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Phase‐field methods are attractive for simulating the mechanical failure of geometrically complex porous microstructures described by 2D/3D x‐ray μ$$ \mu $$CT images in subsurface (e.g., CO 2$$ {}_2 $$ 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 MG$$ {\mathrm{M}}_{\mathrm{G}} $$ and a local smoother ML$$ {\mathrm{M}}_{\mathrm{L}} $$. Together, MG$$ {\mathrm{M}}_{\mathrm{G}} $$ and ML$$ {\mathrm{M}}_{\mathrm{L}} $$ attenuate low‐ and high‐frequency errors simultaneously. The proposed MG$$ {\mathrm{M}}_{\mathrm{G}} $$, 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 ML$$ {\mathrm{M}}_{\mathrm{L}} $$, used in both the momentum and crack‐evolution equations, is built such that it is compatible with MG$$ {\mathrm{M}}_{\mathrm{G}} $$ and more robust and efficient than black‐box smoothers like ILU(k$$ k $$). We test MG$$ {\mathrm{M}}_{\mathrm{G}} $$ and ML$$ {\mathrm{M}}_{\mathrm{L}} $$ 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 MG$$ {\mathrm{M}}_{\mathrm{G}} $$ as cracks evolve and show the associated cost can be minimized if MG$$ {\mathrm{M}}_{\mathrm{G}} $$ is updated adaptively and infrequently. Both MG$$ {\mathrm{M}}_{\mathrm{G}} $$ and ML$$ {\mathrm{M}}_{\mathrm{L}} $$ are scalable on parallel machines and can be implemented non‐intrusively in existing codes. [ABSTRACT FROM AUTHOR]

The shifted fracture method (SFM) is an embedded method that enables sharp crack representations while using mesh‐fitted data structures. In the SFM, the true crack is embedded in the computational grid, but the crack interface conditions are approximated by, or shifted to, a surrogate crack composed of element boundaries (i.e., edges/faces in two/three dimensions). This avoids enriched degrees‐of‐freedom and cut elements, so that data structures and geometrical treatment are much simpler, while still maintaining mesh‐independent and accurate crack approximations. This article presents a continuous‐discontinuous model of fracture based on blending a phase‐field (PF) model with the SFM. The PF tracks the evolution of cracks inside a numerical fracture processing zone: diffuse cracks initiate, propagate, branch, and merge according to the field equations of energy minimization; no ad‐hoc criteria are needed. The PF damage variable is then used to define the geometry of the true crack. With computational efficiency in mind, the PF model is only solved in subdomains where additional crack growth is expected and the SFM representation is used elsewhere. The efficiency and accuracy of the proposed approach in capturing complex crack patterns are illustrated by a representative set of two‐dimensional numerical examples. [ABSTRACT FROM AUTHOR]

Summary: We propose a new embedded finite element method to simulate partial differential equations over domains with internal interfaces. Our approach belongs to the family of surrogate/approximate interface methods and relies on the idea of shifting the location and value of jump interface conditions. This choice has the goal of preserving optimal convergence rates while avoiding small cut cells and related problematic issues, typical of traditional embedded methods. The proposed approach is accurate, robust, efficient, and simple to implement. We apply this concept to internal interface computations in the context of the mixed Poisson problem, also known as the Darcy flow problem. An extensive set of numerical tests is provided to demonstrate the performance of the proposed approach. [ABSTRACT FROM AUTHOR]