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


Geometry
---------


.. image:: images/10000200000001E4000001A6EC7B0AFDC6CE4BE3.png
    :width: 3.3709in
    :height: 2.9425in


.. _RefImage_10000200000001E4000001A6EC7B0AFDC6CE4BE3.png:

Square plate **:**

Length: :math:`l=1.0m`

Thickness: :math:`e=0.1m`


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

Young's module, :math:`E=4.388{10}^{10}N/{m}^{2}`

Poisson's ratio, :math:`\nu =0.0`

Density, :math:`\rho =2500.0\mathrm{kg}/{m}^{3}`


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

**Boundary conditions:**


.. csv-table::

    "**L** location", "**Blocked components**"
    "A1", "DX, DY, DZ, DRX, DRY, DRZ"
    "A", "DX, DZ, DRX, DRY, DRZ"
    "C", "DZ, DRX, DRY, DRZ"


**Loads:**

We apply the linear force on side B in the direction :math:`x`, which depends on time like,

:math:`F(t)={Q}_{0}EKe\mathrm{cos}(\mathrm{Kl})\mathrm{sin}(\omega t)`,

where the following parameters are used:


* :math:`{Q}_{0}` (:math:`={10}^{-4}m`) - load amplitude


* :math:`E` — Young's modulus defined above (in :math:`N/{m}^{2}`)


* :math:`e` — the thickness defined above (in :math:`m`)


* :math:`l` — the plate size defined above (in :math:`m`)


* :math:`K` (:math:`=\frac{\pi }{\mathrm{8l}}`) the wave number of the analytical solution (in :math:`{m}^{-1}`)


* :math:`\omega` — frequency (times :math:`2\pi`), linked to the wave number :math:`K`, :math:`K=\omega /c`, :math:`c` being the speed of the waves in the structure, :math:`c=\sqrt{\frac{E}{\rho }}`

The parameterization introduced makes it possible to apply the load just to obtain the analytical solution, determined simply by the parameters :math:`{Q}_{0}` and :math:`K`, and then by other parameters of the dimensions and material properties of the structure.


Initial conditions
--------------------

Initially, the movements are zero everywhere and the speeds obey the following spatial distribution,

:math:`{v}_{0}(x,y)=\omega {Q}_{0}\mathrm{sin}(\mathrm{K.x})`