Symmetry considerations for topology design in the elastic inverse homogenization problem

Podestá, Juan M.; Méndez, Carlos; Toro, Sebastián; Huespe, Alfredo Edmundo

doi:10.1016/j.jmps.2019.03.018

Cited by 35 publications

(33 citation statements)

References 45 publications

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“…In view of these observations and following similar arguments to the ones given by [13], here, we propose to employ concepts taken from crystallography to define the cell shape. These concepts are intimately related to the symmetry properties of the crystal structures and the elastic target tensors.…”

Section: Microarchitecture Design Using Symmetric Topologiesmentioning

confidence: 83%

“…Furthermore, It has been shown that the realization of certain classes of composites, such as the Vigdergauz microstructures or the microstructures proposed by [10], could be promoted by enforcing some kind of material layout symmetry. These issues have been particularly studied in the 2D elastic material design context by the authors in previous contributions, see [3] and [13].…”

Section: Microarchitecture Design Using Symmetric Topologiesmentioning

confidence: 99%

“…By adopting this approach, it can be guarantee that the homogenized elastic properties of the designed composite will have the same, or higher, symmetry than the target ones. This methodology is a generalization to three-dimensional problems of the one reported in [13] for twodimensional cases.…”

Section: Microarchitecture Design Using Symmetric Topologiesmentioning

confidence: 99%

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Making Use of Symmetries in the Three-Dimensional Elastic Inverse Homogenization Problem

Méndez

Podestá

Toro

et al. 2019

Int J Mult Comp Eng

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The objective of this paper is the design of three-dimensional elastic metamaterials with periodic microarchitectures. The microarchitectures of these materials are attained by following an inverse design technique jointly with an homogenization-based topology optimization algorithm.In this context, we have particularly studied the connection between the symmetry of the material layout at the microscale of 3D periodic composites and the symmetry of the effective elastic properties. We have analyzed some possible Bravais lattices and space groups, which are typically associated with crystallography, to study the way in which the symmetries of these geometrical objects can be usefully used for the microarchitecture design of 3D elastic metamaterial.Following a previous work of the authors for two-dimensional problems, we suggest adopting the design domain of the topology optimization problem coincident with the Wigner-Seitz cells of specific Bravais lattices having the same point group to that of the target elasticity tensor.The numerical assessment described in this papers aims at the design of an extreme material. The solutions obtained with this procedure show that different composite microarchitectures emerge depending on the cell shape selection.

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Section: Microarchitecture Design Using Symmetric Topologiesmentioning

confidence: 83%

Section: Microarchitecture Design Using Symmetric Topologiesmentioning

confidence: 99%

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Making Use of Symmetries in the Three-Dimensional Elastic Inverse Homogenization Problem

Méndez

Podestá

Toro

et al. 2019

Int J Mult Comp Eng

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“…Recently, Wang et al (2019) showed that near-optimal and periodic truss lattice structures could be obtained for multiple load cases by distorting simple Bravais-like lattice structures to a parallelepiped. Besides using a parallelogram in 2D or a parallelepiped in 3D, one can use many more different types of polygons to solve the homogenization equations (Barbarosie et al 2017;Podestá et al 2019). For example, in 2D, a hexagon can be used to describe a periodic isotropic hexagonal microstructure (Sigmund 2000).…”

Section: Unrestricted Unit-cell Designmentioning

confidence: 99%

Topology optimization of multi-scale structures: a review

Sigmund

Groen

2021

Struct Multidisc Optim

344

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Multi-scale structures, as found in nature (e.g., bone and bamboo), hold the promise of achieving superior performance while being intrinsically lightweight, robust, and multi-functional. Recent years have seen a rapid development in topology optimization approaches for designing multi-scale structures, but the field actually dates back to the seminal paper by Bendsøe and Kikuchi from 1988 (Computer Methods in Applied Mechanics and Engineering 71(2): pp. 197–224). In this review, we intend to categorize existing approaches, explain the principles of each category, analyze their strengths and applicabilities, and discuss open research questions. The review and associated analyses will hopefully form a basis for future research and development in this exciting field.

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“…Recently in ref. [69], such kind of 2D cell derived from the hexagonal crystal system and Bravais lattices of periodic pattern were adopted. This method was extended to 3D structures—see ref.…”

Section: Cells Of Periodicity Corresponding To Exact Isotropic and Sqmentioning

confidence: 99%

Recovery of the Auxetic Microstructures Appearing in the Least Compliant Continuum Two‐Dimensional Bodies

Czarnecki

Łukasiak

2020

Physica Status Solidi (b)

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The article is focused on recovering of two-phase microstructures of effective properties for 2D elasticity (see e.g., ref. [1]) predicted by two, stress-based topology optimization methods: isotropic material design (IMD) and cubic material design (CMD)-see e.g., refs. [2-5]. In these articles, the reconstruction of microstructures in subdomains where the auxetic properties [6-13] of the optimal, nonhomogeneous isotropic or cubic elastic material appear has been undertaken. In the present article, being an extension of the previous work, [14] the homogenization method based on parametric description of the shape of the domain filled by the given isotropic material inside the representative volume elements (RVEs), instead of the combinatorial search among the admissible microstructures, is implemented. Within the 2D setting discussed, the local cubic symmetry of the material reduces to the local symmetry of a square. The main goal is to recover isotropic microstructures or microstructures of symmetry of a square at arbitrary points of the body in a relatively simple way, considering the heterogeneity of the distribution of optimal elastic moduli. In the case of an isotropic material, the results obtained in ref. [14] are used as a starting point to the further search of the auxetic microstructures, realizing the Cherkaev-Gibiansky bounds. [15] In the case of 2D models of materials showing auxetic properties, a series of works devoted to their research at the level of regular atomic structures (e.g., hard cyclic hexamers [HCHs] or hard cyclic tetramers and many others) were particularly important in the development of the theory of such materials-see refs. [16-24]. Especially important in this respect is the ref. [16], in which a constant thermodynamic tension Monte Carlo (MC) simulations revealed the first planar, isotropic (and chiral) thermodynamically stable structure of negative Poisson's ratio in tilted phase formed by HCHs. In ref. [17], the exact solution for negative Poisson's ratio at high densities and in the static limit (i.e., at zero temperature) was obtained for planar, isotropic (and chiral) structure of cyclic hexamers interacting through nearest-neighbouring site-site inversepower potential. In ref. [18], MC simulations performed on hard, cyclic tetramers revealed an interesting feature of such a very simple model of molecules composed of four identical hard disks: depending on its shape, all possible material behavioursregarding negative Poisson's ratio could arise and thermodynamically stable auxetic (or partially auxetic) chiral structures of square symmetry could be spontaneously formed. Analyzing in ref. [19], other 2D cyclic trimers model of triatomic molecules in which "atoms" are distributed on vertices of equilateral triangles, the authors have found the exact solution at zero temperature for the planar auxetic structure which is isotropic but not chiral (soft cyclic trimers). The both chiral and nonchiral molecular models were studied in ref. [20], where the influence of molecular geome...

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Symmetry considerations for topology design in the elastic inverse homogenization problem

Cited by 35 publications

References 45 publications

Making Use of Symmetries in the Three-Dimensional Elastic Inverse Homogenization Problem

Making Use of Symmetries in the Three-Dimensional Elastic Inverse Homogenization Problem

Topology optimization of multi-scale structures: a review

Recovery of the Auxetic Microstructures Appearing in the Least Compliant Continuum Two‐Dimensional Bodies

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