2. Benchmark solution

2.1. Calculation method

We want to check that the position of the background after the propagation, calculated by the operator PROPA_FISS, is correct. It is therefore necessary to calculate the theoretical position determined by the advance and the direction of propagation imposed.

As already noted, the crack bottom remains straight throughout the propagation. The background is always perpendicular to the two surfaces of the plate parallel to plane \(\mathit{YZ}\), as shown. The position of the background can therefore be indicated using only the coordinates \(Y\) and \(Z\).

The initial position is as follows:

\({y}_{0}\mathrm{=}{a}_{0}\mathrm{=}2.0\)

\({z}_{0}\mathrm{=}9.0\)

After the \(i\) spread, the new background position can be calculated like this:

\({y}_{i}\mathrm{=}{y}_{i\mathrm{-}1}+\Delta a\mathrm{\cdot }\mathrm{cos}(i\mathrm{\cdot }\beta )\)

\({z}_{i}\mathrm{=}{z}_{i\mathrm{-}1}+\Delta a\mathrm{\cdot }\mathrm{sin}(i\mathrm{\cdot }\beta )\)

2.2. Reference quantities and results

For the three propagations calculated in the tests, the bottom position is as follows:

Propagation	Coord \({y}_{i}\)	Coord \({z}_{i}\)
1	2.34641	9.19999
2	2.54642	9.54640
3	2.54644	9.94640

Table 2.1

In the current version of Code Aster, the coordinates of the crack bottom points are only available in the.mess file and therefore cannot be checked directly in the command file.

However, for this test case, the theoretical position and the shape (a segment) of the bottom of the crack are known. In fact the background is always coincident with the edge that connects the two points \((\mathrm{0,}{y}_{i},{z}_{i})\) and \((\mathrm{1,}{y}_{i},{z}_{i})\). By using MACR_LIGN_COUPE, you can read the values of the level sets at the points of intersection between this edge and the faces of the elements of the mesh. If the position of the bottom after propagation is calculated correctly by PROPA_FISS, the value of the two level sets should be equal to zero for all intersection points found because, by definition, the crack background is formed by all the points where the tangent and normal level set are equal to zero.

This explains why it was decided to give the same advance and direction of propagation at all points at the bottom of the crack. Even if the crack background is located by only two coordinates, the propagation is 3D and the algorithms implemented in PROPA_FISS calculate a 3D propagation.