B modeling ============== Characteristics of modeling ----------------------------------- Meshing with incompressible 3D elements of type TETRA10 only .. image:: images/100147DE000069D5000048C34BB63DF767058DA9.svg :width: 366 :height: 252 .. _RefImage_100147DE000069D5000048C34BB63DF767058DA9.svg: :math:`\mathrm{AB}` is on the :math:`\mathrm{OX}` axis (unlike the :math:`A` modeling). For stripping purposes, we define node :math:`\mathrm{NOEUMI}=A+(0.0.e/4)` where the deformations and stresses are the same as in :math:`A`. **Conditions** limits **:** .. code-block:: text DDL_IMPO = GROUP_NO =' FACSUP ', DZ = 0 GROUP_NO =' FACINF ', DZ = 0 sides :math:`\mathit{AEFD}` (:math:`z\mathrm{=}0` and :math:`z\mathrm{=}0.01`) GROUP_NO =' FACEAB ', DY = 0 side :math:`\mathit{AB}` FACE_IMPO = GROUP_MA =' FACEEF ', DNOR = 0 sides :math:`\mathit{EF}` PRES_REP = GROUP_MA =' FACEAE ', PRES = 60 sided :math:`\mathit{AE}` Characteristics of the mesh ---------------------------- Number of knots: 13907 Number of meshes: 8519 TETRA10 Tested sizes and results ------------------------------ Result at point :math:`A`: * first column 3D_ INCO_UPG without imposing :math:`\mathrm{GONF}=0` * second column 3D_ INCO_UPG by imposing :math:`\mathrm{GONF}=0` * third column 3D_ INCO_UP with quadratic elements * fourth column3D_ INCO_UP with linear elements * fifth column3D_ INCO_UPO with linear elements +---------------------------------------------+------------------+-------------------+-------------+-------------+-------------+-------------+-------------+ |**Identification** |**Reference type**|**Reference value**|**Tolerance** | + + + +-------------+-------------+-------------+-------------+-------------+ | | | |**1** |**2** |**3** |**4** |**5** | +---------------------------------------------+------------------+-------------------+-------------+-------------+-------------+-------------+-------------+ |:math:`u` |'ANALYTIQUE' | #. 10 -5 |0.50% |0.50% |0.50% |0.50% |0.50% | + + + + + + + + + | | | | | | | | | +---------------------------------------------+------------------+-------------------+-------------+-------------+-------------+-------------+-------------+ |:math:`v` |'ANALYTIQUE' | |10-5 |10-5 |10-5 |10-5 |10-5 | +---------------------------------------------+------------------+-------------------+-------------+-------------+-------------+-------------+-------------+ |:math:`{\sigma }_{\mathrm{xx}}` |'ANALYTIQUE' |—60. |1.00% |1.00% |1.00% |4.00% |3.00% | +---------------------------------------------+------------------+-------------------+-------------+-------------+-------------+-------------+-------------+ |:math:`{\sigma }_{\mathrm{yy}}` |'ANALYTIQUE' | |1.00% |1.00% |1.00% |1.50% |1.00% | +---------------------------------------------+------------------+-------------------+-------------+-------------+-------------+-------------+-------------+ |:math:`{\sigma }_{\mathrm{zz}}` |'ANALYTIQUE' | |2.50% |2.50% |2.50% |10.00% |3.00% | +---------------------------------------------+------------------+-------------------+-------------+-------------+-------------+-------------+-------------+ |:math:`{\sigma }_{\mathrm{xy}}` |'ANALYTIQUE' | |2.5 |2.5 |2.5 |2.5 |2.5 | +---------------------------------------------+------------------+-------------------+-------------+-------------+-------------+-------------+-------------+ |:math:`{\varepsilon }_{\mathrm{xx}}` |'ANALYTIQUE' |-6. 