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


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

We use 3D modeling and the CONTINUE contact treatment method.


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

A convergence study is carried out with the fineness of the mesh from the calculated solution to the analytical solution. A series of meshes obtained by uniform refinement using the MACR_ADAP_MAIL command is used:


* mesh 0:1 element of type QUAD8, 12 elements of type TRIA6 and 6 elements of type TETRA10


* mesh 1:1 element of type QUAD8, 48 elements of type TRIA6, and 48 elements of type TETRA10


* mesh 2:1 element of type QUAD8, 192 elements of type TRIA6, and 384 elements of type TETRA10


* mesh 3:1 element of type QUAD8, 768 elements of type TRIA6, and 3072 elements of type TETRA10


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

The convergence speed is tested with the fineness of the mesh from the calculated solution to the analytical solution in standard :math:`{L}_{2}`:


* the largest real :math:`{\mathrm{\alpha }}_{U}>0` such as :math:`{\Vert {\underline{U}}^{\text{calc}}-{\underline{U}}^{\text{ref}}\Vert }_{0,\mathrm{\Omega }}<{C}_{U}\times {h}^{{\mathrm{\alpha }}_{U}}` where :math:`{C}_{U}` is independent of :math:`h` for displacement;


* the largest real :math:`{\mathrm{\alpha }}_{p}>0` such as :math:`{\Vert {p}^{\text{calc}}-{p}^{\text{ref}}\Vert }_{0,{\mathrm{\Gamma }}_{C}}<{C}_{p}\times {h}^{{\mathrm{\alpha }}_{p}}` where :math:`{C}_{p}` is independent of :math:`h` for contact pressure.

The sum of the absolute values of the difference between the calculated solution and the analytical solution for the displacement is also tested.


+-----------------------------------------------------------------------------------------------------------------------------------------------------------------------+------------------+-------------------+
|**Identification**                                                                                                                                                     |**Reference type**|**Reference value**|
+-----------------------------------------------------------------------------------------------------------------------------------------------------------------------+------------------+-------------------+
|:math: `\ sum ^ {\ text {nodes} n}} | {\ underline {U}} | {\ underline {U}}} _ {\ text {calc}} - {\ underline {U}}} _ {\ text {U}}} _ {\ text {U}}}} _ {\ text {n}}} |`|'NON_REGRESSION'  |4.13935026178E-05  |
+-----------------------------------------------------------------------------------------------------------------------------------------------------------------------+------------------+-------------------+
|:math:`{\mathrm{\alpha }}_{U}`                                                                                                                                         |'ANALYTIQUE'      |3.0                |
+-----------------------------------------------------------------------------------------------------------------------------------------------------------------------+------------------+-------------------+

.. csv-table::

    ":math:`{\mathrm{\alpha }}_{p}` ", "'NON_REGRESSION'", "2.534025066720"
    ":math:`{\mathrm{\alpha }}_{p}` ", "'ANALYTIQUE'", "2.5"