A dynamical view of nonlinear conjugate gradient methods with applications to FFT-based computational micromechanics

For fast Fourier transform (FFT)-based computational micromechanics, solvers need to be fast, memory-efficient, and independent of tedious parameter calibration. In this work, we investigate the benefits of nonlinear conjugate gradient (CG) methods in the context of FFT-based computational micromechanics. Traditionally, nonlinear CG methods require dedicated line-search procedures to be efficient, rendering them not competitive in the FFT-based context. We contribute to nonlinear CG methods devoid of line searches by exploiting similarities between nonlinear CG methods and accelerated gradient methods. More precisely, by letting the step-size go to zero, we exhibit the Fletcher–Reeves nonlinear CG as a dynamical system with state-dependent nonlinear damping. We show how to implement nonlinear CG methods for FFT-based computational micromechanics, and demonstrate by numerical experiments that the Fletcher–Reeves nonlinear CG represents a competitive, memory-efficient and parameter-choice free solution method for linear and nonlinear homogenization problems, which, in addition, decreases the residual monotonically

An algorithm for generating microstructures of fiber‐reinforced composites with long fibers

We describe a sequential addition and migration (SAM) algorithm for generating microstructures of fiber-reinforced composites with a direct control of the magnitude of curvature of the fibers. The algorithm permits to generate microstructures with fibers that are significantly longer than the edge lengths of the underlying cell. Industrially processed short and long-fiber composites naturally feature a high volume fraction, which needs to be reflected by state-of-the-art microstructure generation tools. Nowadays, it is well understood that digital twins of the microstructure of composites are essential for reliable computational multiscale methods. The original SAM algorithm was shown to reliably generate microstructures for short and straight cylindrical fibers. Digital volume images reveal, however, that the fibers in such fiber-reinforced composites may show significant curvature, in particular for long fibers. The work at hand introduces an extension of the original SAM approach to curved fibers. More precisely, curved fibers are considered as sequences of straight fibers which are joined at their respective ends and whose level of bending is controlled by the angle between adjacent fiber segments. We discuss how to efficiently implement the novel method and how to select the crucial numerical parameters. We compare the introduced methodology to the original SAM algorithm for short fibers and demonstrate the superiority of the novel strategy for long fibers

A fast Fourier transform based method for computing the effective crack energy of a heterogeneous material on a combinatorially consistent grid

This work is concerned with computing the effective crack energy of periodic and random media which arises in mathematical homogenization results for the Francfort–Marigo model of brittle fracture. A previous solver based on the fast Fourier transform (FFT) led to solution fields with ringing or checkerboard artifacts and was limited in terms of the achievable accuracy. As computing the effective crack energy may be recast as a continuous maximum flow problem, we suggest using the combinatorial continuous maximum flow discretization introduced by Couprie et al. The latter is devoid of artifacts, but lacks an efficient large-scale solution method. We fill this gap and introduce a novel solver which relies upon the FFT and a doubling of the local degrees of freedom which is resolved by the alternating direction method of multipliers (ADMM). Last but not least we provide an adaptive strategy for choosing the ADMM penalty parameter, further speeding up the solution procedure. We demonstrate the salient features of the proposed approach on problems of industrial scale