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


Geometry
---------

The geometry, shown in the following figure, is composed of a triangle with a base :math:`\mathrm{1m}` and a height :math:`\mathrm{1m}`.


.. image:: images/1000000000000584000002CA52D4A07263A5345D.png
    :width: 6.3374in
    :height: 3.2043in


.. _RefImage_1000000000000584000002CA52D4A07263A5345D.png:


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

The mesh is 2D, quadratic and shown by the following figure:


.. image:: images/1000000000000584000002CA12798E944A172410.png
    :width: 6.3374in
    :height: 3.2043in


.. _RefImage_1000000000000584000002CA12798E944A172410.png:

It is composed of 6 knots for a TRIA6 stitch.


Adaptations made and resulting meshes
---------------------------------------------

A field for movement at the nodes is created and the component :math:`\mathit{DX}` is initialized to zero for all nodes, except node :math:`A` (identified in the previous figure) for which :math:`\mathit{DX}\mathrm{=}1`.

Two uniform refinements are made with the displacement field updated. The first update is done with option CH_AUTO and the second with option CH_ISOP2. The expected updated field should therefore not be the same at the end of the two adaptations.

The following figure shows the initial field:


.. image:: images/1000000000000536000002CAD941B5FB4CFC1405.png
    :width: 6.3374in
    :height: 3.3917in


.. _RefImage_1000000000000536000002CAD941B5FB4CFC1405.png:

*Note:*

Validation, based on the updated field, will be performed on nodes 2 and 4. The value of :math:`\mathit{DX}` in :math:`\mathit{N2}` must be equal to 1 while in :math:`\mathit{N4}` it is zero.

The following figure shows the field resulting from the adaptation with option CH_AUTO:


.. image:: images/1000000000000536000002CA2FE42E9CB9FF3249.png
    :width: 6.3374in
    :height: 3.3917in


.. _RefImage_1000000000000536000002CA2FE42E9CB9FF3249.png:

It appears, as expected, that the interpolated field does not respect the extreme values of the initial field (confer U7.03.01, §4.15.5).

*Note:*

Validation, based on the updated field, will be performed on nodes 2, 4 and 8. The values at nodes 2 and 4 should be the same as in the initial field. For node 8, with option CH_AUTO, the shape functions :math:`\mathit{P2}`, :math:`{N}_{Ni}(i\mathrm{\in }\mathrm{[}\mathrm{1,6}\mathrm{]})` (confer R3.01.01 §3.1) from the initial mesh are used for interpolation. So we expect to find: :math:`{\mathit{DX}}_{\mathit{N8}}\mathrm{=}\mathrm{\sum }_{1}^{6}{\mathit{DX}}_{Ni}\mathrm{.}{N}_{Ni}({\xi }_{\mathit{N8}},{\eta }_{\mathit{N8}})` for :math:`{\xi }_{\mathit{N8}}\mathrm{=}1\mathrm{/}4`, :math:`{\eta }_{\mathit{N8}}\mathrm{=}0`, :math:`{\mathit{DX}}_{\mathit{N2}}\mathrm{=}1.` and :math:`{\mathit{DX}}_{Ni}\mathrm{=}0.(i\mathrm{\ne }2)` for,, and i.e.: :math:`{\mathit{DX}}_{\mathit{N8}}\mathrm{=}1.(\mathrm{-}(1.\mathrm{-}0.25)\mathrm{.}(1.\mathrm{-}2.(1.\mathrm{-}0.25)))\mathrm{=}0.375`.

The following figure shows the field resulting from the adaptation with option CH_ISOP2:


.. image:: images/1000000000000536000002CA4B0FFCCEEA2A9CEF.png
    :width: 6.3374in
    :height: 3.3917in


.. _RefImage_1000000000000536000002CA4B0FFCCEEA2A9CEF.png:

It appears, as expected, that with option CH_ISOP2, the interpolated field respects the extreme values of the initial field (confer U7.03.01, §4.15.5).

*Note:*

Nodes 2, 4, and 8 are identified because they are used for validation. The values at nodes 2 and 4 should be the same as in the initial field. For node 8, with option CH_ISOP2, functions of shapes :math:`\mathit{P1}`, :math:`{N}_{Ni}(i\mathrm{\in }\mathrm{[}\mathrm{1,3}\mathrm{]})` (confer R3.01.01 §3.1) expressed on the submeshes of the initial mesh element are used for interpolation. So we expect to find: :math:`{\mathit{DX}}_{\mathit{N8}}\mathrm{=}\mathrm{\sum }_{1}^{3}{\mathit{DX}}_{Ni}\mathrm{.}{N}_{Ni}({\xi }_{\mathit{N8}},{\eta }_{\mathit{N8}})` for :math:`{\xi }_{\mathit{N8}}\mathrm{=}1\mathrm{/}2`, :math:`{\eta }_{\mathit{N8}}\mathrm{=}0`, :math:`{\mathit{DX}}_{\mathit{N2}}\mathrm{=}1.` and :math:`{\mathit{DX}}_{Ni}\mathrm{=}0.(i\mathrm{\ne }2)` for,, and i.e.: :math:`{\mathit{DX}}_{\mathit{N8}}\mathrm{=}1.(1.\mathrm{-}0.5)\mathrm{=}0.5`.


Tested sizes
-----------------

The following quantities are tested:

.. csv-table::

    "Initial Field", "Analytical Values", "Tolerance"
    ":math:`\mathit{N2}` ", "1. ", "1.E-6"
    ":math:`\mathit{N4}` ", "0. ", "1.E-6"



.. csv-table::

    "Field CH_AUTO ", "Analytical Values", "Tolerance"
    ":math:`\mathit{N2}` ", "1. ", "1.E-6"
    ":math:`\mathit{N4}` ", "0. ", "1.E-6"
    ":math:`\mathit{N8}` ", "0.375", "1.E-6"



.. csv-table::

    "Field CH_ISOP2 ", "Analytical Values", "Tolerance"
    ":math:`\mathit{N2}` ", "1. ", "1.E-6"
    ":math:`\mathit{N4}` ", "0. ", "1.E-6"
    ":math:`\mathit{N8}` ", "0.5", "1.E-6"