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.
Highlights
Homogenization theory [1–3] serves as the mathematical basis for understanding the behavior of microstructured and composite materials
To obtain deterministic effective constitutive laws, the cell problem needs to be solved on sufficiently large cells, socalled representative volume elements (RVE) [26,27]
Zeman et al [54] reported that the conjugate gradient method converged when applied to the Lippmann–Schwinger Eq (2.12) and that the produced iterates were identical to those resulting from BiCGStab! The reasons behind this unexpected behavior were uncovered by Vondrejc et al [68], who showed that the operator Id + 0 : C − C0 is symmetric and positive definite on the subspace of compatible strain fields
Summary
Homogenization theory [1–3] serves as the mathematical basis for understanding the behavior of microstructured and composite materials. For special cases, the effective constitutive laws may be expressed in closed form [4], and bounding techniques [5,6] may lack accuracy For these reasons, numerical approaches for solving the cell problem, socalled computational homogenization methods [7], emerged. Numerical approaches for solving the cell problem, socalled computational homogenization methods [7], emerged Such techniques face a number of challenges. Computational homogenization methods need to solve the cell problem for complex microstructures on large volumes, and need to do so repeatedly in order to quantify the randomness. The Moulinec–Suquet approach builds upon the Lippmann–Schwinger equation [35–37], a reformulation of the cell problem as an equivalent volume integral equation It operates directly on voxel data and was formulated for nonlinear material behavior from the very beginning. This article is not intended to provide innovative research, and we claim no originality
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