Making use of Symmetries in the 3D Elastic Inverse Homogenization Problem

Abstract

An analysis of the connection between the symmetry of the material layout at the microscale and the properties of the effective elasticity of 3D periodic composites is performed. We analyze some possible Bravais lattices and spaces groups to study the way in which the symmetries of these objects can be used in the microarchitecture design methodology to find an adequate 3D elastic metamaterial.For obtaining the material architecture at the microscale, we follow an inverse design technique jointly with an homogenization-based topology optimization algorithm. In this context, a procedure is proposed for:- solving the topology optimization algorithm on a conveniently chosen spatial domain;- identifying the space group symmetry of the periodic material topology, at the microscale, to guarantee the correct symmetry of the homogenized elastic properties of the designed material;- building the microcell finite element mesh such that the symmetry of the material layouts, being compatible with different space groups, can be easilyParticularly, related to item i), we suggest adopting the Wigner--Seitz cells of specific Bravais lattices having the same point group of the target elasticity tensor.The numerical assessment of the proposed design procedure is presented. It consists of the searching for a class of extreme materials. The obtained solutions show that different composites architectures emerge depending on the cell shape selection.