Spatial convergence of crack nucleation using a cohesive finite-element model on a pinwheel-based mesh

Abstract

SUMMARYWe consider the use of initially rigid cohesive interface models in a two-dimensional dynamic finiteelement solution of a fracture process. Our focus is on convergence of finite-element solutions to a solution of the undiscretized medium as the mesh spacing x (and therefore time-step t) tends to zero. We propose the use of pinwheel meshes, which possess the 'isoperimetric property' that for any curve C in the computational domain, there is an approximation to C using mesh edges that tends to C including a correct representation of its length, as the grid size tends to zero. We suggest that the isoperimetric property is a necessary condition for any possible spatial convergence proof in the general case that the crack path is not known in advance. Conversely, we establish that if the pinwheel mesh is used, the discrete interface first activated in the finite-element model will converge to the initial crack in the undiscretized medium. Finally, we carry out a mesh refinement experiment to check convergence of both nucleation and propagation. Our results indicate that the crack path computed in the pinwheel mesh is more stable as the mesh is refined compared to other types of meshes.

Adaptive dynamic cohesive fracture simulation using nodal perturbation and edge‐swap operators

“…Ortiz and Pandolfi [23] extrinsic cohesive elements for 3D dynamic fracture problems. Papoulia et al [5,24] addressed time continuity at the time of cohesive surface element activation, and errors in representing cracks in a finite element mesh. Zhou and Molinari [25] and Molinari et al [26] selected cohesive strength based on a modified weakest link Weibull distribution while addressing mesh dependency and energy convergence.…”

“…For instance, Papoulia et al [5] stated that 'no matter how much the mesh is refined, the jagged path will always be approximately 8% longer than the mathematical path' for standard 4k structured meshes. To further illustrate, a simple geometric problem is constructed within the 1 × 1 square domain (O ABC), shown in Figure 2(a).…”

“…The locations of the maximum deviation are described in solid circles. Figure 1(a), obtained from reference [5], illustrates the discrepancy between a mathematical path and a path represented by a 4k structured mesh, where the Hausdorff distance (H (P, Q)) is 0.12156. With the same mesh discretization, the Hausdorff distance can be reduced even further by selecting another discretized shortest path, as shown in Figure 1(b), i.e.…”

Numerical modelling and experimental validation of dynamic fracture events along weak planes

“…Its simplicity makes it amenable to massively parallel simulations. Its main drawback is the fact that the crack path must conform to the mesh geometry (surfaces of decohesion can only appear along finite element faces), although ample experience provides quantitative recommendations on the level of mesh refinement and the type of meshes needed to avoid possible mesh-dependence issues [19,25]. New methods pioneered by Belytschko et al [17], which allow for crack propagation independently of the mesh, have strongly emerged in recent times (XFEM).…”

The cohesive element approach to dynamic fragmentation: the question of energy convergence

“…Recently Papoulia et al have provided computational evidence of mesh convergence of the crack path. Convergence was observed for a special category of meshes referred to as pinwheel-based meshes [26]. Crack path convergence will not be addressed in this paper, which will be solely and fully devoted to the evolution of microcrack density as function of mesh size.…”