B modeling
==============


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

We cut the plate into 100 QUAD4 meshes.

Three models for the plate are used: Q4G, DKT (DKQ), DST (DSQ).

Compared to modeling A, the plate was rotated in plane :math:`(o,x,y)` by an angle of :math:`60°`


.. image:: images/100008E8000016BD0000131FD09459F6F6D69378.svg
    :width: 293
    :height: 246


.. _RefImage_100008E8000016BD0000131FD09459F6F6D69378.svg:

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


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

For the membrane problem, the boundary conditions are:


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


* at node :math:`A`, we block the movement :math:`\mathrm{DY}` in the coordinate system :math:`(A,x\text{'},y\text{'})`,


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


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


.. csv-table::

    "Number of knots: 121", "Number of stitches and types: 100 QUAD4"


The characteristic points of the mesh are as follows:


.. csv-table::

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



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

For flexure modes:


.. csv-table::

    "**Number**
    **of the mode**", "", "**Frequencies**", "", ""
    "", "**Reference**", "**Aster** **** DKQ **", "**% difference**", "**% tolerance**"
    "4", "35.63", "35.359", "—0.760", "0.8"
    "5", "68.51", "67.491", "—1.427", "1.5"
    "6", "109.62", "108.563", "—0.964", "1."
    "7", "123.32", "121.144", "—1.765", "1.8"
    "8", "142.51", "138.402", "—2.882", "2.9"
    "9", "197.32", "188.500", "—4.470", "4.5"
    "", "", "**Aster** **** DSQ **", ""
    "4", "35.63", "35.351", "—0.782", "0.8"
    "5", "68.51", "67.464", "—1.527", "1.6"
    "6", "109.62", "108.494", "—1.027", "1.1"
    "7", "123.32", "121.060", "—1.832", "1.9"
    "8", "142.51", "138.291", "—2.961", "3."
    "9", "197.32", "188.298", "—4.572", "4.6"
    "", "", "**Aster** **Q4G**", "", ""
    "4", "35.63", "36.011", "36.011", "1.068", "1.1"
    "5", "68.51", "70.795", "70.795", "3.5"
    "6", "109.62", "114.593", "4.536", "4.6"
    "7", "123.32", "134.899", "9.39", "9.4"
    "8", "142.51", "142.941", "4.513", "4.6"
    "9", "197.32", "212.045", "7.463", "7.5"


For the membrane problem:


.. csv-table::

    "", "**Reference**", "**Aster**", "**% difference**", "**% tolerance**"
    "DKQ ", "0.14714", "0.14713", "—0.003", "0.1"
    "DSQ ", "0.14714", "0.14714", "—0.002", "0.1"
    "Q4G", "0.14714", "0.14714", "0. ", "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:

.. code-block:: text


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


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