[Paper Review] Dynamic phase-field fracture with a first-order discontinuous Galerkin method for elastic waves

Kerstin Weinberg, Christian Wieners|arXiv (Cornell University)|Jun 29, 2021

Numerical methods in engineering38 references21 citations

TL;DR

This paper proposes a first-order discontinuous Galerkin (DG) method coupled with a staggered time integration scheme for dynamic phase-field fracture in elastic waves. By reformulating the wave equation as a hyperbolic first-order system and using energy-conserving time integration, the method achieves minimal numerical dissipation, enabling accurate simulation of crack initiation and spallation driven by wave superposition and reflection, validated in 2D and 3D benchmarks with convergence on adaptive meshes.

ABSTRACT

We present a new numerical approach for wave induced dynamic fracture. The method is based on a discontinuous Galerkin approximation of the first-order hyperbolic system for elastic waves and a phase-field approximation of brittle fracture driven by the maximum tension. The algorithm is staggered in time and combines an implicit midpoint rule for the wave propagation followed by an implicit Euler step for the phase-field evolution. At fracture, the material is degraded, and the waves are reflected at the diffusive interfaces. Two and three-dimensional examples demonstrate the advantages of the proposed method for the computation of crack growth and spalling initiated by reflected and superposed waves.

Motivation & Objective

• Address the issue of numerical dissipation in existing time-integration schemes for dynamic phase-field fracture, which compromises accuracy in wave-dominated problems.
• Develop a high-fidelity numerical framework for simulating dynamic fracture processes driven by elastic wave interactions, including reflection, superposition, and spallation.
• Enable accurate, energy-conserving simulation of crack evolution in brittle materials under dynamic loading using a first-order hyperbolic formulation.
• Demonstrate convergence and robustness of the proposed method on both 2D and 3D benchmarks with varying mesh refinement.
• Establish a scalable, parallel finite element implementation capable of handling high-resolution simulations of complex crack patterns and wave dynamics.

Proposed method

• Reformulate the second-order wave equation as a first-order hyperbolic system for velocity and stress, enabling stable and accurate DG discretization.
• Apply a first-order discontinuous Galerkin (DG) method with full upwind numerical flux to the hyperbolic system, ensuring stability and low dissipation.
• Use the implicit midpoint rule for time integration of the wave system to preserve energy and minimize numerical dissipation.
• Employ the implicit Euler method for phase-field evolution, ensuring irreversibility and stability of crack growth.
• Implement a staggered time integration scheme with variable time steps, where wave and phase-field updates are solved sequentially.
• Use a stress-based phase-field driving force based on the maximum principal stress, with a regularized crack evolution governed by a variational inequality.

Experimental results

Research questions

• RQ1 Can a first-order DG method for elastic waves significantly reduce numerical dissipation compared to traditional Newmark or finite difference schemes in dynamic fracture simulations?
• RQ2 How does the proposed staggered, energy-conserving scheme perform in capturing wave-induced crack initiation and spallation in 2D and 3D configurations?
• RQ3 To what extent does the method converge with mesh refinement, and can it reproduce consistent crack patterns across different mesh resolutions?
• RQ4 Can the method accurately simulate complex crack patterns arising from wave interference and reflection without geometric crack initialization?
• RQ5 What are the computational demands and scalability of the method in high-resolution 3D simulations of dynamic fracture?

Key findings

• The proposed DG-based method exhibits minimal numerical dissipation, enabling accurate simulation of wave energy transport and reflection critical for spallation phenomena.
• Convergence of the crack path and evolution is demonstrated across different mesh refinements in the 2D configuration, confirming the method's consistency and reliability.
• In 3D, the method successfully captures the initiation and propagation of a secondary crack due to wave superposition, with the crack tip advancing at approximately 50% of the wave speed.
• The final phase-field configuration at t = 0.75 in 3D shows detailed crack patterns with 1,085,825 degrees of freedom on a hexahedral mesh, resolving fine-scale spallation.
• The simulation required approximately 20 hours on 2048 cores using a parallel preconditioned GMRES solver, with each time step solvable in a few seconds.
• The method successfully reproduces wave-induced spallation as a purely dynamic phenomenon driven by stress interaction, without requiring geometric crack seeds.