Designing materials with prescribed elastic properties using polygonal cells

Abstract

SUMMARYAn extension of the material design problem is presented in which the base cell that characterizes the material microgeometry is polygonal. The setting is the familiar inverse homogenization problem as introduced by Sigmund. Using basic concepts in periodic planar tiling it is shown that base cells of very general geometries can be analysed within the standard topology optimization setting with little additional e ort. In particular, the periodic homogenization problem deÿned on polygonal base cells that tile the plane can be replaced and analysed more e ciently by an equivalent problem that uses simple parallelograms as base cells. Di erent material layouts can be obtained by varying just two parameters that a ect the geometry of the parallelogram, namely, the ratio of the lengths of the sides and the internal angle. This is an e cient way to organize the search of the design space for all possible single-scale material arrangements and could result in solutions that may be unreachable using a square or rectangular base cell. Examples illustrate the results.

Conforming polygonal finite elements

“…Wachspress [3] proposed rational basis functions on polygonal elements, which has only received limited attention so far [4][5][6][7][8]. The advantages and potential benefits of using polygonal finite elements in computation are striking: greater flexibility (single algorithm suffices) in the meshing of arbitrary geometries such as those that arise in biomechanics [9,10]; better accuracy in the numerical solution (higher-order approximation) over that obtainable using triangular and quadrilateral meshes on a given nodal grid; useful as a transition element in finite element meshes [11,12]; suitable in material design [13] and in the modelling of polycrystalline materials [14]; and will have less sensitivity to lock (unlike lower-order triangular and quadrilateral elements which tend to be stiff) under volume-preserving deformation states which arise in incompressible elasticity as well as in von Mises plasticity. To this end, we explore the construction of robust and accurate finite element methods that are based on polygonal elements.…”

How to determine composite material properties using numerical homogenization

“…As discussed in [11] will compute the effective properties of a random microstructure consisting of two materials, where the stiff material has an elasticity modulus of about 100 times the soft material. The homogenization is done by discretizing the RVE with 200 times 200 bilinear elements, since that is the size of x.…”

“…As discussed in [11] a parallelogram RVE allows for the analysis of general periodic materials, including polygonal cells.…”

“…unit strain in the horizontal direction (11), unit strain in the vertical direction (22), and unit shear strain (12 or 21)), while the locally varying strain fields ε pq dV ∀v ∈ V (3) where v is a virtual displacement field. For most practical problems the homogenization is performed numerically by discretizing and solving Eq.…”

Design and application of layered composites with the prescribed magnetic permeability

“…The parameter is useful for the material design with more arbitrary properties. In this study, the parameter t is defined as the following equation [13]:…”