Received: 16 July 2019 Revised: 27 November 2019 Accepted: 28 November 2019 DOI: 10.1002/nme.6283 RESEARCH ARTICLE On Quasi-Newton methods in fast Fourier transform-based micromechanics Daniel Wicht Matti Schneider Thomas Böhlke Karlsruhe Institute of Technology, Institute of Engineering Mechanics, Karlsruhe, Germany Correspondence Thomas Böhlke, Karlsruhe Institute of Technology, Institute of Engineering Mechanics, Karlsruhe 76131, Germany. Email: thomas.boehlke@kit.edu Funding information German Research Foundation (DFG), Grant/Award Number: “Integrated engineering of continuous-discontinuous long fiber reinforced polymer structures” (GRK 2078); German Research Foundation (DFG), Grant/Award Number: “Lamellar Fe-Al in situ composite materials” (BO 1466/12-2) SUMMARY 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 mod- ern Quasi-Newton approaches such as the recently introduced Anderson accel- erated basic scheme. In this context, we propose two algorithms based on the Broyden-Fletcher-Goldfarb-Shanno (BFGS) method, one of the most power- ful 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 tech- nique is necessary to ensure global convergence. Specific to the FFT-based context, we promote a Dong-type line search, avoiding function evaluations alto- gether. Furthermore, we investigate the influence of the forcing term, that is, the accuracy for solving the linear system, on the overall performance of inex- act (Quasi-)Newton methods. This work concludes with numerical experiments, comparing the convergence characteristics and runtime of the proposed tech- niques for complex microstructures with nonlinear material behavior and finite as well as infinite material contrast. KEYWORDS BFGS, composites, crystal viscoplasticity, FFT-based micromechanics, homogenization, Quasi-Newton methods 1 INTRODUCTION Fast Fourier transform (FFT)-based solution schemes 5,6 have become an established tool in computational microme- chanics. Their popularity rests on their computational efficiency, the ability to treat nonlinear material behavior and the compatibility to imaging techniques such as microcomputed tomography. Owing to these advantages, FFT-based solvers have found widespread application in various fields, including the homogenization of polycrystals at small 7 and finite strains, 8 damage 9 and fracture mechanics, 10 fatigue prediction, 11 electromechanically coupled materials, 12 and concurrent multiscale simulations. 13,14 [Correction added on 14 February 2020, after first online publication: Funding information has been updated.] This is an open access article under the terms of the Creative Commons Attribution-NonCommercial-NoDerivs License, which permits use and distribution in any medium, provided the original work is properly cited, the use is non-commercial and no modifications or adaptations are made. © 2019 The Authors. International Journal for Numerical Methods in Engineering published by John Wiley & Sons, Ltd. Int J Numer Methods Eng. 2020;121:1665–1694. wileyonlinelibrary.com/journal/nme 1665