10-4 |0.50% |0.50% |0.50% |1.50% |2.0% | +---------------------------------------------+------------------+-------------------+-------------+-------------+-------------+-------------+-------------+ |:math:`{\varepsilon }_{\mathrm{yy}}` |'ANALYTIQUE' | #. 10-4 |0.50% |0.50% |0.50% |1.00% |1.00% | + + + + + + + + + | | | | | | | | | +---------------------------------------------+------------------+-------------------+-------------+-------------+-------------+-------------+-------------+ |:math:`{\varepsilon }_{\mathrm{xy}}` |'ANALYTIQUE' | | #. 10-5 | #. 10-5 | #. 10-5 | #. 10-5 | #. 10-5 | + + + + + + + + + | | | | | | | | | +---------------------------------------------+------------------+-------------------+-------------+-------------+-------------+-------------+-------------+ |:math:`{\varepsilon }_{\mathrm{eq}}` - INVA_2|'ANALYTIQUE' |6.92 10-4 |0.50% |0.50% |0.50% |1.00% |1.50% | +---------------------------------------------+------------------+-------------------+-------------+-------------+-------------+-------------+-------------+ |:math:`{\varepsilon }_{\mathrm{eq}}` - PRIN_1|'ANALYTIQUE' |—6. 10-4 |0.50% |0.50% |0.50% |1.50% |2.0% | +---------------------------------------------+------------------+-------------------+-------------+-------------+-------------+-------------+-------------+ |:math:`{\varepsilon }_{\mathrm{eq}}` - PRIN_2|'ANALYTIQUE' | |10-5 |10-5 |10-5 |10-5 |10-5 | +---------------------------------------------+------------------+-------------------+-------------+-------------+-------------+-------------+-------------+ |:math:`{\varepsilon }_{\mathrm{eq}}` - PRIN_3|'ANALYTIQUE' | #. 10-4 |0.50% |0.50% |0.50% |1.00% |1.0% | + + + + + + + + + | | | | | | | | | +---------------------------------------------+------------------+-------------------+-------------+-------------+-------------+-------------+-------------+ |:math:`{\sigma }_{\mathrm{eq}}` - VMIS |'ANALYTIQUE' |138.56 |1.00% |1.00% |1.00% |1.00% |1.5% | +---------------------------------------------+------------------+-------------------+-------------+-------------+-------------+-------------+-------------+ |:math:`{\sigma }_{\mathrm{eq}}` - TRESCA |'ANALYTIQUE' | |1.00% |1.00% |1.00% |1.00% |1.5% | +---------------------------------------------+------------------+-------------------+-------------+-------------+-------------+-------------+-------------+ |:math:`{\sigma }_{\mathrm{eq}}` - PRIN_1 |'ANALYTIQUE' |-60. |3.00% |3.00% |3.00% |4.00% |3.00% | +---------------------------------------------+------------------+-------------------+-------------+-------------+-------------+-------------+-------------+ |:math:`{\sigma }_{\mathrm{eq}}` - PRIN_2 |'ANALYTIQUE' | |3.00% |3.00% |3.00% |10.00% |3.00% | +---------------------------------------------+------------------+-------------------+-------------+-------------+-------------+-------------+-------------+ |:math:`{\sigma }_{\mathrm{eq}}` - PRIN_3 |'ANALYTIQUE' | |1.00% |1.00% |1.00% |1.50% |1.00% | +---------------------------------------------+------------------+-------------------+-------------+-------------+-------------+-------------+-------------+ |:math:`{\sigma }_{\mathrm{eq}}` - VMIS |'ANALYTIQUE' |138.56 |1.00% |1.00% |1.00% |1.00% |1.5% | +---------------------------------------------+------------------+-------------------+-------------+-------------+-------------+-------------+-------------+ Result at point :math:`F`: * first column 3D_ INCO_UPG without imposing :math:`\mathrm{GONF}=0` * second column 3D_ INCO_UPG by imposing :math:`\mathrm{GONF}=0` * third column 