A review of nonlinear FFT-based computational homogenization methods

Acta Mech https://doi.org/10.1007/s00707-021-02962-1 REVIEW AND PERSPECTIVE IN MECHANICS Matti Schneider A review of nonlinear FFT-based computational homogenization methods Received: 9 December 2020 / Revised: 18 January 2021 / Accepted: 16 February 2021 © The Author(s) 2021 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. Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Discretization schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Lippmann–Schwinger framework for the cell problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 The Moulinec–Suquet discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 The Fourier–Galerkin discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Discretizations based on the Hashin–Shtrikman principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Finite difference, finite element and finite volume discretizations . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Modern nonlinear solution methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 The basic scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Krylov subspace methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Fast gradient methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Newton and Quasi-Newton methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Polarization methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Convergence tests and overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Practical issues and efficient implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 The dual scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Mixed boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Composite voxels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Conductivity and diffusivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Coupled problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Damage and fracture mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Polycrystalline materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Composites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Porous and granular materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Concluding remarks and future directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M. Schneider (B) Karlsruhe Institute of Technology (KIT), Institute of Engineering Mechanics, Karlsruhe, Germany E-mail: M. Schneider 1 Introduction Homogenization theory [1–3] serves as the mathematical basis for understanding the behavior of microstructured and composite materials. For given constitutive laws and an explicit description of the microstructure, the effective constitutive law emerges naturally, based on solving a partial differential equation (PDE), the so-called cell problem. Only 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. Such techniques face a number of challenges. For a start, microstructures of heterogeneous materials may be rather complex, which becomes clear when looking at typical materials to be homogenized: polycrystalline materials, fiber-reinforced composites, rocks, foams, textiles and many more [8]. Deep insights into such microstructures are provided by modern digital volume imaging techniques like serial sectioning [9,10], optical microscopy [11,12], FIB-SEM [13–15], Electron backscatter diffraction [16–18], X-ray diffraction microscopy [19,20] and micro-computed tomography [21,22]. Computational homogenization methods need to handle the complexity of industrial-scale microstructures, and should be compatible to digital volume data. In addition to their inherent complexity, microstructured materials typically show a degree of randomness as a result of the production process [23–25]. To obtain deterministic effective constitutive laws, the cell problem needs to be solved on sufficiently large cells, socalled representative volume elements (RVE) [26,27]. To decide whether a cell is representative (enough), it is necessary to quantify the standard deviation of the computed effective properties for cells of fixed size (the so-called dispersion [28,29]), and to investigate the change in the empirical mean of the computed effective properties for increasing cell size (to quantify the bias [28,29]). Thus, 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. In addition to the geometrical properties of the microstructure, computational homogenization methods should be applicable to a wide range of (...truncated)