C modeling
==============


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

3D modeling and the DISCRETE method of the projected conjugate gradient (GCP) of contact treatment are used.


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 0:14 elements of type TRIA6 and 6 elements of type TETRA10


* mesh 1:50 elements of type TRIA6 and 48 elements of type TETRA10


* mesh 2:194 elements of type TRIA6 and 384 elements of type TETRA10


* mesh 3:770 elements of type TRIA6 and 3072 elements of type TETRA10

Note that, compared to models A and B, the base is meshed with 2 TRIA6 instead of QUAD8. Indeed, discrete methods are not adapted to the use of these elements (see [:ref:`R5.03.50 <R5.03.50>`]).


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 biggest 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 moving in the field :math:`\mathrm{\Omega }`;


* the largest real :math:`{\alpha }_{s}>0` such as :math:`{\mathrm{\parallel }{U}^{\text{calc}}\mathrm{-}{U}^{\text{ref}}\mathrm{\parallel }}_{\mathrm{0,}{\Gamma }_{C}}<{C}_{s}\mathrm{\times }{h}^{{\alpha }_{s}}` where :math:`{C}_{s}` is independent of :math:`h` for movement on the surface :math:`{\mathrm{\Gamma }}_{C}`.

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


.. csv-table::

    "**Identification**", "**Reference type**", "**Reference value**"
    ":math:`\sum ^{\text{noeuds}n}\mid {\underline{U}}_{\text{n}}^{\text{calc}}-{\underline{U}}_{\text{n}}^{\text{ref}}\mid` ", "'NON_REGRESSION'", "4.25881911029E-05", "4.25881911029E-05"
    ":math:`{\alpha }_{U}` ", "'ANALYTIQUE'", "3.0"


.. csv-table::

    ":math:`{\alpha }_{p}` ", "'ANALYTIQUE'", "3.5"