Adaptive Finite Element Method for Phase Field Fracture Models Based on Recovery Error Estimates

TL;DR

This paper presents a theoretically grounded adaptive finite element method for phase field fracture models that automatically refines meshes based on recovery error estimates, improving crack propagation simulation accuracy.

Contribution

It introduces a recovery error estimate-based adaptive FEM for phase field fracture models that eliminates empirical refinement parameters.

Findings

Accurate simulation of crack propagation in 2D and 3D examples.

Robustness demonstrated through classical brittle fracture cases.

Efficient mesh adaptation without empirical parameters.

Abstract

The phase field model is a widely used mathematical approach for describing crack propagation in continuum damage fractures. In the context of phase field fracture simulations, adaptive finite element methods (AFEM) are often employed to address the mesh size dependency of the model. However, existing AFEM approaches for this application frequently rely on heuristic adjustments and empirical parameters for mesh refinement. In this paper, we introduce an adaptive finite element method based on a recovery type posteriori error estimates approach grounded in theoretical analysis. This method transforms the gradient of the numerical solution into a smoother function space, using the difference between the recovered gradient and the original numerical gradient as an error indicator for adaptive mesh refinement. This enables the automatic capture of crack propagation directions without the need for empirical parameters. We have implemented this adaptive method for the Hybrid formulation of the phase field model using the open-source software package FEALPy. The accuracy and efficiency of the proposed approach are demonstrated through simulations of classical 2D and 3D brittle fracture examples, validating the robustness and effectiveness of our implementation.