3D_ INCO_UP with quadratic elements * fourth column3D_ INCO_UP with linear elements * fifth column3D_ INCO_UPO with linear elements +------------------------------------+------------------+-------------------+-------------+-----+-----+-----+-----+ |**Identification** |**Reference type**|**Reference value**|**Tolerance** | + + + +-------------+-----+-----+-----+-----+ | | | |**1** |**2**|**3**|**4**|**5**| +------------------------------------+------------------+-------------------+-------------+-----+-----+-----+-----+ |:math:`u` |'ANALYTIQUE' |2.12 10-5 |0.50% |0.50%|0.50%|0.50%|0.50%| +------------------------------------+------------------+-------------------+-------------+-----+-----+-----+-----+ |:math:`v` |'ANALYTIQUE' |2.12 10-5 |0.50% |0.50%|0.50%|0.50%|0.50%| +------------------------------------+------------------+-------------------+-------------+-----+-----+-----+-----+ |:math:`{\sigma }_{\mathrm{xx}}` |'ANALYTIQUE' | |1.00% |1.00%|1.00%|1.50%|1.50%| +------------------------------------+------------------+-------------------+-------------+-----+-----+-----+-----+ |:math:`{\sigma }_{\mathrm{yy}}` |'ANALYTIQUE' | |1.00% |1.00%|1.00%|1.00%|1.00%| +------------------------------------+------------------+-------------------+-------------+-----+-----+-----+-----+ |:math:`{\sigma }_{\mathrm{zz}}` |'ANALYTIQUE' | |1.00% |1.00%|1.00%|1.50%|1.00%| +------------------------------------+------------------+-------------------+-------------+-----+-----+-----+-----+ |:math:`{\sigma }_{\mathrm{xy}}` |'ANALYTIQUE' |-20. |1.00% |1.00%|1.00%|1.00%|1.00%| +------------------------------------+------------------+-------------------+-------------+-----+-----+-----+-----+ |:math:`{\varepsilon }_{\mathrm{xx}}`|'ANALYTIQUE' | |10-5 |10-5 |10-5 |10-5 |10-5 | +------------------------------------+------------------+-------------------+-------------+-----+-----+-----+-----+ |:math:`{\varepsilon }_{\mathrm{yy}}`|'ANALYTIQUE' | |10-5 |10-5 |10-5 |10-5 |10-5 | +------------------------------------+------------------+-------------------+-------------+-----+-----+-----+-----+ |:math:`{\varepsilon }_{\mathrm{xy}}`|'ANALYTIQUE' |-1.5 10-4 |0.50% |0.50%|0.50%|1.00%|1.00%| +------------------------------------+------------------+-------------------+-------------+-----+-----+-----+-----+ Checking the transition to the nodes for the middle nodes (only for the result obtained without imposing GONF = 0) - value at node :math:`\mathrm{NOEUMI}`: .. csv-table:: "**Identification**", "**Reference type**", "**Reference value**", "**Tolerance (** :math:`\text{\%}` **)**" ":math:`{\sigma }_{\mathrm{xx}}` ", "'ANALYTIQUE'", "-60. "," 1.70%" ":math:`{\sigma }_{\mathrm{yy}}` ", "'ANALYTIQUE'", "100. "," 0.60%" ":math:`{\sigma }_{\mathrm{zz}}` ", "'ANALYTIQUE'", "20. "," 3.50%" ":math:`{\varepsilon }_{\mathrm{xx}}` ", "'ANALYTIQUE'", "-6. 10-4"," 0.50%" ":math:`{\varepsilon }_{\mathrm{yy}}` ", "'ANALYTIQUE'", "6. 10-4"," 0.50%" notes --------- The results obtained here are a bit worse than in the case of modeling :math:`A`, but the discretization is coarser since there are about 2 times fewer nodes in this test case. The results are still satisfactory since the differences are less than :math:`\text{0.2 \%}` for the movements, less than :math:`\text{0.5 \%}` for the deformations and less than :math:`\text{1 \%}` for the constraints. We note again that there is no significant improvement in the result when :math:`\mathrm{tr}\varepsilon =0` is explicitly imposed.