Schur‐type preconditioning of a phase‐field fracture model in mixed form

Heister, Timo; Mang, Katrin; Wick, Thomas

doi:10.1002/pamm.202100065

Cited by 4 publications

(5 citation statements)

References 19 publications

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“…Preconditioning the (3×3) linear system. The system matrix M mixed of the mixed phase-field fracture from the modified mixed problem formulation has the following block structure [24]:…”

Section: 21mentioning

confidence: 99%

“…These are the spatial discretization parameter h and the Poisson ratio ν (related to the Lamé coefficient λ) up to the incompressible limit ν = 0.5, and the regularization parameter κ and the crack bandwidth . We note that the basis of this work was developed in Section 6 of the PhD thesis of the second author [39] and some preliminary results were published in [24].…”

Section: Introductionmentioning

confidence: 99%

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Robust preconditioning for a mixed formulation of phase-field fracture problems

Heister¹,

Mang²,

Wick³

2022

Preprint

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In this work, we consider fracture propagation in nearly incompressible and (fully) incompressible materials using a phase-field formulation. We use a mixed form of the elasticity equation to overcome volume locking effects and develop a robust, nonlinear and linear solver scheme and preconditioner for the resulting system. The coupled variational inequality system, which is solved monolithically, consists of three unknowns: displacements, pressure, and phase-field. Nonlinearities due to coupling, constitutive laws, and crack irreversibility are solved using a combined Newton algorithm for the nonlinearities in the partial differential equation and employing a primal-dual active set strategy for the crack irreverrsibility constraint. The linear system in each Newton step is solved iteratively with a flexible generalized minimal residual method (GMRES). The key contribution of this work is the development of a problem-specific preconditioner that leverages the saddle-point structure of the displacement and pressure variable. Four numerical examples in pure solids and pressure-driven fractures are conducted on uniformly and locally refined meshes to investigate the robustness of the solver concerning the Poisson ratio as well as the discretization and regularization parameters.

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“…Preconditioning the (3×3) linear system. The system matrix M mixed of the mixed phase-field fracture from the modified mixed problem formulation has the following block structure [24]:…”

Section: 21mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Robust preconditioning for a mixed formulation of phase-field fracture problems

Heister¹,

Mang²,

Wick³

2022

Preprint

Self Cite

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“…Work on preconditioning for three field problems arising in the context of phase-field modeling of fracture in incompressible materials can be found in Ref. 20 In Ref.,21 block-diagonal preconditioning is used in matrix-free approaches for the solution of phase-field fracture problems, and nonlinear preconditioning for phase-field modeling of fracture is proposed in Ref. 22 Tailored preconditioners for specific multifield problems therefore represent a vast area of research, and the present paper aims at contributing to this field of study for the special case of generalized continuum models.…”

Section: Introductionmentioning

confidence: 99%

“…In Ref.,19 the performance of algebraic multigrid preconditioning for phase‐field models is investigated. Work on preconditioning for three field problems arising in the context of phase‐field modeling of fracture in incompressible materials can be found in Ref 20. In Ref.,21 block‐diagonal preconditioning is used in matrix‐free approaches for the solution of phase‐field fracture problems, and nonlinear preconditioning for phase‐field modeling of fracture is proposed in Ref 22…”

Section: Introductionmentioning

confidence: 99%

Block preconditioning strategies for generalized continuum models with micropolar and nonlocal damage formulations

Alkmim,

Gamnitzer,

Neuner

et al. 2024

Num Anal Meth Geomechanics

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In this work, preconditioning strategies are developed in the context of generalized continuum formulations used to regularize multifield models for simulating localized failure of quasi‐brittle materials. Specifically, a micropolar continuum extended by a nonlocal damage formulation is considered for regularizing both, shear dominated failure and tensile cracking. For such models, additional microrotation and nonlocal damage fields, and their interactions, increase the complexity and size of the arising linear systems. This increases the demand for specialized preconditioning strategies when iterative solvers are adopted. Herein, a block preconditioning strategy, employing algebraic multigrid methods (AMG) for approximating the application of sub‐block inverses, is developed and tested in three steps. First, a block preconditioner is introduced for linear systems resulting from micropolar models. For this case, a simple sparse Schur complement approximation, which is practical to compute, is proposed and analyzed. It is tested for three different discretizations. Second, the developed preconditioner is extended to reflect the additional nonlocal damage field. This extended three‐field preconditioner is tested on the simulation of a compression test on a sandstone sample. All numerical tests show an improved performance of the block preconditioning approach in comparison to a black‐box monolithic AMG approach. Finally, a problem‐adapted preconditioner setup strategy is proposed, which involves a reuse of the multigrid hierarchy during nonlinear iterations, and additionally accounts for the different stages occurring in the simulation of localized failure. The problem‐adapted preconditioning strategy has the potential to further reduce the total computation time.

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“…We propose a multiscale preconditioner that applies sequentially to the linear(ized) systems of the momentum and crack‐evolution equations in both static‐ and evolving‐crack problems. We remark in passing that fully coupled (or monolithic) preconditioners for mixed formulations of the phase‐field equations have also been proposed 23 . Here, for the momentum equation, we combine a global preconditioner, M G , with a local smoother, M L , to simultaneously attenuate low‐ and high‐frequency errors.…”

Section: Introductionmentioning

confidence: 99%

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

Li,

Mehmani

2024

Numerical Meth Engineering

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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 CT images in subsurface (e.g., CO 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 and a local smoother . Together, and attenuate low‐ and high‐frequency errors simultaneously. The proposed , 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 , used in both the momentum and crack‐evolution equations, is built such that it is compatible with and more robust and efficient than black‐box smoothers like ILU(). We test and 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 as cracks evolve and show the associated cost can be minimized if is updated adaptively and infrequently. Both and are scalable on parallel machines and can be implemented non‐intrusively in existing codes.

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Schur‐type preconditioning of a phase‐field fracture model in mixed form

Cited by 4 publications

References 19 publications

Robust preconditioning for a mixed formulation of phase-field fracture problems

Robust preconditioning for a mixed formulation of phase-field fracture problems

Block preconditioning strategies for generalized continuum models with micropolar and nonlocal damage formulations

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

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