On Quasi-Newton methods in fast Fourier transform-based micromechanics

This work is devoted to investigating the computational power of Quasi-Newton methods in the context of fast Fourier transform (FFT)-based computational micromechanics. We revisit (FFT)-based Newton-Krylov solvers as well as modern Quasi-Newton approaches such as the recently introduced Anderson accelerated basic scheme. In this context, we propose two algorithms based on the Broyden-Fletcher-Goldfarb-Shanno (BFGS) method, one of the most powerful Quasi-Newton schemes. To be specific, we use the BFGS update formula to approximate the global Hessian or, alternatively, the local material tangent stiffness. Both for Newton and Quasi-Newton methods, a globalization technique is necessary to ensure global convergence. Specific to the FFT-based context, we promote a Dong-type line search, avoiding function evaluations altogether. Furthermore, we investigate the influence of the forcing term, that is, the accuracy for solving the linear system, on the overall performance of inexact (Quasi-)Newton methods. This work concludes with numerical experiments, comparing the convergence characteristics and runtime of the proposed techniques for complex microstructures with nonlinear material behavior and finite as well as infinite material contrast.