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Matrix-Free Geometric Multigrid Preconditioning Of Combined Newton-GMRES For Solving Phase-Field Fracture With Local Mesh Refinement

Leon Maximilian Kolditz, Thomas Wick

TL;DR

The paper develops a matrix-free, locally refined Geometric Multigrid preconditioned GMRES framework for solving the inner linear systems arising in a quasi-static phase-field fracture model with crack irreversibility enforced via a primal-dual active set method. It combines a line-search Newton solver with a matrix-free GMG preconditioner on locally refined meshes to efficiently handle the coupled displacement $u$ and phase-field $\varphi$ variables, including the inequality constraint. Key contributions include the integration of a Chebyshev-Jacobi smoother, a robust transfer of the active set across multigrid levels, and a fully matrix-free deal.II implementation. Numerical tests on Sneddon’s benchmark demonstrate convergence toward reference solutions with mesh refinement and robustness of the linear solver, despite occasional spikes tied to active-set predictions.

Abstract

In this work, the matrix-free solution of quasi-static phase-field fracture problems is further investigated. More specifically, we consider a quasi-monolithic formulation in which the irreversibility constraint is imposed with a primal-dual active set method. The resulting nonlinear problem is solved with a line-search assisted Newton method. Therein, the arising linear equation systems are solved with a generalized minimal residual method (GMRES), which is preconditioned with a matrix-free geometric multigrid method including geometric local mesh refinement. Our solver is substantiated with a numerical test on locally refined meshes.

Matrix-Free Geometric Multigrid Preconditioning Of Combined Newton-GMRES For Solving Phase-Field Fracture With Local Mesh Refinement

TL;DR

The paper develops a matrix-free, locally refined Geometric Multigrid preconditioned GMRES framework for solving the inner linear systems arising in a quasi-static phase-field fracture model with crack irreversibility enforced via a primal-dual active set method. It combines a line-search Newton solver with a matrix-free GMG preconditioner on locally refined meshes to efficiently handle the coupled displacement and phase-field variables, including the inequality constraint. Key contributions include the integration of a Chebyshev-Jacobi smoother, a robust transfer of the active set across multigrid levels, and a fully matrix-free deal.II implementation. Numerical tests on Sneddon’s benchmark demonstrate convergence toward reference solutions with mesh refinement and robustness of the linear solver, despite occasional spikes tied to active-set predictions.

Abstract

In this work, the matrix-free solution of quasi-static phase-field fracture problems is further investigated. More specifically, we consider a quasi-monolithic formulation in which the irreversibility constraint is imposed with a primal-dual active set method. The resulting nonlinear problem is solved with a line-search assisted Newton method. Therein, the arising linear equation systems are solved with a generalized minimal residual method (GMRES), which is preconditioned with a matrix-free geometric multigrid method including geometric local mesh refinement. Our solver is substantiated with a numerical test on locally refined meshes.
Paper Structure (7 sections, 16 equations, 2 figures, 2 tables, 2 algorithms)

This paper contains 7 sections, 16 equations, 2 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Problem Formulation: The Phase-Field Model
  3. Nonlinear Solution With Inner Linear GMRES Iterative Solver And Matrix-Free Geometric Multigrid Preconditioning
  4. A Combined Newton Type Algorithm
  5. A Geometric Multigrid Block Preconditioner For GMRES
  6. The Matrix-Free Approach And Final Algorithm
  7. Numerical Test: Sneddon's Benchmark

Figures (2)

  • Figure 1: Left: Geometry two dimensional Sneddon test. Right: Geometrically locally refined mesh.
  • Figure 2: Visualization of the $\operatorname{COD}$-values for different $h$. The corresponding exact $\operatorname{TCV}$ values are given in Table \ref{['tab:2']}.