A review of nonlinear FFT-based computational homogenization methods

Abstract

Since their inception, computational homogenization methods based on the fast Fourier transform (FFT) have grown in popularity, establishing themselves as a powerful tool applicable to complex, digitized microstructures. At the same time, the understanding of the underlying principles has grown, in terms of both discretization schemes and solution methods, leading to improvements of the original approach and extending the applications. This article provides a condensed overview of results scattered throughout the literature and guides the reader to the current state of the art in nonlinear computational homogenization methods using the fast Fourier transform.

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

“…As a typical model for computational homogenization 1,2 on a periodic cell Q and at small strains, we are concerned with the optimization problem…”

“…The basic scheme (2) is the prototype of a primal method, that is, a method where the kinematic constraints are satisfied at every iteration. Extensions of the basic scheme lead to fast and robust solution methods for FFT-based computational micromechanics.…”

“…To complete the section on the implementation, let us specify that we use a discretization on a staggered grid, 36 see Schneider 2 (sec.2.5) for the efficient evaluation of the ‐operator. Please note, however, that the implementation is suitable for alternative discretizations, as well, see the overview article 2 . (sec.2)…”

“…As a typical model for computational homogenization 1,2 on a periodic cell Q and at small strains, we are concerned with the optimization problem seeking the total strain field on the cell Q minimizing the average elastic energy under the kinematic constraints which encode compatibility and fixed macroscopic strain . A prototype for primal methods for solving the problem (1) is projected gradient descent, which iterates for step sizes , where denotes the gradient of W and refers to the orthogonal projection onto the kinematic constraint .…”

Computing the effective crack energy of heterogeneous and anisotropic microstructures via anisotropic minimal surfaces

“…Ernesti and Schneider [53] proposed to solve the maximum flow problem in the combinatorial continuous maximum flow (CCMF) discretization [56] by FFT-based methods [57]. The formulation is based on doubling the degrees of freedom.…”