Towards an Extrinsic, CG-XFEM Approach Based on Hierarchical Enrichments for Modeling Progressive Fracture

TL;DR

This paper introduces a hierarchical, extrinsic CG-XFEM method for 3D fracture modeling, enabling complex crack surfaces and progressive fracture simulation with efficient DoF management and visualization tools.

Contribution

It generalizes existing CG-XFEM by allowing multiple hierarchical enrichments per element, supporting complex 3D crack evolution and scalable implementation for progressive fracture modeling.

Findings

Supports arbitrary enrichments within elements

Enables efficient modeling of complex 3D crack surfaces

Provides visualization and comparison tools for enrichment hierarchies

Abstract

We propose an extrinsic, continuous-Galerkin (CG), extended finite element method (XFEM) that generalizes the work of Hansbo and Hansbo to allow multiple Heaviside enrichments within a single element in a hierarchical manner. This approach enables complex, evolving XFEM surfaces in 3D that cannot be captured using existing CG-XFEM approaches. We describe an implementation of the method for 3D static elasticity with linearized strain for modeling open cracks as a salient step towards modeling progressive fracture. The implementation includes a description of the finite element model, hybrid implicit/explicit representation of enrichments, numerical integration method, and novel degree-of-freedom (DoF) enumeration algorithm. This algorithm supports an arbitrary number of enrichments within an element, while simultaneously maintaining a CG solution across elements. Additionally, our approach easily allows an implementation suitable for distributed computing systems. Enabled by the DoF enumeration algorithm, the proposed method lays the groundwork for a computational tool that efficiently models progressive fracture. To facilitate a discussion of the complex enrichment hierarchies, we develop enrichment diagrams to succinctly describe and visualize the relationships between the enrichments (and the fields they create) within an element. This also provides a unified language for discussing extrinsic XFEM methods in the literature. We compare several methods, relying on the enrichment diagrams to highlight their nuanced differences.