5. C modeling#

5.1. Characteristics of modeling#

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

5.2. 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 [R5.03.50]).

5.3. 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 \({L}_{2}\):

  • the biggest real \({\mathrm{\alpha }}_{U}>0\) such as \({\Vert {\underline{U}}^{\text{calc}}-{\underline{U}}^{\text{ref}}\Vert }_{0,\mathrm{\Omega }}<{C}_{U}\times {h}^{{\mathrm{\alpha }}_{U}}\) where \({C}_{U}\) is independent of \(h\) for moving in the field \(\mathrm{\Omega }\);

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

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

Identification

Reference type

Reference value

\(\sum ^{\text{noeuds}n}\mid {\underline{U}}_{\text{n}}^{\text{calc}}-{\underline{U}}_{\text{n}}^{\text{ref}}\mid\)

“NON_REGRESSION”

4.25881911029E-05

4.25881911029E-05

\({\alpha }_{U}\)

“ANALYTIQUE”

3.0

\({\alpha }_{p}\)

“ANALYTIQUE”

3.5