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A convergent adaptive finite element method for a phase-field model of dynamic fracture

Ram Manohar, S. M. Mallikarjuaniah

TL;DR

The paper addresses dynamic brittle fracture using a phase-field approach that couples elastodynamics with the Ambrosio–Tortorelli regularization of fracture energy. It develops an adaptive finite element method with a residual-based a posteriori estimator and a staggered time-stepping scheme that enforces damage irreversibility, proving convergence of the discrete solutions to a critical point of the total energy. Numerical experiments in 2D demonstrate the method's ability to capture complex fracture patterns such as crack branching and tortuosity while achieving substantial computational savings over uniform refinement. The work provides a rigorous, extensible framework for simulating dynamic fracture and lays a foundation for future multiphysics extensions.

Abstract

We propose and analyze an adaptive finite element method for a phase-field model of dynamic brittle fracture. The model couples a second-order hyperbolic equation for elastodynamics with the Ambrosio-Tortorelli regularization of the Francfort-Marigo variational fracture energy, which circumvents the need for explicit crack tracking. Our numerical scheme combines a staggered time-stepping algorithm with a variational inequality formulation to strictly enforce the irreversibility of damage. The mesh adaptation is driven by a residual-based a posteriori-type estimator, enabling efficient resolution of the evolving fracture process zone. The main theoretical contribution is a rigorous convergence analysis, where we prove that the sequence of discrete solutions generated by the AFEM converges (up to a tolerance) to a critical point of the governing energy functional. Numerical experiments for a two-dimensional domain containing an edge-crack under dynamic anti-plane shear loading demonstrate our method's capability of autonomously capturing complex phenomena, including crack branching and tortuosity, with significant computational savings over uniform refinement.

A convergent adaptive finite element method for a phase-field model of dynamic fracture

TL;DR

The paper addresses dynamic brittle fracture using a phase-field approach that couples elastodynamics with the Ambrosio–Tortorelli regularization of fracture energy. It develops an adaptive finite element method with a residual-based a posteriori estimator and a staggered time-stepping scheme that enforces damage irreversibility, proving convergence of the discrete solutions to a critical point of the total energy. Numerical experiments in 2D demonstrate the method's ability to capture complex fracture patterns such as crack branching and tortuosity while achieving substantial computational savings over uniform refinement. The work provides a rigorous, extensible framework for simulating dynamic fracture and lays a foundation for future multiphysics extensions.

Abstract

We propose and analyze an adaptive finite element method for a phase-field model of dynamic brittle fracture. The model couples a second-order hyperbolic equation for elastodynamics with the Ambrosio-Tortorelli regularization of the Francfort-Marigo variational fracture energy, which circumvents the need for explicit crack tracking. Our numerical scheme combines a staggered time-stepping algorithm with a variational inequality formulation to strictly enforce the irreversibility of damage. The mesh adaptation is driven by a residual-based a posteriori-type estimator, enabling efficient resolution of the evolving fracture process zone. The main theoretical contribution is a rigorous convergence analysis, where we prove that the sequence of discrete solutions generated by the AFEM converges (up to a tolerance) to a critical point of the governing energy functional. Numerical experiments for a two-dimensional domain containing an edge-crack under dynamic anti-plane shear loading demonstrate our method's capability of autonomously capturing complex phenomena, including crack branching and tortuosity, with significant computational savings over uniform refinement.

Paper Structure

This paper contains 6 sections, 8 theorems, 74 equations, 5 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Variational formulation
  3. Discrete version of the model
  4. Convergence Analysis
  5. Computational results
  6. Conclusion

Key Result

Proposition 2.1

Let $u \in \widehat{\mathbb{U}}(0,T_f)$ be the solution of the problem contwave-contcond. Further, we assume that $v\in \widehat{\mathbb{V}}(0,T_f)$ is the minimizer of the functional reg:energy, then $v(x,t)$ satisfies the condition $0 \leq v(x,t) \leq 1$ for a.e.$x\in \Omega$, $t\in [0,T_f]$.

Figures (5)

  • Figure 1: A domain and the boundary indicators.
  • Figure 2: Illustration of the computational mesh, with the initial slit highlighted by the red line.
  • Figure 3: Computational results at the final time step. Left: Final phase-field ($v$), indicating the fracture path. Right: Adapted mesh showing the contour for $0 \leq v \leq 0.01$.
  • Figure 4: Left: Regularized stress field. Right: Evolution of kinetic, strain, and surface energies as a function of time.
  • Figure 5: Left: Number of degrees of freedoms (#DOF), and Right: Minimum and Maximum values of estimator $\mathpzc{R}_h$ on different time steps.

Theorems & Definitions (13)

  • Proposition 2.1
  • Definition 3.1
  • Proposition 3.2
  • Lemma 4.1
  • Lemma 4.2
  • Lemma 4.3
  • Proof 1
  • Lemma 4.4
  • Proof 2
  • Theorem 4.5: Main results
  • ...and 3 more