A robust monolithic solver for phase-field fracture integrated with fracture energy based arc-length method and under-relaxation
Journal article, 2022

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 experiments presented in the manuscript demonstrate 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).

Arc length method

Phase-field fracture

Monolithic solver


Variational damage

Brittle material


Ritukesh Bharali

Chalmers, Industrial and Materials Science, Material and Computational Mechanics

Somdatta Goswami

Brown University

Cosmin Anitescu

Bauhaus-Universität Weimar

Timon Rabczuk

Bauhaus-Universität Weimar

Computer Methods in Applied Mechanics and Engineering

0045-7825 (ISSN)

Vol. 394 114927

Subject Categories


Applied Mechanics

Computational Mathematics



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