Reference problem ===================== Modeling A --------------- Geometry ~~~~~~~~ .. image:: images/10000201000003C0000002D0F82204E22C8FE72B.png :width: 1.6598in :height: 2.1161in .. _RefImage_10000201000003C0000002D0F82204E22C8FE72B.png: +-----------------------------------------------------------------------------------------------------------------------------+-----------------------------------------------------------------------------------------------------------------------------+-----------------------------------------------------------------------------------------------------------------------------+ | | | | + .. image:: images/10000200000000D3000000D32FBE11A4E7E36C91.png + .. image:: images/10000200000000D3000000D8861E09A0EFFAA42B.png + .. image:: images/10000200000001390000013A06F9796DE21D81A3.png + | :width: 1.2571in | :width: 1.2154in | :width: 1.2228in | + :height: 1.4693in + :height: 1.4693in + :height: 1.4453in + | | | | + + + + | | | | +-----------------------------------------------------------------------------------------------------------------------------+-----------------------------------------------------------------------------------------------------------------------------+-----------------------------------------------------------------------------------------------------------------------------+ :math:`\mathrm{Carre1}/\mathrm{carre2}` :math:`\mathrm{AC}` :math:`\mathrm{cube1}` Coordinates of points :math:`(m)`: :math:`A:(0.,0.)` :math:`B:(1.,0.)` :math:`C:(0.,0.)` :math:`D:(0.,1.)` Cube geometry: Center: :math:`(0.,0.,-0.5)` Side: :math:`L=1` Mesh group: :math:`\mathrm{carre}1` surface :math:`A,B,C,D` :math:`\mathrm{carre}2` surface :math:`A,B,C,D` :math:`\mathrm{AC}` segment :math:`\mathrm{AC}` :math:`\mathrm{cube1}` volume Material properties ~~~~~~~~~~~~~~~~~~~~~~~~~ Not applicable Boundary conditions and loads ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Not applicable Benchmark solution --------------------- Calculation method ~~~~~~~~~~~~~~~~~~~~ **Calculating the reference for the temperature field** The field that is projected from one model to the other is an analytical temperature field whose evolution is as follows: :math:`T=3.+X+Y` The reference solution is identical to the projected analytical field. **Calculating the reference for the constraint field** The objective is to change the coordinate system, after projecting a known stress field onto a mesh into a 3D mesh. Transition from a cylindrical coordinate system (:math:`\mathrm{XOY}`) to a Cartesian coordinate system :math:`\mathrm{3D}` (:math:`\mathrm{XYZ}`). The constraint field :math:`(N/{m}^{2})` in the axisymmetric coordinate system (axis :math:`\mathrm{OY}`) is as follows: * :math:`\mathrm{SIXX}=2` * :math:`\mathrm{SIYY}=y` * :math:`\mathrm{SIZZ}=1` * :math:`\mathrm{SIXY}=0.` The constraint field in the Cartesian coordinate system (:math:`\mathrm{3D}`) is obtained by performing: * A projection of the stress tensor evaluated on mesh :math:`\mathrm{2D}\mathrm{axis}` onto the 3D mesh. * Change of frame of the stress tensor :math:`[{\sigma }_{\mathrm{3D}}]=[P][{\sigma }_{\mathrm{cyl}}]{[P]}^{T}` or :math:`[P]` represents the change of frame matrix. The numerical results are as follows: .. csv-table:: "NOEUD "," :math:`X` "," :math:`Y` "," :math:`Z` "," "," :math:`\mathrm{SIXX}` "," :math:`\mathrm{SIYY}` "," :math:`\mathrm{SIZZ}`" "N258", "-1/3", "-1/3", "-1/3", "1/3", "1/3", "1/3", "1/3", "1/3", "1/3", "1/3", "1/3", "1/3", "1/6" "N33", "-1/3", "0. ", "1. ", "2. ", "1. ", "1." "N108", "0. ", "1/2", "2/3", "1. ", "2. ", "2/3." Benchmark results ~~~~~~~~~~~~~~~~~~~~~~~~~ +------------------------------------------+---------------------+---------------------+---------------+ |Projection type |Point |Size :math:`(°C)` |Reference | +------------------------------------------+---------------------+---------------------+---------------+ |:math:`\mathrm{carre1}\to \mathrm{carre2}`|:math:`A` |:math:`\mathrm{TEMP}`|:math:`3` | + +---------------------+---------------------+---------------+ | |:math:`B` |:math:`\mathrm{TEMP}`|:math:`4` | + +---------------------+---------------------+---------------+ | |:math:`C` |:math:`\mathrm{TEMP}`|:math:`5` | + +---------------------+---------------------+---------------+ | |:math:`\mathrm{N364}`|:math:`\mathrm{TEMP}`|:math:`4.66` | +------------------------------------------+---------------------+---------------------+---------------+ |:math:`\mathrm{carre2}\to \mathrm{carre1}`|:math:`A` |:math:`\mathrm{TEMP}`|:math:`3` | + +---------------------+---------------------+---------------+ | |:math:`B` |:math:`\mathrm{TEMP}`|:math:`4` | + +---------------------+---------------------+---------------+ | |:math:`C` |:math:`\mathrm{TEMP}`|:math:`5` | + +---------------------+---------------------+---------------+ | |:math:`\mathrm{N355}`|:math:`\mathrm{TEMP}`|:math:`3.75` | +------------------------------------------+---------------------+---------------------+---------------+ |:math:`\mathrm{carre2}\to \mathrm{AC}` |:math:`A` |:math:`\mathrm{TEMP}`|:math:`3` | + +---------------------+---------------------+---------------+ | |:math:`C` |:math:`\mathrm{TEMP}`|:math:`5` | + +---------------------+---------------------+---------------+ | |:math:`\mathrm{N356}`|:math:`\mathrm{TEMP}`|:math:`4` | +------------------------------------------+---------------------+---------------------+---------------+ |:math:`\mathrm{AC}\to \mathrm{carre2}` |:math:`A` |:math:`\mathrm{TEMP}`|:math:`3` | + +---------------------+---------------------+---------------+ | |:math:`B` |:math:`\mathrm{TEMP}`|:math:`4` | +------------------------------------------+---------------------+---------------------+---------------+ |:math:`\mathrm{carre2}\to \mathrm{cube1}` |:math:`C` |:math:`\mathrm{TEMP}`|:math:`5` | + +---------------------+---------------------+---------------+ | |:math:`\mathrm{N363}`|:math:`\mathrm{TEMP}`|:math:`3.33` | + +---------------------+---------------------+---------------+ | |:math:`\mathrm{N341}`|:math:`\mathrm{TEMP}`|:math:`3.69371`| +------------------------------------------+---------------------+---------------------+---------------+ +-----------------------------------------+---------------------+------------------------+------------+ |Projection type |Point |Size :math:`(N/{m}^{2})`|Reference | +-----------------------------------------+---------------------+------------------------+------------+ |:math:`\mathrm{carre2}\to \mathrm{cube1}`|:math:`\mathrm{N258}`|:math:`\mathrm{SIXX}` |:math:`1.5` | + +---------------------+------------------------+------------+ | |:math:`\mathrm{N258}`|:math:`\mathrm{SIYY}` |:math:`1.5` | + +---------------------+------------------------+------------+ | |:math:`\mathrm{N258}`|:math:`\mathrm{SIZZ}` |:math:`0.16`| + +---------------------+------------------------+------------+ | |:math:`\mathrm{N33}` |:math:`\mathrm{SIXX}` |:math:`2` | + +---------------------+------------------------+------------+ | |:math:`\mathrm{N33}` |:math:`\mathrm{SIYY}` |:math:`1` | + +---------------------+------------------------+------------+ | |:math:`\mathrm{N33}` |:math:`\mathrm{SIZZ}` |:math:`1` | + +---------------------+------------------------+------------+ | |:math:`\mathrm{N108}`|:math:`\mathrm{SIXX}` |:math:`1` | + +---------------------+------------------------+------------+ | |:math:`\mathrm{N108}`|:math:`\mathrm{SIYY}` |:math:`2` | + +---------------------+------------------------+------------+ | |:math:`\mathrm{N108}`|:math:`\mathrm{SIZZ}` |:math:`0.66`| +-----------------------------------------+---------------------+------------------------+------------+ B modeling -------------- Geometry ~~~~~~~~ .. image:: images/Shape1.gif .. _RefSchema_Shape1.gif: Cube side: :math:`L=2` Material properties ~~~~~~~~~~~~~~~~~~~~~~~~~ * * * :math:`E=2N/{m}^{2}` * :math:`\nu =0.` Boundary conditions and loads ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Compulsory trips * plan :math:`z=\mathrm{1m}` :math:`\mathrm{DX}=0.=\mathrm{DY}=0.=\mathrm{DZ}=0.` * plan :math:`z=3m` :math:`\mathrm{DZ}=\mathrm{2.m}` Benchmark solution --------------------- Calculation method ~~~~~~~~~~~~~~~~~~~~ The Poisson ratio is zero :math:`\nu =0` which gives us :math:`{\sigma }_{\mathrm{xx}}={\sigma }_{\mathrm{yy}}={\sigma }_{\mathrm{xy}}={\sigma }_{\mathrm{xz}}={\sigma }_{\mathrm{yz}}=0` :math:`{\sigma }_{\mathrm{zz}}=E\epsilon =E\frac{\mathrm{DZ}}{L}` Benchmark results ~~~~~~~~~~~~~~~~~~~~~~~~~ :math:`\mathrm{SIZZ}=2N/{m}^{2}` At any point in the cube