Preconditioning strategies for vectorial finite element linear systems arising from phase-field models for fracture mechanics

Abstract Phase-field models are frequently adopted to simulate fracture mechanics problems in the context of the finite element method. To depict fracture, this method involves solving a coupled set of Helmholtz-like damage-field equation and augmented linear momentum balance equation. Solutions to these coupled equations are then used as descriptions of crack propagation phenomena within solids. However, this method imposes a constrain of using extremely fine meshing for properly predicting cracks. For practical problems of interest, this very often leads to linear systems with large sizes that have to be repetitively assembled and solved. As such, iterative solution procedures such as the Krylov subspace-based methods for solving these large linear systems within the framework of serial/parallel computing environments become mandatory to obtain results in a feasible time. In this work, the vectorial finite discretization for a hybrid phase-field formulation – a monolithic solving scheme – is presented. The underlying nonlinearity present in the coupled set of equations of the hybrid phase-field model is dealt through Picard iteration that helps to preserve the symmetry of the linearized system to solve. Due to the symmetric positive definite nature of the finite element linear systems obtained for this problem, the conjugate gradient method makes a standard choice of iterative solution algorithm. In this article, to improve convergence rates, consequently time to solution, of the conjugate gradient method applied to crack propagation problems, different preconditioning strategies are analyzed, tuned, and discussed. Brittle fracture benchmarks are used to measure the performance of preconditioners which are then applied to massively parallel simulations with millions of unknowns. A series of numerical experiments show that the algebraic multigrid preconditioner is well suited for solving the phase-field model for fracture, being superior to the Jacobi and the block Jacobi preconditioning in all regards: ease of solving the problem, iterations to converge, time to solution, and parallel scaling on more than a thousand processes.