Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Tian Tian, Chen Chunyu, He Liang, Wei Huayi

TL;DR

This work tackles mesh sensitivity in phase-field fracture simulations by introducing an adaptive finite element method driven by recovery-type posteriori error estimates. It recovers the gradient of the phase-field in a smoother space and uses the discrepancy between the recovered and computed gradients as an error indicator to guide refinement, eliminating the need for empirical parameters. Implemented within FEALPy for the Hybrid phase-field model, the approach enables automatic crack-path capture in both 2D and 3D with substantial efficiency gains, including GPU-accelerated solvers. Numerical experiments demonstrate robust accuracy and significant reductions in element counts and computation time, validating the method's practicality for large-scale brittle fracture problems.

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.

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

TL;DR

This work tackles mesh sensitivity in phase-field fracture simulations by introducing an adaptive finite element method driven by recovery-type posteriori error estimates. It recovers the gradient of the phase-field in a smoother space and uses the discrepancy between the recovered and computed gradients as an error indicator to guide refinement, eliminating the need for empirical parameters. Implemented within FEALPy for the Hybrid phase-field model, the approach enables automatic crack-path capture in both 2D and 3D with substantial efficiency gains, including GPU-accelerated solvers. Numerical experiments demonstrate robust accuracy and significant reductions in element counts and computation time, validating the method's practicality for large-scale brittle fracture problems.

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.
Paper Structure (17 sections, 35 equations, 17 figures, 1 table, 1 algorithm)

This paper contains 17 sections, 35 equations, 17 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Mathematical model
  3. Crack energy function
  4. Strain energy function with crack propagation
  5. Governing equations
  6. Hybrid phase field model
  7. Algorithm design
  8. Nonlinear finite element discretization
  9. An adaptive method based on recovery type posterior error estimates
  10. Overall Algorithm Process
  11. Algorithm validation and application
  12. Square Model with a Circular Notch: Upward Stretch on the Upper Boundary
  13. Model with a Notch: Upward Stretch on the Upper Boundary in the $y$ Direction
  14. Model with a Notch: Shearing Stretch on the Upper Boundary in the x-Direction
  15. The L-shaped plate (Stretch and compression mixing)
  16. ...and 2 more sections

Figures (17)

  • Figure 1: Algorithm Flow
  • Figure 2: Geometry and boundary conditions of the square model with a circular notch (the unit is mm)
  • Figure 3: The final results of the square model with a circular notch. The upper figures show the results of adaptive mesh refinement (initial mesh size h=0.05), and the lower figures show the results without using adaptive (mesh size h=0.01). The left figures show the final phase field values, and the right figures show the final mesh situation.
  • Figure 4: Residual froce with load-displacement curves of the square model with a circular notch, adaptive and non-adaptive
  • Figure 5: Geometry and boundary conditions of the model with a notch (the unit is mm) and initial mesh generation of the model with a notch
  • ...and 12 more figures