A multiscale preconditioner for crack evolution in porous microstructures: Accelerating phase-field methods

Phase-field methods are attractive for simulating the mechanical failure of geometrically complex porous microstructures described by 2D/3D x-ray (Formula presented.) CT images in subsurface (e.g., CO (Formula presented.) storage) and manufacturing (e.g., Li-ion battery) applications. They capture the nucleation, growth, and branching of fractures without prior knowledge of the propagation path or having to remesh the domain. Their drawback lies in the high computational cost for the typical domain sizes encountered in practice. We present a multiscale preconditioner that significantly accelerates the convergence of Krylov solvers in computing solutions of linear(ized) systems arising from the sequential discretization of the momentum and crack-evolution equations in phase-field methods. The preconditioner is an algebraic reformulation of a recent pore-level multiscale method (PLMM) by the authors and consists of a global preconditioner (Formula presented.) and a local smoother (Formula presented.). Together, (Formula presented.) and (Formula presented.) attenuate low- and high-frequency errors simultaneously. The proposed (Formula presented.), used in the momentum equation only, is a simplification of a recent variant proposed by the authors that is much cheaper and easier to deploy in existing solvers. The smoother (Formula presented.), used in both the momentum and crack-evolution equations, is built such that it is compatible with (Formula presented.) and more robust and efficient than black-box smoothers like ILU((Formula presented.)). We test (Formula presented.) and (Formula presented.) systematically for static- and evolving-crack problems on complex 2D/3D porous microstructures, and show that they outperform existing algebraic multigrid solvers. We also probe different strategies for updating (Formula presented.) as cracks evolve and show the associated cost can be minimized if (Formula presented.) is updated adaptively and infrequently. Both (Formula presented.) and (Formula presented.) are scalable on parallel machines and can be implemented non-intrusively in existing codes.