An advanced approach for the generation of complex cellular material representative volume elements using distance fields and level sets

Rypl

et al. 2020

Computer-Aided Design

Microstructural geometry plays a critical role in the response of heterogeneous materials. Consequently, methods for generating microstructural samples are increasingly crucial to advanced numerical analyses. We extend Sonon et al.'s unified framework, developed originally for generating particulate and foam-like microstructural geometries of Periodic Unit Cells, to non-periodic microstructural representations based on the formalism of Wang tiles. This formalism has been recently proposed in order to generalize the Periodic Unit Cell approach, enabling a fast synthesis of arbitrarily large, stochastic microstructural samples from a handful of domains with predefined microstructural compatibility constraints. However, a robust procedure capable of designing complex, three-dimensional, foam-like and cellular morphologies of Wang tiles has not yet been proposed. This contribution fills the gap by significantly broadening the applicability of the tiling concept.Since the original Sonon et al.'s framework builds on a random sequential addition of particles enhanced with an implicit representation of particle boundaries by the level-set field, we first devise an analysis based on a connectivity graph of a tile set, resolving the question where a particle should be copied when it intersects a tile boundary. Next, we introduce several modifications to the original algorithm that are necessary to ensure microstructural compatibility in the generalized periodicity setting of Wang tiles. Having established a universal procedure for generating tile morphologies, we compare strictly aperiodic and stochastic sets with the same cardinality in terms of reducing the artificial periodicity in reconstructed microstructural samples. We demonstrate the superiority of the vertex-defined tile sets for two-dimensional problems and illustrate the capabilities of the algorithm with two-and three-dimensional examples.

Section: State-of-the-art In Modelling Random Microstructuresmentioning

confidence: 99%

Section: Wang Tiling In Microstructure Modellingmentioning

confidence: 99%

Level-set Based Design of Wang Tiles for Modelling Complex Microstructures

Doškář

Rypl

et al. 2020

Computer-Aided Design

“…The system was compressed into the same tile set, in terms of tile code definition, as the previous microstructure. The internal geometry of tiles was artificially designed with a modified version of the levelset based approach developed by Sonon et al [76]. The compressed geometry, displayed in Fig.…”

Section: Foammentioning

confidence: 99%

Wang tiling aided statistical determination of the Representative Volume Element size of random heterogeneous materials

Doškář

European Journal of Mechanics - A/Solids

Jarušková

et al. 2018

Wang tile based representation of a heterogeneous material facilitates fast synthesis of non-periodic microstructure realizations. In this paper, we apply the tiling approach in numerical homogenization to determine the Representative Volume Element size related to the user-defined significance level and the discrepancy between bounds on the apparent properties. First, the tiling concept is employed to efficiently generate arbitrarily large, statistically consistent realizations of investigated microstructures. Second, benefiting from the regular structure inherent to the tiling concept, the Partition theorem, and statistical sampling, we construct confidence intervals of the apparent properties related to the size of a microstructure specimen. Based on the interval width and the upper and lower bounds on the apparent properties, we adaptively generate additional microstructure realizations in order to arrive at an RVE satisfying the prescribed tolerance. The methodology is illustrated with the homogenization of thermo-mechanical properties of three two-dimensional microstructure models: a microstructure with mono-disperse elliptic inclusions, foam, and sandstone.

“…Various experimental and simulation techniques, such as serial sectioning [1], computed tomography [2], statistical reconstruction [3], or digital models [4] are currently available to characterize microstructures of heterogeneous materials in a degree of realism not possible before. When combined with the tools of homogenization theories, e.g.…”

Section: Introductionmentioning

confidence: 99%

A comparative study on low-memory iterative solvers for FFT-based homogenization of periodic media

Mishra

Vondřejc

Journal of Computational Physics

2016

In this paper, we assess the performance of four iterative algorithms for solving nonsymmetric rank-deficient linear systems arising in the FFT-based homogenization of heterogeneous materials defined by digital images. Our framework is based on the FourierGalerkin method with exact and approximate integrations that has recently been shown to generalize the Lippmann-Schwinger setting of the original work by Moulinec and Suquet from 1994. It follows from this variational format that the ensuing system of linear equations can be solved by general-purpose iterative algorithms for symmetric positivedefinite systems, such as the Richardson, the Conjugate gradient, and the Chebyshev algorithms, that are compared here to the Eyre-Milton scheme -the most efficient specialized method currently available. Our numerical experiments, carried out for twodimensional elliptic problems, reveal that the Conjugate gradient algorithm is the most efficient option, while the Eyre-Milton method performs comparably to the Chebyshev semi-iteration. The Richardson algorithm, equivalent to the still widely used original Moulinec-Suquet solver, exhibits the slowest convergence. Besides this, we hope that our study highlights the potential of the well-established techniques of numerical linear algebra to further increase the efficiency of FFT-based homogenization methods.