J modeling
==============


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

3D modeling is used with element type PYRAM5 and the LAC contact processing 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 as in the G modeling for elements HEXA8.

However, by using the option DECOUPE_HEXA = 'PYRA' in the CREA_MAILLAGE command, you can treat the slave meshes with the element type PYRAM5 for contact processing by the LAC method:


* mesh 0:4 elements of type TRIA3, 5 elements of type QUAD4, 5 elements of type PYRAM5 and 0 elements of type HEXA8


* mesh 1:16 elements of type TRIA3, 20 elements of type QUAD4, 20 elements of type PYRAM5et 4 element of type HEXA8


* mesh 2:64 elements of type TRIA3, 80 elements of type QUAD4, 80 elements of type PYRAM5et, 48 elements of type HEXA8


* mesh 3:256 elements of type TRIA3, 320 elements of type QUAD4, 320 elements of type PYRAM5et, 448 elements of type HEXA8


* mesh 4:1024 elements of type TRIA3, 1280 elements of type QUAD4, 1280 elements of type PYRAM5et, 3840 element of type HEXA8



We note that, compared to the E and F models, the curved surface :math:`\text{MAITRE}` is meshed with 2 TRIA6au instead of a single QUAD8.


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:`{\alpha }_{U}>0` such as :math:`{\mathrm{\parallel }{\underline{U}}^{\text{calc}}\mathrm{-}{\underline{U}}^{\text{ref}}\mathrm{\parallel }}_{\mathrm{0,}\Omega }<{C}_{U}\mathrm{\times }{h}^{{\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.


.. csv-table::

    "**Identification**", "**Reference type**", "**Reference value**"
    ":math:`{\mathrm{\alpha }}_{U}` ", "'ANALYTIQUE'", "2.0"


.. csv-table::

    ":math:`{\mathrm{\alpha }}_{p}` ", "'ANALYTIQUE'", "0.5"