An implicit geometrical approach to level sets update for 3D non planar X-FEM crack propagation

In the X-FEM framework the crack geometry is described independently of the structural mesh, usually by means of two level set functions, thanks to the versatility and wide applicability of the level set methodology. The main difficulty in the adoption of the level set approach is the formulation of an efficient and robust update scheme allowing to calculate the new level set values associated to the new crack configuration after a propagation step. In this paper an implicit geometrical methodology is formulated in order to overcome the stability and performance issues affecting the resolution of the Hamilton–Jacobi partial differential equations governing the level set update. The calculation of the new level sets values is based on geometrical considerations only, which enormously reduces the computational time and increases the robustness of the update calculations. In contrast to other geometrical approaches, this methodology can be applied both to 2D and 3D problems and the crack front and surface are always implicitly described by means of the two level sets. This avoids both the need to describe somehow the crack surface and front and the creation of algorithms to track their evolution, which improves further the performance of the methodology. The numerical tests reported in the paper show that the proposed methodology is 30 times faster in 2D and 3–6 times faster in 3D than classic approaches based on the resolution of the Hamilton–Jacobi equations, and 2–7 times faster than explicit geometrical approaches. The proposed methodology can also manage the separation of a crack front in several fronts during the propagation and the merging of several fronts into one single front.