[Paper Review] A robust monolithic solver for phase-field fracture integrated with fracture energy based arc-length method and under-relaxation

Ritukesh Bharali, Somdatta Goswami+2 more|arXiv (Cornell University)|Nov 20, 2021

Numerical methods in engineering60 references67 citations

TL;DR

This paper presents a robust monolithic solver for phase-field fracture that combines a fracture energy-based arc-length method and adaptive under-relaxation to accurately capture snap-back behavior and improve convergence. The method integrates isogeometric analysis with PHT-spline-based adaptive mesh refinement, enabling efficient, localized resolution of crack propagation without prior knowledge of fracture paths.

ABSTRACT

The phase-field fracture free-energy functional is non-convex with respect to the displacement and the phase field. This results in a poor performance of the conventional monolithic solvers like the Newton-Raphson method. In order to circumvent this issue, researchers opt for the alternate minimization (staggered) solvers. Staggered solvers are robust for the phase-field based fracture simulations as the displacement and the phase-field sub-problems are convex in nature. Nevertheless, the staggered solver requires very large number of iterations (of the order of thousands) to converge. In this work, a robust monolithic solver is presented for the phase-field fracture problem. The solver adopts a fracture energy-based arc-length method and an adaptive under-relaxation scheme. The arc-length method enables the simulation to overcome critical points (snap-back, snap-through instabilities) during the loading of a specimen. The use of an under-relaxation scheme stabilizes the solver by preventing the divergence due to an ill-behaving stiffness matrix. The efficiency of the proposed solver is further amplified with an adaptive mesh refinement scheme based on PHT-splines within the framework of isogeometric analysis. The numerical examples presented in the manuscript demonstrates the efficacy of the solver. All the codes and data-sets accompanying this work will be made available on GitHub (https://github.com/rbharali/IGAFrac).

Motivation & Objective

• To develop a robust monolithic solver for phase-field fracture that overcomes convergence issues from non-convex energy functional and ill-conditioned stiffness matrices.
• To accurately capture post-peak snap-back behavior during crack propagation, which is often missed by staggered solvers.
• To integrate adaptive mesh refinement using PHT-splines within isogeometric analysis (IGA) for efficient local resolution of high-gradient zones.
• To ensure variational consistency and stability through a fracture energy-based arc-length constraint and adaptive under-relaxation.
• To enable natural crack nucleation and propagation without pre-defined crack paths, enhancing realism in brittle fracture simulations.

Proposed method

• Employs a monolithic solution scheme for coupled displacement and phase-field variables, solving the full system simultaneously.
• Introduces a fracture energy-based arc-length method to control the loading path and enable large displacement steps during snap-back events.
• Applies adaptive under-relaxation to stabilize convergence by damping oscillations due to erratic Jacobian behavior.
• Uses PHT-splines within the isogeometric analysis (IGA) framework for efficient, local h-refinement that adapts to crack evolution.
• Implements a penalization approach to enforce the fracture irreversibility constraint via variational inequality.
• Adopts a dynamic time-stepping strategy based on the arc-length constraint, allowing adaptive energy dissipation and improved stability.

Experimental results

Research questions

• RQ1 Can a monolithic phase-field fracture solver with arc-length control and under-relaxation effectively capture snap-back behavior in post-peak loading?
• RQ2 How does the combination of adaptive under-relaxation and fracture energy-based arc-length improve convergence and robustness compared to standard monolithic or staggered solvers?
• RQ3 To what extent does PHT-spline-based adaptive mesh refinement enhance computational efficiency and accuracy in phase-field fracture simulations?
• RQ4 Can the proposed solver naturally capture crack nucleation and propagation without pre-defined crack paths or remeshing?
• RQ5 How does the choice of degradation function (quadratic vs. cubic) influence the load-displacement response and crack topology in post-peak regimes?

Key findings

• The proposed monolithic solver successfully captures snap-back behavior in both single-edge notched tension (SENT) and shear (SENS) specimens, which is not achievable with alternate minimization or quasi-Newton Raphson solvers.
• For the SENT case, the solver achieves peak load and pre-peak response comparable to other methods, but uniquely captures the post-peak discontinuity and negative slope region.
• The cubic degradation function with s = 0.1 exhibits a linear pre-peak stage, while the quadratic and cubic (s=1.0) functions show no such stage, indicating sensitivity to degradation model choice.
• Adaptive mesh refinement using PHT-splines yields load-displacement curves equivalent to those on uniformly refined meshes, confirming accuracy and efficiency.
• The crack path evolves naturally from nucleation without initialization, and the final fracture topology differs from prior work due to the use of the Rankine criterion instead of spectral decomposition-based strain energy.
• The solver demonstrates robust convergence across all test cases, with adaptive time-stepping and under-relaxation preventing divergence even in highly nonlinear regimes.