A novel hybrid approach for level set characterization and tracking of non-planar 3D cracks in the extended finite element method

Abstract

In this paper, the limitations associated with implicit and explicit representations of cracks in the extended finite element method (XFEM) is recapitulated via numerical explorations followed by the development of a novel hybrid approach for the characterization of non-planar 3D cracks along with its capability demonstrations. In the XFEM, the crack geometry is independent of the structural mesh, and is often described implicitly by means of two level set functions. The implicit representation is very convenient for purposes of computing the crack front velocity and for handling situations where crack front is concave and the velocity vectors may cross. The main difficulty of this implicit description is the formulation of an efficient and robust update scheme for the level set values after a propagation step. On the other hand, the crack geometry can be described by an explicit triangulated mesh which can be easily updated after a propagation step. The explicit representation has its own shortcomings, e.g., difficulties in handling crack overlaps and extraction of crack local coordinates. Given the difficulties associated with the use of either implicit or explicit method for the geometric description of complex crack geometry, a novel hybrid method is developed by a combination of an implicit level set representation of the crack and an explicit triangulated mesh representation. In the hybrid approach, the implicit representation is updated after each propagation step and disconnected crack surfaces are removed using a paint-fill algorithm based on the current explicit representation of the crack. Then, an updated explicit representation is constructed based on the updated implicit representation using the marching cubes algorithm. Finally, the implicit representation is rebuilt from the explicit representation. The use of the explicit representation ensures that the data in the level set representation is generated from a consistent crack description. The effectiveness of the developed hybrid approach is demonstrated by analyzing several 3D crack propagation problems including a quarter-circular crack in a complex helicopter component, a u-shaped crack and an inclined elliptical crack in cuboids, and an inclined edge crack in a three-point bending beam.

Stable 3D XFEM/vector level sets for non‐planar 3D crack propagation and comparison of enrichment schemes

“…The above data are obtained from the element partitioning algorithm used for numerical integration . The predicted crack path lies in good agreement with the experimental results of Buchholz et al, as well as the numerical results of Sadeghirad et al…”

“…The next example, also studied by Sadeghirad et al, involves an edge crack in a beam under 3 point bending for which experimental results are also available . Two different angles of the crack are considered, as in Figures 20 and 23, which result in different crack paths.…”

“…The geometrical parameters of the problem are L =260 mm, L 1 =240 mm, H =60 mm, d =10 mm, α =20 mm, and β =0°, 45°, while the material parameters are E =2.1×10 5 N/mm 2 , ν =0.3, and the applied load P =2 kN/mm. The above values have been chosen to be identical to the ones of previous works to facilitate comparisons.…”

“…Similarly, in the 2D version of the vector level set method, which could also be considered as a hybrid method, crack increments are represented as linear segments and the level set functions are obtained by computing the distances of points from those segments and the crack tip using only geometric (vector) operations. In the work of Sadeghirad et al, an explicit representation is constructed from the implicit representation to remove disconnected parts of the crack and subsequently consistent level sets are recomputed using the explicit representation.…”

Challenges, Tools and Applications of Tracking Algorithms in the Numerical Modelling of Cracks in Concrete and Masonry Structures

“…When the crack-extension is identified, then the level sets are updated to include the new crack-path. Similar approaches have been recently presented by Holl et al [190] and Sadeghirad et al [191].…”