Abstract
AbstractClassical solution methods in fast Fourier transform‐based computational micromechanics operate on, either, compatible strain fields or equilibrated stress fields. By contrast, polarization schemes are primal‐dual methods whose iterates are neither compatible nor equilibrated. Recently, it was demonstrated that polarization schemes may outperform the classical methods. Unfortunately, their computational power critically depends on a judicious choice of numerical parameters. In this work, we investigate the extension of polarization methods by Anderson acceleration and demonstrate that this combination leads to robust and fast general‐purpose solvers for computational micromechanics. We discuss the (theoretically) optimum parameter choice for polarization methods, describe how Anderson acceleration fits into the picture, and exhibit the characteristics of the newly designed methods for problems of industrial scale and interest.
Highlights
For solving the corrector problem arising in periodic and stochastic homogenization problems, methods based on the fast Fourier transform (FFT), pioneered by Moulinec and Suquet,[1,2] were identified as fast, robust, and versatile
We study the combination of polarization methods and Anderson acceleration, producing a fast, flexible and versatile general-purpose FFT-based solver
As an alternative strategy for computing the step size s, we investigated the approach of Schneider et al.,[44] where μ and L are computed in every iteration and the step size s is subsequently updated based on the minimum value of μ and the maximum value of L over all past iterates
Summary
For solving the (mechanical) corrector problem arising in periodic and stochastic homogenization problems, methods based on the fast Fourier transform (FFT), pioneered by Moulinec and Suquet,[1,2] were identified as fast, robust, and versatile As they operate on a regular voxel grid, such schemes are directly compatible to modern imaging techniques, such as microcomputed tomography, and do not depend on creating interface-conforming meshes for complex microstructures. Modern FFT-based methods have to be flexible enough to support a wide variety of (nonlinear) material models, and to work robustly for different classes of microstructures In this context, research on FFT-based algorithms has mostly focused on two key areas. Polarization-based schemes are highly sensitive to the choice of algorithmic parameters, limiting their suitability as general-purpose solvers
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