1. Reference problem#

1.1. Geometry#

Structure \(\mathrm{2d}\) is a unit square: \(\mathit{LX}\mathrm{=}1m\) and \(\mathrm{LY}=1m\) (Figure).

_images/10000000000001D9000001C359B1BA019214653F.png

Figure 1.1-1: Healthy Plate Geometry

 _images/100000000000020E000001C38B9C795AD10E9DE2.png

Figure 1.1-2: Plate geometry with fictional crack

For the purposes of the test case, the same domain is also considered in the presence of a fictional crack, straight and emerging on the right, located halfway up (Figure). Note \({P}^{\text{+}}\) the coordinate point \(\left(\mathit{LX},{\mathit{LY}}^{\text{+}}/2\right)\) (located on the upper lip), \({P}^{\text{-}}\) the coordinate point \(\left(\mathit{LX},{\mathit{LY}}^{\text{-}}/2\right)\) (located on the lower lip), and \(Q\) the coordinate point \(\left(\mathit{LX}/\mathrm{2,}\mathit{LY}/2\right)\) (located at the tip of the crack).

Note:

This crack is qualified as fictional because it does not constitute a step in the domain’s border, and therefore has no impact on the solution of the reference problem. Its presence is only due to the computer validation of the tested functionality.

1.2. Material properties#

Thermal conductivity: \(\lambda =1{\mathit{W.m}}^{\text{-1}}\mathrm{.}{K}^{\text{-1}}\)

Calorific volume capacity: \(\rho {C}_{p}=2\mathit{J.m}-{\mathrm{3.K}}^{-1}\)

1.3. Boundary conditions and loads#

We impose a temperature \(\text{}\stackrel{̄}{T}{\text{}}^{\text{inf}}=10°C\) on the nodes of segment \(\mathit{AB}\) and \(\text{}\stackrel{̄}{T}{\text{}}^{\text{sup}}=20°C\) on the nodes of segment \(\mathit{CD}\) (see Figure).

1.4. Initial conditions#

None (the problem is stationary)