Linear and nonlinear solvers for variational phase-field models of brittle fracture

Patrick Farrell,Corrado Maurini

doi:10.1002/nme.5300

Abstract

The variational approach to fracture is effective for simulating the nucleation and propagation of complex crack patterns, but is computationally demanding. The model is a strongly nonlinear non-convex variational inequality that demands the resolution of small length scales. The current standard algorithm for its solution, alternate minimization, is robust but converges slowly and demands the solution of large, ill-conditioned linear subproblems. In this paper, we propose several advances in the numerical solution of this model that improve its computational efficiency. We reformulate alternate minimization as a nonlinear Gauss-Seidel iteration and employ over-relaxation to accelerate its convergence; we compose this accelerated alternate minimization with Newton's method, to further reduce the time to solution; and we formulate efficient preconditioners for the solution of the linear subproblems arising in both alternate minimization and in Newton's method. We investigate the improvements in efficiency on several examples from the literature; the new solver is 5–6× faster on a majority of the test cases considered.

Highlights

Cracks may be regarded as surfaces where the displacement field may be discontinuous
We present here the results of the numerical experiments that were performed to assess the performance of the proposed solvers
We proposed several improvements to the current standard algorithm for solving variational fracture models

Summary

Introduction

Cracks may be regarded as surfaces where the displacement field may be discontinuous. Fracture mechanics studies the nucleation and propagation of cracks inside a solid structure. Variational formulations recast this fundamental and difficult problem of solid mechanics as an optimization problem. The variational framework naturally leads to regularized phase-field formulations based on a smeared description of the discontinuities. These methods are attracting an increasing interest in computational mechanics. The aim of our work is to propose several improvements in the linear and nonlinear solvers used in this framework

Objectives

Results

Conclusion