Martin Doškář

A jigsaw puzzle framework for homogenization of high porosity foams

The Wang tiling technique for generating large ensembles of arbitrarily sized RVEs.First order homogenization via macroscopic degrees of freedom for wire-frame models.The lower bound obtained by employing the minimal Kinematic Boundary Conditions.A discussion on validity of two-dimensional models for foam materials. An approach to homogenization of high porosity metallic foams is explored. The emphasis is on the Alporas? foam and its representation by means of two-dimensional wire-frame models. The guaranteed upper and lower bounds on the effective stiffness coefficients are derived by the first-order homogenization with the Uniform and Minimal Kinematic Boundary Conditions at heart. This is combined with the method of Wang tilings to generate sufficiently large material samples along with their finite element discretization. The obtained results are compared to experimental and numerical data available in literature and the suitability of the two-dimensional setting itself is discussed.

Aperiodic compression and reconstruction of real-world material systems based on Wang tiles.

The paper presents a concept to compress and synthesize complex material morphologies that is based on Wang tilings. Specifically, a microstructure is stored in a set of Wang tiles and its reconstruction is performed by means of a stochastic tiling algorithm. A substantial part of the study is devoted to the setup of optimal parameters of the automatic tile design by means of parametric studies with statistical descriptors at heart. The performance of the method is demonstrated on four two-dimensional two-phase target systems, monodisperse media with hard and soft disks, sandstone, and high porosity metallic foam.

Compression of Heterogeneous Material Systems Based on Wang Tilings

The present study is on the concept of modeling of heterogeneous materials by means of Wang tilings. The central idea is to store a microstructural information within a finite set of Wang Tiles, which allow for reconstructing heterogeneity patterns of random media in planar domains of arbitrary sizes. The particular objective of presented work is our automatic tile set designer in conjunction with stochastic tiling synthesis algorithm. The proposed methodology is demonstrated on different examples. The proximity of synthesized microstructures to reference media is explored by statistical descriptors and discussed in terms of parasitic spatial orientation orders that may occur.

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

Abstract 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.

A jigsaw puzzle metamaterial concept

Abstract A concept of a planar modular mechanical metamaterial inspired by the principle of local adaptivity is proposed. The metamaterial consists of identical pieces similar to jigsaw puzzle tiles. Their rotation within assembly provides a substantial flexibility in terms of structural behavior, whereas mechanical interlocks enable reassembly. The tile design with a diagonal elliptical opening allows us to vary elastic properties from stiff to compliant, with positive, zero, or negative Poisson’s ratio. The outcomes of experimental testing on additively manufactured specimens confirm that the assembly properties can be accurately designed using optimization approaches with finite element analysis at heart.

SYNTHESIZED ENRICHMENT FUNCTIONS FOR EXTENDED FINITE ELEMENT ANALYSES WITH FULLY RESOLVED MICROSTRUCTURE

Inspired by the first order numerical homogenization, we present a method for extracting continuous fluctuation fields from the Wang tile based compression of a material microstructure. The fluctuation fields are then used as enrichment basis in Extended Finite Element Method (XFEM) to reduce number of unknowns in problems with fully resolved microstructural geometry synthesized by means of the tiling concept. In addition, the XFEM basis functions are taken as reduced modes of a detailed discretization in order to circumvent the need for non-standard numerical quadratures. The methodology is illustrated with a scalar steady-state problem.

WANG TILES LOCAL TILINGS CREATED BY MERGING EDGE-COMPATIBLE FINITE ELEMENT MESHES

In this contribution, we present the concept of Wang Tiles as a surrogate of the periodic unit cell method (PUC) for modelling of materials with heterogeneous microstructures and for synthesis of micro-mechanical fields. The concept is based on a set of specifically designed cells that compresses the stochastic microstructure into a small set of statistical volume elements – tiles. Tiles are placed side by side according to matching edges like in a game of domino. Opposite to the repeating pattern of PUC the Wang Tiles method with the stochastic tiling algorithm preserves the randomness for reconstructed microstructures. The same process is applied to obtain the micro-mechanical response of domains where the evaluation as one piece would be time consuming. Therefore the micro-mechanical quantities are evaluated only on tiles (with surrounding layers of tiles of each addressed tile included into the evaluation) and then synthesized to the micro-mechanical field of whole domain.

An Additively Manufactured Modular Metamaterial Composed of a Single Cell

Preliminary outcomes on a novel modular additive manufacturing technology are presentedin this contribution. The capabilities of the proposed system to synthesize materials with conventionalor meta properties are explored. Two limiting and a single mixed compositions are tested in com-pression in order to reveal both, global and local behaviours of the conglomerates. The experimentalmeasurements are supported by DIC (Digital Image Correlation) observations.

Wang Tiling for Microstructure-Informed Enrichment Functions: A One-Dimensional Study

With the current contribution we continue in our long-term ambition to develop an efficient numerical method that incorporates detailed knowledge of material microstructure into upper-scale response. In particular, we investigate potential of the Wang tiling concept to provide microstructure-informed enrichment functions for the eXtended/Generalized Finite Element Method. Preliminary results for a one-dimensional linear elasticity problem are presented, covering two modes of loading and their combination. The performance of the proposed methodology is quantified by means of

Comparison of DEM-Based Wang Tilings and PUC

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