1. Reference problem#

1.1. Geometry#

We consider a square plate \(\mathrm{2D}\) with a unitary side, centered at the origin.

This plate has 2 cracks and 2 inclusions (interfaces):

  • a notch-type inclusion of length \({L}_{e}\mathrm{=}\mathrm{0,6}\) and radius \({R}_{e}\mathrm{=}\mathrm{0,05}\), centered at \(E(0;\mathrm{-}\mathrm{0,3})\),

  • a circular inclusion with radius \({R}_{c}\mathrm{=}\mathrm{0,05}\) centered at the origin,

  • a rectilinear horizontal non-through crack of length \({L}_{A}\mathrm{=}\mathrm{0,2}\) centered in \(A(\mathrm{0,3 };\mathrm{0,25})\),

  • a rectilinear horizontal non-through crack of length \({L}_{B}\mathrm{=}\mathrm{0,2}\) centered in \(B(-\mathrm{0,2 };\mathrm{0,25})\).

_images/10000000000003F6000003907CE32725EF206DCB.png

Figure 1.1-1 : diagram of the multi-cracked plate

1.2. Material properties#

The material has the following properties:

  • Young’s modulus: \(E=205000\mathrm{MPa}\)

  • Poisson’s ratio: \(\nu =\mathrm{0,3}\)

1.3. Boundary conditions and loads#

Mechanical calculation does not matter in this test because we only want to test the refinement criterion a priori. The mechanical calculation is given only for illustrative purposes in modeling A.

We consider a loading of traction of \(p\mathrm{=}1\mathit{MPa}\) on the left, right and upper faces of the plate. The underside is embedded.

1.4. Brief description of the various models#

All models fall within the framework of the representation of cracks and interfaces by the X- FEM method associated with level sets.

Modeling A calculates a distance indicator (fields at the nodes). The test will focus on the value of this indicator in different nodes. The mesh refinement methodology used is the simplest (criterion in percentage of refined meshes), but is not necessarily optimal in terms of number of cells.

Modeling B calculates a distance indicator (fields at the nodes). The advantage of this modeling is to show a more optimal methodology in terms of number of cells. The test will focus on the mesh size in the refined area.

C modeling calculates one indicator per area (by cells). The methodology presented is the most advanced and the most robust. It is this indicator and this methodology that we recommend.