Direct estimation of generalized stress intensity factors using a multigrid XFEM
Résumé
Over the 15 last years, several numerical methods have been devised to make the simulation of crack propagation be more and more convenient, flexible, and easy to implement. Among them, the eXtended Finite Element Method (X-FEM [1]) has allowed a significant step in this direction. With only a few additional degrees of freedom (DOF), it is possible to use meshes that do not conform to the crack shape, and so there is no need to remesh the structure at every propagation step. The introduction of singular enrichments in the discrete field interpolation is also a major asset, because a standard FE convergence can be reached even when using standard linear elements in the vicinity of the crack tip. In this context, an alternative enrichment technique was proposed recently [2] which has the advantage to estimate directly and accurately the generalized Stress Intensity Factors (SIF) \emph{without any post processing}. The model starts form a standard X-FEM description, but the singular enrichment differ. Indeed, near the crack tip, the finite elements are replaced by analytical linear elastic crack tip asymptotic fields [3] for the interpolation of the unknown displacement field. The two overlapping descriptions are coupled thanks to the Arlequin Method. In this work, a non-overlapping variant of this method is proposed. The coupling is performed thanks to a Mortar technique on the interface. We will show that this variant, although easier to implement and to use (since there are less parameters), provides comparable efficiency and accuracy. However, the methods discussed herein, as well as X-FEM, remain FE-based method, and thus require a sufficiently fine mesh in the zones of interests. In particular, if the mesh is such that its elements are larger than the scale of the crack, then it may not be adapted to take into account properly the effect of the crack in the simulation, even with discontinuous and singular enrichments. Another example concerns the simulation of curvilinear crack propagation. Sometimes, the radius of curvature of the crack may not be large enough so that we can no longer assume that the crack is right at least locally around the tip. In this situation, the standard singular enrichments functions, which are build for a straight crack, may not be appropriate. We choose not to resort to more sophisticated enrichment functions taking into account the curvature of the crack in order to have to handle a unique set of predefined modes. In these two previous situations, the problem can be solved by refining the mesh, to make it more suited to the crack geometry, despite the non-compatibility condition of X-FEM. Besides the computational cost of remeshing, the refinement lead to a larger algebraic system which can yield a significant additional cost, which may, in some tridimensional complex cases, become prohibitive. In a second part of the talk, we will present a multiscale method that attempts to circumvent the above limitations. A multigrid solver inspired from the multigrid-X-FEM method of [4] is developed and coupled to the non-overlapping method presented in the first section. The proposed method [5] bridges three characteristic length scales that can be present in fracture mechanics: the structure, the crack and the singularity at the crack tip. For each of them, a relevant model is proposed. First, a truncated analytical reduced-order model based on Williams' expansion is used to describe the singularity at the tip. Then, it is coupled with a standard extended finite element (FE) method model which is known to be appropriate to the scale of the crack. A multigrid solver finally bridges the scale of the crack to that of the structure for which a standard FE model is often accurate enough. Dedicated coupling algorithms are presented and the effects of their parameters are discussed. The efficiency and accuracy of this new approach will be exemplified using several reference benchmarks.
