


.. image:: images/1000049A0000102000000C4D64BE3B3C9CDD093A.svg
    :width: 273
    :height: 409


.. _RefImage_1000049A0000102000000C4D64BE3B3C9CDD093A.svg:

**Figure 1.1-b: Reference cell** :math:`Y` ****- Magnification of** :math:`\frac{1}{\varepsilon }=10`


* Characteristics of the period:


* Dimensions:

:math:`\varepsilon Y=(0.21m,0.21m)`

:math:`a=1.5m`

:math:`e=0.3m`


* Characteristics of each beam:


* Section:

:math:`A\mathrm{=}{(\varepsilon \mathrm{\times }a)}^{2}\mathrm{=}{(0.1\mathrm{\times }1.5)}^{2}\mathrm{=}0.0225{m}^{2}`


* Length:

:math:`L=4.1m`


* Bending moment of inertia:

:math:`{I}_{x}={I}_{y}={(\varepsilon \times a)}^{4}/12{m}^{4}`


Material properties
-----------------------

Isotropic linear elastic material:

:math:`E={10}^{9}\mathrm{Pa}`

:math:`\mathrm{NU}=0.3`

Density:

Beam:

:math:`\mathrm{Rho}=7641\mathrm{kg}/{m}^{3}`

Fluid:

:math:`\mathit{Rho}\mathrm{=}0\mathit{kg}\mathrm{/}{m}^{3}` (case without fluid)

:math:`\mathrm{Rho}=1000\mathrm{kg}/{m}^{3}` (case with fluid)


Corrective terms
------------------

The correction terms are calculated on the reference cell :math:`Y` [:ref:`fig 1.1-b <fig 1.1-b>`].


.. csv-table::

    "B_T= :math:`0.79{m}^{2}`"
    "B_N= :math:`0.79{m}^{2}`"
    "B_TN= :math:`0{m}^{2}`"
    "A_ FLUI = :math:`2.16{m}^{2}`"
    "A_ CELL = :math:`2.25{m}^{2}`"
    "COEF_ECHELLE =10"



Boundary conditions and loads
-------------------------------------

**Case without fluid:**

Bottom surface :math:`\mathrm{Sb}`: recessed

All degrees of freedom are blocked.

Top surface :math:`\mathrm{Sh}`: recessed

All degrees of freedom are blocked.

**Case with fluid:**

Bottom surface :math:`\mathrm{Sb}`: recessed

All degrees of freedom are blocked.

Top surface :math:`\mathrm{Sh}`: flat support (bilateral connection)

All rotations are locked.

Longitudinal displacement :math:`\mathrm{DZ}` is blocked.

All the nodes in :math:`\mathrm{Sh}` have the same transverse displacement :math:`\mathrm{DX}` and the same normal displacement :math:`\mathrm{DY}`.