2. Benchmark solution#

2.1. Calculation method used for the reference solution#

The reference solution is analytical:

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

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

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

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

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

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

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

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

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

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

2.2. 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 \(A\), \(C\),, \(J\) and \(L\), the following flow \(X\) and \(Y\) by element at nodes \(A\), \(C\), \(J\) and \(L\) and the flow at the first Gauss point of the same elements,

  • With the quadratic temperature field: the temperature at nodes \(A\), \(C\),,, \(J\), and \(L\), and the following flow \(X\) and \(Y\) by element at nodes \(A\), \(C\), \(J\), and \(L\).

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

  • With the linear temperature field: the temperature at nodes \({D}_{i}\), the next flow \(X\) and \(Y\) per element at nodes \({D}_{i}\), and the flow at the first Gauss point of the same elements,

  • With the quadratic temperature field: the temperature at nodes \({D}_{i}\), and the following flow \(X\) and \(Y\) per element at nodes \({D}_{i}\).

With \(i\mathrm{=}1\) to \(9\) for 3D modeling and \(i\mathrm{=}1\) to \(4\) for 3D modeling_ DIAG.