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

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).