Benchmark solution
=====================


Calculation method used for the reference solution
--------------------------------------------------------

The reference solution is analytical:


.. image:: images/Object_2.svg
    :width: 225
    :height: 133


.. _RefImage_Object_2.svg:

Given the value of :math:`\lambda` (:math:`1`), the flow obtained for each configuration is therefore:

For models 2 DPLAN and PLAN_DIAG (models A and B):


* :math:`T(X,Y)\mathrm{=}\mathrm{2.X}+\mathrm{3.Y}` is :math:`{\Phi }_{x}(X,Y)\mathrm{=}–2` and :math:`{\Phi }_{y}(X,Y)\mathrm{=}–3`


* :math:`T(X,Y)\mathrm{=}{\mathrm{2.X}}^{2}+{\mathrm{3.Y}}^{2}` is :math:`{\Phi }_{x}(X,Y)\mathrm{=}–\mathrm{4.X}` and :math:`{\Phi }_{y}(X,Y)\mathrm{=}–\mathrm{6.Y}`

For models 2 DAXIS and AXIS_DIAG (models C and D):


* :math:`T(X,Y,Z)\mathrm{=}\mathrm{2.R}+\mathrm{3.Y}` (:math:`R\mathrm{=}X` in the :math:`(\mathit{OXY})` plan) i.e. :math:`{\Phi }_{x}(X,Y,Z)\mathrm{=}–2` and :math:`{\Phi }_{y}(X,Y,Z)\mathrm{=}–3`


* :math:`T(X,Y,Z)\mathrm{=}{\mathrm{2.R}}^{2}+{\mathrm{3.Y}}^{2}` (:math:`R\mathrm{=}X` in the :math:`(\mathit{OXY})` plan) i.e. :math:`{\Phi }_{x}(X,Y,Z)\mathrm{=}–\mathrm{4.X}` and :math:`{\Phi }_{y}(X,Y,Z)\mathrm{=}–\mathrm{6.Y}`

For 3D and 3D models_ DIAG (E and F models):


* :math:`T(X,Y,Z)\mathrm{=}\mathrm{2.X}+\mathrm{3.Y}+\mathrm{4.Z}` is :math:`{\Phi }_{x}(X,Y,Z)\mathrm{=}–2`, :math:`{\Phi }_{y}(X,Y,Z)\mathrm{=}–3` and :math:`{\Phi }_{z}(X,Y,Z)\mathrm{=}–4`


* :math:`T(X,Y,Z)\mathrm{=}{\mathrm{2.X}}^{2}+{\mathrm{3.Y}}^{2}+{\mathrm{4Z}}^{2}` is :math:`{\Phi }_{x}(X,Y,Z)\mathrm{=}–\mathrm{4.X}`, :math:`{\Phi }_{y}(X,Y,Z)\mathrm{=}–\mathrm{6.Y}` and :math:`{\Phi }_{z}(X,Y,Z)\mathrm{=}–\mathrm{8.Z}`


Benchmark results
----------------------

For 2D models (PLAN, PLAN_DIAG, AXIS and AXIS_DIAG), the values tested are:


* With the linear temperature field: the temperature at nodes :math:`A`, :math:`C`,, :math:`J` and :math:`L`, the following flow :math:`X` and :math:`Y` by element at nodes :math:`A`, :math:`C`, :math:`J` and :math:`L` and the flow at the first Gauss point of the same elements,


* With the quadratic temperature field: the temperature at nodes :math:`A`, :math:`C`,,, :math:`J`, and :math:`L`, and the following flow :math:`X` and :math:`Y` by element at nodes :math:`A`, :math:`C`, :math:`J`, and :math:`L`.

For 3D models (3D and 3D_ DIAG), the values tested are:


* With the linear temperature field: the temperature at nodes :math:`{D}_{i}`, the next flow :math:`X` and :math:`Y` per element at nodes :math:`{D}_{i}`, and the flow at the first Gauss point of the same elements,


* With the quadratic temperature field: the temperature at nodes :math:`{D}_{i}`, and the following flow :math:`X` and :math:`Y` per element at nodes :math:`{D}_{i}`.

With :math:`i\mathrm{=}1` to :math:`9` for 3D modeling and :math:`i\mathrm{=}1` to :math:`4` for 3D modeling_ DIAG.