Multi‐level hp‐adaptivity for cohesive fracture modeling

Abstract

Discretization induced oscillations in the load-displacement curve are a well known problem for simulations of cohesive crack growth with finite elements. The problem results from an insufficient resolution of the complex stress state within the cohesive zone ahead of the crack tip. This work demonstrates that the hp-version of the finite element method is ideally suited to resolve this complex and localized solution characteristic with high accuracy and low computational effort. To this end, we formulate a local and hierarchic mesh refinement scheme that follows dynamically the propagating crack tip. In this way, the usually applied static a priori mesh refinement along the complete potential crack path is avoided, which significantly reduces the size of the numerical problem. Studying systematically the influence of h-, p-and hp-refinement, we demonstrate why the suggested hp-formulation allows to capture accurately the complex stress state at the crack front preventing artificial snapthrough and snap-back effects. This allows to decrease significantly the number of degrees of freedom and the simulation runtime. Furthermore, we show that by combining this idea with the finite cell method, the crack propagation within complex domains can be simulated efficiently without resolving the geometry by the mesh.Keywords: cohesive fracture, automatic hp-adaptivity, arbitrary hanging nodes, dynamic meshes, finite cell method Changes to the manuscript resulting from remarks of Reviewer one are marked in red. Changes resulting from remarks of Reviewer two are marked in blue. Changes resulting from remarks of both reviewers are marked in green. Other textual changes are marked in brown.

Adaptive hierarchical refinement of NURBS in cohesive fracture analysis

“…38,39 However, when the crack path is known a priori as, for instance, in delamination of composite structures, interface elements can be embedded in the continuum at predefined locations, thus leading to a relatively straightforward discretisation. [40][41][42][43][44][45] In a cohesive zone model, a crack is represented as an interface Γ c in the physical domain Ω, Figure 1. In this contribution, the interface Γ c is assumed to be predefined, as is the case of crack propagation along a material interface.…”

Parallelization of the multi-level hp-adaptive finite cell method

“…The need for complex data structures and algorithms to consolidate and constrain hanging nodes is alleviated as hanging nodes are avoided by construction. The approach is shown to achieve exponential convergence even in the presence of singularities and was first introduced for the two-dimensional case, extended in [5] to three dimensions and applied to cohesive fracture modeling in [16].…”

“…The aforementioned simplicity of the shared mesh data structure coupled with a priori hp-refinement, yields a fast, easy to implement hp-scheme as shown by the numerical examples in Section 4. The data structures used are implemented in terms of pointers as described in [1,16], where the mesh stores elements as pointers. Mesh elements in turn, hold pointers to their sub-elements/children, allowing for easy navigation through the mesh hierarchy.…”

High order transition elements: The xy-element concept—Part I: Statics

“…With this approach, the discontinuities in the displacement field caused by material inclusions can be easily captured. Yet, the most general approach is the multi-level hp-FEM, where several high order (hierarchical) overlay meshes are introduced [14,20,26,27]. In this method, an optimal combination of high order base elements and high or low order overlay elements can be set up.…”