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


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

The plate was cut into 200 TRIA3 meshes. Two models for the plate are used: DKT and DST.

For the flexure problem, the boundary conditions are as follows:


* in all knots on the edge: :math:`\mathit{DZ}\mathrm{=}0`

For the membrane problem, the boundary conditions are:


* in all the nodes of the mesh: :math:`\mathit{DZ}\mathrm{=}0` :math:`\mathit{DRX}\mathrm{=}\mathit{DRY}\mathrm{=}\mathit{DRZ}\mathrm{=}0`,


* in all the nodes on sides :math:`\mathit{AB}` and :math:`\mathit{BC}`: :math:`\mathit{DY}\mathrm{=}0`


* At points :math:`A,B,C,D`, discrete stiffness elements are added (direction :math:`x`).


.. image:: images/10001A5E00001D8F0000110E1F11F87DCB2C3E07.svg
    :width: 381
    :height: 220


.. _RefImage_10001A5E00001D8F0000110E1F11F87DCB2C3E07.svg:


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

Number of knots: 121

Number of meshes and types: 200 TRIA3

The characteristic points of the mesh are as follows:


.. csv-table::

    "Dot :math:`A` = :math:`\mathit{N1}` ", "Dot :math:`C` = :math:`\mathit{N121}`"
    "Dot :math:`B` = :math:`\mathit{N111}` ", "Dot :math:`D` = :math:`\mathit{N11}`"



Tested sizes and results
------------------------------

For flexure modes:


.. csv-table::

    "**Number**
    **of the mode**", "", "", "", "**Frequencies**"
    "", "**Reference**", "**Aster** **** DKT **", "**% difference**", "**% tolerance**"
    "4", "35.63", "35.46", "—0.477", "0.5"
    "5", "68.51", "67.82", "—1.003", "1.1"
    "6", "109.62", "108.67", "—0.867", "0.9"
    "7", "123.32", "121.90", "—1.150", "1.2"
    "8", "142.51", "139.99", "—1.761", "1.8"
    "9", "197.32", "191.70", "—2.846", "2.9"



.. csv-table::

    "**Aster** **DST**", "**% difference**", "**% tolerance**"
    "35.45", "—0.492", "0.5"
    "67.80", "—1.030", "1.1"
    "108.62", "—0.910", "1."
    "121.84", "—1.199", "1.3"
    "139.92", "—1.815", "1.9"
    "191.57", "—2.912", "3."


For the membrane problem:


.. csv-table::

    "**Reference**", "**Aster** **** DKT **", "**% difference**", "**% tolerance**"
    "0.14714", "0.147136", "—0.002", "0.1"
    "", "**Aster** **** DST **", "**% difference**", "**% tolerance**"
    "", "0.147136", "—0.001", "0.1"



notes
---------

For the flexure problem, the modal position of the first mode found in band :math:`(5.,200.)` is the fourth, because there are three solid body modes at the zero frequency:


* translation modes :math:`u` and :math:`v` in the plane,


* rotation mode around the :math:`z` axis.