hp‐Generalized FEM and crack surface representation for non‐planar 3‐D cracks

Abstract

SUMMARYA high-order generalized finite element method (GFEM) for non-planar three-dimensional crack surfaces is presented. Discontinuous p-hierarchical enrichment functions are applied to strongly graded tetrahedral meshes automatically created around crack fronts. The GFEM is able to model a crack arbitrarily located within a finite element (FE) mesh and thus the proposed method allows fully automated fracture analysis using an existing FE discretization without cracks. We also propose a crack surface representation that is independent of the underlying GFEM discretization and controlled only by the physics of the problem. The representation preserves continuity of the crack surface while being able to represent non-planar, non-smooth, crack surfaces inside of elements of any size. The proposed representation also provides support for the implementation of accurate, robust, and computationally efficient numerical integration of the weak form over elements cut by the crack surface. Numerical simulations using the proposed GFEM show high convergence rates of extracted stress intensity factors along non-planar curved crack fronts and the robustness of the method.

High order X-FEM and levelsets for complex microstructures: Uncoupling geometry and approximation

“…Earlier attempts have been made on this subject in the context of the generalized finite element method (G-FEM) [18,12], but in the case where the interface is represented explicitly. Here, an Comparison between p refinement and h extension.…”

“…The other possibility would consist in vectorizing the assembly process, as proposed in [52]. We are currently improving the numerical efficiency of the proposed approach by replacing on the fly the integration cells by means of a local Delaunay tetrahedralization, following [18]. Concerning non-linear material behaviour, Nitsche's stabilization parameter should be recomputed at each iteration, which could be cumbersome.…”

“…Earlier works conducted by Strouboulis et al [12] considered the material interfaces explicitly which enabled to define their degree of smoothness independently of the density of the mesh used for the approximation. A similar approach was also considered by Peirera et al [18] in the context of 3D fracture mechanics where the authors proposed strategies for the generation of the sub-elements that are needed for the integration of the weak form. In the context of the Finite Cell Method, Düster et al [7,8] proposed an integration of the weak form on a set of cubic sub-cells that approximates the geometrical interfaces as "voxelized" surfaces (with a selective use of the integration points).…”

“…While the geometry is defined explicitly in the GFEM, it is defined implicitly with the X-FEM by means of Level Sets [15]. Both approaches were applied in a wide variety of applications from fracture mechanics [16,17,18] to the treatment of physical interfaces [19,20,21,22]. The reader can refer to two recent reviews on this topic [23,24].…”

“…The idea of defining the approximation on a coarse mesh with high order hierarchical polynomials seems to be promising, provided that the interfaces are correctly represented. Here, the objective is to use a high order approximation and a C 0 representation of the interface as in [12,18], but with an implicit representation of the interfaces which leads to extra difficulties. In addition, focus is put on the optimal convergence of the approach.…”