Anderson‐accelerated polarization schemes for fast Fourier transform‐based computational homogenization

Abstract

Classical 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.

On non‐stationary polarization methods in FFT‐based computational micromechanics

“…Under a geometric assumption on the solid‐pore interface, 49 the convergence is linear.Still, optimized primal solvers outperformed the considered primal‐dual solvers. It remains to investigate whether an improved selection strategy for the reference material permits outperforming primal solvers for porous microstructures, as can be done for finite material contrast 28,42 We devoted attention to adaptive choices of the reference material in the damped ADMM.…”

“…We devoted attention to adaptive choices of the reference material in the damped ADMM. Such adaptive choices were shown to represent memory‐efficient alternatives to the Anderson‐accelerated polarization methods, 42 sometimes even improving upon their performance. For the ADMM, the residual‐balancing strategy 45 is a classic, and widely used. We discussed the strategy for the problem at hand, see Section 2.3.…”

“…Still, optimized primal solvers outperformed the considered primal‐dual solvers. It remains to investigate whether an improved selection strategy for the reference material permits outperforming primal solvers for porous microstructures, as can be done for finite material contrast 28,42 …”

“…Anderson acceleration 40,41 is a Quasi‐Newton method applicable to generic fixed point schemes that was successfully applied to primal FFT‐based methods 13‐15 . Combined with polarization schemes, the resulting Anderson‐accelerated polarization schemes 42 showed robust and fast convergence, also for porous microstructures. Unfortunately, Anderson acceleration comes with a severe memory demand.…”

An FFT-based approach for Bloch wave analysis: application to polycrystals

“…The main advantages of FFT-based approaches are their very efficient numerical performance (computational cost only grows a n log n, where n is the number of discretization points) and the possibility of using direct input from images of the material, avoiding complex meshing procedures. In recent years, several modifications of the original FFT-based approach have been proposed to improve the convergence of the iterative method for high mechanical contrast [25][26][27][28][29][30][31][32][33]. The FFT-based method, originally proposed and implemented for composites, was extended to linear elastic and non-linear polycrystals (e.g.…”