Superaccurate effective elastic moduli via postprocessing in computational homogenization

With the complexity of modern microstructured materials, computational homogenization methods have been shown to provide accurate estimates of their effective mechanical properties, reducing the involved experimental effort considerably. After solving the balance of linear momentum on the microscale, the effective stress is traditionally computed through volume averaging the microscopic stress field. In the work at hand, we exploit the idea that averaging the elastic energy may lead to much more accurate effective elastic properties than through stress averaging. We show that the accuracy is roughly doubled when using energy equivalence instead of strain equivalence for compatible iterates of iterative schemes. Thus, to achieve a prescribed accuracy, the necessary effort is roughly reduced by a factor of two. In addition to the theory, we provide a handbook for utilizing these ideas for modern solvers prominent in FFT-based micromechanics. We demonstrate the superiority of energy averaging through computational examples, discuss the peculiarities of polarization methods with their non-compatible iterates and expose a superaccuracy phenomenon occurring for the linear conjugate gradient method

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

On the effectiveness of the Moulinec–Suquet discretization for composite materials

Moulinec and Suquet introduced a method for computational homogenization based on the fast Fourier transform which turned out to be rather computationally efficient. The underlying discretization scheme was subsequently identified as an approach based on trigonometric polynomials, coupled to the trapezoidal rule to substitute full integration. For problems with smooth solutions, the power of spectral methods is well-known. However, for heterogeneous microstructures, there are jumps in the coefficients, and the solution fields are not smooth enough due to discontinuities across material interfaces. Previous convergence results only provided convergence of the discretization per se, that is, without explicit rates, and could not explain the effectiveness of the discretization observed in practice. In this work, we provide such explicit convergence rates for the local strain as well as the stress field and the effective stresses based on more refined techniques. More precisely, we consider a class of industrially relevant, discontinuous elastic moduli separated by sufficiently smooth interfaces and show rates which are known to be sharp from numerical experiments. The applied techniques are of independent interest, that is, we employ a local smoothing strategy, utilize Féjer means as well as Bernstein estimates and rely upon recently established superconvergence results for the effective elastic energy in the Galerkin setting. The presented results shed theoretical light on the effectiveness of the Moulinec–Suquet discretization in practice. Indeed, the obtained convergence rates coincide with those obtained for voxel finite element methods, which typically require higher computational effort

A review of nonlinear FFT-based computational homogenization methods

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

Voxel‐based finite elements with hourglass control in fast Fourier transform‐based computational homogenization

The power of fast Fourier transform (FFT)-based methods in computational micromechanics critically depends on a seamless integration of discretization scheme and solution method. In contrast to solution methods, where options are available that are fast, robust and memory-efficient at the same time, choosing the underlying discretization scheme still requires the user to make compromises. Discretizations with trigonometric polynomials suffer from spurious oscillations in the solution fields and lead to ill-conditioned systems for complex porous materials, but come with rather accurate effective properties for finitely contrasted materials. The staggered grid discretization, a finite-volume scheme, is devoid of bulk artifacts in the solution fields and works robustly for porous materials, but does not handle anisotropic materials in a natural way. Fully integrated finite-element discretizations share the advantages of the staggered grid, but involve a higher memory footprint, require a higher computational effort due to the increased number of integration points and typically overestimate the effective properties. Most widely used is the rotated staggered grid discretization, which may also be viewed as an underintegrated trilinear finite element discretization, which does not impose restrictions on the constitutive law, has fewer artifacts than Fourier-type discretizations and leads to rather accurate effective properties. However, this discretization comes with two downsides. For a start, checkerboard artifacts are still present. Second, convergence problems occur for complex porous microstructures. The work at hand introduces FFT-based solution techniques for underintegrated trilinear finite elements with hourglass control. The latter approach permits to suppress local hourglass modes, which stabilizes the convergence behavior of the solvers for porous materials and removes the checkerboards from the local solution field. Moreover, the hourglass-control parameter can be adjusted to “soften” the material response compared to fully integrated elements, using only a single integration point for nonlinear analyses at the same time. To be effective, the introduced technology requires a displacement-based implementation. The article exposes an efficient way for doing so, providing minimal interfaces to the most commonly used solution techniques and the appropriate convergence criterion

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